Physlib.Electromagnetism.Vacuum.IsPlaneWave — Physlib

Imports

/-
Copyright (c) 2025 Zhi Kai Pong. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhi Kai Pong, Joseph Tooby-Smith
-/
module

public import Physlib.ClassicalMechanics.WaveEquation.Basic
public import Physlib.Electromagnetism.Dynamics.IsExtrema

Electromagnetic wave equation

i. Overview

In this module we define a proposition IsPlaneWave on electromagnetic potentials which is true if the potential corresponds to a plane wave. From this we derive various properties of plane waves including the orthogonality of the electric field, magnetic field and direction of propagation, in general dimensions.

ii. Key results

IsPlaneWave : The proposition defining plane waves.

IsPlaneWave.electricFunction : The electric function corresponding to a plane wave.

IsPlaneWave.magneticFunction : The magnetic function corresponding to a plane wave.

IsPlaneWave.magneticFieldMatrix_eq_propogator_cross_electricField : The magnetic field expressed in terms of the electric field and direction of propagation.

IsPlaneWave.electricField_eq_propogator_cross_magneticFieldMatrix : The electric field expressed in terms of the magnetic field and direction of propagation.

iii. Table of contents

A. The property of being a plane wave

A.1. The electric and magnetic functions from a plane wave

A.1.1. Electric function and magnetic function in terms of E and B fields

A.1.2. Uniqueness of the electric function

A.1.3. Uniqueness of the magnetic function

A.2. Differentiability conditions

A.3. Time derivative of electric and magnetic fields of a plane wave

A.4. Space derivative of electric and magnetic fields of a plane wave

A.5. Space derivative in terms of time derivative

B. The magnetic field in terms of the electric field

B.1. Time derivative of the magnetic field in terms of electric field

B.2. Space derivative of the magnetic field in terms of electric field

B.3. Magnetic field equal propogator cross electric field up to constant

C. The electric field in terms of the magnetic field

C.1. The time derivative of the electric field in terms of magnetic field

C.2. The space derivative of the electric field in terms of magnetic field

C.3. Electric field equal propogator cross magnetic field up to constant

iv. References

@[expose] public section

A. The property of being a plane wave

The proposition on a electromagnetic potential which is true if it corresponds to a plane wave.

def IsPlaneWave {d : ℕ} (𝓕 : FreeSpace)
    (A : ElectromagneticPotential d) (s : Direction d) : Prop :=
  (∃ E₀, A.electricField 𝓕.c = planeWave E₀ 𝓕.c s) ∧
  (∃ (B₀ : ℝ → Fin d × Fin d → ℝ), ∀ t x, A.magneticFieldMatrix 𝓕.c t x =
    B₀ (⟪x, s.unit⟫_ℝ - 𝓕.c * t))

A.1. The electric and magnetic functions from a plane wave

lemma electricField_eq_electricFunction {d : ℕ} {𝓕 : FreeSpace}
    {A : ElectromagneticPotential d} {s : Direction d}
    (P : IsPlaneWave 𝓕 A s) (t : Time) (x : Space d) :
    A.electricField 𝓕.c t x =
    P.electricFunction (⟪x, s.unit⟫_ℝ - 𝓕.c * t) :=
  congrFun (congrFun (Classical.choose_spec P.1) t) x

lemma magneticFieldMatrix_eq_magneticFunction {d : ℕ}
    {𝓕 : FreeSpace} {A : ElectromagneticPotential d} {s : Direction d}
    (P : IsPlaneWave 𝓕 A s) (t : Time) (x : Space d) :
    A.magneticFieldMatrix 𝓕.c t x =
    P.magneticFunction (⟪x, s.unit⟫_ℝ - 𝓕.c * t) :=
  Classical.choose_spec P.2 t x

A.1.1. Electric function and magnetic function in terms of E and B fields



lemma electricFunction_eq_electricField {d : ℕ}
    {𝓕 : FreeSpace} {A : ElectromagneticPotential d}
    {s : Direction d} (P : IsPlaneWave 𝓕 A s) :
    P.electricFunction = fun u =>
    A.electricField 𝓕.c ⟨(- u)/𝓕.c.1⟩ (0 : Space d) := byd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A s⊢ P.electricFunction = fun u => electricField 𝓕.c A { val := -u / 𝓕.c.val } 0
  funext ud:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A su:ℝ⊢ P.electricFunction u = electricField 𝓕.c A { val := -u / 𝓕.c.val } 0
  rw [P.electricField_eq_electricFunction,d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A su:ℝ⊢ P.electricFunction u = P.electricFunction (⟪0, s.unit⟫_ℝ - 𝓕.c.val * { val := -u / 𝓕.c.val }.val)d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A su:ℝ⊢ P.electricFunction u = P.electricFunction (-(𝓕.c.val * { val := -u / 𝓕.c.val }.val)) inner_zero_left,d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A su:ℝ⊢ P.electricFunction u = P.electricFunction (0 - 𝓕.c.val * { val := -u / 𝓕.c.val }.val)d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A su:ℝ⊢ P.electricFunction u = P.electricFunction (-(𝓕.c.val * { val := -u / 𝓕.c.val }.val)) zero_subd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A su:ℝ⊢ P.electricFunction u = P.electricFunction (-(𝓕.c.val * { val := -u / 𝓕.c.val }.val))d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A su:ℝ⊢ P.electricFunction u = P.electricFunction (-(𝓕.c.val * { val := -u / 𝓕.c.val }.val))]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A su:ℝ⊢ P.electricFunction u = P.electricFunction (-(𝓕.c.val * { val := -u / 𝓕.c.val }.val))
  congr 1e_ad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A su:ℝ⊢ u = -(𝓕.c.val * { val := -u / 𝓕.c.val }.val)
  field_simpAll goals completed! 🐙



lemma magneticFunction_eq_magneticFieldMatrix {d : ℕ}
    {𝓕 : FreeSpace} {A : ElectromagneticPotential d}
    {s : Direction d} (P : IsPlaneWave 𝓕 A s) :
    P.magneticFunction = fun u =>
    A.magneticFieldMatrix 𝓕.c ⟨(- u)/𝓕.c.1⟩ (0 : Space d) := byd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A s⊢ P.magneticFunction = fun u => magneticFieldMatrix 𝓕.c A { val := -u / 𝓕.c.val } 0
  funext ud:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A su:ℝ⊢ P.magneticFunction u = magneticFieldMatrix 𝓕.c A { val := -u / 𝓕.c.val } 0
  rw [P.magneticFieldMatrix_eq_magneticFunction,d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A su:ℝ⊢ P.magneticFunction u = P.magneticFunction (⟪0, s.unit⟫_ℝ - 𝓕.c.val * { val := -u / 𝓕.c.val }.val)d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A su:ℝ⊢ P.magneticFunction u = P.magneticFunction (-(𝓕.c.val * { val := -u / 𝓕.c.val }.val)) inner_zero_left,d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A su:ℝ⊢ P.magneticFunction u = P.magneticFunction (0 - 𝓕.c.val * { val := -u / 𝓕.c.val }.val)d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A su:ℝ⊢ P.magneticFunction u = P.magneticFunction (-(𝓕.c.val * { val := -u / 𝓕.c.val }.val)) zero_subd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A su:ℝ⊢ P.magneticFunction u = P.magneticFunction (-(𝓕.c.val * { val := -u / 𝓕.c.val }.val))d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A su:ℝ⊢ P.magneticFunction u = P.magneticFunction (-(𝓕.c.val * { val := -u / 𝓕.c.val }.val))]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A su:ℝ⊢ P.magneticFunction u = P.magneticFunction (-(𝓕.c.val * { val := -u / 𝓕.c.val }.val))
  congr 1e_ad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A su:ℝ⊢ u = -(𝓕.c.val * { val := -u / 𝓕.c.val }.val)
  field_simpAll goals completed! 🐙

A.1.2. Uniqueness of the electric function



lemma electricFunction_unique {d : ℕ} {𝓕 : FreeSpace}
    {A : ElectromagneticPotential d} {s : Direction d}
    (P : IsPlaneWave 𝓕 A s) (E1 : ℝ → EuclideanSpace ℝ (Fin d))
    (hE₁ : A.electricField 𝓕.c = planeWave E1 𝓕.c s) :
    E1 = P.electricFunction := byd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A sE1:ℝ → EuclideanSpace ℝ (Fin d)hE₁:electricField 𝓕.c A = planeWave E1 𝓕.c.val s⊢ E1 = P.electricFunction
  funext xd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A sE1:ℝ → EuclideanSpace ℝ (Fin d)hE₁:electricField 𝓕.c A = planeWave E1 𝓕.c.val sx:ℝ⊢ E1 x = P.electricFunction x
  obtain ⟨t, rfl⟩ : ∃ t, x = ⟪0, s.unit⟫_ℝ - 𝓕.c * t := byd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A sE1:ℝ → EuclideanSpace ℝ (Fin d)hE₁:electricField 𝓕.c A = planeWave E1 𝓕.c.val sx:ℝ⊢ ∃ t, x = ⟪0, s.unit⟫_ℝ - 𝓕.c.val * td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A sE1:ℝ → EuclideanSpace ℝ (Fin d)hE₁:electricField 𝓕.c A = planeWave E1 𝓕.c.val st:ℝ⊢ E1 (⟪0, s.unit⟫_ℝ - 𝓕.c.val * t) = P.electricFunction (⟪0, s.unit⟫_ℝ - 𝓕.c.val * t) use (- x/𝓕.c)hd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A sE1:ℝ → EuclideanSpace ℝ (Fin d)hE₁:electricField 𝓕.c A = planeWave E1 𝓕.c.val sx:ℝ⊢ x = ⟪0, s.unit⟫_ℝ - 𝓕.c.val * (-x / 𝓕.c.val)d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A sE1:ℝ → EuclideanSpace ℝ (Fin d)hE₁:electricField 𝓕.c A = planeWave E1 𝓕.c.val st:ℝ⊢ E1 (⟪0, s.unit⟫_ℝ - 𝓕.c.val * t) = P.electricFunction (⟪0, s.unit⟫_ℝ - 𝓕.c.val * t); field_simphd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A sE1:ℝ → EuclideanSpace ℝ (Fin d)hE₁:electricField 𝓕.c A = planeWave E1 𝓕.c.val sx:ℝ⊢ x = ⟪0, s.unit⟫_ℝ - -xd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A sE1:ℝ → EuclideanSpace ℝ (Fin d)hE₁:electricField 𝓕.c A = planeWave E1 𝓕.c.val st:ℝ⊢ E1 (⟪0, s.unit⟫_ℝ - 𝓕.c.val * t) = P.electricFunction (⟪0, s.unit⟫_ℝ - 𝓕.c.val * t); simpd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A sE1:ℝ → EuclideanSpace ℝ (Fin d)hE₁:electricField 𝓕.c A = planeWave E1 𝓕.c.val st:ℝ⊢ E1 (⟪0, s.unit⟫_ℝ - 𝓕.c.val * t) = P.electricFunction (⟪0, s.unit⟫_ℝ - 𝓕.c.val * t)d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A sE1:ℝ → EuclideanSpace ℝ (Fin d)hE₁:electricField 𝓕.c A = planeWave E1 𝓕.c.val st:ℝ⊢ E1 (⟪0, s.unit⟫_ℝ - 𝓕.c.val * t) = P.electricFunction (⟪0, s.unit⟫_ℝ - 𝓕.c.val * t)
  rw [← P.electricField_eq_electricFunction,d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A sE1:ℝ → EuclideanSpace ℝ (Fin d)hE₁:electricField 𝓕.c A = planeWave E1 𝓕.c.val st:ℝ⊢ E1 (⟪0, s.unit⟫_ℝ - 𝓕.c.val * t) = electricField 𝓕.c A { val := t } 0d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A sE1:ℝ → EuclideanSpace ℝ (Fin d)hE₁:electricField 𝓕.c A = planeWave E1 𝓕.c.val st:ℝ⊢ E1 (⟪0, s.unit⟫_ℝ - 𝓕.c.val * t) = planeWave E1 𝓕.c.val s { val := t } 0 hE₁d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A sE1:ℝ → EuclideanSpace ℝ (Fin d)hE₁:electricField 𝓕.c A = planeWave E1 𝓕.c.val st:ℝ⊢ E1 (⟪0, s.unit⟫_ℝ - 𝓕.c.val * t) = planeWave E1 𝓕.c.val s { val := t } 0d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A sE1:ℝ → EuclideanSpace ℝ (Fin d)hE₁:electricField 𝓕.c A = planeWave E1 𝓕.c.val st:ℝ⊢ E1 (⟪0, s.unit⟫_ℝ - 𝓕.c.val * t) = planeWave E1 𝓕.c.val s { val := t } 0]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A sE1:ℝ → EuclideanSpace ℝ (Fin d)hE₁:electricField 𝓕.c A = planeWave E1 𝓕.c.val st:ℝ⊢ E1 (⟪0, s.unit⟫_ℝ - 𝓕.c.val * t) = planeWave E1 𝓕.c.val s { val := t } 0
  rflAll goals completed! 🐙

A.1.3. Uniqueness of the magnetic function



lemma magneticFunction_unique {d : ℕ} {𝓕 : FreeSpace}
    {A : ElectromagneticPotential d} {s : Direction d}
    (P : IsPlaneWave 𝓕 A s)
    (B1 : ℝ → Fin d × Fin d → ℝ)
    (hB₁ : ∀ t x, A.magneticFieldMatrix 𝓕.c t x =
      B1 (⟪x, s.unit⟫_ℝ - 𝓕.c * t)) :
    B1 = P.magneticFunction := byd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A sB1:ℝ → Fin d × Fin d → ℝhB₁:∀ (t : Time) (x : Space d), magneticFieldMatrix 𝓕.c A t x = B1 (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)⊢ B1 = P.magneticFunction
  funext xd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A sB1:ℝ → Fin d × Fin d → ℝhB₁:∀ (t : Time) (x : Space d), magneticFieldMatrix 𝓕.c A t x = B1 (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)x:ℝ⊢ B1 x = P.magneticFunction x
  obtain ⟨t, rfl⟩ : ∃ t, x = ⟪0, s.unit⟫_ℝ - 𝓕.c * t := byd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A sB1:ℝ → Fin d × Fin d → ℝhB₁:∀ (t : Time) (x : Space d), magneticFieldMatrix 𝓕.c A t x = B1 (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)x:ℝ⊢ ∃ t, x = ⟪0, s.unit⟫_ℝ - 𝓕.c.val * td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A sB1:ℝ → Fin d × Fin d → ℝhB₁:∀ (t : Time) (x : Space d), magneticFieldMatrix 𝓕.c A t x = B1 (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)t:ℝ⊢ B1 (⟪0, s.unit⟫_ℝ - 𝓕.c.val * t) = P.magneticFunction (⟪0, s.unit⟫_ℝ - 𝓕.c.val * t) use (- x/𝓕.c)hd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A sB1:ℝ → Fin d × Fin d → ℝhB₁:∀ (t : Time) (x : Space d), magneticFieldMatrix 𝓕.c A t x = B1 (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)x:ℝ⊢ x = ⟪0, s.unit⟫_ℝ - 𝓕.c.val * (-x / 𝓕.c.val)d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A sB1:ℝ → Fin d × Fin d → ℝhB₁:∀ (t : Time) (x : Space d), magneticFieldMatrix 𝓕.c A t x = B1 (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)t:ℝ⊢ B1 (⟪0, s.unit⟫_ℝ - 𝓕.c.val * t) = P.magneticFunction (⟪0, s.unit⟫_ℝ - 𝓕.c.val * t); field_simphd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A sB1:ℝ → Fin d × Fin d → ℝhB₁:∀ (t : Time) (x : Space d), magneticFieldMatrix 𝓕.c A t x = B1 (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)x:ℝ⊢ x = ⟪0, s.unit⟫_ℝ - -xd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A sB1:ℝ → Fin d × Fin d → ℝhB₁:∀ (t : Time) (x : Space d), magneticFieldMatrix 𝓕.c A t x = B1 (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)t:ℝ⊢ B1 (⟪0, s.unit⟫_ℝ - 𝓕.c.val * t) = P.magneticFunction (⟪0, s.unit⟫_ℝ - 𝓕.c.val * t); simpd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A sB1:ℝ → Fin d × Fin d → ℝhB₁:∀ (t : Time) (x : Space d), magneticFieldMatrix 𝓕.c A t x = B1 (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)t:ℝ⊢ B1 (⟪0, s.unit⟫_ℝ - 𝓕.c.val * t) = P.magneticFunction (⟪0, s.unit⟫_ℝ - 𝓕.c.val * t)d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A sB1:ℝ → Fin d × Fin d → ℝhB₁:∀ (t : Time) (x : Space d), magneticFieldMatrix 𝓕.c A t x = B1 (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)t:ℝ⊢ B1 (⟪0, s.unit⟫_ℝ - 𝓕.c.val * t) = P.magneticFunction (⟪0, s.unit⟫_ℝ - 𝓕.c.val * t)
  rw [← P.magneticFieldMatrix_eq_magneticFunction,d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A sB1:ℝ → Fin d × Fin d → ℝhB₁:∀ (t : Time) (x : Space d), magneticFieldMatrix 𝓕.c A t x = B1 (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)t:ℝ⊢ B1 (⟪0, s.unit⟫_ℝ - 𝓕.c.val * t) = magneticFieldMatrix 𝓕.c A { val := t } 0All goals completed! 🐙 hB₁d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A sB1:ℝ → Fin d × Fin d → ℝhB₁:∀ (t : Time) (x : Space d), magneticFieldMatrix 𝓕.c A t x = B1 (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)t:ℝ⊢ B1 (⟪0, s.unit⟫_ℝ - 𝓕.c.val * t) = B1 (⟪0, s.unit⟫_ℝ - 𝓕.c.val * { val := t }.val)All goals completed! 🐙]All goals completed! 🐙

