Physlib.Electromagnetism.Dynamics.IsExtrema

Imports

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

public import Physlib.Electromagnetism.Dynamics.Lagrangian

Extrema of the Lagrangian density

i. Overview

In this module we define what it means for an electromagnetic potential to be an extremum of the Lagrangian density in presence of a Lorentz current density.

This is equivalent to the electromagnetic potential satisfying Maxwell's equations with sources, i.e. Gauss's law and Ampère's law.

ii. Key results

IsExtrema : The condition on an electromagnetic potential to be an extrema of the lagrangian.

isExtrema_iff_gauss_ampere_magneticFieldMatrix : The electromagnetic potential is an extrema of the lagrangian if and only if Gauss's law and Ampère's law hold (in terms of the magnetic field matrix).

time_deriv_time_deriv_magneticFieldMatrix_of_isExtrema : A wave-like equation for the magnetic field matrix from the extrema condition.

time_deriv_time_deriv_electricField_of_isExtrema : A wave-like equation for the electric field from the extrema condition.

iii. Table of contents

A. The condition for an extrema of the Lagrangian density

A.1. Extrema condition in terms of the field strength matrix

A.2. Extrema condition in terms of tensors

A.3. Equivariance of the extrema condition

B. Gauss's law and Ampère's law and the extrema condition

C. Time derivatives from the extrema condition

D. Second time derivatives from the extrema condition

D.1. Second time derivatives of the magnetic field from the extrema condition

D.2. Second time derivatives of the electric field from the extrema condition

iv. References

              @[expose] public sectionattribute [-simp] Fintype.sum_sum_typeattribute [-simp] Nat.succ_eq_add_one

A. The condition for an extrema of the Lagrangian density

The condition on an electromagnetic potential to be an extrema of the lagrangian.

def IsExtrema {d} (𝓕 : FreeSpace) (A : ElectromagneticPotential d)
    (J : LorentzCurrentDensity d) : Prop :=
  gradLagrangian 𝓕 A J = 0

lemma isExtrema_iff_gradLagrangian {𝓕 : FreeSpace} (A : ElectromagneticPotential d)
    (J : LorentzCurrentDensity d) :
    IsExtrema 𝓕 A J ↔ A.gradLagrangian 𝓕 J = 0 := byd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dJ:LorentzCurrentDensity d⊢ IsExtrema 𝓕 A J ↔ gradLagrangian 𝓕 A J = 0 rflAll goals completed! 🐙