A.2. Differentiability conditions



lemma electricFunction_differentiable {d : ℕ}
    {𝓕 : FreeSpace} {A : ElectromagneticPotential d}
    {s : Direction d} (P : IsPlaneWave 𝓕 A s) (hA : ContDiff ℝ 2 A) :
    Differentiable ℝ P.electricFunction := byd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.val⊢ Differentiable ℝ P.electricFunction
  rw [electricFunction_eq_electricFieldd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.val⊢ Differentiable ℝ fun u => electricField 𝓕.c A { val := -u / 𝓕.c.val } 0d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.val⊢ Differentiable ℝ fun u => electricField 𝓕.c A { val := -u / 𝓕.c.val } 0]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.val⊢ Differentiable ℝ fun u => electricField 𝓕.c A { val := -u / 𝓕.c.val } 0
  change Differentiable ℝ (↿(electricField 𝓕.c A) ∘ fun u => ({ val := -u / 𝓕.c.val }, 0))d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.val⊢ Differentiable ℝ (↿(electricField 𝓕.c A) ∘ fun u => ({ val := -u / 𝓕.c.val }, 0))
  apply (electricField_differentiable hA).compd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.val⊢ Differentiable ℝ fun u => ({ val := -u / 𝓕.c.val }, 0)
  refine Differentiable.prodMk ?_ ?_refine_1d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.val⊢ Differentiable ℝ fun u => { val := -u / 𝓕.c.val }refine_2d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.val⊢ Differentiable ℝ fun u => 0
  ·refine_1d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.val⊢ Differentiable ℝ fun u => { val := -u / 𝓕.c.val } change Differentiable ℝ (Time.toRealCLE.symm ∘ fun u => -u / 𝓕.c.val)refine_1d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.val⊢ Differentiable ℝ (⇑toRealCLE.symm ∘ fun u => -u / 𝓕.c.val)
    fun_propAll goals completed! 🐙
  ·refine_2d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.val⊢ Differentiable ℝ fun u => 0 fun_propAll goals completed! 🐙



lemma magneticFunction_differentiable {d : ℕ}
    {𝓕 : FreeSpace} {A : ElectromagneticPotential d}
    {s : Direction d} (P : IsPlaneWave 𝓕 A s) (hA : ContDiff ℝ 2 A)
    (ij : Fin d × Fin d) :
    Differentiable ℝ (fun u => P.magneticFunction u ij) := byd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valij:Fin d × Fin d⊢ Differentiable ℝ fun u => P.magneticFunction u ij
  rw [magneticFunction_eq_magneticFieldMatrixd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valij:Fin d × Fin d⊢ Differentiable ℝ fun u => (fun u => magneticFieldMatrix 𝓕.c A { val := -u / 𝓕.c.val } 0) u ijd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valij:Fin d × Fin d⊢ Differentiable ℝ fun u => (fun u => magneticFieldMatrix 𝓕.c A { val := -u / 𝓕.c.val } 0) u ij]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valij:Fin d × Fin d⊢ Differentiable ℝ fun u => (fun u => magneticFieldMatrix 𝓕.c A { val := -u / 𝓕.c.val } 0) u ij
  simp onlyd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valij:Fin d × Fin d⊢ Differentiable ℝ fun u => magneticFieldMatrix 𝓕.c A { val := -u / 𝓕.c.val } 0 ij
  change Differentiable ℝ (↿(fun t x => A.magneticFieldMatrix 𝓕.c t x ij) ∘
    fun u => ({ val := -u / 𝓕.c.val }, 0))d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valij:Fin d × Fin d⊢ Differentiable ℝ ((↿fun t x => magneticFieldMatrix 𝓕.c A t x ij) ∘ fun u => ({ val := -u / 𝓕.c.val }, 0))
  apply (magneticFieldMatrix_differentiable A hA ij).compd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valij:Fin d × Fin d⊢ Differentiable ℝ fun u => ({ val := -u / 𝓕.c.val }, 0)
  refine Differentiable.prodMk ?_ ?_refine_1d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valij:Fin d × Fin d⊢ Differentiable ℝ fun u => { val := -u / 𝓕.c.val }refine_2d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valij:Fin d × Fin d⊢ Differentiable ℝ fun u => 0
  ·refine_1d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valij:Fin d × Fin d⊢ Differentiable ℝ fun u => { val := -u / 𝓕.c.val } change Differentiable ℝ (Time.toRealCLE.symm ∘ fun u => -u / 𝓕.c.val)refine_1d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valij:Fin d × Fin d⊢ Differentiable ℝ (⇑toRealCLE.symm ∘ fun u => -u / 𝓕.c.val)
    fun_propAll goals completed! 🐙
  ·refine_2d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valij:Fin d × Fin d⊢ Differentiable ℝ fun u => 0 fun_propAll goals completed! 🐙

A.3. Time derivative of electric and magnetic fields of a plane wave



lemma electricField_time_deriv {d : ℕ}
    {𝓕 : FreeSpace} {A : ElectromagneticPotential d}
    {s : Direction d} (P : IsPlaneWave 𝓕 A s) (hA : ContDiff ℝ 2 A) (t : Time)
    (x : Space d) :
    ∂ₜ (A.electricField 𝓕.c · x) t = - 𝓕.c.val •
    fderiv ℝ P.electricFunction (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val) 1 := byd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space d⊢ ∂ₜ (fun x_1 => electricField 𝓕.c A x_1 x) t =
  -𝓕.c.val • (fderiv ℝ P.electricFunction (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)) 1
  have h : A.electricField 𝓕.c = planeWave P.electricFunction 𝓕.c.val s :=
    Classical.choose_spec P.1d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space dh:electricField 𝓕.c A = planeWave P.electricFunction 𝓕.c.val s⊢ ∂ₜ (fun x_1 => electricField 𝓕.c A x_1 x) t =
  -𝓕.c.val • (fderiv ℝ P.electricFunction (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)) 1
  rw [h,d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space dh:electricField 𝓕.c A = planeWave P.electricFunction 𝓕.c.val s⊢ ∂ₜ (fun x_1 => planeWave P.electricFunction 𝓕.c.val s x_1 x) t =
  -𝓕.c.val • (fderiv ℝ P.electricFunction (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)) 1d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space dh:electricField 𝓕.c A = planeWave P.electricFunction 𝓕.c.val s⊢ (-𝓕.c.val • fun t => planeWave (fun x => (fderiv ℝ P.electricFunction x) 1) 𝓕.c.val s t x) t =
  -𝓕.c.val • (fderiv ℝ P.electricFunction (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)) 1 planeWave_time_deriv (P.electricFunction_differentiable hA)d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space dh:electricField 𝓕.c A = planeWave P.electricFunction 𝓕.c.val s⊢ (-𝓕.c.val • fun t => planeWave (fun x => (fderiv ℝ P.electricFunction x) 1) 𝓕.c.val s t x) t =
  -𝓕.c.val • (fderiv ℝ P.electricFunction (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)) 1d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space dh:electricField 𝓕.c A = planeWave P.electricFunction 𝓕.c.val s⊢ (-𝓕.c.val • fun t => planeWave (fun x => (fderiv ℝ P.electricFunction x) 1) 𝓕.c.val s t x) t =
  -𝓕.c.val • (fderiv ℝ P.electricFunction (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)) 1]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space dh:electricField 𝓕.c A = planeWave P.electricFunction 𝓕.c.val s⊢ (-𝓕.c.val • fun t => planeWave (fun x => (fderiv ℝ P.electricFunction x) 1) 𝓕.c.val s t x) t =
  -𝓕.c.val • (fderiv ℝ P.electricFunction (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)) 1
  simp [planeWave_eq]All goals completed! 🐙



lemma magneticFieldMatrix_time_deriv {d : ℕ}
    {𝓕 : FreeSpace} {A : ElectromagneticPotential d}
    {s : Direction d} (P : IsPlaneWave 𝓕 A s) (hA : ContDiff ℝ 2 A) (t : Time)
    (x : Space d) (i j : Fin d) :
    ∂ₜ (A.magneticFieldMatrix 𝓕.c · x (i, j)) t = - 𝓕.c.val •
    fderiv ℝ (fun u => P.magneticFunction u (i, j)) (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val) 1 := byd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin d⊢ ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, j)) t =
  -𝓕.c.val • (fderiv ℝ (fun u => P.magneticFunction u (i, j)) (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)) 1
  conv_lhs =>
    enter [1, t]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt✝:Timex:Space di:Fin dj:Fin dt:Time| magneticFieldMatrix 𝓕.c A t x (i, j)
    rw [P.magneticFieldMatrix_eq_magneticFunction]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt✝:Timex:Space di:Fin dj:Fin dt:Time| P.magneticFunction (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val) (i, j)
  rw [Time.deriv_eqd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin d⊢ (fderiv ℝ (fun t => P.magneticFunction (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val) (i, j)) t) 1 =
  -𝓕.c.val • (fderiv ℝ (fun u => P.magneticFunction u (i, j)) (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)) 1d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin d⊢ (fderiv ℝ (fun t => P.magneticFunction (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val) (i, j)) t) 1 =
  -𝓕.c.val • (fderiv ℝ (fun u => P.magneticFunction u (i, j)) (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)) 1]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin d⊢ (fderiv ℝ (fun t => P.magneticFunction (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val) (i, j)) t) 1 =
  -𝓕.c.val • (fderiv ℝ (fun u => P.magneticFunction u (i, j)) (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)) 1
  change fderiv ℝ ((fun u => P.magneticFunction u (i, j)) ∘
    fun t => ⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val) t 1 = _d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin d⊢ (fderiv ℝ ((fun u => P.magneticFunction u (i, j)) ∘ fun t => ⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val) t) 1 =
  -𝓕.c.val • (fderiv ℝ (fun u => P.magneticFunction u (i, j)) (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)) 1
  rw [fderiv_comp _ (magneticFunction_differentiable P hA (i, j)).differentiableAt (byd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin d⊢ DifferentiableAt ℝ (fun t => ⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin d⊢ (fderiv ℝ (fun u => P.magneticFunction u (i, j)) (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val) ∘SL
      (fderiv ℝ (fun t => ⟪x, s.unit⟫_ℝ) t - 𝓕.c.val • fderiv ℝ Time.val t))
    1 =
  -𝓕.c.val • (fderiv ℝ (fun u => P.magneticFunction u (i, j)) (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)) 1 fun_propAll goals completed! 🐙d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin d⊢ (fderiv ℝ (fun u => P.magneticFunction u (i, j)) (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val) ∘SL
      (fderiv ℝ (fun t => ⟪x, s.unit⟫_ℝ) t - 𝓕.c.val • fderiv ℝ Time.val t))
    1 =
  -𝓕.c.val • (fderiv ℝ (fun u => P.magneticFunction u (i, j)) (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)) 1),
    fderiv_fun_sub (byd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin d⊢ DifferentiableAt ℝ (fun t => ⟪x, s.unit⟫_ℝ) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin d⊢ (fderiv ℝ (fun u => P.magneticFunction u (i, j)) (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val) ∘SL
      (fderiv ℝ (fun t => ⟪x, s.unit⟫_ℝ) t - 𝓕.c.val • fderiv ℝ Time.val t))
    1 =
  -𝓕.c.val • (fderiv ℝ (fun u => P.magneticFunction u (i, j)) (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)) 1 fun_propAll goals completed! 🐙d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin d⊢ (fderiv ℝ (fun u => P.magneticFunction u (i, j)) (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val) ∘SL
      (fderiv ℝ (fun t => ⟪x, s.unit⟫_ℝ) t - 𝓕.c.val • fderiv ℝ Time.val t))
    1 =
  -𝓕.c.val • (fderiv ℝ (fun u => P.magneticFunction u (i, j)) (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)) 1) (byd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin d⊢ DifferentiableAt ℝ (fun t => 𝓕.c.val * t.val) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin d⊢ (fderiv ℝ (fun u => P.magneticFunction u (i, j)) (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val) ∘SL
      (fderiv ℝ (fun t => ⟪x, s.unit⟫_ℝ) t - 𝓕.c.val • fderiv ℝ Time.val t))
    1 =
  -𝓕.c.val • (fderiv ℝ (fun u => P.magneticFunction u (i, j)) (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)) 1 fun_propAll goals completed! 🐙d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin d⊢ (fderiv ℝ (fun u => P.magneticFunction u (i, j)) (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val) ∘SL
      (fderiv ℝ (fun t => ⟪x, s.unit⟫_ℝ) t - 𝓕.c.val • fderiv ℝ Time.val t))
    1 =
  -𝓕.c.val • (fderiv ℝ (fun u => P.magneticFunction u (i, j)) (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)) 1), fderiv_const_mul (byd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin d⊢ DifferentiableAt ℝ Time.val td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin d⊢ (fderiv ℝ (fun u => P.magneticFunction u (i, j)) (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val) ∘SL
      (fderiv ℝ (fun t => ⟪x, s.unit⟫_ℝ) t - 𝓕.c.val • fderiv ℝ Time.val t))
    1 =
  -𝓕.c.val • (fderiv ℝ (fun u => P.magneticFunction u (i, j)) (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)) 1 fun_propAll goals completed! 🐙d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin d⊢ (fderiv ℝ (fun u => P.magneticFunction u (i, j)) (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val) ∘SL
      (fderiv ℝ (fun t => ⟪x, s.unit⟫_ℝ) t - 𝓕.c.val • fderiv ℝ Time.val t))
    1 =
  -𝓕.c.val • (fderiv ℝ (fun u => P.magneticFunction u (i, j)) (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)) 1)]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin d⊢ (fderiv ℝ (fun u => P.magneticFunction u (i, j)) (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val) ∘SL
      (fderiv ℝ (fun t => ⟪x, s.unit⟫_ℝ) t - 𝓕.c.val • fderiv ℝ Time.val t))
    1 =
  -𝓕.c.val • (fderiv ℝ (fun u => P.magneticFunction u (i, j)) (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)) 1
  simp [Time.fderiv_val]All goals completed! 🐙

A.4. Space derivative of electric and magnetic fields of a plane wave



lemma electricField_space_deriv {d : ℕ}
    {𝓕 : FreeSpace} {A : ElectromagneticPotential d}
    {s : Direction d} (P : IsPlaneWave 𝓕 A s) (hA : ContDiff ℝ 2 A) (t : Time)
    (x : Space d) (i : Fin d) :
    ∂[i] (A.electricField 𝓕.c t ·) x = s.unit i •
    fderiv ℝ P.electricFunction (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val) 1 := byd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin d⊢ Space.deriv i (fun x => electricField 𝓕.c A t x) x =
  s.unit.val i • (fderiv ℝ P.electricFunction (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)) 1
  have h : A.electricField 𝓕.c = planeWave P.electricFunction 𝓕.c.val s :=
    Classical.choose_spec P.1d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dh:electricField 𝓕.c A = planeWave P.electricFunction 𝓕.c.val s⊢ Space.deriv i (fun x => electricField 𝓕.c A t x) x =
  s.unit.val i • (fderiv ℝ P.electricFunction (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)) 1
  rw [hd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dh:electricField 𝓕.c A = planeWave P.electricFunction 𝓕.c.val s⊢ Space.deriv i (fun x => planeWave P.electricFunction 𝓕.c.val s t x) x =
  s.unit.val i • (fderiv ℝ P.electricFunction (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)) 1d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dh:electricField 𝓕.c A = planeWave P.electricFunction 𝓕.c.val s⊢ Space.deriv i (fun x => planeWave P.electricFunction 𝓕.c.val s t x) x =
  s.unit.val i • (fderiv ℝ P.electricFunction (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)) 1]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dh:electricField 𝓕.c A = planeWave P.electricFunction 𝓕.c.val s⊢ Space.deriv i (fun x => planeWave P.electricFunction 𝓕.c.val s t x) x =
  s.unit.val i • (fderiv ℝ P.electricFunction (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)) 1
  simp only [planeWave_space_deriv (P.electricFunction_differentiable hA)]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dh:electricField 𝓕.c A = planeWave P.electricFunction 𝓕.c.val s⊢ (s.unit.val i • fun x => planeWave (fun x => (fderiv ℝ P.electricFunction x) 1) 𝓕.c.val s t x) x =
  s.unit.val i • (fderiv ℝ P.electricFunction (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)) 1
  simp [planeWave_eq]All goals completed! 🐙



lemma magneticFieldMatrix_space_deriv {d : ℕ}
    {𝓕 : FreeSpace} {A : ElectromagneticPotential d}
    {s : Direction d} (P : IsPlaneWave 𝓕 A s) (hA : ContDiff ℝ 2 A) (t : Time)
    (x : Space d) (i j : Fin d) (k : Fin d) :
    ∂[k] (A.magneticFieldMatrix 𝓕.c t · (i, j)) x = s.unit k •
    fderiv ℝ (fun u => P.magneticFunction u (i, j))
      (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val) 1 := byd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ Space.deriv k (fun x => magneticFieldMatrix 𝓕.c A t x (i, j)) x =
  s.unit.val k • (fderiv ℝ (fun u => P.magneticFunction u (i, j)) (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)) 1
  conv_lhs =>
    enter [2, t]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt✝:Timex:Space di:Fin dj:Fin dk:Fin dt:Space d| magneticFieldMatrix 𝓕.c A t✝ t (i, j)
    rw [P.magneticFieldMatrix_eq_magneticFunction]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt✝:Timex:Space di:Fin dj:Fin dk:Fin dt:Space d| P.magneticFunction (⟪t, s.unit⟫_ℝ - 𝓕.c.val * t✝.val) (i, j)
  rw [Space.deriv_eq_fderiv_basisd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ (fderiv ℝ (fun t_1 => P.magneticFunction (⟪t_1, s.unit⟫_ℝ - 𝓕.c.val * t.val) (i, j)) x) (Space.basis k) =
  s.unit.val k • (fderiv ℝ (fun u => P.magneticFunction u (i, j)) (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)) 1d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ (fderiv ℝ (fun t_1 => P.magneticFunction (⟪t_1, s.unit⟫_ℝ - 𝓕.c.val * t.val) (i, j)) x) (Space.basis k) =
  s.unit.val k • (fderiv ℝ (fun u => P.magneticFunction u (i, j)) (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)) 1]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ (fderiv ℝ (fun t_1 => P.magneticFunction (⟪t_1, s.unit⟫_ℝ - 𝓕.c.val * t.val) (i, j)) x) (Space.basis k) =
  s.unit.val k • (fderiv ℝ (fun u => P.magneticFunction u (i, j)) (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)) 1
  change fderiv ℝ ((fun u => P.magneticFunction u (i, j)) ∘
    fun x => ⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val) x _ = _d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ (fderiv ℝ ((fun u => P.magneticFunction u (i, j)) ∘ fun x => ⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val) x) (Space.basis k) =
  s.unit.val k • (fderiv ℝ (fun u => P.magneticFunction u (i, j)) (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)) 1
  rw [fderiv_comp _ (magneticFunction_differentiable P hA (i, j)).differentiableAt (byd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => ⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val) xd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ (fderiv ℝ (fun u => P.magneticFunction u (i, j)) (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val) ∘SL
      (fderiv ℝ (fun x => ⟪x, s.unit⟫_ℝ) x - fderiv ℝ (fun x => 𝓕.c.val * t.val) x))
    (Space.basis k) =
  s.unit.val k • (fderiv ℝ (fun u => P.magneticFunction u (i, j)) (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)) 1 fun_propAll goals completed! 🐙d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ (fderiv ℝ (fun u => P.magneticFunction u (i, j)) (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val) ∘SL
      (fderiv ℝ (fun x => ⟪x, s.unit⟫_ℝ) x - fderiv ℝ (fun x => 𝓕.c.val * t.val) x))
    (Space.basis k) =
  s.unit.val k • (fderiv ℝ (fun u => P.magneticFunction u (i, j)) (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)) 1),
    fderiv_fun_sub (byd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => ⟪x, s.unit⟫_ℝ) xd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ (fderiv ℝ (fun u => P.magneticFunction u (i, j)) (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val) ∘SL
      (fderiv ℝ (fun x => ⟪x, s.unit⟫_ℝ) x - fderiv ℝ (fun x => 𝓕.c.val * t.val) x))
    (Space.basis k) =
  s.unit.val k • (fderiv ℝ (fun u => P.magneticFunction u (i, j)) (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)) 1 fun_propAll goals completed! 🐙d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ (fderiv ℝ (fun u => P.magneticFunction u (i, j)) (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val) ∘SL
      (fderiv ℝ (fun x => ⟪x, s.unit⟫_ℝ) x - fderiv ℝ (fun x => 𝓕.c.val * t.val) x))
    (Space.basis k) =
  s.unit.val k • (fderiv ℝ (fun u => P.magneticFunction u (i, j)) (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)) 1) (byd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => 𝓕.c.val * t.val) xd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ (fderiv ℝ (fun u => P.magneticFunction u (i, j)) (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val) ∘SL
      (fderiv ℝ (fun x => ⟪x, s.unit⟫_ℝ) x - fderiv ℝ (fun x => 𝓕.c.val * t.val) x))
    (Space.basis k) =
  s.unit.val k • (fderiv ℝ (fun u => P.magneticFunction u (i, j)) (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)) 1 fun_propAll goals completed! 🐙d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ (fderiv ℝ (fun u => P.magneticFunction u (i, j)) (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val) ∘SL
      (fderiv ℝ (fun x => ⟪x, s.unit⟫_ℝ) x - fderiv ℝ (fun x => 𝓕.c.val * t.val) x))
    (Space.basis k) =
  s.unit.val k • (fderiv ℝ (fun u => P.magneticFunction u (i, j)) (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)) 1)]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ (fderiv ℝ (fun u => P.magneticFunction u (i, j)) (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val) ∘SL
      (fderiv ℝ (fun x => ⟪x, s.unit⟫_ℝ) x - fderiv ℝ (fun x => 𝓕.c.val * t.val) x))
    (Space.basis k) =
  s.unit.val k • (fderiv ℝ (fun u => P.magneticFunction u (i, j)) (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)) 1
  simp only [ContinuousLinearMap.coe_comp, Function.comp_apply, fderiv_eq_smul_deriv, smul_eq_mul,
    one_mul, mul_eq_mul_right_iff, fderiv_fun_const, Pi.zero_apply, sub_zero]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ (fderiv ℝ (fun x => ⟪x, s.unit⟫_ℝ) x) (Space.basis k) = s.unit.val k ∨
  _root_.deriv (fun u => P.magneticFunction u (i, j)) (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val) = 0
  rw [← Space.deriv_eq_fderiv_basisd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ Space.deriv k (fun x => ⟪x, s.unit⟫_ℝ) x = s.unit.val k ∨
  _root_.deriv (fun u => P.magneticFunction u (i, j)) (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val) = 0d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ Space.deriv k (fun x => ⟪x, s.unit⟫_ℝ) x = s.unit.val k ∨
  _root_.deriv (fun u => P.magneticFunction u (i, j)) (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val) = 0]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ Space.deriv k (fun x => ⟪x, s.unit⟫_ℝ) x = s.unit.val k ∨
  _root_.deriv (fun u => P.magneticFunction u (i, j)) (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val) = 0
  simp only [deriv_inner_left, true_or]All goals completed! 🐙