A.1. Extrema condition in terms of the field strength matrix



lemma isExtrema_iff_fieldStrengthMatrix {𝓕 : FreeSpace}
    (A : ElectromagneticPotential d)
    (hA : ContDiff ℝ ∞ A) (J : LorentzCurrentDensity d) (hJ : ContDiff ℝ ∞ J) :
    IsExtrema 𝓕 A J ↔
    ∀ x, ∀ ν, ∑ μ, ∂_ μ (A.fieldStrengthMatrix · (μ, ν)) x = 𝓕.μ₀ * J x ν := byd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ J⊢ IsExtrema 𝓕 A J ↔
  ∀ (x : SpaceTime d) (ν : Fin 1 ⊕ Fin d), ∑ μ, ∂_ μ (fun x => (A.fieldStrengthMatrix x) (μ, ν)) x = 𝓕.μ₀ * J x ν
  rw [isExtrema_iff_gradLagrangian,d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ J⊢ gradLagrangian 𝓕 A J = 0 ↔
  ∀ (x : SpaceTime d) (ν : Fin 1 ⊕ Fin d), ∑ μ, ∂_ μ (fun x => (A.fieldStrengthMatrix x) (μ, ν)) x = 𝓕.μ₀ * J x νd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ J⊢ (∀ (x : SpaceTime d),
    ∑ ν, η ν ν • (1 / 𝓕.μ₀ * ∑ μ, ∂_ μ (fun x => (A.fieldStrengthMatrix x) (μ, ν)) x - J x ν) • Lorentz.Vector.basis ν =
      0 x) ↔
  ∀ (x : SpaceTime d) (ν : Fin 1 ⊕ Fin d), ∑ μ, ∂_ μ (fun x => (A.fieldStrengthMatrix x) (μ, ν)) x = 𝓕.μ₀ * J x ν gradLagrangian_eq_sum_fieldStrengthMatrix A hA J hJ,d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ J⊢ (fun x =>
      ∑ ν,
        η ν ν • (1 / 𝓕.μ₀ * ∑ μ, ∂_ μ (fun x => (A.fieldStrengthMatrix x) (μ, ν)) x - J x ν) • Lorentz.Vector.basis ν) =
    0 ↔
  ∀ (x : SpaceTime d) (ν : Fin 1 ⊕ Fin d), ∑ μ, ∂_ μ (fun x => (A.fieldStrengthMatrix x) (μ, ν)) x = 𝓕.μ₀ * J x νd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ J⊢ (∀ (x : SpaceTime d),
    ∑ ν, η ν ν • (1 / 𝓕.μ₀ * ∑ μ, ∂_ μ (fun x => (A.fieldStrengthMatrix x) (μ, ν)) x - J x ν) • Lorentz.Vector.basis ν =
      0 x) ↔
  ∀ (x : SpaceTime d) (ν : Fin 1 ⊕ Fin d), ∑ μ, ∂_ μ (fun x => (A.fieldStrengthMatrix x) (μ, ν)) x = 𝓕.μ₀ * J x ν funext_iffd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ J⊢ (∀ (x : SpaceTime d),
    ∑ ν, η ν ν • (1 / 𝓕.μ₀ * ∑ μ, ∂_ μ (fun x => (A.fieldStrengthMatrix x) (μ, ν)) x - J x ν) • Lorentz.Vector.basis ν =
      0 x) ↔
  ∀ (x : SpaceTime d) (ν : Fin 1 ⊕ Fin d), ∑ μ, ∂_ μ (fun x => (A.fieldStrengthMatrix x) (μ, ν)) x = 𝓕.μ₀ * J x νd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ J⊢ (∀ (x : SpaceTime d),
    ∑ ν, η ν ν • (1 / 𝓕.μ₀ * ∑ μ, ∂_ μ (fun x => (A.fieldStrengthMatrix x) (μ, ν)) x - J x ν) • Lorentz.Vector.basis ν =
      0 x) ↔
  ∀ (x : SpaceTime d) (ν : Fin 1 ⊕ Fin d), ∑ μ, ∂_ μ (fun x => (A.fieldStrengthMatrix x) (μ, ν)) x = 𝓕.μ₀ * J x ν]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ J⊢ (∀ (x : SpaceTime d),
    ∑ ν, η ν ν • (1 / 𝓕.μ₀ * ∑ μ, ∂_ μ (fun x => (A.fieldStrengthMatrix x) (μ, ν)) x - J x ν) • Lorentz.Vector.basis ν =
      0 x) ↔
  ∀ (x : SpaceTime d) (ν : Fin 1 ⊕ Fin d), ∑ μ, ∂_ μ (fun x => (A.fieldStrengthMatrix x) (μ, ν)) x = 𝓕.μ₀ * J x ν
  conv_lhs =>
    enter [x, 1, 2, ν]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jx:SpaceTime dν:Fin 1 ⊕ Fin d| η ν ν • (1 / 𝓕.μ₀ * ∑ μ, ∂_ μ (fun x => (A.fieldStrengthMatrix x) (μ, ν)) x - J x ν) • Lorentz.Vector.basis ν
    rw [smul_smul]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jx:SpaceTime dν:Fin 1 ⊕ Fin d| (η ν ν * (1 / 𝓕.μ₀ * ∑ μ, ∂_ μ (fun x => (A.fieldStrengthMatrix x) (μ, ν)) x - J x ν)) • Lorentz.Vector.basis ν
  conv_lhs =>
    enter [x]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jx:SpaceTime d| ∑ ν, (η ν ν * (1 / 𝓕.μ₀ * ∑ μ, ∂_ μ (fun x => (A.fieldStrengthMatrix x) (μ, ν)) x - J x ν)) • Lorentz.Vector.basis ν =
  0 x
    simp only [one_div, Pi.zero_apply]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jx:SpaceTime d| ∑ x_1,
    (η x_1 x_1 * (𝓕.μ₀⁻¹ * ∑ μ, ∂_ μ (fun x => (A.fieldStrengthMatrix x) (μ, x_1)) x - J x x_1)) •
      Lorentz.Vector.basis x_1 =
  0
    rw [Lorentz.Vector.sum_basis_eq_zero_iff]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jx:SpaceTime d| ∀ (μ : Fin 1 ⊕ Fin d), η μ μ * (𝓕.μ₀⁻¹ * ∑ μ_1, ∂_ μ_1 (fun x => (A.fieldStrengthMatrix x) (μ_1, μ)) x - J x μ) = 0
  apply Iff.intrompd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ J⊢ (∀ (x : SpaceTime d) (μ : Fin 1 ⊕ Fin d),
    η μ μ * (𝓕.μ₀⁻¹ * ∑ μ_1, ∂_ μ_1 (fun x => (A.fieldStrengthMatrix x) (μ_1, μ)) x - J x μ) = 0) →
  ∀ (x : SpaceTime d) (ν : Fin 1 ⊕ Fin d), ∑ μ, ∂_ μ (fun x => (A.fieldStrengthMatrix x) (μ, ν)) x = 𝓕.μ₀ * J x νmprd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ J⊢ (∀ (x : SpaceTime d) (ν : Fin 1 ⊕ Fin d), ∑ μ, ∂_ μ (fun x => (A.fieldStrengthMatrix x) (μ, ν)) x = 𝓕.μ₀ * J x ν) →
  ∀ (x : SpaceTime d) (μ : Fin 1 ⊕ Fin d),
    η μ μ * (𝓕.μ₀⁻¹ * ∑ μ_1, ∂_ μ_1 (fun x => (A.fieldStrengthMatrix x) (μ_1, μ)) x - J x μ) = 0
  ·mpd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ J⊢ (∀ (x : SpaceTime d) (μ : Fin 1 ⊕ Fin d),
    η μ μ * (𝓕.μ₀⁻¹ * ∑ μ_1, ∂_ μ_1 (fun x => (A.fieldStrengthMatrix x) (μ_1, μ)) x - J x μ) = 0) →
  ∀ (x : SpaceTime d) (ν : Fin 1 ⊕ Fin d), ∑ μ, ∂_ μ (fun x => (A.fieldStrengthMatrix x) (μ, ν)) x = 𝓕.μ₀ * J x ν intro h x νmpd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:∀ (x : SpaceTime d) (μ : Fin 1 ⊕ Fin d),
  η μ μ * (𝓕.μ₀⁻¹ * ∑ μ_1, ∂_ μ_1 (fun x => (A.fieldStrengthMatrix x) (μ_1, μ)) x - J x μ) = 0x:SpaceTime dν:Fin 1 ⊕ Fin d⊢ ∑ μ, ∂_ μ (fun x => (A.fieldStrengthMatrix x) (μ, ν)) x = 𝓕.μ₀ * J x ν
    specialize h x νmpd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jx:SpaceTime dν:Fin 1 ⊕ Fin dh:η ν ν * (𝓕.μ₀⁻¹ * ∑ μ, ∂_ μ (fun x => (A.fieldStrengthMatrix x) (μ, ν)) x - J x ν) = 0⊢ ∑ μ, ∂_ μ (fun x => (A.fieldStrengthMatrix x) (μ, ν)) x = 𝓕.μ₀ * J x ν
    simp at hmpd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jx:SpaceTime dν:Fin 1 ⊕ Fin dh:η ν ν = 0 ∨ 𝓕.μ₀⁻¹ * ∑ μ, ∂_ μ (fun x => (A.fieldStrengthMatrix x) (μ, ν)) x - J x ν = 0⊢ ∑ μ, ∂_ μ (fun x => (A.fieldStrengthMatrix x) (μ, ν)) x = 𝓕.μ₀ * J x ν
    have h' : η ν ν ≠ 0 := η_diag_ne_zerompd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jx:SpaceTime dν:Fin 1 ⊕ Fin dh:η ν ν = 0 ∨ 𝓕.μ₀⁻¹ * ∑ μ, ∂_ μ (fun x => (A.fieldStrengthMatrix x) (μ, ν)) x - J x ν = 0h':η ν ν ≠ 0⊢ ∑ μ, ∂_ μ (fun x => (A.fieldStrengthMatrix x) (μ, ν)) x = 𝓕.μ₀ * J x ν
    simp_allmpd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jx:SpaceTime dν:Fin 1 ⊕ Fin dh:𝓕.μ₀⁻¹ * ∑ μ, ∂_ μ (fun x => (A.fieldStrengthMatrix x) (μ, ν)) x - J x ν = 0h':¬η ν ν = 0⊢ ∑ μ, ∂_ μ (fun x => (A.fieldStrengthMatrix x) (μ, ν)) x = 𝓕.μ₀ * J x ν
    linear_combination (norm := field_simpad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jx:SpaceTime dν:Fin 1 ⊕ Fin dh:𝓕.μ₀⁻¹ * ∑ μ, ∂_ μ (fun x => (A.fieldStrengthMatrix x) (μ, ν)) x - J x ν = 0h':¬η ν ν = 0⊢ ∑ μ, ∂_ μ (fun x => (A.fieldStrengthMatrix x) (μ, ν)) x + 𝓕.μ₀ * 0 -
    (𝓕.μ₀ * J x ν + (∑ μ, ∂_ μ (fun x => (A.fieldStrengthMatrix x) (μ, ν)) x - 𝓕.μ₀ * J x ν)) =
  0) 𝓕.μ₀ * h
    ringAll goals completed! 🐙
  ·mprd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ J⊢ (∀ (x : SpaceTime d) (ν : Fin 1 ⊕ Fin d), ∑ μ, ∂_ μ (fun x => (A.fieldStrengthMatrix x) (μ, ν)) x = 𝓕.μ₀ * J x ν) →
  ∀ (x : SpaceTime d) (μ : Fin 1 ⊕ Fin d),
    η μ μ * (𝓕.μ₀⁻¹ * ∑ μ_1, ∂_ μ_1 (fun x => (A.fieldStrengthMatrix x) (μ_1, μ)) x - J x μ) = 0 intro h x νmprd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:∀ (x : SpaceTime d) (ν : Fin 1 ⊕ Fin d), ∑ μ, ∂_ μ (fun x => (A.fieldStrengthMatrix x) (μ, ν)) x = 𝓕.μ₀ * J x νx:SpaceTime dν:Fin 1 ⊕ Fin d⊢ η ν ν * (𝓕.μ₀⁻¹ * ∑ μ, ∂_ μ (fun x => (A.fieldStrengthMatrix x) (μ, ν)) x - J x ν) = 0
    specialize h x νmprd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jx:SpaceTime dν:Fin 1 ⊕ Fin dh:∑ μ, ∂_ μ (fun x => (A.fieldStrengthMatrix x) (μ, ν)) x = 𝓕.μ₀ * J x ν⊢ η ν ν * (𝓕.μ₀⁻¹ * ∑ μ, ∂_ μ (fun x => (A.fieldStrengthMatrix x) (μ, ν)) x - J x ν) = 0
    simp only [mul_eq_zero]mprd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jx:SpaceTime dν:Fin 1 ⊕ Fin dh:∑ μ, ∂_ μ (fun x => (A.fieldStrengthMatrix x) (μ, ν)) x = 𝓕.μ₀ * J x ν⊢ η ν ν = 0 ∨ 𝓕.μ₀⁻¹ * ∑ μ, ∂_ μ (fun x => (A.fieldStrengthMatrix x) (μ, ν)) x - J x ν = 0
    rightmprd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jx:SpaceTime dν:Fin 1 ⊕ Fin dh:∑ μ, ∂_ μ (fun x => (A.fieldStrengthMatrix x) (μ, ν)) x = 𝓕.μ₀ * J x ν⊢ 𝓕.μ₀⁻¹ * ∑ μ, ∂_ μ (fun x => (A.fieldStrengthMatrix x) (μ, ν)) x - J x ν = 0
    linear_combination (norm := field_simpad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jx:SpaceTime dν:Fin 1 ⊕ Fin dh:∑ μ, ∂_ μ (fun x => (A.fieldStrengthMatrix x) (μ, ν)) x = 𝓕.μ₀ * J x ν⊢ ∑ μ, ∂_ μ (fun x => (A.fieldStrengthMatrix x) (μ, ν)) x - 𝓕.μ₀ * J x ν + 𝓕.μ₀ * J x ν -
    (𝓕.μ₀ * 0 + ∑ μ, ∂_ μ (fun x => (A.fieldStrengthMatrix x) (μ, ν)) x) =
  𝓕.μ₀ * 0) 𝓕.μ₀⁻¹ * h
    ringAll goals completed! 🐙