A.5. Space derivative in terms of time derivative



lemma electricField_space_deriv_eq_time_deriv {d : ℕ} {𝓕 : FreeSpace}
    {A : ElectromagneticPotential d}
    {s : Direction d} (P : IsPlaneWave 𝓕 A s) (hA : ContDiff ℝ 2 A) (t : Time)
    (x : Space d) (i : Fin d) (k : Fin d) :
    ∂[k] (A.electricField 𝓕.c t · i) x = - (s.unit k / 𝓕.c.val) •
    ∂ₜ (A.electricField 𝓕.c · x i) t := byd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dk:Fin d⊢ Space.deriv k (fun x => (electricField 𝓕.c A t x).ofLp i) x =
  -(s.unit.val k / 𝓕.c.val) • ∂ₜ (fun x_1 => (electricField 𝓕.c A x_1 x).ofLp i) t
  rw [Space.deriv_euclid,d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dk:Fin d⊢ (Space.deriv k (fun x => electricField 𝓕.c A t x) x).ofLp i =
  -(s.unit.val k / 𝓕.c.val) • ∂ₜ (fun x_1 => (electricField 𝓕.c A x_1 x).ofLp i) thfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dk:Fin d⊢ Differentiable ℝ (electricField 𝓕.c A t)d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dk:Fin d⊢ (s.unit.val k • (fderiv ℝ P.electricFunction (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)) 1).ofLp i =
  -(s.unit.val k / 𝓕.c.val) • (-𝓕.c.val • (fderiv ℝ P.electricFunction (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)) 1).ofLp ihfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dk:Fin d⊢ Differentiable ℝ fun x_1 => electricField 𝓕.c A x_1 xhfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dk:Fin d⊢ Differentiable ℝ (electricField 𝓕.c A t) IsPlaneWave.electricField_space_deriv P hA t x k,d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dk:Fin d⊢ (s.unit.val k • (fderiv ℝ P.electricFunction (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)) 1).ofLp i =
  -(s.unit.val k / 𝓕.c.val) • ∂ₜ (fun x_1 => (electricField 𝓕.c A x_1 x).ofLp i) thfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dk:Fin d⊢ Differentiable ℝ (electricField 𝓕.c A t)d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dk:Fin d⊢ (s.unit.val k • (fderiv ℝ P.electricFunction (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)) 1).ofLp i =
  -(s.unit.val k / 𝓕.c.val) • (-𝓕.c.val • (fderiv ℝ P.electricFunction (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)) 1).ofLp ihfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dk:Fin d⊢ Differentiable ℝ fun x_1 => electricField 𝓕.c A x_1 xhfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dk:Fin d⊢ Differentiable ℝ (electricField 𝓕.c A t) Time.deriv_euclid,d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dk:Fin d⊢ (s.unit.val k • (fderiv ℝ P.electricFunction (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)) 1).ofLp i =
  -(s.unit.val k / 𝓕.c.val) • (∂ₜ (fun t => electricField 𝓕.c A t x) t).ofLp ihfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dk:Fin d⊢ Differentiable ℝ fun x_1 => electricField 𝓕.c A x_1 xhfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dk:Fin d⊢ Differentiable ℝ (electricField 𝓕.c A t)d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dk:Fin d⊢ (s.unit.val k • (fderiv ℝ P.electricFunction (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)) 1).ofLp i =
  -(s.unit.val k / 𝓕.c.val) • (-𝓕.c.val • (fderiv ℝ P.electricFunction (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)) 1).ofLp ihfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dk:Fin d⊢ Differentiable ℝ fun x_1 => electricField 𝓕.c A x_1 xhfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dk:Fin d⊢ Differentiable ℝ (electricField 𝓕.c A t)
    IsPlaneWave.electricField_time_deriv P hA t xd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dk:Fin d⊢ (s.unit.val k • (fderiv ℝ P.electricFunction (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)) 1).ofLp i =
  -(s.unit.val k / 𝓕.c.val) • (-𝓕.c.val • (fderiv ℝ P.electricFunction (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)) 1).ofLp ihfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dk:Fin d⊢ Differentiable ℝ fun x_1 => electricField 𝓕.c A x_1 xhfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dk:Fin d⊢ Differentiable ℝ (electricField 𝓕.c A t)d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dk:Fin d⊢ (s.unit.val k • (fderiv ℝ P.electricFunction (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)) 1).ofLp i =
  -(s.unit.val k / 𝓕.c.val) • (-𝓕.c.val • (fderiv ℝ P.electricFunction (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)) 1).ofLp ihfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dk:Fin d⊢ Differentiable ℝ fun x_1 => electricField 𝓕.c A x_1 xhfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dk:Fin d⊢ Differentiable ℝ (electricField 𝓕.c A t)]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dk:Fin d⊢ (s.unit.val k • (fderiv ℝ P.electricFunction (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)) 1).ofLp i =
  -(s.unit.val k / 𝓕.c.val) • (-𝓕.c.val • (fderiv ℝ P.electricFunction (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)) 1).ofLp ihfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dk:Fin d⊢ Differentiable ℝ fun x_1 => electricField 𝓕.c A x_1 xhfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dk:Fin d⊢ Differentiable ℝ (electricField 𝓕.c A t)
  simp only [fderiv_eq_smul_deriv, one_smul, PiLp.smul_apply, smul_eq_mul, neg_mul, mul_neg,
    neg_neg]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dk:Fin d⊢ s.unit.val k * (_root_.deriv P.electricFunction (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)).ofLp i =
  s.unit.val k / 𝓕.c.val * (𝓕.c.val * (_root_.deriv P.electricFunction (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)).ofLp i)hfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dk:Fin d⊢ Differentiable ℝ fun x_1 => electricField 𝓕.c A x_1 xhfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dk:Fin d⊢ Differentiable ℝ (electricField 𝓕.c A t)
  field_simphfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dk:Fin d⊢ Differentiable ℝ fun x_1 => electricField 𝓕.c A x_1 xhfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dk:Fin d⊢ Differentiable ℝ (electricField 𝓕.c A t)
  ·hfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dk:Fin d⊢ Differentiable ℝ fun x_1 => electricField 𝓕.c A x_1 x exact electricField_differentiable_time hA xAll goals completed! 🐙
  ·hfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dk:Fin d⊢ Differentiable ℝ (electricField 𝓕.c A t) exact electricField_differentiable_space hA tAll goals completed! 🐙



lemma magneticFieldMatrix_space_deriv_eq_time_deriv {d : ℕ}
    {𝓕 : FreeSpace} {A : ElectromagneticPotential d}
    {s : Direction d} (P : IsPlaneWave 𝓕 A s) (hA : ContDiff ℝ 2 A) (t : Time)
    (x : Space d) (i j : Fin d) (k : Fin d) :
    ∂[k] (A.magneticFieldMatrix 𝓕.c t · (i, j)) x = - (s.unit k / 𝓕.c.val) •
    ∂ₜ (A.magneticFieldMatrix 𝓕.c · x (i, j)) t := byd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ Space.deriv k (fun x => magneticFieldMatrix 𝓕.c A t x (i, j)) x =
  -(s.unit.val k / 𝓕.c.val) • ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, j)) t
  rw [IsPlaneWave.magneticFieldMatrix_space_deriv P hA t x i j k,d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ s.unit.val k • (fderiv ℝ (fun u => P.magneticFunction u (i, j)) (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)) 1 =
  -(s.unit.val k / 𝓕.c.val) • ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, j)) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ s.unit.val k • (fderiv ℝ (fun u => P.magneticFunction u (i, j)) (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)) 1 =
  -(s.unit.val k / 𝓕.c.val) •
    -𝓕.c.val • (fderiv ℝ (fun u => P.magneticFunction u (i, j)) (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)) 1
    IsPlaneWave.magneticFieldMatrix_time_deriv P hA t x i jd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ s.unit.val k • (fderiv ℝ (fun u => P.magneticFunction u (i, j)) (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)) 1 =
  -(s.unit.val k / 𝓕.c.val) •
    -𝓕.c.val • (fderiv ℝ (fun u => P.magneticFunction u (i, j)) (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)) 1d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ s.unit.val k • (fderiv ℝ (fun u => P.magneticFunction u (i, j)) (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)) 1 =
  -(s.unit.val k / 𝓕.c.val) •
    -𝓕.c.val • (fderiv ℝ (fun u => P.magneticFunction u (i, j)) (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)) 1]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ s.unit.val k • (fderiv ℝ (fun u => P.magneticFunction u (i, j)) (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)) 1 =
  -(s.unit.val k / 𝓕.c.val) •
    -𝓕.c.val • (fderiv ℝ (fun u => P.magneticFunction u (i, j)) (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val)) 1
  simp only [fderiv_eq_smul_deriv, smul_eq_mul, one_mul, neg_mul, mul_neg, neg_neg]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ s.unit.val k * _root_.deriv (fun u => P.magneticFunction u (i, j)) (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val) =
  s.unit.val k / 𝓕.c.val *
    (𝓕.c.val * _root_.deriv (fun u => P.magneticFunction u (i, j)) (⟪x, s.unit⟫_ℝ - 𝓕.c.val * t.val))
  field_simpAll goals completed! 🐙

B. The magnetic field in terms of the electric field

B.1. Time derivative of the magnetic field in terms of electric field



lemma time_deriv_magneticFieldMatrix_eq_electricField_mul_propogator {d : ℕ}
    {𝓕 : FreeSpace} {A : ElectromagneticPotential d}
    {s : Direction d} (P : IsPlaneWave 𝓕 A s) (hA : ContDiff ℝ 2 A)
    (t : Time) (x : Space d) (i j : Fin d) :
    ∂ₜ (A.magneticFieldMatrix 𝓕.c · x (i, j)) t =
    ∂ₜ (fun t => s.unit j / 𝓕.c * A.electricField 𝓕.c t x i
    - s.unit i / 𝓕.c * A.electricField 𝓕.c t x j) t := byd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin d⊢ ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, j)) t =
  ∂ₜ
    (fun t =>
      s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i -
        s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j)
    t
  have he : ∀ k, DifferentiableAt ℝ (fun t => A.electricField 𝓕.c t x k) t :=
    fun k => (electricField_apply_differentiable_time hA x k).differentiableAtd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dhe:∀ (k : Fin d), DifferentiableAt ℝ (fun t => (electricField 𝓕.c A t x).ofLp k) t⊢ ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, j)) t =
  ∂ₜ
    (fun t =>
      s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i -
        s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j)
    t
  rw [time_deriv_magneticFieldMatrix A hA,d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dhe:∀ (k : Fin d), DifferentiableAt ℝ (fun t => (electricField 𝓕.c A t x).ofLp k) t⊢ Space.deriv i (fun x => (electricField 𝓕.c A t x).ofLp j) x -
    Space.deriv j (fun x => (electricField 𝓕.c A t x).ofLp i) x =
  ∂ₜ
    (fun t =>
      s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i -
        s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j)
    td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dhe:∀ (k : Fin d), DifferentiableAt ℝ (fun t => (electricField 𝓕.c A t x).ofLp k) t⊢ -(s.unit.val i / 𝓕.c.val) • ∂ₜ (fun x_1 => (electricField 𝓕.c A x_1 x).ofLp j) t -
    -(s.unit.val j / 𝓕.c.val) • ∂ₜ (fun x_1 => (electricField 𝓕.c A x_1 x).ofLp i) t =
  ∂ₜ
    (fun t =>
      s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i -
        s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j)
    t P.electricField_space_deriv_eq_time_deriv hA,d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dhe:∀ (k : Fin d), DifferentiableAt ℝ (fun t => (electricField 𝓕.c A t x).ofLp k) t⊢ -(s.unit.val i / 𝓕.c.val) • ∂ₜ (fun x_1 => (electricField 𝓕.c A x_1 x).ofLp j) t -
    Space.deriv j (fun x => (electricField 𝓕.c A t x).ofLp i) x =
  ∂ₜ
    (fun t =>
      s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i -
        s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j)
    td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dhe:∀ (k : Fin d), DifferentiableAt ℝ (fun t => (electricField 𝓕.c A t x).ofLp k) t⊢ -(s.unit.val i / 𝓕.c.val) • ∂ₜ (fun x_1 => (electricField 𝓕.c A x_1 x).ofLp j) t -
    -(s.unit.val j / 𝓕.c.val) • ∂ₜ (fun x_1 => (electricField 𝓕.c A x_1 x).ofLp i) t =
  ∂ₜ
    (fun t =>
      s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i -
        s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j)
    t
    P.electricField_space_deriv_eq_time_deriv hAd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dhe:∀ (k : Fin d), DifferentiableAt ℝ (fun t => (electricField 𝓕.c A t x).ofLp k) t⊢ -(s.unit.val i / 𝓕.c.val) • ∂ₜ (fun x_1 => (electricField 𝓕.c A x_1 x).ofLp j) t -
    -(s.unit.val j / 𝓕.c.val) • ∂ₜ (fun x_1 => (electricField 𝓕.c A x_1 x).ofLp i) t =
  ∂ₜ
    (fun t =>
      s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i -
        s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j)
    td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dhe:∀ (k : Fin d), DifferentiableAt ℝ (fun t => (electricField 𝓕.c A t x).ofLp k) t⊢ -(s.unit.val i / 𝓕.c.val) • ∂ₜ (fun x_1 => (electricField 𝓕.c A x_1 x).ofLp j) t -
    -(s.unit.val j / 𝓕.c.val) • ∂ₜ (fun x_1 => (electricField 𝓕.c A x_1 x).ofLp i) t =
  ∂ₜ
    (fun t =>
      s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i -
        s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j)
    t]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dhe:∀ (k : Fin d), DifferentiableAt ℝ (fun t => (electricField 𝓕.c A t x).ofLp k) t⊢ -(s.unit.val i / 𝓕.c.val) • ∂ₜ (fun x_1 => (electricField 𝓕.c A x_1 x).ofLp j) t -
    -(s.unit.val j / 𝓕.c.val) • ∂ₜ (fun x_1 => (electricField 𝓕.c A x_1 x).ofLp i) t =
  ∂ₜ
    (fun t =>
      s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i -
        s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j)
    t
  conv_rhs =>
    rw [Time.deriv_eq, fderiv_fun_sub ((he i).const_mul _) ((he j).const_mul _),
      fderiv_const_mul (he i), fderiv_const_mul (he j)]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dhe:∀ (k : Fin d), DifferentiableAt ℝ (fun t => (electricField 𝓕.c A t x).ofLp k) t| ((s.unit.val j / 𝓕.c.val) • fderiv ℝ (fun t => (electricField 𝓕.c A t x).ofLp i) t -
    (s.unit.val i / 𝓕.c.val) • fderiv ℝ (fun t => (electricField 𝓕.c A t x).ofLp j) t)
  1
  simp [← Time.deriv_eq]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dhe:∀ (k : Fin d), DifferentiableAt ℝ (fun t => (electricField 𝓕.c A t x).ofLp k) t⊢ -(s.unit.val i / 𝓕.c.val * ∂ₜ (fun x_1 => (electricField 𝓕.c A x_1 x).ofLp j) t) +
    s.unit.val j / 𝓕.c.val * ∂ₜ (fun x_1 => (electricField 𝓕.c A x_1 x).ofLp i) t =
  s.unit.val j / 𝓕.c.val * ∂ₜ (fun x_1 => (electricField 𝓕.c A x_1 x).ofLp i) t -
    s.unit.val i / 𝓕.c.val * ∂ₜ (fun x_1 => (electricField 𝓕.c A x_1 x).ofLp j) t
  field_simpd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dhe:∀ (k : Fin d), DifferentiableAt ℝ (fun t => (electricField 𝓕.c A t x).ofLp k) t⊢ -(s.unit.val i * ∂ₜ (fun x_1 => (electricField 𝓕.c A x_1 x).ofLp j) t) +
    s.unit.val j * ∂ₜ (fun x_1 => (electricField 𝓕.c A x_1 x).ofLp i) t =
  s.unit.val j * ∂ₜ (fun x_1 => (electricField 𝓕.c A x_1 x).ofLp i) t -
    s.unit.val i * ∂ₜ (fun x_1 => (electricField 𝓕.c A x_1 x).ofLp j) t
  ringAll goals completed! 🐙

B.2. Space derivative of the magnetic field in terms of electric field



lemma space_deriv_magneticFieldMatrix_eq_electricField_mul_propogator {d : ℕ}
    {𝓕 : FreeSpace} {A : ElectromagneticPotential d}
    {s : Direction d} (P : IsPlaneWave 𝓕 A s) (hA : ContDiff ℝ 2 A)
    (t : Time) (x : Space d) (i j k : Fin d) :
    ∂[k] (A.magneticFieldMatrix 𝓕.c t · (i, j)) x =
    ∂[k] (fun x => s.unit j / 𝓕.c * A.electricField 𝓕.c t x i
    - s.unit i / 𝓕.c * A.electricField 𝓕.c t x j) x := byd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ Space.deriv k (fun x => magneticFieldMatrix 𝓕.c A t x (i, j)) x =
  Space.deriv k
    (fun x =>
      s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i -
        s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j)
    x
  rw [P.magneticFieldMatrix_space_deriv_eq_time_deriv hA,d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ -(s.unit.val k / 𝓕.c.val) • ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, j)) t =
  Space.deriv k
    (fun x =>
      s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i -
        s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j)
    xd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ -(s.unit.val k / 𝓕.c.val) •
    ∂ₜ
      (fun t =>
        s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i -
          s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j)
      t =
  ((s.unit.val j / 𝓕.c.val) • fderiv ℝ (fun x => (electricField 𝓕.c A t x).ofLp i) x -
      (s.unit.val i / 𝓕.c.val) • fderiv ℝ (fun x => (electricField 𝓕.c A t x).ofLp j) x)
    (Space.basis k)had:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => (electricField 𝓕.c A t x).ofLp j) xhad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => (electricField 𝓕.c A t x).ofLp i) xhfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i) xhgd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j) x
    P.time_deriv_magneticFieldMatrix_eq_electricField_mul_propogator hA,d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ -(s.unit.val k / 𝓕.c.val) •
    ∂ₜ
      (fun t =>
        s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i -
          s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j)
      t =
  Space.deriv k
    (fun x =>
      s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i -
        s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j)
    xd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ -(s.unit.val k / 𝓕.c.val) •
    ∂ₜ
      (fun t =>
        s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i -
          s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j)
      t =
  ((s.unit.val j / 𝓕.c.val) • fderiv ℝ (fun x => (electricField 𝓕.c A t x).ofLp i) x -
      (s.unit.val i / 𝓕.c.val) • fderiv ℝ (fun x => (electricField 𝓕.c A t x).ofLp j) x)
    (Space.basis k)had:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => (electricField 𝓕.c A t x).ofLp j) xhad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => (electricField 𝓕.c A t x).ofLp i) xhfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i) xhgd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j) x
    Space.deriv_eq_fderiv_basis,d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ -(s.unit.val k / 𝓕.c.val) •
    ∂ₜ
      (fun t =>
        s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i -
          s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j)
      t =
  (fderiv ℝ
      (fun x =>
        s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i -
          s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j)
      x)
    (Space.basis k)d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ -(s.unit.val k / 𝓕.c.val) •
    ∂ₜ
      (fun t =>
        s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i -
          s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j)
      t =
  ((s.unit.val j / 𝓕.c.val) • fderiv ℝ (fun x => (electricField 𝓕.c A t x).ofLp i) x -
      (s.unit.val i / 𝓕.c.val) • fderiv ℝ (fun x => (electricField 𝓕.c A t x).ofLp j) x)
    (Space.basis k)had:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => (electricField 𝓕.c A t x).ofLp j) xhad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => (electricField 𝓕.c A t x).ofLp i) xhfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i) xhgd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j) x fderiv_fun_sub,d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ -(s.unit.val k / 𝓕.c.val) •
    ∂ₜ
      (fun t =>
        s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i -
          s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j)
      t =
  (fderiv ℝ (fun x => s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i) x -
      fderiv ℝ (fun x => s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j) x)
    (Space.basis k)hfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i) xhgd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j) xd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ -(s.unit.val k / 𝓕.c.val) •
    ∂ₜ
      (fun t =>
        s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i -
          s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j)
      t =
  ((s.unit.val j / 𝓕.c.val) • fderiv ℝ (fun x => (electricField 𝓕.c A t x).ofLp i) x -
      (s.unit.val i / 𝓕.c.val) • fderiv ℝ (fun x => (electricField 𝓕.c A t x).ofLp j) x)
    (Space.basis k)had:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => (electricField 𝓕.c A t x).ofLp j) xhad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => (electricField 𝓕.c A t x).ofLp i) xhfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i) xhgd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j) x fderiv_const_mul,d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ -(s.unit.val k / 𝓕.c.val) •
    ∂ₜ
      (fun t =>
        s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i -
          s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j)
      t =
  ((s.unit.val j / 𝓕.c.val) • fderiv ℝ (fun x => (electricField 𝓕.c A t x).ofLp i) x -
      fderiv ℝ (fun x => s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j) x)
    (Space.basis k)had:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => (electricField 𝓕.c A t x).ofLp i) xhfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i) xhgd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j) xd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ -(s.unit.val k / 𝓕.c.val) •
    ∂ₜ
      (fun t =>
        s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i -
          s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j)
      t =
  ((s.unit.val j / 𝓕.c.val) • fderiv ℝ (fun x => (electricField 𝓕.c A t x).ofLp i) x -
      (s.unit.val i / 𝓕.c.val) • fderiv ℝ (fun x => (electricField 𝓕.c A t x).ofLp j) x)
    (Space.basis k)had:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => (electricField 𝓕.c A t x).ofLp j) xhad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => (electricField 𝓕.c A t x).ofLp i) xhfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i) xhgd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j) x fderiv_const_muld:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ -(s.unit.val k / 𝓕.c.val) •
    ∂ₜ
      (fun t =>
        s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i -
          s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j)
      t =
  ((s.unit.val j / 𝓕.c.val) • fderiv ℝ (fun x => (electricField 𝓕.c A t x).ofLp i) x -
      (s.unit.val i / 𝓕.c.val) • fderiv ℝ (fun x => (electricField 𝓕.c A t x).ofLp j) x)
    (Space.basis k)had:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => (electricField 𝓕.c A t x).ofLp j) xhad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => (electricField 𝓕.c A t x).ofLp i) xhfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i) xhgd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j) xd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ -(s.unit.val k / 𝓕.c.val) •
    ∂ₜ
      (fun t =>
        s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i -
          s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j)
      t =
  ((s.unit.val j / 𝓕.c.val) • fderiv ℝ (fun x => (electricField 𝓕.c A t x).ofLp i) x -
      (s.unit.val i / 𝓕.c.val) • fderiv ℝ (fun x => (electricField 𝓕.c A t x).ofLp j) x)
    (Space.basis k)had:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => (electricField 𝓕.c A t x).ofLp j) xhad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => (electricField 𝓕.c A t x).ofLp i) xhfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i) xhgd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j) x]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ -(s.unit.val k / 𝓕.c.val) •
    ∂ₜ
      (fun t =>
        s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i -
          s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j)
      t =
  ((s.unit.val j / 𝓕.c.val) • fderiv ℝ (fun x => (electricField 𝓕.c A t x).ofLp i) x -
      (s.unit.val i / 𝓕.c.val) • fderiv ℝ (fun x => (electricField 𝓕.c A t x).ofLp j) x)
    (Space.basis k)had:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => (electricField 𝓕.c A t x).ofLp j) xhad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => (electricField 𝓕.c A t x).ofLp i) xhfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i) xhgd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j) x
  simp [← Space.deriv_eq_fderiv_basis]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ -(s.unit.val k / 𝓕.c.val *
      ∂ₜ
        (fun t =>
          s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i -
            s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j)
        t) =
  s.unit.val j / 𝓕.c.val * Space.deriv k (fun x => (electricField 𝓕.c A t x).ofLp i) x -
    s.unit.val i / 𝓕.c.val * Space.deriv k (fun x => (electricField 𝓕.c A t x).ofLp j) xhad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => (electricField 𝓕.c A t x).ofLp j) xhad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => (electricField 𝓕.c A t x).ofLp i) xhfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i) xhgd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j) x
  rw [Time.deriv_eq,d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ -(s.unit.val k / 𝓕.c.val *
      (fderiv ℝ
          (fun t =>
            s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i -
              s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j)
          t)
        1) =
  s.unit.val j / 𝓕.c.val * Space.deriv k (fun x => (electricField 𝓕.c A t x).ofLp i) x -
    s.unit.val i / 𝓕.c.val * Space.deriv k (fun x => (electricField 𝓕.c A t x).ofLp j) xhad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => (electricField 𝓕.c A t x).ofLp j) xhad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => (electricField 𝓕.c A t x).ofLp i) xhfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i) xhgd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j) xd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ -(s.unit.val k / 𝓕.c.val *
      ((s.unit.val j / 𝓕.c.val) • fderiv ℝ (fun t => (electricField 𝓕.c A t x).ofLp i) t -
          (s.unit.val i / 𝓕.c.val) • fderiv ℝ (fun t => (electricField 𝓕.c A t x).ofLp j) t)
        1) =
  s.unit.val j / 𝓕.c.val * Space.deriv k (fun x => (electricField 𝓕.c A t x).ofLp i) x -
    s.unit.val i / 𝓕.c.val * Space.deriv k (fun x => (electricField 𝓕.c A t x).ofLp j) xhad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun t => (electricField 𝓕.c A t x).ofLp j) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun t => (electricField 𝓕.c A t x).ofLp i) thfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun t => s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i) thgd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun t => s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => (electricField 𝓕.c A t x).ofLp j) xhad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => (electricField 𝓕.c A t x).ofLp i) xhfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i) xhgd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j) x fderiv_fun_sub,d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ -(s.unit.val k / 𝓕.c.val *
      (fderiv ℝ (fun t => s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i) t -
          fderiv ℝ (fun t => s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j) t)
        1) =
  s.unit.val j / 𝓕.c.val * Space.deriv k (fun x => (electricField 𝓕.c A t x).ofLp i) x -
    s.unit.val i / 𝓕.c.val * Space.deriv k (fun x => (electricField 𝓕.c A t x).ofLp j) xhfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun t => s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i) thgd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun t => s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => (electricField 𝓕.c A t x).ofLp j) xhad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => (electricField 𝓕.c A t x).ofLp i) xhfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i) xhgd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j) xd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ -(s.unit.val k / 𝓕.c.val *
      ((s.unit.val j / 𝓕.c.val) • fderiv ℝ (fun t => (electricField 𝓕.c A t x).ofLp i) t -
          (s.unit.val i / 𝓕.c.val) • fderiv ℝ (fun t => (electricField 𝓕.c A t x).ofLp j) t)
        1) =
  s.unit.val j / 𝓕.c.val * Space.deriv k (fun x => (electricField 𝓕.c A t x).ofLp i) x -
    s.unit.val i / 𝓕.c.val * Space.deriv k (fun x => (electricField 𝓕.c A t x).ofLp j) xhad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun t => (electricField 𝓕.c A t x).ofLp j) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun t => (electricField 𝓕.c A t x).ofLp i) thfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun t => s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i) thgd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun t => s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => (electricField 𝓕.c A t x).ofLp j) xhad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => (electricField 𝓕.c A t x).ofLp i) xhfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i) xhgd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j) x fderiv_const_mul,d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ -(s.unit.val k / 𝓕.c.val *
      ((s.unit.val j / 𝓕.c.val) • fderiv ℝ (fun t => (electricField 𝓕.c A t x).ofLp i) t -
          fderiv ℝ (fun t => s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j) t)
        1) =
  s.unit.val j / 𝓕.c.val * Space.deriv k (fun x => (electricField 𝓕.c A t x).ofLp i) x -
    s.unit.val i / 𝓕.c.val * Space.deriv k (fun x => (electricField 𝓕.c A t x).ofLp j) xhad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun t => (electricField 𝓕.c A t x).ofLp i) thfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun t => s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i) thgd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun t => s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => (electricField 𝓕.c A t x).ofLp j) xhad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => (electricField 𝓕.c A t x).ofLp i) xhfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i) xhgd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j) xd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ -(s.unit.val k / 𝓕.c.val *
      ((s.unit.val j / 𝓕.c.val) • fderiv ℝ (fun t => (electricField 𝓕.c A t x).ofLp i) t -
          (s.unit.val i / 𝓕.c.val) • fderiv ℝ (fun t => (electricField 𝓕.c A t x).ofLp j) t)
        1) =
  s.unit.val j / 𝓕.c.val * Space.deriv k (fun x => (electricField 𝓕.c A t x).ofLp i) x -
    s.unit.val i / 𝓕.c.val * Space.deriv k (fun x => (electricField 𝓕.c A t x).ofLp j) xhad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun t => (electricField 𝓕.c A t x).ofLp j) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun t => (electricField 𝓕.c A t x).ofLp i) thfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun t => s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i) thgd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun t => s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => (electricField 𝓕.c A t x).ofLp j) xhad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => (electricField 𝓕.c A t x).ofLp i) xhfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i) xhgd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j) x fderiv_const_muld:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ -(s.unit.val k / 𝓕.c.val *
      ((s.unit.val j / 𝓕.c.val) • fderiv ℝ (fun t => (electricField 𝓕.c A t x).ofLp i) t -
          (s.unit.val i / 𝓕.c.val) • fderiv ℝ (fun t => (electricField 𝓕.c A t x).ofLp j) t)
        1) =
  s.unit.val j / 𝓕.c.val * Space.deriv k (fun x => (electricField 𝓕.c A t x).ofLp i) x -
    s.unit.val i / 𝓕.c.val * Space.deriv k (fun x => (electricField 𝓕.c A t x).ofLp j) xhad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun t => (electricField 𝓕.c A t x).ofLp j) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun t => (electricField 𝓕.c A t x).ofLp i) thfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun t => s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i) thgd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun t => s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => (electricField 𝓕.c A t x).ofLp j) xhad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => (electricField 𝓕.c A t x).ofLp i) xhfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i) xhgd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j) xd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ -(s.unit.val k / 𝓕.c.val *
      ((s.unit.val j / 𝓕.c.val) • fderiv ℝ (fun t => (electricField 𝓕.c A t x).ofLp i) t -
          (s.unit.val i / 𝓕.c.val) • fderiv ℝ (fun t => (electricField 𝓕.c A t x).ofLp j) t)
        1) =
  s.unit.val j / 𝓕.c.val * Space.deriv k (fun x => (electricField 𝓕.c A t x).ofLp i) x -
    s.unit.val i / 𝓕.c.val * Space.deriv k (fun x => (electricField 𝓕.c A t x).ofLp j) xhad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun t => (electricField 𝓕.c A t x).ofLp j) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun t => (electricField 𝓕.c A t x).ofLp i) thfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun t => s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i) thgd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun t => s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => (electricField 𝓕.c A t x).ofLp j) xhad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => (electricField 𝓕.c A t x).ofLp i) xhfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i) xhgd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j) x]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ -(s.unit.val k / 𝓕.c.val *
      ((s.unit.val j / 𝓕.c.val) • fderiv ℝ (fun t => (electricField 𝓕.c A t x).ofLp i) t -
          (s.unit.val i / 𝓕.c.val) • fderiv ℝ (fun t => (electricField 𝓕.c A t x).ofLp j) t)
        1) =
  s.unit.val j / 𝓕.c.val * Space.deriv k (fun x => (electricField 𝓕.c A t x).ofLp i) x -
    s.unit.val i / 𝓕.c.val * Space.deriv k (fun x => (electricField 𝓕.c A t x).ofLp j) xhad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun t => (electricField 𝓕.c A t x).ofLp j) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun t => (electricField 𝓕.c A t x).ofLp i) thfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun t => s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i) thgd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun t => s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => (electricField 𝓕.c A t x).ofLp j) xhad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => (electricField 𝓕.c A t x).ofLp i) xhfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i) xhgd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j) x
  simp [← Time.deriv_eq]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ -(s.unit.val k / 𝓕.c.val *
      (s.unit.val j / 𝓕.c.val * ∂ₜ (fun t => (electricField 𝓕.c A t x).ofLp i) t -
        s.unit.val i / 𝓕.c.val * ∂ₜ (fun t => (electricField 𝓕.c A t x).ofLp j) t)) =
  s.unit.val j / 𝓕.c.val * Space.deriv k (fun x => (electricField 𝓕.c A t x).ofLp i) x -
    s.unit.val i / 𝓕.c.val * Space.deriv k (fun x => (electricField 𝓕.c A t x).ofLp j) xhad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun t => (electricField 𝓕.c A t x).ofLp j) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun t => (electricField 𝓕.c A t x).ofLp i) thfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun t => s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i) thgd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun t => s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => (electricField 𝓕.c A t x).ofLp j) xhad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => (electricField 𝓕.c A t x).ofLp i) xhfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i) xhgd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j) x
  rw [P.electricField_space_deriv_eq_time_deriv hA,d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ -(s.unit.val k / 𝓕.c.val *
      (s.unit.val j / 𝓕.c.val * ∂ₜ (fun t => (electricField 𝓕.c A t x).ofLp i) t -
        s.unit.val i / 𝓕.c.val * ∂ₜ (fun t => (electricField 𝓕.c A t x).ofLp j) t)) =
  s.unit.val j / 𝓕.c.val * -(s.unit.val k / 𝓕.c.val) • ∂ₜ (fun x_1 => (electricField 𝓕.c A x_1 x).ofLp i) t -
    s.unit.val i / 𝓕.c.val * Space.deriv k (fun x => (electricField 𝓕.c A t x).ofLp j) xhad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun t => (electricField 𝓕.c A t x).ofLp j) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun t => (electricField 𝓕.c A t x).ofLp i) thfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun t => s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i) thgd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun t => s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => (electricField 𝓕.c A t x).ofLp j) xhad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => (electricField 𝓕.c A t x).ofLp i) xhfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i) xhgd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j) xd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ -(s.unit.val k / 𝓕.c.val *
      (s.unit.val j / 𝓕.c.val * ∂ₜ (fun t => (electricField 𝓕.c A t x).ofLp i) t -
        s.unit.val i / 𝓕.c.val * ∂ₜ (fun t => (electricField 𝓕.c A t x).ofLp j) t)) =
  s.unit.val j / 𝓕.c.val * -(s.unit.val k / 𝓕.c.val) • ∂ₜ (fun x_1 => (electricField 𝓕.c A x_1 x).ofLp i) t -
    s.unit.val i / 𝓕.c.val * -(s.unit.val k / 𝓕.c.val) • ∂ₜ (fun x_1 => (electricField 𝓕.c A x_1 x).ofLp j) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun t => (electricField 𝓕.c A t x).ofLp j) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun t => (electricField 𝓕.c A t x).ofLp i) thfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun t => s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i) thgd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun t => s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => (electricField 𝓕.c A t x).ofLp j) xhad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => (electricField 𝓕.c A t x).ofLp i) xhfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i) xhgd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j) x P.electricField_space_deriv_eq_time_deriv hAd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ -(s.unit.val k / 𝓕.c.val *
      (s.unit.val j / 𝓕.c.val * ∂ₜ (fun t => (electricField 𝓕.c A t x).ofLp i) t -
        s.unit.val i / 𝓕.c.val * ∂ₜ (fun t => (electricField 𝓕.c A t x).ofLp j) t)) =
  s.unit.val j / 𝓕.c.val * -(s.unit.val k / 𝓕.c.val) • ∂ₜ (fun x_1 => (electricField 𝓕.c A x_1 x).ofLp i) t -
    s.unit.val i / 𝓕.c.val * -(s.unit.val k / 𝓕.c.val) • ∂ₜ (fun x_1 => (electricField 𝓕.c A x_1 x).ofLp j) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun t => (electricField 𝓕.c A t x).ofLp j) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun t => (electricField 𝓕.c A t x).ofLp i) thfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun t => s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i) thgd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun t => s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => (electricField 𝓕.c A t x).ofLp j) xhad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => (electricField 𝓕.c A t x).ofLp i) xhfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i) xhgd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j) xd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ -(s.unit.val k / 𝓕.c.val *
      (s.unit.val j / 𝓕.c.val * ∂ₜ (fun t => (electricField 𝓕.c A t x).ofLp i) t -
        s.unit.val i / 𝓕.c.val * ∂ₜ (fun t => (electricField 𝓕.c A t x).ofLp j) t)) =
  s.unit.val j / 𝓕.c.val * -(s.unit.val k / 𝓕.c.val) • ∂ₜ (fun x_1 => (electricField 𝓕.c A x_1 x).ofLp i) t -
    s.unit.val i / 𝓕.c.val * -(s.unit.val k / 𝓕.c.val) • ∂ₜ (fun x_1 => (electricField 𝓕.c A x_1 x).ofLp j) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun t => (electricField 𝓕.c A t x).ofLp j) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun t => (electricField 𝓕.c A t x).ofLp i) thfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun t => s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i) thgd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun t => s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => (electricField 𝓕.c A t x).ofLp j) xhad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => (electricField 𝓕.c A t x).ofLp i) xhfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i) xhgd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j) x]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ -(s.unit.val k / 𝓕.c.val *
      (s.unit.val j / 𝓕.c.val * ∂ₜ (fun t => (electricField 𝓕.c A t x).ofLp i) t -
        s.unit.val i / 𝓕.c.val * ∂ₜ (fun t => (electricField 𝓕.c A t x).ofLp j) t)) =
  s.unit.val j / 𝓕.c.val * -(s.unit.val k / 𝓕.c.val) • ∂ₜ (fun x_1 => (electricField 𝓕.c A x_1 x).ofLp i) t -
    s.unit.val i / 𝓕.c.val * -(s.unit.val k / 𝓕.c.val) • ∂ₜ (fun x_1 => (electricField 𝓕.c A x_1 x).ofLp j) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun t => (electricField 𝓕.c A t x).ofLp j) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun t => (electricField 𝓕.c A t x).ofLp i) thfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun t => s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i) thgd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun t => s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => (electricField 𝓕.c A t x).ofLp j) xhad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => (electricField 𝓕.c A t x).ofLp i) xhfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i) xhgd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j) x
  simp only [smul_eq_mul, neg_mul, mul_neg, sub_neg_eq_add]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ -(s.unit.val k / 𝓕.c.val *
      (s.unit.val j / 𝓕.c.val * ∂ₜ (fun t => (electricField 𝓕.c A t x).ofLp i) t -
        s.unit.val i / 𝓕.c.val * ∂ₜ (fun t => (electricField 𝓕.c A t x).ofLp j) t)) =
  -(s.unit.val j / 𝓕.c.val * (s.unit.val k / 𝓕.c.val * ∂ₜ (fun t => (electricField 𝓕.c A t x).ofLp i) t)) +
    s.unit.val i / 𝓕.c.val * (s.unit.val k / 𝓕.c.val * ∂ₜ (fun t => (electricField 𝓕.c A t x).ofLp j) t)had:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun t => (electricField 𝓕.c A t x).ofLp j) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun t => (electricField 𝓕.c A t x).ofLp i) thfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun t => s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i) thgd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun t => s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => (electricField 𝓕.c A t x).ofLp j) xhad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => (electricField 𝓕.c A t x).ofLp i) xhfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i) xhgd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j) x
  field_simpd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ -(s.unit.val k *
      (s.unit.val j * ∂ₜ (fun t => (electricField 𝓕.c A t x).ofLp i) t -
        s.unit.val i * ∂ₜ (fun t => (electricField 𝓕.c A t x).ofLp j) t)) =
  s.unit.val k *
    (-(s.unit.val j * ∂ₜ (fun t => (electricField 𝓕.c A t x).ofLp i) t) +
      s.unit.val i * ∂ₜ (fun t => (electricField 𝓕.c A t x).ofLp j) t)had:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun t => (electricField 𝓕.c A t x).ofLp j) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun t => (electricField 𝓕.c A t x).ofLp i) thfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun t => s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i) thgd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun t => s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => (electricField 𝓕.c A t x).ofLp j) xhad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => (electricField 𝓕.c A t x).ofLp i) xhfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i) xhgd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j) x
  ringhad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun t => (electricField 𝓕.c A t x).ofLp j) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun t => (electricField 𝓕.c A t x).ofLp i) thfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun t => s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i) thgd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun t => s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => (electricField 𝓕.c A t x).ofLp j) xhad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => (electricField 𝓕.c A t x).ofLp i) xhfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i) xhgd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ DifferentiableAt ℝ (fun x => s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j) x
  any_goals apply Differentiable.differentiableAthgd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ Differentiable ℝ fun x => s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j
  any_goals apply Differentiable.const_mulhgd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ Differentiable ℝ fun y => (electricField 𝓕.c A t y).ofLp j
  any_goals exact electricField_apply_differentiable_time hA x _hgd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.valt:Timex:Space di:Fin dj:Fin dk:Fin d⊢ Differentiable ℝ fun y => (electricField 𝓕.c A t y).ofLp j
  any_goals exact electricField_apply_differentiable_space hA t _All goals completed! 🐙