A.2. Extrema condition in terms of tensors

The electromagnetic potential is an exterma of the lagrangian if and only if

$$\frac{1}{\mu_0} \partial_\mu F^{\mu \nu} - J^{\nu} = 0.$$

attribute [-simp] Nat.reduceAdd Nat.reduceSucc Fin.isValue



lemma isExtrema_iff_tensors {𝓕 : FreeSpace}
    (A : ElectromagneticPotential d)
    (hA : ContDiff ℝ ∞ A) (J : LorentzCurrentDensity d) (hJ : ContDiff ℝ ∞ J) :
    IsExtrema 𝓕 A J ↔ ∀ x,
    {((1/ 𝓕.μ₀ : ℝ) • tensorDeriv A.toFieldStrength x | κ κ ν') + - (J x | ν')}ᵀ = 0 := byd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ J⊢ IsExtrema 𝓕 A J ↔
  ∀ (x : SpaceTime d),
    (contrT 1 0 1 ⋯) (Tensorial.toTensor ((1 / 𝓕.μ₀) • tensorDeriv A.toFieldStrength x)) +
        (permT ![0] ⋯) (-Tensorial.toTensor (J x)) =
      0
  apply Iff.intrompd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ J⊢ IsExtrema 𝓕 A J →
  ∀ (x : SpaceTime d),
    (contrT 1 0 1 ⋯) (Tensorial.toTensor ((1 / 𝓕.μ₀) • tensorDeriv A.toFieldStrength x)) +
        (permT ![0] ⋯) (-Tensorial.toTensor (J x)) =
      0mprd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ J⊢ (∀ (x : SpaceTime d),
    (contrT 1 0 1 ⋯) (Tensorial.toTensor ((1 / 𝓕.μ₀) • tensorDeriv A.toFieldStrength x)) +
        (permT ![0] ⋯) (-Tensorial.toTensor (J x)) =
      0) →
  IsExtrema 𝓕 A J
  ·mpd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ J⊢ IsExtrema 𝓕 A J →
  ∀ (x : SpaceTime d),
    (contrT 1 0 1 ⋯) (Tensorial.toTensor ((1 / 𝓕.μ₀) • tensorDeriv A.toFieldStrength x)) +
        (permT ![0] ⋯) (-Tensorial.toTensor (J x)) =
      0 intro hmpd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A J⊢ ∀ (x : SpaceTime d),
  (contrT 1 0 1 ⋯) (Tensorial.toTensor ((1 / 𝓕.μ₀) • tensorDeriv A.toFieldStrength x)) +
      (permT ![0] ⋯) (-Tensorial.toTensor (J x)) =
    0
    simp only [IsExtrema] at hmpd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:gradLagrangian 𝓕 A J = 0⊢ ∀ (x : SpaceTime d),
  (contrT 1 0 1 ⋯) (Tensorial.toTensor ((1 / 𝓕.μ₀) • tensorDeriv A.toFieldStrength x)) +
      (permT ![0] ⋯) (-Tensorial.toTensor (J x)) =
    0
    intro xmpd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:gradLagrangian 𝓕 A J = 0x:SpaceTime d⊢ (contrT 1 0 1 ⋯) (Tensorial.toTensor ((1 / 𝓕.μ₀) • tensorDeriv A.toFieldStrength x)) +
    (permT ![0] ⋯) (-Tensorial.toTensor (J x)) =
  0
    have h1 : ((Tensorial.toTensor (M := Lorentz.Vector d)).symm
        (permT id (IsReindexing.auto) {((1/ 𝓕.μ₀ : ℝ) • tensorDeriv A.toFieldStrength x | κ κ ν') +
        - (J x | ν')}ᵀ)) = 0 := byd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ J⊢ IsExtrema 𝓕 A J ↔
  ∀ (x : SpaceTime d),
    (contrT 1 0 1 ⋯) (Tensorial.toTensor ((1 / 𝓕.μ₀) • tensorDeriv A.toFieldStrength x)) +
        (permT ![0] ⋯) (-Tensorial.toTensor (J x)) =
      0mpd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:gradLagrangian 𝓕 A J = 0x:SpaceTime dh1:Tensorial.toTensor.symm
    ((permT id ⋯)
      ((contrT 1 0 1 ⋯) (Tensorial.toTensor ((1 / 𝓕.μ₀) • tensorDeriv A.toFieldStrength x)) +
        (permT ![0] ⋯) (-Tensorial.toTensor (J x)))) =
  0⊢ (contrT 1 0 1 ⋯) (Tensorial.toTensor ((1 / 𝓕.μ₀) • tensorDeriv A.toFieldStrength x)) +
    (permT ![0] ⋯) (-Tensorial.toTensor (J x)) =
  0
      funext νd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:gradLagrangian 𝓕 A J = 0x:SpaceTime dν:Fin 1 ⊕ Fin d⊢ Tensorial.toTensor.symm
    ((permT id ⋯)
      ((contrT 1 0 1 ⋯) (Tensorial.toTensor ((1 / 𝓕.μ₀) • tensorDeriv A.toFieldStrength x)) +
        (permT ![0] ⋯) (-Tensorial.toTensor (J x))))
    ν =
  0 νmpd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:gradLagrangian 𝓕 A J = 0x:SpaceTime dh1:Tensorial.toTensor.symm
    ((permT id ⋯)
      ((contrT 1 0 1 ⋯) (Tensorial.toTensor ((1 / 𝓕.μ₀) • tensorDeriv A.toFieldStrength x)) +
        (permT ![0] ⋯) (-Tensorial.toTensor (J x)))) =
  0⊢ (contrT 1 0 1 ⋯) (Tensorial.toTensor ((1 / 𝓕.μ₀) • tensorDeriv A.toFieldStrength x)) +
    (permT ![0] ⋯) (-Tensorial.toTensor (J x)) =
  0
      have h2 : gradLagrangian 𝓕 A J x ν = 0 := byd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ J⊢ IsExtrema 𝓕 A J ↔
  ∀ (x : SpaceTime d),
    (contrT 1 0 1 ⋯) (Tensorial.toTensor ((1 / 𝓕.μ₀) • tensorDeriv A.toFieldStrength x)) +
        (permT ![0] ⋯) (-Tensorial.toTensor (J x)) =