B.3. Magnetic field equal propogator cross electric field up to constant



lemma magneticFieldMatrix_eq_propogator_cross_electricField {d : ℕ}
    {𝓕 : FreeSpace} {A : ElectromagneticPotential d}
    {s : Direction d} (P : IsPlaneWave 𝓕 A s) (hA : ContDiff ℝ 2 A) (i j : Fin d) :
    ∃ C, ∀ t x, A.magneticFieldMatrix 𝓕.c t x (i, j) =
    1/ 𝓕.c * (s.unit j * A.electricField 𝓕.c t x i -
      s.unit i * A.electricField 𝓕.c t x j) + C := byd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.vali:Fin dj:Fin d⊢ ∃ C,
  ∀ (t : Time) (x : Space d),
    magneticFieldMatrix 𝓕.c A t x (i, j) =
      1 / 𝓕.c.val *
          (s.unit.val j * (electricField 𝓕.c A t x).ofLp i - s.unit.val i * (electricField 𝓕.c A t x).ofLp j) +
        C
  apply Space.equal_up_to_const_of_deriv_eqhfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.vali:Fin dj:Fin d⊢ Differentiable ℝ ↿fun t x => magneticFieldMatrix 𝓕.c A t x (i, j)hgd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.vali:Fin dj:Fin d⊢ Differentiable ℝ
  ↿fun t x =>
    1 / 𝓕.c.val * (s.unit.val j * (electricField 𝓕.c A t x).ofLp i - s.unit.val i * (electricField 𝓕.c A t x).ofLp j)h₁d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.vali:Fin dj:Fin d⊢ ∀ (t : Time) (x : Space d),
  ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, j)) t =
    ∂ₜ
      (fun x_1 =>
        1 / 𝓕.c.val *
          (s.unit.val j * (electricField 𝓕.c A x_1 x).ofLp i - s.unit.val i * (electricField 𝓕.c A x_1 x).ofLp j))
      th₂d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.vali:Fin dj:Fin d⊢ ∀ (t : Time) (x : Space d) (i_1 : Fin d),
  Space.deriv i_1 (fun x => magneticFieldMatrix 𝓕.c A t x (i, j)) x =
    Space.deriv i_1
      (fun x =>
        1 / 𝓕.c.val *
          (s.unit.val j * (electricField 𝓕.c A t x).ofLp i - s.unit.val i * (electricField 𝓕.c A t x).ofLp j))
      x
  ·hfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.vali:Fin dj:Fin d⊢ Differentiable ℝ ↿fun t x => magneticFieldMatrix 𝓕.c A t x (i, j) exact magneticFieldMatrix_differentiable A hA (i, j)All goals completed! 🐙
  ·hgd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.vali:Fin dj:Fin d⊢ Differentiable ℝ
  ↿fun t x =>
    1 / 𝓕.c.val * (s.unit.val j * (electricField 𝓕.c A t x).ofLp i - s.unit.val i * (electricField 𝓕.c A t x).ofLp j) exact (((electricField_apply_differentiable hA).const_mul _).sub
      ((electricField_apply_differentiable hA).const_mul _)).const_mul _All goals completed! 🐙
  ·h₁d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.vali:Fin dj:Fin d⊢ ∀ (t : Time) (x : Space d),
  ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, j)) t =
    ∂ₜ
      (fun x_1 =>
        1 / 𝓕.c.val *
          (s.unit.val j * (electricField 𝓕.c A x_1 x).ofLp i - s.unit.val i * (electricField 𝓕.c A x_1 x).ofLp j))
      t intro t xh₁d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.vali:Fin dj:Fin dt:Timex:Space d⊢ ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, j)) t =
  ∂ₜ
    (fun x_1 =>
      1 / 𝓕.c.val *
        (s.unit.val j * (electricField 𝓕.c A x_1 x).ofLp i - s.unit.val i * (electricField 𝓕.c A x_1 x).ofLp j))
    t
    rw [P.time_deriv_magneticFieldMatrix_eq_electricField_mul_propogator hA t x i jh₁d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.vali:Fin dj:Fin dt:Timex:Space d⊢ ∂ₜ
    (fun t =>
      s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i -
        s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j)
    t =
  ∂ₜ
    (fun x_1 =>
      1 / 𝓕.c.val *
        (s.unit.val j * (electricField 𝓕.c A x_1 x).ofLp i - s.unit.val i * (electricField 𝓕.c A x_1 x).ofLp j))
    th₁d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.vali:Fin dj:Fin dt:Timex:Space d⊢ ∂ₜ
    (fun t =>
      s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i -
        s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j)
    t =
  ∂ₜ
    (fun x_1 =>
      1 / 𝓕.c.val *
        (s.unit.val j * (electricField 𝓕.c A x_1 x).ofLp i - s.unit.val i * (electricField 𝓕.c A x_1 x).ofLp j))
    t]h₁d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.vali:Fin dj:Fin dt:Timex:Space d⊢ ∂ₜ
    (fun t =>
      s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i -
        s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j)
    t =
  ∂ₜ
    (fun x_1 =>
      1 / 𝓕.c.val *
        (s.unit.val j * (electricField 𝓕.c A x_1 x).ofLp i - s.unit.val i * (electricField 𝓕.c A x_1 x).ofLp j))
    t
    congrh₁.e_fd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.vali:Fin dj:Fin dt:Timex:Space d⊢ (fun t =>
    s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i -
      s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j) =
  fun x_1 =>
  1 / 𝓕.c.val * (s.unit.val j * (electricField 𝓕.c A x_1 x).ofLp i - s.unit.val i * (electricField 𝓕.c A x_1 x).ofLp j)
    funext th₁.e_fd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.vali:Fin dj:Fin dt✝:Timex:Space dt:Time⊢ s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i - s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j =
  1 / 𝓕.c.val * (s.unit.val j * (electricField 𝓕.c A t x).ofLp i - s.unit.val i * (electricField 𝓕.c A t x).ofLp j)
    field_simpAll goals completed! 🐙
  ·h₂d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.vali:Fin dj:Fin d⊢ ∀ (t : Time) (x : Space d) (i_1 : Fin d),
  Space.deriv i_1 (fun x => magneticFieldMatrix 𝓕.c A t x (i, j)) x =
    Space.deriv i_1
      (fun x =>
        1 / 𝓕.c.val *
          (s.unit.val j * (electricField 𝓕.c A t x).ofLp i - s.unit.val i * (electricField 𝓕.c A t x).ofLp j))
      x intro t x kh₂d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.vali:Fin dj:Fin dt:Timex:Space dk:Fin d⊢ Space.deriv k (fun x => magneticFieldMatrix 𝓕.c A t x (i, j)) x =
  Space.deriv k
    (fun x =>
      1 / 𝓕.c.val * (s.unit.val j * (electricField 𝓕.c A t x).ofLp i - s.unit.val i * (electricField 𝓕.c A t x).ofLp j))
    x
    rw [P.space_deriv_magneticFieldMatrix_eq_electricField_mul_propogator hA t x i jh₂d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.vali:Fin dj:Fin dt:Timex:Space dk:Fin d⊢ Space.deriv k
    (fun x =>
      s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i -
        s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j)
    x =
  Space.deriv k
    (fun x =>
      1 / 𝓕.c.val * (s.unit.val j * (electricField 𝓕.c A t x).ofLp i - s.unit.val i * (electricField 𝓕.c A t x).ofLp j))
    xh₂d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.vali:Fin dj:Fin dt:Timex:Space dk:Fin d⊢ Space.deriv k
    (fun x =>
      s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i -
        s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j)
    x =
  Space.deriv k
    (fun x =>
      1 / 𝓕.c.val * (s.unit.val j * (electricField 𝓕.c A t x).ofLp i - s.unit.val i * (electricField 𝓕.c A t x).ofLp j))
    x]h₂d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.vali:Fin dj:Fin dt:Timex:Space dk:Fin d⊢ Space.deriv k
    (fun x =>
      s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i -
        s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j)
    x =
  Space.deriv k
    (fun x =>
      1 / 𝓕.c.val * (s.unit.val j * (electricField 𝓕.c A t x).ofLp i - s.unit.val i * (electricField 𝓕.c A t x).ofLp j))
    x
    congrh₂.e_fd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.vali:Fin dj:Fin dt:Timex:Space dk:Fin d⊢ (fun x =>
    s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i -
      s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j) =
  fun x =>
  1 / 𝓕.c.val * (s.unit.val j * (electricField 𝓕.c A t x).ofLp i - s.unit.val i * (electricField 𝓕.c A t x).ofLp j)
    funext xh₂.e_fd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ 2 A.vali:Fin dj:Fin dt:Timex✝:Space dk:Fin dx:Space d⊢ s.unit.val j / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp i - s.unit.val i / 𝓕.c.val * (electricField 𝓕.c A t x).ofLp j =
  1 / 𝓕.c.val * (s.unit.val j * (electricField 𝓕.c A t x).ofLp i - s.unit.val i * (electricField 𝓕.c A t x).ofLp j)
    field_simpAll goals completed! 🐙