      0d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:gradLagrangian 𝓕 A J = 0x:SpaceTime dν:Fin 1 ⊕ Fin dh2:gradLagrangian 𝓕 A J x ν = 0⊢ Tensorial.toTensor.symm
    ((permT id ⋯)
      ((contrT 1 0 1 ⋯) (Tensorial.toTensor ((1 / 𝓕.μ₀) • tensorDeriv A.toFieldStrength x)) +
        (permT ![0] ⋯) (-Tensorial.toTensor (J x))))
    ν =
  0 νmpd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:gradLagrangian 𝓕 A J = 0x:SpaceTime dh1:Tensorial.toTensor.symm
    ((permT id ⋯)
      ((contrT 1 0 1 ⋯) (Tensorial.toTensor ((1 / 𝓕.μ₀) • tensorDeriv A.toFieldStrength x)) +
        (permT ![0] ⋯) (-Tensorial.toTensor (J x)))) =
  0⊢ (contrT 1 0 1 ⋯) (Tensorial.toTensor ((1 / 𝓕.μ₀) • tensorDeriv A.toFieldStrength x)) +
    (permT ![0] ⋯) (-Tensorial.toTensor (J x)) =
  0 simp [h]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:gradLagrangian 𝓕 A J = 0x:SpaceTime dν:Fin 1 ⊕ Fin dh2:gradLagrangian 𝓕 A J x ν = 0⊢ Tensorial.toTensor.symm
    ((permT id ⋯)
      ((contrT 1 0 1 ⋯) (Tensorial.toTensor ((1 / 𝓕.μ₀) • tensorDeriv A.toFieldStrength x)) +
        (permT ![0] ⋯) (-Tensorial.toTensor (J x))))
    ν =
  0 νmpd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:gradLagrangian 𝓕 A J = 0x:SpaceTime dh1:Tensorial.toTensor.symm
    ((permT id ⋯)
      ((contrT 1 0 1 ⋯) (Tensorial.toTensor ((1 / 𝓕.μ₀) • tensorDeriv A.toFieldStrength x)) +
        (permT ![0] ⋯) (-Tensorial.toTensor (J x)))) =
  0⊢ (contrT 1 0 1 ⋯) (Tensorial.toTensor ((1 / 𝓕.μ₀) • tensorDeriv A.toFieldStrength x)) +
    (permT ![0] ⋯) (-Tensorial.toTensor (J x)) =
  0d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:gradLagrangian 𝓕 A J = 0x:SpaceTime dν:Fin 1 ⊕ Fin dh2:gradLagrangian 𝓕 A J x ν = 0⊢ Tensorial.toTensor.symm
    ((permT id ⋯)
      ((contrT 1 0 1 ⋯) (Tensorial.toTensor ((1 / 𝓕.μ₀) • tensorDeriv A.toFieldStrength x)) +
        (permT ![0] ⋯) (-Tensorial.toTensor (J x))))
    ν =
  0 νmpd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:gradLagrangian 𝓕 A J = 0x:SpaceTime dh1:Tensorial.toTensor.symm
    ((permT id ⋯)
      ((contrT 1 0 1 ⋯) (Tensorial.toTensor ((1 / 𝓕.μ₀) • tensorDeriv A.toFieldStrength x)) +
        (permT ![0] ⋯) (-Tensorial.toTensor (J x)))) =
  0⊢ (contrT 1 0 1 ⋯) (Tensorial.toTensor ((1 / 𝓕.μ₀) • tensorDeriv A.toFieldStrength x)) +
    (permT ![0] ⋯) (-Tensorial.toTensor (J x)) =
  0
      rw [gradLagrangian_eq_tensor A hA J hJd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:gradLagrangian 𝓕 A J = 0x:SpaceTime dν:Fin 1 ⊕ Fin dh2:η ν ν *
    Tensorial.toTensor.symm
      ((permT id ⋯)
        ((contrT 1 0 1 ⋯) (Tensorial.toTensor ((1 / 𝓕.μ₀) • tensorDeriv A.toFieldStrength x)) +
          (permT ![0] ⋯) (-Tensorial.toTensor (J x))))
      ν =
  0⊢ Tensorial.toTensor.symm
    ((permT id ⋯)
      ((contrT 1 0 1 ⋯) (Tensorial.toTensor ((1 / 𝓕.μ₀) • tensorDeriv A.toFieldStrength x)) +
        (permT ![0] ⋯) (-Tensorial.toTensor (J x))))
    ν =
  0 νd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:gradLagrangian 𝓕 A J = 0x:SpaceTime dν:Fin 1 ⊕ Fin dh2:η ν ν *
    Tensorial.toTensor.symm
      ((permT id ⋯)
        ((contrT 1 0 1 ⋯) (Tensorial.toTensor ((1 / 𝓕.μ₀) • tensorDeriv A.toFieldStrength x)) +
          (permT ![0] ⋯) (-Tensorial.toTensor (J x))))
      ν =
  0⊢ Tensorial.toTensor.symm
    ((permT id ⋯)
      ((contrT 1 0 1 ⋯) (Tensorial.toTensor ((1 / 𝓕.μ₀) • tensorDeriv A.toFieldStrength x)) +
        (permT ![0] ⋯) (-Tensorial.toTensor (J x))))
    ν =
  0 νmpd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:gradLagrangian 𝓕 A J = 0x:SpaceTime dh1:Tensorial.toTensor.symm
    ((permT id ⋯)
      ((contrT 1 0 1 ⋯) (Tensorial.toTensor ((1 / 𝓕.μ₀) • tensorDeriv A.toFieldStrength x)) +
        (permT ![0] ⋯) (-Tensorial.toTensor (J x)))) =
  0⊢ (contrT 1 0 1 ⋯) (Tensorial.toTensor ((1 / 𝓕.μ₀) • tensorDeriv A.toFieldStrength x)) +
    (permT ![0] ⋯) (-Tensorial.toTensor (J x)) =
  0] at h2d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:gradLagrangian 𝓕 A J = 0x:SpaceTime dν:Fin 1 ⊕ Fin dh2:η ν ν *
    Tensorial.toTensor.symm
      ((permT id ⋯)
        ((contrT 1 0 1 ⋯) (Tensorial.toTensor ((1 / 𝓕.μ₀) • tensorDeriv A.toFieldStrength x)) +
          (permT ![0] ⋯) (-Tensorial.toTensor (J x))))
      ν =
  0⊢ Tensorial.toTensor.symm
    ((permT id ⋯)
      ((contrT 1 0 1 ⋯) (Tensorial.toTensor ((1 / 𝓕.μ₀) • tensorDeriv A.toFieldStrength x)) +
        (permT ![0] ⋯) (-Tensorial.toTensor (J x))))
    ν =
  0 νmpd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:gradLagrangian 𝓕 A J = 0x:SpaceTime dh1:Tensorial.toTensor.symm
    ((permT id ⋯)
      ((contrT 1 0 1 ⋯) (Tensorial.toTensor ((1 / 𝓕.μ₀) • tensorDeriv A.toFieldStrength x)) +
        (permT ![0] ⋯) (-Tensorial.toTensor (J x)))) =
  0⊢ (contrT 1 0 1 ⋯) (Tensorial.toTensor ((1 / 𝓕.μ₀) • tensorDeriv A.toFieldStrength x)) +
    (permT ![0] ⋯) (-Tensorial.toTensor (J x)) =
  0
      simp only [one_div, map_smul, map_neg, map_add,
        permT_permT, CompTriple.comp_eq, apply_add, apply_smul, Lorentz.Vector.neg_apply,
        mul_eq_zero] at h2d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:gradLagrangian 𝓕 A J = 0x:SpaceTime dν:Fin 1 ⊕ Fin dh2:η ν ν = 0 ∨
  𝓕.μ₀⁻¹ *
        Tensorial.toTensor.symm ((permT id ⋯) ((contrT 1 0 1 ⋯) (Tensorial.toTensor (tensorDeriv A.toFieldStrength x))))
          ν +
      -Tensorial.toTensor.symm ((permT ![0] ⋯) (Tensorial.toTensor (J x))) ν =
    0⊢ Tensorial.toTensor.symm
    ((permT id ⋯)
      ((contrT 1 0 1 ⋯) (Tensorial.toTensor ((1 / 𝓕.μ₀) • tensorDeriv A.toFieldStrength x)) +
        (permT ![0] ⋯) (-Tensorial.toTensor (J x))))
    ν =
  0 νmpd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:gradLagrangian 𝓕 A J = 0x:SpaceTime dh1:Tensorial.toTensor.symm
    ((permT id ⋯)
      ((contrT 1 0 1 ⋯) (Tensorial.toTensor ((1 / 𝓕.μ₀) • tensorDeriv A.toFieldStrength x)) +
        (permT ![0] ⋯) (-Tensorial.toTensor (J x)))) =
  0⊢ (contrT 1 0 1 ⋯) (Tensorial.toTensor ((1 / 𝓕.μ₀) • tensorDeriv A.toFieldStrength x)) +
    (permT ![0] ⋯) (-Tensorial.toTensor (J x)) =
  0
      have hn : η ν ν ≠ 0 := η_diag_ne_zerod:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:gradLagrangian 𝓕 A J = 0x:SpaceTime dν:Fin 1 ⊕ Fin dh2:η ν ν = 0 ∨
  𝓕.μ₀⁻¹ *
        Tensorial.toTensor.symm ((permT id ⋯) ((contrT 1 0 1 ⋯) (Tensorial.toTensor (tensorDeriv A.toFieldStrength x))))
          ν +
      -Tensorial.toTensor.symm ((permT ![0] ⋯) (Tensorial.toTensor (J x))) ν =
    0hn:η ν ν ≠ 0⊢ Tensorial.toTensor.symm
    ((permT id ⋯)
      ((contrT 1 0 1 ⋯) (Tensorial.toTensor ((1 / 𝓕.μ₀) • tensorDeriv A.toFieldStrength x)) +
        (permT ![0] ⋯) (-Tensorial.toTensor (J x))))
    ν =
  0 νmpd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:gradLagrangian 𝓕 A J = 0x:SpaceTime dh1:Tensorial.toTensor.symm
    ((permT id ⋯)
      ((contrT 1 0 1 ⋯) (Tensorial.toTensor ((1 / 𝓕.μ₀) • tensorDeriv A.toFieldStrength x)) +
        (permT ![0] ⋯) (-Tensorial.toTensor (J x)))) =
  0⊢ (contrT 1 0 1 ⋯) (Tensorial.toTensor ((1 / 𝓕.μ₀) • tensorDeriv A.toFieldStrength x)) +
    (permT ![0] ⋯) (-Tensorial.toTensor (J x)) =
  0
      simp_all only [false_or, ne_eq, one_div, map_smul,
        map_neg, map_add, permT_permT, CompTriple.comp_eq, apply_add, apply_smul,
        Lorentz.Vector.neg_apply, Lorentz.Vector.zero_apply]mpd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:gradLagrangian 𝓕 A J = 0x:SpaceTime dh1:Tensorial.toTensor.symm
    ((permT id ⋯)
      ((contrT 1 0 1 ⋯) (Tensorial.toTensor ((1 / 𝓕.μ₀) • tensorDeriv A.toFieldStrength x)) +
        (permT ![0] ⋯) (-Tensorial.toTensor (J x)))) =
  0⊢ (contrT 1 0 1 ⋯) (Tensorial.toTensor ((1 / 𝓕.μ₀) • tensorDeriv A.toFieldStrength x)) +
    (permT ![0] ⋯) (-Tensorial.toTensor (J x)) =
  0mpd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:gradLagrangian 𝓕 A J = 0x:SpaceTime dh1:Tensorial.toTensor.symm
    ((permT id ⋯)
      ((contrT 1 0 1 ⋯) (Tensorial.toTensor ((1 / 𝓕.μ₀) • tensorDeriv A.toFieldStrength x)) +
        (permT ![0] ⋯) (-Tensorial.toTensor (J x)))) =
  0⊢ (contrT 1 0 1 ⋯) (Tensorial.toTensor ((1 / 𝓕.μ₀) • tensorDeriv A.toFieldStrength x)) +
    (permT ![0] ⋯) (-Tensorial.toTensor (J x)) =
  0
    generalize {((1/ 𝓕.μ₀ : ℝ) • tensorDeriv A.toFieldStrength x | κ κ ν') +
        - (J x | ν')}ᵀ = V at *mpd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:gradLagrangian 𝓕 A J = 0x:SpaceTime dV:(realLorentzTensor d).Tensor (Fin.append ![Color.down] (Fin.append ![Color.up] ![Color.up]) ∘ Fin.succSuccAbove 0 1)h1:Tensorial.toTensor.symm ((permT id ⋯) V) = 0⊢ V = 0