C. The electric field in terms of the magnetic field

C.1. The time derivative of the electric field in terms of magnetic field



lemma time_deriv_electricField_eq_magneticFieldMatrix {d : ℕ}
    {𝓕 : FreeSpace} {A : ElectromagneticPotential d}
    {s : Direction d} (P : IsPlaneWave 𝓕 A s) (hA : ContDiff ℝ ∞ A)
    (h : IsExtrema 𝓕 A 0)
    (t : Time) (x : Space d) (i : Fin d) :
    ∂ₜ (A.electricField 𝓕.c · x i) t =
    ∂ₜ (fun t => 𝓕.c * ∑ j, A.magneticFieldMatrix 𝓕.c t x (i, j) * s.unit j) t := byd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin d⊢ ∂ₜ (fun x_1 => (electricField 𝓕.c A x_1 x).ofLp i) t =
  ∂ₜ (fun t => 𝓕.c.val * ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t
  have hBd : ∀ k, Differentiable ℝ (fun t => A.magneticFieldMatrix 𝓕.c t x (i, k)) :=
    fun k => magneticFieldMatrix_differentiable_time A (hA.of_le ENat.LEInfty.out) x (i, k)d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ ∂ₜ (fun x_1 => (electricField 𝓕.c A x_1 x).ofLp i) t =
  ∂ₜ (fun t => 𝓕.c.val * ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t
  rw [Time.deriv_euclid,d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ (∂ₜ (fun t => electricField 𝓕.c A t x) t).ofLp i =
  ∂ₜ (fun t => 𝓕.c.val * ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) thfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ Differentiable ℝ fun x_1 => electricField 𝓕.c A x_1 xd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ 1 / (𝓕.μ₀ * 𝓕.ε₀) * ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x -
    1 / 𝓕.ε₀ * (LorentzCurrentDensity.currentDensity 𝓕.c 0 t x).ofLp i =
  ∂ₜ (fun t => 𝓕.c.val * ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ ContDiff ℝ ∞ 0hfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ Differentiable ℝ fun x_1 => electricField 𝓕.c A x_1 x time_deriv_electricField_of_isExtrema hA 0 _ h t x id:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ 1 / (𝓕.μ₀ * 𝓕.ε₀) * ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x -
    1 / 𝓕.ε₀ * (LorentzCurrentDensity.currentDensity 𝓕.c 0 t x).ofLp i =
  ∂ₜ (fun t => 𝓕.c.val * ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ ContDiff ℝ ∞ 0hfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ Differentiable ℝ fun x_1 => electricField 𝓕.c A x_1 xd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ 1 / (𝓕.μ₀ * 𝓕.ε₀) * ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x -
    1 / 𝓕.ε₀ * (LorentzCurrentDensity.currentDensity 𝓕.c 0 t x).ofLp i =
  ∂ₜ (fun t => 𝓕.c.val * ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ ContDiff ℝ ∞ 0hfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ Differentiable ℝ fun x_1 => electricField 𝓕.c A x_1 x]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ 1 / (𝓕.μ₀ * 𝓕.ε₀) * ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x -
    1 / 𝓕.ε₀ * (LorentzCurrentDensity.currentDensity 𝓕.c 0 t x).ofLp i =
  ∂ₜ (fun t => 𝓕.c.val * ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ ContDiff ℝ ∞ 0hfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ Differentiable ℝ fun x_1 => electricField 𝓕.c A x_1 x
  simp only [one_div, _root_.mul_inv_rev, LorentzCurrentDensity.currentDensity_zero, Pi.zero_apply,
    PiLp.zero_apply, mul_zero, sub_zero]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ 𝓕.ε₀⁻¹ * 𝓕.μ₀⁻¹ * ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x =
  ∂ₜ (fun t => 𝓕.c.val * ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ ContDiff ℝ ∞ 0hfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ Differentiable ℝ fun x_1 => electricField 𝓕.c A x_1 x
  conv_lhs =>
    enter [2, 2, i]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di✝:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)i:Fin d| Space.deriv i (fun x => magneticFieldMatrix 𝓕.c A t x (i, i✝)) x;
    rw [magneticFieldMatrix_space_deriv_eq_time_deriv P (hA.of_le ENat.LEInfty.out) t x i]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di✝:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)i:Fin d| -(s.unit.val i / 𝓕.c.val) • ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, i✝)) t
  rw [Time.deriv_eq,d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ 𝓕.ε₀⁻¹ * 𝓕.μ₀⁻¹ * ∑ i_1, -(s.unit.val i_1 / 𝓕.c.val) • ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i_1, i)) t =
  (fderiv ℝ (fun t => 𝓕.c.val * ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t) 1d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ ContDiff ℝ ∞ 0hfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ Differentiable ℝ fun x_1 => electricField 𝓕.c A x_1 xd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ 𝓕.ε₀⁻¹ * 𝓕.μ₀⁻¹ * ∑ i_1, -(s.unit.val i_1 / 𝓕.c.val) • ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i_1, i)) t =
  (𝓕.c.val • fderiv ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t) 1had:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ ContDiff ℝ ∞ 0hfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ Differentiable ℝ fun x_1 => electricField 𝓕.c A x_1 x fderiv_const_muld:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ 𝓕.ε₀⁻¹ * 𝓕.μ₀⁻¹ * ∑ i_1, -(s.unit.val i_1 / 𝓕.c.val) • ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i_1, i)) t =
  (𝓕.c.val • fderiv ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t) 1had:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ ContDiff ℝ ∞ 0hfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ Differentiable ℝ fun x_1 => electricField 𝓕.c A x_1 xd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ 𝓕.ε₀⁻¹ * 𝓕.μ₀⁻¹ * ∑ i_1, -(s.unit.val i_1 / 𝓕.c.val) • ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i_1, i)) t =
  (𝓕.c.val • fderiv ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t) 1had:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ ContDiff ℝ ∞ 0hfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ Differentiable ℝ fun x_1 => electricField 𝓕.c A x_1 x]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ 𝓕.ε₀⁻¹ * 𝓕.μ₀⁻¹ * ∑ i_1, -(s.unit.val i_1 / 𝓕.c.val) • ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i_1, i)) t =
  (𝓕.c.val • fderiv ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t) 1had:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ ContDiff ℝ ∞ 0hfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ Differentiable ℝ fun x_1 => electricField 𝓕.c A x_1 x
  simp [← Time.deriv_eq]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ -(𝓕.ε₀⁻¹ * 𝓕.μ₀⁻¹ * ∑ x_1, s.unit.val x_1 / 𝓕.c.val * ∂ₜ (fun x_2 => magneticFieldMatrix 𝓕.c A x_2 x (x_1, i)) t) =
  𝓕.c.val * ∂ₜ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ ContDiff ℝ ∞ 0hfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ Differentiable ℝ fun x_1 => electricField 𝓕.c A x_1 x
  have h1 : ∂ₜ (fun t => ∑ j, A.magneticFieldMatrix 𝓕.c t x (i, j) * s.unit j) t
    = ∑ j, ∂ₜ (A.magneticFieldMatrix 𝓕.c · x (i, j)) t * s.unit j := byd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin d⊢ ∂ₜ (fun x_1 => (electricField 𝓕.c A x_1 x).ofLp i) t =
  ∂ₜ (fun t => 𝓕.c.val * ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)h1:∂ₜ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t =
  ∑ j, ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, j)) t * s.unit.val j⊢ -(𝓕.ε₀⁻¹ * 𝓕.μ₀⁻¹ * ∑ x_1, s.unit.val x_1 / 𝓕.c.val * ∂ₜ (fun x_2 => magneticFieldMatrix 𝓕.c A x_2 x (x_1, i)) t) =
  𝓕.c.val * ∂ₜ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ ContDiff ℝ ∞ 0hfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ Differentiable ℝ fun x_1 => electricField 𝓕.c A x_1 x
    rw [Time.deriv_eq,d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ (fderiv ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t) 1 =
  ∑ j, ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, j)) t * s.unit.val jd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ (∑ i_1, fderiv ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1) t) 1 =
  ∑ j, ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, j)) t * s.unit.val jd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ ∀ i_1 ∈ Finset.univ, DifferentiableAt ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)h1:∂ₜ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t =
  ∑ j, ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, j)) t * s.unit.val j⊢ -(𝓕.ε₀⁻¹ * 𝓕.μ₀⁻¹ * ∑ x_1, s.unit.val x_1 / 𝓕.c.val * ∂ₜ (fun x_2 => magneticFieldMatrix 𝓕.c A x_2 x (x_1, i)) t) =
  𝓕.c.val * ∂ₜ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ ContDiff ℝ ∞ 0hfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ Differentiable ℝ fun x_1 => electricField 𝓕.c A x_1 x fderiv_fun_sumd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ (∑ i_1, fderiv ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1) t) 1 =
  ∑ j, ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, j)) t * s.unit.val jd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ ∀ i_1 ∈ Finset.univ, DifferentiableAt ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ (∑ i_1, fderiv ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1) t) 1 =
  ∑ j, ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, j)) t * s.unit.val jd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ ∀ i_1 ∈ Finset.univ, DifferentiableAt ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)h1:∂ₜ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t =
  ∑ j, ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, j)) t * s.unit.val j⊢ -(𝓕.ε₀⁻¹ * 𝓕.μ₀⁻¹ * ∑ x_1, s.unit.val x_1 / 𝓕.c.val * ∂ₜ (fun x_2 => magneticFieldMatrix 𝓕.c A x_2 x (x_1, i)) t) =
  𝓕.c.val * ∂ₜ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ ContDiff ℝ ∞ 0hfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ Differentiable ℝ fun x_1 => electricField 𝓕.c A x_1 x]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ (∑ i_1, fderiv ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1) t) 1 =
  ∑ j, ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, j)) t * s.unit.val jd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ ∀ i_1 ∈ Finset.univ, DifferentiableAt ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)h1:∂ₜ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t =
  ∑ j, ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, j)) t * s.unit.val j⊢ -(𝓕.ε₀⁻¹ * 𝓕.μ₀⁻¹ * ∑ x_1, s.unit.val x_1 / 𝓕.c.val * ∂ₜ (fun x_2 => magneticFieldMatrix 𝓕.c A x_2 x (x_1, i)) t) =
  𝓕.c.val * ∂ₜ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ ContDiff ℝ ∞ 0hfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ Differentiable ℝ fun x_1 => electricField 𝓕.c A x_1 x
    simp only [FunLike.coe_sum, Finset.sum_apply]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ ∑ c, (fderiv ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, c) * s.unit.val c) t) 1 =
  ∑ j, ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, j)) t * s.unit.val jd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ ∀ i_1 ∈ Finset.univ, DifferentiableAt ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)h1:∂ₜ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t =
  ∑ j, ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, j)) t * s.unit.val j⊢ -(𝓕.ε₀⁻¹ * 𝓕.μ₀⁻¹ * ∑ x_1, s.unit.val x_1 / 𝓕.c.val * ∂ₜ (fun x_2 => magneticFieldMatrix 𝓕.c A x_2 x (x_1, i)) t) =
  𝓕.c.val * ∂ₜ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ ContDiff ℝ ∞ 0hfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ Differentiable ℝ fun x_1 => electricField 𝓕.c A x_1 x
    conv_lhs =>
      enter [2, k]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)k:Fin d| (fderiv ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, k) * s.unit.val k) t) 1d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)h1:∂ₜ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t =
  ∑ j, ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, j)) t * s.unit.val j⊢ -(𝓕.ε₀⁻¹ * 𝓕.μ₀⁻¹ * ∑ x_1, s.unit.val x_1 / 𝓕.c.val * ∂ₜ (fun x_2 => magneticFieldMatrix 𝓕.c A x_2 x (x_1, i)) t) =
  𝓕.c.val * ∂ₜ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ ContDiff ℝ ∞ 0hfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ Differentiable ℝ fun x_1 => electricField 𝓕.c A x_1 x
      rw [fderiv_mul_const (hBd _).differentiableAt]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)k:Fin d| (s.unit.val k • fderiv ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, k)) t) 1d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)h1:∂ₜ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t =
  ∑ j, ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, j)) t * s.unit.val j⊢ -(𝓕.ε₀⁻¹ * 𝓕.μ₀⁻¹ * ∑ x_1, s.unit.val x_1 / 𝓕.c.val * ∂ₜ (fun x_2 => magneticFieldMatrix 𝓕.c A x_2 x (x_1, i)) t) =
  𝓕.c.val * ∂ₜ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ ContDiff ℝ ∞ 0hfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ Differentiable ℝ fun x_1 => electricField 𝓕.c A x_1 x
    simp only [FunLike.coe_smul, Pi.smul_apply, smul_eq_mul]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ ∑ x_1, s.unit.val x_1 * (fderiv ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, x_1)) t) 1 =
  ∑ j, ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, j)) t * s.unit.val jd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ ∀ i_1 ∈ Finset.univ, DifferentiableAt ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)h1:∂ₜ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t =
  ∑ j, ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, j)) t * s.unit.val j⊢ -(𝓕.ε₀⁻¹ * 𝓕.μ₀⁻¹ * ∑ x_1, s.unit.val x_1 / 𝓕.c.val * ∂ₜ (fun x_2 => magneticFieldMatrix 𝓕.c A x_2 x (x_1, i)) t) =
  𝓕.c.val * ∂ₜ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ ContDiff ℝ ∞ 0hfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ Differentiable ℝ fun x_1 => electricField 𝓕.c A x_1 x
    congre_fd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ (fun x_1 => s.unit.val x_1 * (fderiv ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, x_1)) t) 1) = fun j =>
  ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, j)) t * s.unit.val jd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ ∀ i_1 ∈ Finset.univ, DifferentiableAt ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)h1:∂ₜ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t =
  ∑ j, ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, j)) t * s.unit.val j⊢ -(𝓕.ε₀⁻¹ * 𝓕.μ₀⁻¹ * ∑ x_1, s.unit.val x_1 / 𝓕.c.val * ∂ₜ (fun x_2 => magneticFieldMatrix 𝓕.c A x_2 x (x_1, i)) t) =
  𝓕.c.val * ∂ₜ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ ContDiff ℝ ∞ 0hfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ Differentiable ℝ fun x_1 => electricField 𝓕.c A x_1 x
    funext ie_fd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di✝:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)i:Fin d⊢ s.unit.val i * (fderiv ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i✝, i)) t) 1 =
  ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i✝, i)) t * s.unit.val id:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ ∀ i_1 ∈ Finset.univ, DifferentiableAt ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)h1:∂ₜ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t =
  ∑ j, ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, j)) t * s.unit.val j⊢ -(𝓕.ε₀⁻¹ * 𝓕.μ₀⁻¹ * ∑ x_1, s.unit.val x_1 / 𝓕.c.val * ∂ₜ (fun x_2 => magneticFieldMatrix 𝓕.c A x_2 x (x_1, i)) t) =
  𝓕.c.val * ∂ₜ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ ContDiff ℝ ∞ 0hfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ Differentiable ℝ fun x_1 => electricField 𝓕.c A x_1 x
    ring_nfe_fd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di✝:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)i:Fin d⊢ s.unit.val i * (fderiv ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i✝, i)) t) 1 =
  s.unit.val i * ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i✝, i)) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ ∀ i_1 ∈ Finset.univ, DifferentiableAt ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)h1:∂ₜ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t =
  ∑ j, ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, j)) t * s.unit.val j⊢ -(𝓕.ε₀⁻¹ * 𝓕.μ₀⁻¹ * ∑ x_1, s.unit.val x_1 / 𝓕.c.val * ∂ₜ (fun x_2 => magneticFieldMatrix 𝓕.c A x_2 x (x_1, i)) t) =
  𝓕.c.val * ∂ₜ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ ContDiff ℝ ∞ 0hfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ Differentiable ℝ fun x_1 => electricField 𝓕.c A x_1 x
    rfld:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ ∀ i_1 ∈ Finset.univ, DifferentiableAt ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)h1:∂ₜ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t =
  ∑ j, ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, j)) t * s.unit.val j⊢ -(𝓕.ε₀⁻¹ * 𝓕.μ₀⁻¹ * ∑ x_1, s.unit.val x_1 / 𝓕.c.val * ∂ₜ (fun x_2 => magneticFieldMatrix 𝓕.c A x_2 x (x_1, i)) t) =
  𝓕.c.val * ∂ₜ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ ContDiff ℝ ∞ 0hfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ Differentiable ℝ fun x_1 => electricField 𝓕.c A x_1 x
    ·d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ ∀ i_1 ∈ Finset.univ, DifferentiableAt ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)h1:∂ₜ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t =
  ∑ j, ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, j)) t * s.unit.val j⊢ -(𝓕.ε₀⁻¹ * 𝓕.μ₀⁻¹ * ∑ x_1, s.unit.val x_1 / 𝓕.c.val * ∂ₜ (fun x_2 => magneticFieldMatrix 𝓕.c A x_2 x (x_1, i)) t) =
  𝓕.c.val * ∂ₜ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ ContDiff ℝ ∞ 0hfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ Differentiable ℝ fun x_1 => electricField 𝓕.c A x_1 x intro k _d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)k:Fin da✝:k ∈ Finset.univ⊢ DifferentiableAt ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, k) * s.unit.val k) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)h1:∂ₜ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t =
  ∑ j, ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, j)) t * s.unit.val j⊢ -(𝓕.ε₀⁻¹ * 𝓕.μ₀⁻¹ * ∑ x_1, s.unit.val x_1 / 𝓕.c.val * ∂ₜ (fun x_2 => magneticFieldMatrix 𝓕.c A x_2 x (x_1, i)) t) =
  𝓕.c.val * ∂ₜ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ ContDiff ℝ ∞ 0hfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ Differentiable ℝ fun x_1 => electricField 𝓕.c A x_1 x
      apply DifferentiableAt.mul_constd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)k:Fin da✝:k ∈ Finset.univ⊢ DifferentiableAt ℝ (fun y => magneticFieldMatrix 𝓕.c A y x (i, k)) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)h1:∂ₜ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t =
  ∑ j, ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, j)) t * s.unit.val j⊢ -(𝓕.ε₀⁻¹ * 𝓕.μ₀⁻¹ * ∑ x_1, s.unit.val x_1 / 𝓕.c.val * ∂ₜ (fun x_2 => magneticFieldMatrix 𝓕.c A x_2 x (x_1, i)) t) =
  𝓕.c.val * ∂ₜ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ ContDiff ℝ ∞ 0hfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ Differentiable ℝ fun x_1 => electricField 𝓕.c A x_1 x
      exact (hBd k).differentiableAtd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)h1:∂ₜ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t =
  ∑ j, ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, j)) t * s.unit.val j⊢ -(𝓕.ε₀⁻¹ * 𝓕.μ₀⁻¹ * ∑ x_1, s.unit.val x_1 / 𝓕.c.val * ∂ₜ (fun x_2 => magneticFieldMatrix 𝓕.c A x_2 x (x_1, i)) t) =
  𝓕.c.val * ∂ₜ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ ContDiff ℝ ∞ 0hfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ Differentiable ℝ fun x_1 => electricField 𝓕.c A x_1 xd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)h1:∂ₜ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t =
  ∑ j, ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, j)) t * s.unit.val j⊢ -(𝓕.ε₀⁻¹ * 𝓕.μ₀⁻¹ * ∑ x_1, s.unit.val x_1 / 𝓕.c.val * ∂ₜ (fun x_2 => magneticFieldMatrix 𝓕.c A x_2 x (x_1, i)) t) =
  𝓕.c.val * ∂ₜ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ ContDiff ℝ ∞ 0hfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ Differentiable ℝ fun x_1 => electricField 𝓕.c A x_1 x