    simp only [EmbeddingLike.map_eq_zero_iff] at h1mpd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:gradLagrangian 𝓕 A J = 0x:SpaceTime dV:(realLorentzTensor d).Tensor (Fin.append ![Color.down] (Fin.append ![Color.up] ![Color.up]) ∘ Fin.succSuccAbove 0 1)h1:(permT id ⋯) V = 0⊢ V = 0
    rw [permT_eq_zero_iffmpd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:gradLagrangian 𝓕 A J = 0x:SpaceTime dV:(realLorentzTensor d).Tensor (Fin.append ![Color.down] (Fin.append ![Color.up] ![Color.up]) ∘ Fin.succSuccAbove 0 1)h1:V = 0⊢ V = 0mpd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:gradLagrangian 𝓕 A J = 0x:SpaceTime dV:(realLorentzTensor d).Tensor (Fin.append ![Color.down] (Fin.append ![Color.up] ![Color.up]) ∘ Fin.succSuccAbove 0 1)h1:V = 0⊢ V = 0] at h1mpd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:gradLagrangian 𝓕 A J = 0x:SpaceTime dV:(realLorentzTensor d).Tensor (Fin.append ![Color.down] (Fin.append ![Color.up] ![Color.up]) ∘ Fin.succSuccAbove 0 1)h1:V = 0⊢ V = 0
    exact h1All goals completed! 🐙
  ·mprd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ J⊢ (∀ (x : SpaceTime d),
    (contrT 1 0 1 ⋯) (Tensorial.toTensor ((1 / 𝓕.μ₀) • tensorDeriv A.toFieldStrength x)) +
        (permT ![0] ⋯) (-Tensorial.toTensor (J x)) =
      0) →
  IsExtrema 𝓕 A J intro hmprd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:∀ (x : SpaceTime d),
  (contrT 1 0 1 ⋯) (Tensorial.toTensor ((1 / 𝓕.μ₀) • tensorDeriv A.toFieldStrength x)) +
      (permT ![0] ⋯) (-Tensorial.toTensor (J x)) =
    0⊢ IsExtrema 𝓕 A J
    simp only [IsExtrema]mprd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:∀ (x : SpaceTime d),
  (contrT 1 0 1 ⋯) (Tensorial.toTensor ((1 / 𝓕.μ₀) • tensorDeriv A.toFieldStrength x)) +
      (permT ![0] ⋯) (-Tensorial.toTensor (J x)) =
    0⊢ gradLagrangian 𝓕 A J = 0
    funext xmprd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:∀ (x : SpaceTime d),
  (contrT 1 0 1 ⋯) (Tensorial.toTensor ((1 / 𝓕.μ₀) • tensorDeriv A.toFieldStrength x)) +
      (permT ![0] ⋯) (-Tensorial.toTensor (J x)) =
    0x:SpaceTime d⊢ gradLagrangian 𝓕 A J x = 0 x
    funext νmprd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:∀ (x : SpaceTime d),
  (contrT 1 0 1 ⋯) (Tensorial.toTensor ((1 / 𝓕.μ₀) • tensorDeriv A.toFieldStrength x)) +
      (permT ![0] ⋯) (-Tensorial.toTensor (J x)) =
    0x:SpaceTime dν:Fin 1 ⊕ Fin d⊢ gradLagrangian 𝓕 A J x ν = 0 x ν
    rw [gradLagrangian_eq_tensor A hA J hJ,mprd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:∀ (x : SpaceTime d),
  (contrT 1 0 1 ⋯) (Tensorial.toTensor ((1 / 𝓕.μ₀) • tensorDeriv A.toFieldStrength x)) +
      (permT ![0] ⋯) (-Tensorial.toTensor (J x)) =
    0x:SpaceTime dν:Fin 1 ⊕ Fin d⊢ η ν ν *
    Tensorial.toTensor.symm
      ((permT id ⋯)
        ((contrT 1 0 1 ⋯) (Tensorial.toTensor ((1 / 𝓕.μ₀) • tensorDeriv A.toFieldStrength x)) +
          (permT ![0] ⋯) (-Tensorial.toTensor (J x))))
      ν =
  0 x νmprd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:∀ (x : SpaceTime d),
  (contrT 1 0 1 ⋯) (Tensorial.toTensor ((1 / 𝓕.μ₀) • tensorDeriv A.toFieldStrength x)) +
      (permT ![0] ⋯) (-Tensorial.toTensor (J x)) =
    0x:SpaceTime dν:Fin 1 ⊕ Fin d⊢ η ν ν * Tensorial.toTensor.symm ((permT id ⋯) 0) ν = 0 x ν hmprd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:∀ (x : SpaceTime d),
  (contrT 1 0 1 ⋯) (Tensorial.toTensor ((1 / 𝓕.μ₀) • tensorDeriv A.toFieldStrength x)) +
      (permT ![0] ⋯) (-Tensorial.toTensor (J x)) =
    0x:SpaceTime dν:Fin 1 ⊕ Fin d⊢ η ν ν * Tensorial.toTensor.symm ((permT id ⋯) 0) ν = 0 x νmprd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:∀ (x : SpaceTime d),
  (contrT 1 0 1 ⋯) (Tensorial.toTensor ((1 / 𝓕.μ₀) • tensorDeriv A.toFieldStrength x)) +
      (permT ![0] ⋯) (-Tensorial.toTensor (J x)) =
    0x:SpaceTime dν:Fin 1 ⊕ Fin d⊢ η ν ν * Tensorial.toTensor.symm ((permT id ⋯) 0) ν = 0 x ν]mprd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:∀ (x : SpaceTime d),
  (contrT 1 0 1 ⋯) (Tensorial.toTensor ((1 / 𝓕.μ₀) • tensorDeriv A.toFieldStrength x)) +
      (permT ![0] ⋯) (-Tensorial.toTensor (J x)) =
    0x:SpaceTime dν:Fin 1 ⊕ Fin d⊢ η ν ν * Tensorial.toTensor.symm ((permT id ⋯) 0) ν = 0 x ν
    simp only [map_zero, Lorentz.Vector.zero_apply, mul_zero, Pi.zero_apply]All goals completed! 🐙