  rw [h1,d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)h1:∂ₜ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t =
  ∑ j, ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, j)) t * s.unit.val j⊢ -(𝓕.ε₀⁻¹ * 𝓕.μ₀⁻¹ * ∑ x_1, s.unit.val x_1 / 𝓕.c.val * ∂ₜ (fun x_2 => magneticFieldMatrix 𝓕.c A x_2 x (x_1, i)) t) =
  𝓕.c.val * ∑ j, ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, j)) t * s.unit.val jhad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ ContDiff ℝ ∞ 0hfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ Differentiable ℝ fun x_1 => electricField 𝓕.c A x_1 xd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)h1:∂ₜ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t =
  ∑ j, ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, j)) t * s.unit.val j⊢ ∑ x_1, -(𝓕.ε₀⁻¹ * 𝓕.μ₀⁻¹ * (s.unit.val x_1 / 𝓕.c.val * ∂ₜ (fun x_2 => magneticFieldMatrix 𝓕.c A x_2 x (x_1, i)) t)) =
  ∑ i_1, 𝓕.c.val * (∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, i_1)) t * s.unit.val i_1)had:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ ContDiff ℝ ∞ 0hfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ Differentiable ℝ fun x_1 => electricField 𝓕.c A x_1 x Finset.mul_sum,d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)h1:∂ₜ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t =
  ∑ j, ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, j)) t * s.unit.val j⊢ -∑ i_1, 𝓕.ε₀⁻¹ * 𝓕.μ₀⁻¹ * (s.unit.val i_1 / 𝓕.c.val * ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i_1, i)) t) =
  𝓕.c.val * ∑ j, ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, j)) t * s.unit.val jhad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ ContDiff ℝ ∞ 0hfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ Differentiable ℝ fun x_1 => electricField 𝓕.c A x_1 xd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)h1:∂ₜ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t =
  ∑ j, ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, j)) t * s.unit.val j⊢ ∑ x_1, -(𝓕.ε₀⁻¹ * 𝓕.μ₀⁻¹ * (s.unit.val x_1 / 𝓕.c.val * ∂ₜ (fun x_2 => magneticFieldMatrix 𝓕.c A x_2 x (x_1, i)) t)) =
  ∑ i_1, 𝓕.c.val * (∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, i_1)) t * s.unit.val i_1)had:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ ContDiff ℝ ∞ 0hfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ Differentiable ℝ fun x_1 => electricField 𝓕.c A x_1 x Finset.mul_sum,d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)h1:∂ₜ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t =
  ∑ j, ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, j)) t * s.unit.val j⊢ -∑ i_1, 𝓕.ε₀⁻¹ * 𝓕.μ₀⁻¹ * (s.unit.val i_1 / 𝓕.c.val * ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i_1, i)) t) =
  ∑ i_1, 𝓕.c.val * (∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, i_1)) t * s.unit.val i_1)had:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ ContDiff ℝ ∞ 0hfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ Differentiable ℝ fun x_1 => electricField 𝓕.c A x_1 xd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)h1:∂ₜ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t =
  ∑ j, ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, j)) t * s.unit.val j⊢ ∑ x_1, -(𝓕.ε₀⁻¹ * 𝓕.μ₀⁻¹ * (s.unit.val x_1 / 𝓕.c.val * ∂ₜ (fun x_2 => magneticFieldMatrix 𝓕.c A x_2 x (x_1, i)) t)) =
  ∑ i_1, 𝓕.c.val * (∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, i_1)) t * s.unit.val i_1)had:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ ContDiff ℝ ∞ 0hfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ Differentiable ℝ fun x_1 => electricField 𝓕.c A x_1 x← Finset.sum_neg_distribd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)h1:∂ₜ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t =
  ∑ j, ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, j)) t * s.unit.val j⊢ ∑ x_1, -(𝓕.ε₀⁻¹ * 𝓕.μ₀⁻¹ * (s.unit.val x_1 / 𝓕.c.val * ∂ₜ (fun x_2 => magneticFieldMatrix 𝓕.c A x_2 x (x_1, i)) t)) =
  ∑ i_1, 𝓕.c.val * (∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, i_1)) t * s.unit.val i_1)had:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ ContDiff ℝ ∞ 0hfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ Differentiable ℝ fun x_1 => electricField 𝓕.c A x_1 xd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)h1:∂ₜ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t =
  ∑ j, ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, j)) t * s.unit.val j⊢ ∑ x_1, -(𝓕.ε₀⁻¹ * 𝓕.μ₀⁻¹ * (s.unit.val x_1 / 𝓕.c.val * ∂ₜ (fun x_2 => magneticFieldMatrix 𝓕.c A x_2 x (x_1, i)) t)) =
  ∑ i_1, 𝓕.c.val * (∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, i_1)) t * s.unit.val i_1)had:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ ContDiff ℝ ∞ 0hfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ Differentiable ℝ fun x_1 => electricField 𝓕.c A x_1 x]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)h1:∂ₜ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t =
  ∑ j, ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, j)) t * s.unit.val j⊢ ∑ x_1, -(𝓕.ε₀⁻¹ * 𝓕.μ₀⁻¹ * (s.unit.val x_1 / 𝓕.c.val * ∂ₜ (fun x_2 => magneticFieldMatrix 𝓕.c A x_2 x (x_1, i)) t)) =
  ∑ i_1, 𝓕.c.val * (∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, i_1)) t * s.unit.val i_1)had:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ ContDiff ℝ ∞ 0hfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ Differentiable ℝ fun x_1 => electricField 𝓕.c A x_1 x
  field_simpd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)h1:∂ₜ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t =
  ∑ j, ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, j)) t * s.unit.val j⊢ ∑ x_1, -(s.unit.val x_1 * ∂ₜ (fun x_2 => magneticFieldMatrix 𝓕.c A x_2 x (x_1, i)) t / (𝓕.ε₀ * 𝓕.μ₀ * 𝓕.c.val)) =
  ∑ x_1, 𝓕.c.val * ∂ₜ (fun x_2 => magneticFieldMatrix 𝓕.c A x_2 x (i, x_1)) t * s.unit.val x_1had:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ ContDiff ℝ ∞ 0hfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ Differentiable ℝ fun x_1 => electricField 𝓕.c A x_1 x
  congre_fd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)h1:∂ₜ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t =
  ∑ j, ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, j)) t * s.unit.val j⊢ (fun x_1 => -(s.unit.val x_1 * ∂ₜ (fun x_2 => magneticFieldMatrix 𝓕.c A x_2 x (x_1, i)) t / (𝓕.ε₀ * 𝓕.μ₀ * 𝓕.c.val))) =
  fun x_1 => 𝓕.c.val * ∂ₜ (fun x_2 => magneticFieldMatrix 𝓕.c A x_2 x (i, x_1)) t * s.unit.val x_1had:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ ContDiff ℝ ∞ 0hfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ Differentiable ℝ fun x_1 => electricField 𝓕.c A x_1 x
  funext ke_fd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)h1:∂ₜ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t =
  ∑ j, ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, j)) t * s.unit.val jk:Fin d⊢ -(s.unit.val k * ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (k, i)) t / (𝓕.ε₀ * 𝓕.μ₀ * 𝓕.c.val)) =
  𝓕.c.val * ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, k)) t * s.unit.val khad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ ContDiff ℝ ∞ 0hfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ Differentiable ℝ fun x_1 => electricField 𝓕.c A x_1 x
  field_simpe_fd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)h1:∂ₜ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t =
  ∑ j, ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, j)) t * s.unit.val jk:Fin d⊢ -(s.unit.val k * ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (k, i)) t) =
  s.unit.val k * 𝓕.ε₀ * 𝓕.μ₀ * 𝓕.c.val ^ 2 * ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, k)) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ ContDiff ℝ ∞ 0hfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ Differentiable ℝ fun x_1 => electricField 𝓕.c A x_1 x
  simp [𝓕.c_sq]e_fd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)h1:∂ₜ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t =
  ∑ j, ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, j)) t * s.unit.val jk:Fin d⊢ -(s.unit.val k * ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (k, i)) t) =
  s.unit.val k * 𝓕.ε₀ * 𝓕.μ₀ * (𝓕.μ₀⁻¹ * 𝓕.ε₀⁻¹) * ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, k)) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ ContDiff ℝ ∞ 0hfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ Differentiable ℝ fun x_1 => electricField 𝓕.c A x_1 x
  field_simpe_fd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)h1:∂ₜ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t =
  ∑ j, ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, j)) t * s.unit.val jk:Fin d⊢ -(s.unit.val k * ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (k, i)) t) =
  s.unit.val k * ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, k)) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ ContDiff ℝ ∞ 0hfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ Differentiable ℝ fun x_1 => electricField 𝓕.c A x_1 x
  conv_lhs =>
    enter [1, 2, 1, t]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t✝:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)h1:∂ₜ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t =
  ∑ j, ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, j)) t * s.unit.val jk:Fin dt:Time| magneticFieldMatrix 𝓕.c A t x (k, i)
    rw [magneticFieldMatrix_antisymm]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t✝:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)h1:∂ₜ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t =
  ∑ j, ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, j)) t * s.unit.val jk:Fin dt:Time| -magneticFieldMatrix 𝓕.c A t x (i, k)
  rw [Time.deriv_eq,e_fd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)h1:∂ₜ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t =
  ∑ j, ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, j)) t * s.unit.val jk:Fin d⊢ -(s.unit.val k * (fderiv ℝ (fun t => -magneticFieldMatrix 𝓕.c A t x (i, k)) t) 1) =
  s.unit.val k * ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, k)) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ ContDiff ℝ ∞ 0hfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ Differentiable ℝ fun x_1 => electricField 𝓕.c A x_1 xe_fd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)h1:∂ₜ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t =
  ∑ j, ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, j)) t * s.unit.val jk:Fin d⊢ -(s.unit.val k * (-fderiv ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, k)) t) 1) =
  s.unit.val k * ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, k)) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ ContDiff ℝ ∞ 0hfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ Differentiable ℝ fun x_1 => electricField 𝓕.c A x_1 x fderiv_fun_nege_fd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)h1:∂ₜ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t =
  ∑ j, ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, j)) t * s.unit.val jk:Fin d⊢ -(s.unit.val k * (-fderiv ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, k)) t) 1) =
  s.unit.val k * ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, k)) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ ContDiff ℝ ∞ 0hfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ Differentiable ℝ fun x_1 => electricField 𝓕.c A x_1 xe_fd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)h1:∂ₜ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t =
  ∑ j, ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, j)) t * s.unit.val jk:Fin d⊢ -(s.unit.val k * (-fderiv ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, k)) t) 1) =
  s.unit.val k * ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, k)) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ ContDiff ℝ ∞ 0hfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ Differentiable ℝ fun x_1 => electricField 𝓕.c A x_1 x]e_fd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)h1:∂ₜ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t =
  ∑ j, ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, j)) t * s.unit.val jk:Fin d⊢ -(s.unit.val k * (-fderiv ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, k)) t) 1) =
  s.unit.val k * ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, k)) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ ContDiff ℝ ∞ 0hfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ Differentiable ℝ fun x_1 => electricField 𝓕.c A x_1 x
  simp [← Time.deriv_eq]had:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ ContDiff ℝ ∞ 0hfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ Differentiable ℝ fun x_1 => electricField 𝓕.c A x_1 x
  ·had:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t refine DifferentiableAt.fun_sum ?_had:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ ∀ i_1 ∈ Finset.univ, DifferentiableAt ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1) t
    intro k _had:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)k:Fin da✝:k ∈ Finset.univ⊢ DifferentiableAt ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, k) * s.unit.val k) t
    apply DifferentiableAt.mul_consthad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)k:Fin da✝:k ∈ Finset.univ⊢ DifferentiableAt ℝ (fun y => magneticFieldMatrix 𝓕.c A y x (i, k)) t
    exact (hBd k).differentiableAtAll goals completed! 🐙
  ·d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ ContDiff ℝ ∞ 0 change ContDiff ℝ ∞ (fun _ => 0)d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ ContDiff ℝ ∞ fun x => 0
    fun_propAll goals completed! 🐙
  ·hfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dhBd:∀ (k : Fin d), Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, k)⊢ Differentiable ℝ fun x_1 => electricField 𝓕.c A x_1 x exact electricField_differentiable_time (hA.of_le (ENat.LEInfty.out)) xAll goals completed! 🐙

C.2. The space derivative of the electric field in terms of magnetic field



lemma space_deriv_electricField_eq_magneticFieldMatrix {d : ℕ}
    {𝓕 : FreeSpace} {A : ElectromagneticPotential d}
    {s : Direction d} (P : IsPlaneWave 𝓕 A s) (hA : ContDiff ℝ ∞ A)
    (h : IsExtrema 𝓕 A 0)
    (t : Time) (x : Space d) (i k : Fin d) :
    ∂[k] (A.electricField 𝓕.c t · i) x =
    ∂[k] (fun x => 𝓕.c * ∑ j, A.magneticFieldMatrix 𝓕.c t x (i, j) * s.unit j) x := byd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin d⊢ Space.deriv k (fun x => (electricField 𝓕.c A t x).ofLp i) x =
  Space.deriv k (fun x => 𝓕.c.val * ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) x
  have hA2 : ContDiff ℝ 2 A := hA.of_le ENat.LEInfty.outd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ Space.deriv k (fun x => (electricField 𝓕.c A t x).ofLp i) x =
  Space.deriv k (fun x => 𝓕.c.val * ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) x
  rw [P.electricField_space_deriv_eq_time_deriv hA2 t x i k,d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ -(s.unit.val k / 𝓕.c.val) • ∂ₜ (fun x_1 => (electricField 𝓕.c A x_1 x).ofLp i) t =
  Space.deriv k (fun x => 𝓕.c.val * ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) xd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ -(s.unit.val k / 𝓕.c.val) •
    (𝓕.c.val • ∑ i_1, fderiv ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1) t) 1 =
  Space.deriv k (fun x => 𝓕.c.val * ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) xd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ ∀ i_1 ∈ Finset.univ, DifferentiableAt ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t
    P.time_deriv_electricField_eq_magneticFieldMatrix hA h t x i,d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ -(s.unit.val k / 𝓕.c.val) • ∂ₜ (fun t => 𝓕.c.val * ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t =
  Space.deriv k (fun x => 𝓕.c.val * ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) xd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ -(s.unit.val k / 𝓕.c.val) •
    (𝓕.c.val • ∑ i_1, fderiv ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1) t) 1 =
  Space.deriv k (fun x => 𝓕.c.val * ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) xd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ ∀ i_1 ∈ Finset.univ, DifferentiableAt ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t Time.deriv_eq,d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ -(s.unit.val k / 𝓕.c.val) •
    (fderiv ℝ (fun t => 𝓕.c.val * ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t) 1 =
  Space.deriv k (fun x => 𝓕.c.val * ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) xd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ -(s.unit.val k / 𝓕.c.val) •
    (𝓕.c.val • ∑ i_1, fderiv ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1) t) 1 =
  Space.deriv k (fun x => 𝓕.c.val * ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) xd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ ∀ i_1 ∈ Finset.univ, DifferentiableAt ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t fderiv_const_mul,d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ -(s.unit.val k / 𝓕.c.val) •
    (𝓕.c.val • fderiv ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t) 1 =
  Space.deriv k (fun x => 𝓕.c.val * ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) xhad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ -(s.unit.val k / 𝓕.c.val) •
    (𝓕.c.val • ∑ i_1, fderiv ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1) t) 1 =
  Space.deriv k (fun x => 𝓕.c.val * ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) xd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ ∀ i_1 ∈ Finset.univ, DifferentiableAt ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t
    fderiv_fun_sumd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ -(s.unit.val k / 𝓕.c.val) •
    (𝓕.c.val • ∑ i_1, fderiv ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1) t) 1 =
  Space.deriv k (fun x => 𝓕.c.val * ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) xd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ ∀ i_1 ∈ Finset.univ, DifferentiableAt ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ -(s.unit.val k / 𝓕.c.val) •
    (𝓕.c.val • ∑ i_1, fderiv ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1) t) 1 =
  Space.deriv k (fun x => 𝓕.c.val * ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) xd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ ∀ i_1 ∈ Finset.univ, DifferentiableAt ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ -(s.unit.val k / 𝓕.c.val) •
    (𝓕.c.val • ∑ i_1, fderiv ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1) t) 1 =
  Space.deriv k (fun x => 𝓕.c.val * ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) xd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ ∀ i_1 ∈ Finset.univ, DifferentiableAt ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t
  simp [Finset.mul_sum, - Finset.sum_neg_distrib]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ ∑ x_1,
    -(s.unit.val k / 𝓕.c.val *
        (𝓕.c.val * (fderiv ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, x_1) * s.unit.val x_1) t) 1)) =
  Space.deriv k (fun x => ∑ i_1, 𝓕.c.val * (magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1)) xd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ ∀ i_1 ∈ Finset.univ, DifferentiableAt ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t
  rw [Space.deriv_eq_fderiv_basis,d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ ∑ x_1,
    -(s.unit.val k / 𝓕.c.val *
        (𝓕.c.val * (fderiv ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, x_1) * s.unit.val x_1) t) 1)) =
  (fderiv ℝ (fun x => ∑ i_1, 𝓕.c.val * (magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1)) x) (Space.basis k)d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ ∀ i_1 ∈ Finset.univ, DifferentiableAt ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ ∑ x_1,
    -(s.unit.val k / 𝓕.c.val *
        (𝓕.c.val * (fderiv ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, x_1) * s.unit.val x_1) t) 1)) =
  (∑ i_1, fderiv ℝ (fun x => 𝓕.c.val * (magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1)) x) (Space.basis k)d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ ∀ i_1 ∈ Finset.univ, DifferentiableAt ℝ (fun x => 𝓕.c.val * (magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1)) xd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ ∀ i_1 ∈ Finset.univ, DifferentiableAt ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t fderiv_fun_sumd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ ∑ x_1,
    -(s.unit.val k / 𝓕.c.val *
        (𝓕.c.val * (fderiv ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, x_1) * s.unit.val x_1) t) 1)) =
  (∑ i_1, fderiv ℝ (fun x => 𝓕.c.val * (magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1)) x) (Space.basis k)d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ ∀ i_1 ∈ Finset.univ, DifferentiableAt ℝ (fun x => 𝓕.c.val * (magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1)) xd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ ∀ i_1 ∈ Finset.univ, DifferentiableAt ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ ∑ x_1,
    -(s.unit.val k / 𝓕.c.val *
        (𝓕.c.val * (fderiv ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, x_1) * s.unit.val x_1) t) 1)) =
  (∑ i_1, fderiv ℝ (fun x => 𝓕.c.val * (magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1)) x) (Space.basis k)d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ ∀ i_1 ∈ Finset.univ, DifferentiableAt ℝ (fun x => 𝓕.c.val * (magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1)) xd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ ∀ i_1 ∈ Finset.univ, DifferentiableAt ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ ∑ x_1,
    -(s.unit.val k / 𝓕.c.val *
        (𝓕.c.val * (fderiv ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, x_1) * s.unit.val x_1) t) 1)) =
  (∑ i_1, fderiv ℝ (fun x => 𝓕.c.val * (magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1)) x) (Space.basis k)d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ ∀ i_1 ∈ Finset.univ, DifferentiableAt ℝ (fun x => 𝓕.c.val * (magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1)) xd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ ∀ i_1 ∈ Finset.univ, DifferentiableAt ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t
  simp [- Finset.sum_neg_distrib]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ ∑ x_1,
    -(s.unit.val k / 𝓕.c.val *
        (𝓕.c.val * (fderiv ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, x_1) * s.unit.val x_1) t) 1)) =
  ∑ i_1, (fderiv ℝ (fun x => 𝓕.c.val * (magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1)) x) (Space.basis k)d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ ∀ i_1 ∈ Finset.univ, DifferentiableAt ℝ (fun x => 𝓕.c.val * (magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1)) xd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ ∀ i_1 ∈ Finset.univ, DifferentiableAt ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t
  congre_fd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ (fun x_1 =>
    -(s.unit.val k / 𝓕.c.val *
        (𝓕.c.val * (fderiv ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, x_1) * s.unit.val x_1) t) 1))) =
  fun i_1 => (fderiv ℝ (fun x => 𝓕.c.val * (magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1)) x) (Space.basis k)d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ ∀ i_1 ∈ Finset.univ, DifferentiableAt ℝ (fun x => 𝓕.c.val * (magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1)) xd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ ∀ i_1 ∈ Finset.univ, DifferentiableAt ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t
  funext je_fd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.valj:Fin d⊢ -(s.unit.val k / 𝓕.c.val * (𝓕.c.val * (fderiv ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t) 1)) =
  (fderiv ℝ (fun x => 𝓕.c.val * (magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j)) x) (Space.basis k)d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ ∀ i_1 ∈ Finset.univ, DifferentiableAt ℝ (fun x => 𝓕.c.val * (magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1)) xd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ ∀ i_1 ∈ Finset.univ, DifferentiableAt ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t
  rw [fderiv_mul_const,e_fd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.valj:Fin d⊢ -(s.unit.val k / 𝓕.c.val * (𝓕.c.val * (s.unit.val j • fderiv ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, j)) t) 1)) =
  (fderiv ℝ (fun x => 𝓕.c.val * (magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j)) x) (Space.basis k)e_f.hcd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.valj:Fin d⊢ DifferentiableAt ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, j)) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ ∀ i_1 ∈ Finset.univ, DifferentiableAt ℝ (fun x => 𝓕.c.val * (magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1)) xd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ ∀ i_1 ∈ Finset.univ, DifferentiableAt ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) te_fd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.valj:Fin d⊢ -(s.unit.val k / 𝓕.c.val * (𝓕.c.val * (s.unit.val j • fderiv ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, j)) t) 1)) =
  (𝓕.c.val • s.unit.val j • fderiv ℝ (fun x => magneticFieldMatrix 𝓕.c A t x (i, j)) x) (Space.basis k)e_f.hcd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.valj:Fin d⊢ DifferentiableAt ℝ (fun x => magneticFieldMatrix 𝓕.c A t x (i, j)) xe_f.had:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.valj:Fin d⊢ DifferentiableAt ℝ (fun x => magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) xe_f.hcd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.valj:Fin d⊢ DifferentiableAt ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, j)) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ ∀ i_1 ∈ Finset.univ, DifferentiableAt ℝ (fun x => 𝓕.c.val * (magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1)) xd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ ∀ i_1 ∈ Finset.univ, DifferentiableAt ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t fderiv_const_mul,e_fd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.valj:Fin d⊢ -(s.unit.val k / 𝓕.c.val * (𝓕.c.val * (s.unit.val j • fderiv ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, j)) t) 1)) =
  (𝓕.c.val • fderiv ℝ (fun x => magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) x) (Space.basis k)e_f.had:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.valj:Fin d⊢ DifferentiableAt ℝ (fun x => magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) xe_f.hcd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.valj:Fin d⊢ DifferentiableAt ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, j)) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ ∀ i_1 ∈ Finset.univ, DifferentiableAt ℝ (fun x => 𝓕.c.val * (magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1)) xd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ ∀ i_1 ∈ Finset.univ, DifferentiableAt ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) te_fd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.valj:Fin d⊢ -(s.unit.val k / 𝓕.c.val * (𝓕.c.val * (s.unit.val j • fderiv ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, j)) t) 1)) =
  (𝓕.c.val • s.unit.val j • fderiv ℝ (fun x => magneticFieldMatrix 𝓕.c A t x (i, j)) x) (Space.basis k)e_f.hcd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.valj:Fin d⊢ DifferentiableAt ℝ (fun x => magneticFieldMatrix 𝓕.c A t x (i, j)) xe_f.had:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.valj:Fin d⊢ DifferentiableAt ℝ (fun x => magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) xe_f.hcd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.valj:Fin d⊢ DifferentiableAt ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, j)) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ ∀ i_1 ∈ Finset.univ, DifferentiableAt ℝ (fun x => 𝓕.c.val * (magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1)) xd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ ∀ i_1 ∈ Finset.univ, DifferentiableAt ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t fderiv_mul_conste_fd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.valj:Fin d⊢ -(s.unit.val k / 𝓕.c.val * (𝓕.c.val * (s.unit.val j • fderiv ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, j)) t) 1)) =
  (𝓕.c.val • s.unit.val j • fderiv ℝ (fun x => magneticFieldMatrix 𝓕.c A t x (i, j)) x) (Space.basis k)e_f.hcd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.valj:Fin d⊢ DifferentiableAt ℝ (fun x => magneticFieldMatrix 𝓕.c A t x (i, j)) xe_f.had:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.valj:Fin d⊢ DifferentiableAt ℝ (fun x => magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) xe_f.hcd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.valj:Fin d⊢ DifferentiableAt ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, j)) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ ∀ i_1 ∈ Finset.univ, DifferentiableAt ℝ (fun x => 𝓕.c.val * (magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1)) xd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ ∀ i_1 ∈ Finset.univ, DifferentiableAt ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) te_fd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.valj:Fin d⊢ -(s.unit.val k / 𝓕.c.val * (𝓕.c.val * (s.unit.val j • fderiv ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, j)) t) 1)) =
  (𝓕.c.val • s.unit.val j • fderiv ℝ (fun x => magneticFieldMatrix 𝓕.c A t x (i, j)) x) (Space.basis k)e_f.hcd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.valj:Fin d⊢ DifferentiableAt ℝ (fun x => magneticFieldMatrix 𝓕.c A t x (i, j)) xe_f.had:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.valj:Fin d⊢ DifferentiableAt ℝ (fun x => magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) xe_f.hcd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.valj:Fin d⊢ DifferentiableAt ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, j)) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ ∀ i_1 ∈ Finset.univ, DifferentiableAt ℝ (fun x => 𝓕.c.val * (magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1)) xd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ ∀ i_1 ∈ Finset.univ, DifferentiableAt ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t]e_fd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.valj:Fin d⊢ -(s.unit.val k / 𝓕.c.val * (𝓕.c.val * (s.unit.val j • fderiv ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, j)) t) 1)) =
  (𝓕.c.val • s.unit.val j • fderiv ℝ (fun x => magneticFieldMatrix 𝓕.c A t x (i, j)) x) (Space.basis k)e_f.hcd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.valj:Fin d⊢ DifferentiableAt ℝ (fun x => magneticFieldMatrix 𝓕.c A t x (i, j)) xe_f.had:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.valj:Fin d⊢ DifferentiableAt ℝ (fun x => magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) xe_f.hcd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.valj:Fin d⊢ DifferentiableAt ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, j)) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ ∀ i_1 ∈ Finset.univ, DifferentiableAt ℝ (fun x => 𝓕.c.val * (magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1)) xd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ ∀ i_1 ∈ Finset.univ, DifferentiableAt ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t
  simp only [FunLike.coe_smul, Pi.smul_apply, smul_eq_mul]e_fd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.valj:Fin d⊢ -(s.unit.val k / 𝓕.c.val *
      (𝓕.c.val * (s.unit.val j * (fderiv ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, j)) t) 1))) =
  𝓕.c.val * (s.unit.val j * (fderiv ℝ (fun x => magneticFieldMatrix 𝓕.c A t x (i, j)) x) (Space.basis k))e_f.hcd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.valj:Fin d⊢ DifferentiableAt ℝ (fun x => magneticFieldMatrix 𝓕.c A t x (i, j)) xe_f.had:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.valj:Fin d⊢ DifferentiableAt ℝ (fun x => magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) xe_f.hcd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.valj:Fin d⊢ DifferentiableAt ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, j)) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ ∀ i_1 ∈ Finset.univ, DifferentiableAt ℝ (fun x => 𝓕.c.val * (magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1)) xd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ ∀ i_1 ∈ Finset.univ, DifferentiableAt ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t
  rw [← Space.deriv_eq_fderiv_basis,e_fd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.valj:Fin d⊢ -(s.unit.val k / 𝓕.c.val *
      (𝓕.c.val * (s.unit.val j * (fderiv ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, j)) t) 1))) =
  𝓕.c.val * (s.unit.val j * Space.deriv k (fun x => magneticFieldMatrix 𝓕.c A t x (i, j)) x)e_f.hcd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.valj:Fin d⊢ DifferentiableAt ℝ (fun x => magneticFieldMatrix 𝓕.c A t x (i, j)) xe_f.had:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.valj:Fin d⊢ DifferentiableAt ℝ (fun x => magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) xe_f.hcd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.valj:Fin d⊢ DifferentiableAt ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, j)) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ ∀ i_1 ∈ Finset.univ, DifferentiableAt ℝ (fun x => 𝓕.c.val * (magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1)) xd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ ∀ i_1 ∈ Finset.univ, DifferentiableAt ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) te_fd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.valj:Fin d⊢ -(s.unit.val k / 𝓕.c.val *
      (𝓕.c.val * (s.unit.val j * (fderiv ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, j)) t) 1))) =
  𝓕.c.val * (s.unit.val j * -(s.unit.val k / 𝓕.c.val) • ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, j)) t)e_f.hcd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.valj:Fin d⊢ DifferentiableAt ℝ (fun x => magneticFieldMatrix 𝓕.c A t x (i, j)) xe_f.had:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.valj:Fin d⊢ DifferentiableAt ℝ (fun x => magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) xe_f.hcd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.valj:Fin d⊢ DifferentiableAt ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, j)) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ ∀ i_1 ∈ Finset.univ, DifferentiableAt ℝ (fun x => 𝓕.c.val * (magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1)) xd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ ∀ i_1 ∈ Finset.univ, DifferentiableAt ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t P.magneticFieldMatrix_space_deriv_eq_time_deriv hA2 t x i j ke_fd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.valj:Fin d⊢ -(s.unit.val k / 𝓕.c.val *
      (𝓕.c.val * (s.unit.val j * (fderiv ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, j)) t) 1))) =
  𝓕.c.val * (s.unit.val j * -(s.unit.val k / 𝓕.c.val) • ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, j)) t)e_f.hcd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.valj:Fin d⊢ DifferentiableAt ℝ (fun x => magneticFieldMatrix 𝓕.c A t x (i, j)) xe_f.had:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.valj:Fin d⊢ DifferentiableAt ℝ (fun x => magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) xe_f.hcd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.valj:Fin d⊢ DifferentiableAt ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, j)) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ ∀ i_1 ∈ Finset.univ, DifferentiableAt ℝ (fun x => 𝓕.c.val * (magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1)) xd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ ∀ i_1 ∈ Finset.univ, DifferentiableAt ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) te_fd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.valj:Fin d⊢ -(s.unit.val k / 𝓕.c.val *
      (𝓕.c.val * (s.unit.val j * (fderiv ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, j)) t) 1))) =
  𝓕.c.val * (s.unit.val j * -(s.unit.val k / 𝓕.c.val) • ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, j)) t)e_f.hcd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.valj:Fin d⊢ DifferentiableAt ℝ (fun x => magneticFieldMatrix 𝓕.c A t x (i, j)) xe_f.had:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.valj:Fin d⊢ DifferentiableAt ℝ (fun x => magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) xe_f.hcd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.valj:Fin d⊢ DifferentiableAt ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, j)) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ ∀ i_1 ∈ Finset.univ, DifferentiableAt ℝ (fun x => 𝓕.c.val * (magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1)) xd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ ∀ i_1 ∈ Finset.univ, DifferentiableAt ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t]e_fd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.valj:Fin d⊢ -(s.unit.val k / 𝓕.c.val *
      (𝓕.c.val * (s.unit.val j * (fderiv ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, j)) t) 1))) =
  𝓕.c.val * (s.unit.val j * -(s.unit.val k / 𝓕.c.val) • ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, j)) t)e_f.hcd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.valj:Fin d⊢ DifferentiableAt ℝ (fun x => magneticFieldMatrix 𝓕.c A t x (i, j)) xe_f.had:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.valj:Fin d⊢ DifferentiableAt ℝ (fun x => magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) xe_f.hcd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.valj:Fin d⊢ DifferentiableAt ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, j)) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ ∀ i_1 ∈ Finset.univ, DifferentiableAt ℝ (fun x => 𝓕.c.val * (magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1)) xd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ ∀ i_1 ∈ Finset.univ, DifferentiableAt ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t
  simp [← Time.deriv_eq]e_fd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.valj:Fin d⊢ s.unit.val k / 𝓕.c.val * (𝓕.c.val * (s.unit.val j * ∂ₜ (fun t => magneticFieldMatrix 𝓕.c A t x (i, j)) t)) =
  𝓕.c.val * (s.unit.val j * (s.unit.val k / 𝓕.c.val * ∂ₜ (fun t => magneticFieldMatrix 𝓕.c A t x (i, j)) t))e_f.hcd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.valj:Fin d⊢ DifferentiableAt ℝ (fun x => magneticFieldMatrix 𝓕.c A t x (i, j)) xe_f.had:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.valj:Fin d⊢ DifferentiableAt ℝ (fun x => magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) xe_f.hcd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.valj:Fin d⊢ DifferentiableAt ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, j)) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ ∀ i_1 ∈ Finset.univ, DifferentiableAt ℝ (fun x => 𝓕.c.val * (magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1)) xd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ ∀ i_1 ∈ Finset.univ, DifferentiableAt ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t
  field_simpe_f.hcd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.valj:Fin d⊢ DifferentiableAt ℝ (fun x => magneticFieldMatrix 𝓕.c A t x (i, j)) xe_f.had:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.valj:Fin d⊢ DifferentiableAt ℝ (fun x => magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) xe_f.hcd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.valj:Fin d⊢ DifferentiableAt ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, j)) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ ∀ i_1 ∈ Finset.univ, DifferentiableAt ℝ (fun x => 𝓕.c.val * (magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1)) xd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ ∀ i_1 ∈ Finset.univ, DifferentiableAt ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1) thad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ DifferentiableAt ℝ (fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t
  any_goals apply Differentiable.differentiableAthad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ Differentiable ℝ fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j
  ·e_f.hcd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.valj:Fin d⊢ Differentiable ℝ fun x => magneticFieldMatrix 𝓕.c A t x (i, j) exact fieldStrengthMatrix_differentiable_space hA2 tAll goals completed! 🐙
  ·e_f.had:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.valj:Fin d⊢ Differentiable ℝ fun x => magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j apply Differentiable.mul_conste_f.had:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.valj:Fin d⊢ Differentiable ℝ fun y => magneticFieldMatrix 𝓕.c A t y (i, j)
    exact fieldStrengthMatrix_differentiable_space hA2 tAll goals completed! 🐙
  ·e_f.hcd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.valj:Fin d⊢ Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i, j) exact fieldStrengthMatrix_differentiable_time hA2 xAll goals completed! 🐙
  ·d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ ∀ i_1 ∈ Finset.univ, DifferentiableAt ℝ (fun x => 𝓕.c.val * (magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1)) x intro i _d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di✝:Fin dk:Fin dhA2:ContDiff ℝ 2 A.vali:Fin da✝:i ∈ Finset.univ⊢ DifferentiableAt ℝ (fun x => 𝓕.c.val * (magneticFieldMatrix 𝓕.c A t x (i✝, i) * s.unit.val i)) x
    apply Differentiable.differentiableAtd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di✝:Fin dk:Fin dhA2:ContDiff ℝ 2 A.vali:Fin da✝:i ∈ Finset.univ⊢ Differentiable ℝ fun x => 𝓕.c.val * (magneticFieldMatrix 𝓕.c A t x (i✝, i) * s.unit.val i)
    apply Differentiable.const_muld:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di✝:Fin dk:Fin dhA2:ContDiff ℝ 2 A.vali:Fin da✝:i ∈ Finset.univ⊢ Differentiable ℝ fun y => magneticFieldMatrix 𝓕.c A t y (i✝, i) * s.unit.val i
    apply Differentiable.mul_constd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di✝:Fin dk:Fin dhA2:ContDiff ℝ 2 A.vali:Fin da✝:i ∈ Finset.univ⊢ Differentiable ℝ fun y => magneticFieldMatrix 𝓕.c A t y (i✝, i)
    exact fieldStrengthMatrix_differentiable_space hA2 tAll goals completed! 🐙
  ·d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ ∀ i_1 ∈ Finset.univ, DifferentiableAt ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i, i_1) * s.unit.val i_1) t intro i _d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di✝:Fin dk:Fin dhA2:ContDiff ℝ 2 A.vali:Fin da✝:i ∈ Finset.univ⊢ DifferentiableAt ℝ (fun t => magneticFieldMatrix 𝓕.c A t x (i✝, i) * s.unit.val i) t
    apply Differentiable.differentiableAtd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di✝:Fin dk:Fin dhA2:ContDiff ℝ 2 A.vali:Fin da✝:i ∈ Finset.univ⊢ Differentiable ℝ fun t => magneticFieldMatrix 𝓕.c A t x (i✝, i) * s.unit.val i
    apply Differentiable.mul_constd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di✝:Fin dk:Fin dhA2:ContDiff ℝ 2 A.vali:Fin da✝:i ∈ Finset.univ⊢ Differentiable ℝ fun y => magneticFieldMatrix 𝓕.c A y x (i✝, i)
    exact fieldStrengthMatrix_differentiable_time hA2 xAll goals completed! 🐙
  ·had:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ Differentiable ℝ fun t => ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j apply Differentiable.fun_sumhad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di:Fin dk:Fin dhA2:ContDiff ℝ 2 A.val⊢ ∀ i_1 ∈ Finset.univ, Differentiable ℝ fun y => magneticFieldMatrix 𝓕.c A y x (i, i_1) * s.unit.val i_1
    intro i _had:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di✝:Fin dk:Fin dhA2:ContDiff ℝ 2 A.vali:Fin da✝:i ∈ Finset.univ⊢ Differentiable ℝ fun y => magneticFieldMatrix 𝓕.c A y x (i✝, i) * s.unit.val i
    apply Differentiable.mul_consthad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0t:Timex:Space di✝:Fin dk:Fin dhA2:ContDiff ℝ 2 A.vali:Fin da✝:i ∈ Finset.univ⊢ Differentiable ℝ fun y => magneticFieldMatrix 𝓕.c A y x (i✝, i)
    exact fieldStrengthMatrix_differentiable_time hA2 xAll goals completed! 🐙

C.3. Electric field equal propogator cross magnetic field up to constant



lemma electricField_eq_propogator_cross_magneticFieldMatrix {d : ℕ}
    {𝓕 : FreeSpace} {A : ElectromagneticPotential d}
    {s : Direction d} (P : IsPlaneWave 𝓕 A s) (hA : ContDiff ℝ ∞ A)
    (h : IsExtrema 𝓕 A 0) (i : Fin d) :
    ∃ C, ∀ t x, A.electricField 𝓕.c t x i =
    𝓕.c * ∑ j, A.magneticFieldMatrix 𝓕.c t x (i, j) * s.unit j + C := byd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0i:Fin d⊢ ∃ C,
  ∀ (t : Time) (x : Space d),
    (electricField 𝓕.c A t x).ofLp i = 𝓕.c.val * ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j + C
  have hA2 : ContDiff ℝ 2 A := hA.of_le ENat.LEInfty.outd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0i:Fin dhA2:ContDiff ℝ 2 A.val⊢ ∃ C,
  ∀ (t : Time) (x : Space d),
    (electricField 𝓕.c A t x).ofLp i = 𝓕.c.val * ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j + C
  apply Space.equal_up_to_const_of_deriv_eqhfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0i:Fin dhA2:ContDiff ℝ 2 A.val⊢ Differentiable ℝ ↿fun t x => (electricField 𝓕.c A t x).ofLp ihgd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0i:Fin dhA2:ContDiff ℝ 2 A.val⊢ Differentiable ℝ ↿fun t x => 𝓕.c.val * ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val jh₁d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0i:Fin dhA2:ContDiff ℝ 2 A.val⊢ ∀ (t : Time) (x : Space d),
  ∂ₜ (fun x_1 => (electricField 𝓕.c A x_1 x).ofLp i) t =
    ∂ₜ (fun x_1 => 𝓕.c.val * ∑ j, magneticFieldMatrix 𝓕.c A x_1 x (i, j) * s.unit.val j) th₂d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0i:Fin dhA2:ContDiff ℝ 2 A.val⊢ ∀ (t : Time) (x : Space d) (i_1 : Fin d),
  Space.deriv i_1 (fun x => (electricField 𝓕.c A t x).ofLp i) x =
    Space.deriv i_1 (fun x => 𝓕.c.val * ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) x
  ·hfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0i:Fin dhA2:ContDiff ℝ 2 A.val⊢ Differentiable ℝ ↿fun t x => (electricField 𝓕.c A t x).ofLp i exact electricField_apply_differentiable hA2All goals completed! 🐙
  ·hgd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0i:Fin dhA2:ContDiff ℝ 2 A.val⊢ Differentiable ℝ ↿fun t x => 𝓕.c.val * ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j exact (Differentiable.fun_sum fun j _ =>
      (magneticFieldMatrix_differentiable A hA2 (i, j)).mul_const _).const_mul _All goals completed! 🐙
  ·h₁d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0i:Fin dhA2:ContDiff ℝ 2 A.val⊢ ∀ (t : Time) (x : Space d),
  ∂ₜ (fun x_1 => (electricField 𝓕.c A x_1 x).ofLp i) t =
    ∂ₜ (fun x_1 => 𝓕.c.val * ∑ j, magneticFieldMatrix 𝓕.c A x_1 x (i, j) * s.unit.val j) t intro t xh₁d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0i:Fin dhA2:ContDiff ℝ 2 A.valt:Timex:Space d⊢ ∂ₜ (fun x_1 => (electricField 𝓕.c A x_1 x).ofLp i) t =
  ∂ₜ (fun x_1 => 𝓕.c.val * ∑ j, magneticFieldMatrix 𝓕.c A x_1 x (i, j) * s.unit.val j) t
    rw [P.time_deriv_electricField_eq_magneticFieldMatrix hA _ t x ih₁d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0i:Fin dhA2:ContDiff ℝ 2 A.valt:Timex:Space d⊢ ∂ₜ (fun t => 𝓕.c.val * ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) t =
  ∂ₜ (fun x_1 => 𝓕.c.val * ∑ j, magneticFieldMatrix 𝓕.c A x_1 x (i, j) * s.unit.val j) td:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0i:Fin dhA2:ContDiff ℝ 2 A.valt:Timex:Space d⊢ IsExtrema 𝓕 A 0d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0i:Fin dhA2:ContDiff ℝ 2 A.valt:Timex:Space d⊢ IsExtrema 𝓕 A 0]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0i:Fin dhA2:ContDiff ℝ 2 A.valt:Timex:Space d⊢ IsExtrema 𝓕 A 0
    congrAll goals completed! 🐙
  ·h₂d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0i:Fin dhA2:ContDiff ℝ 2 A.val⊢ ∀ (t : Time) (x : Space d) (i_1 : Fin d),
  Space.deriv i_1 (fun x => (electricField 𝓕.c A t x).ofLp i) x =
    Space.deriv i_1 (fun x => 𝓕.c.val * ∑ j, magneticFieldMatrix 𝓕.c A t x (i, j) * s.unit.val j) x intro t x ih₂d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0i✝:Fin dhA2:ContDiff ℝ 2 A.valt:Timex:Space di:Fin d⊢ Space.deriv i (fun x => (electricField 𝓕.c A t x).ofLp i✝) x =
  Space.deriv i (fun x => 𝓕.c.val * ∑ j, magneticFieldMatrix 𝓕.c A t x (i✝, j) * s.unit.val j) x
    rw [P.space_deriv_electricField_eq_magneticFieldMatrix hAh₂d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0i✝:Fin dhA2:ContDiff ℝ 2 A.valt:Timex:Space di:Fin d⊢ Space.deriv i (fun x => 𝓕.c.val * ∑ j, magneticFieldMatrix 𝓕.c A t x (i✝, j) * s.unit.val j) x =
  Space.deriv i (fun x => 𝓕.c.val * ∑ j, magneticFieldMatrix 𝓕.c A t x (i✝, j) * s.unit.val j) xh₂.hd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0i✝:Fin dhA2:ContDiff ℝ 2 A.valt:Timex:Space di:Fin d⊢ IsExtrema 𝓕 A 0h₂.hd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0i✝:Fin dhA2:ContDiff ℝ 2 A.valt:Timex:Space di:Fin d⊢ IsExtrema 𝓕 A 0]h₂.hd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential ds:Direction dP:IsPlaneWave 𝓕 A shA:ContDiff ℝ ∞ A.valh:IsExtrema 𝓕 A 0i✝:Fin dhA2:ContDiff ℝ 2 A.valt:Timex:Space di:Fin d⊢ IsExtrema 𝓕 A 0
    congrAll goals completed! 🐙