A.3. Equivariance of the extrema condition

If A is an extrema of the lagrangian with current density J, then the Lorentz transformation Λ • A (Λ⁻¹ • x) is an extrema of the lagrangian with current density Λ • J (Λ⁻¹ • x).

Combined with time_deriv_time_deriv_electricField_of_isExtrema, this shows that the speed with which an electromagnetic wave propagates is invariant under Lorentz transformations.



set_option maxHeartbeats 600000 in
set_option backward.isDefEq.respectTransparency false in
lemma isExtrema_lorentzGroup_apply_iff {𝓕 : FreeSpace}
    (A : ElectromagneticPotential d)
    (hA : ContDiff ℝ ∞ A) (J : LorentzCurrentDensity d) (hJ : ContDiff ℝ ∞ J)
    (Λ : LorentzGroup d) :
    IsExtrema 𝓕 (Λ • A) (fun x => Λ • J (Λ⁻¹ • x)) ↔
    IsExtrema 𝓕 A J := byd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ JΛ:↑(LorentzGroup d)⊢ (IsExtrema 𝓕 (Λ • A) fun x => Λ • J (Λ⁻¹ • x)) ↔ IsExtrema 𝓕 A J
  rw [isExtrema_iff_tensorsd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ JΛ:↑(LorentzGroup d)⊢ (∀ (x : SpaceTime d),
    (contrT 1 0 1 ⋯) (Tensorial.toTensor ((1 / 𝓕.μ₀) • tensorDeriv (Λ • A).toFieldStrength x)) +
        (permT ![0] ⋯) (-Tensorial.toTensor (Λ • J (Λ⁻¹ • x))) =
      0) ↔
  IsExtrema 𝓕 A JhAd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ JΛ:↑(LorentzGroup d)⊢ ContDiff ℝ ∞ (Λ • A).valhJd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ JΛ:↑(LorentzGroup d)⊢ ContDiff ℝ ∞ fun x => Λ • J (Λ⁻¹ • x)d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ JΛ:↑(LorentzGroup d)⊢ (∀ (x : SpaceTime d),
    (contrT 1 0 1 ⋯) (Tensorial.toTensor ((1 / 𝓕.μ₀) • tensorDeriv (Λ • A).toFieldStrength x)) +
        (permT ![0] ⋯) (-Tensorial.toTensor (Λ • J (Λ⁻¹ • x))) =
      0) ↔
  IsExtrema 𝓕 A JhAd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ JΛ:↑(LorentzGroup d)⊢ ContDiff ℝ ∞ (Λ • A).valhJd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ JΛ:↑(LorentzGroup d)⊢ ContDiff ℝ ∞ fun x => Λ • J (Λ⁻¹ • x)]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ JΛ:↑(LorentzGroup d)⊢ (∀ (x : SpaceTime d),
    (contrT 1 0 1 ⋯) (Tensorial.toTensor ((1 / 𝓕.μ₀) • tensorDeriv (Λ • A).toFieldStrength x)) +
        (permT ![0] ⋯) (-Tensorial.toTensor (Λ • J (Λ⁻¹ • x))) =
      0) ↔
  IsExtrema 𝓕 A JhAd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ JΛ:↑(LorentzGroup d)⊢ ContDiff ℝ ∞ (Λ • A).valhJd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ JΛ:↑(LorentzGroup d)⊢ ContDiff ℝ ∞ fun x => Λ • J (Λ⁻¹ • x)
  conv_lhs =>
    enter [x, 1, 1, 2, 2, 2]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ JΛ:↑(LorentzGroup d)x:SpaceTime d| tensorDeriv (Λ • A).toFieldStrength x
    change tensorDeriv (fun x => toFieldStrength (Λ • A) x) xd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ JΛ:↑(LorentzGroup d)x:SpaceTime d| tensorDeriv (fun x => (Λ • A).toFieldStrength x) x
    enter [1,x]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ JΛ:↑(LorentzGroup d)x✝:SpaceTime dx:SpaceTime d| (Λ • A).toFieldStrength x
    rw [toFieldStrength_equivariant _ _ (hA.differentiable (byd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ JΛ:↑(LorentzGroup d)x✝:SpaceTime dx:SpaceTime d⊢ ∞ ≠ 0 simpAll goals completed! 🐙))]
  conv_lhs =>
    enter [x]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ JΛ:↑(LorentzGroup d)x:SpaceTime d| (contrT 1 0 1 ⋯) (Tensorial.toTensor ((1 / 𝓕.μ₀) • tensorDeriv (fun x => Λ • A.toFieldStrength (Λ⁻¹ • x)) x)) +
    (permT ![0] ⋯) (-Tensorial.toTensor (Λ • J (Λ⁻¹ • x))) =
  0
    rw [tensorDeriv_equivariant _ _ _ (differentiable_toFieldStrength_of_smooth hA)]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ JΛ:↑(LorentzGroup d)x:SpaceTime d| (contrT 1 0 1 ⋯) (Tensorial.toTensor ((1 / 𝓕.μ₀) • Λ • tensorDeriv A.toFieldStrength (Λ⁻¹ • x))) +
    (permT ![0] ⋯) (-Tensorial.toTensor (Λ • J (Λ⁻¹ • x))) =
  0
    rw [smul_comm]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ JΛ:↑(LorentzGroup d)x:SpaceTime d| (contrT 1 0 1 ⋯) (Tensorial.toTensor (Λ • (1 / 𝓕.μ₀) • tensorDeriv A.toFieldStrength (Λ⁻¹ • x))) +
    (permT ![0] ⋯) (-Tensorial.toTensor (Λ • J (Λ⁻¹ • x))) =
  0
    rw [Tensorial.toTensor_smul, Tensorial.toTensor_smul]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ JΛ:↑(LorentzGroup d)x:SpaceTime d| (contrT 1 0 1 ⋯) (Λ • Tensorial.toTensor ((1 / 𝓕.μ₀) • tensorDeriv A.toFieldStrength (Λ⁻¹ • x))) +
    (permT ![0] ⋯) (-(Λ • Tensorial.toTensor (J (Λ⁻¹ • x)))) =
  0
    simp only [one_div, map_smul, actionT_smul,
      contrT_equivariant, map_neg, permT_equivariant]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ JΛ:↑(LorentzGroup d)x:SpaceTime d| 𝓕.μ₀⁻¹ • Λ • (contrT 1 0 1 ⋯) (Tensorial.toTensor (tensorDeriv A.toFieldStrength (Λ⁻¹ • x))) +
    -(Λ • (permT ![0] ⋯) (Tensorial.toTensor (J (Λ⁻¹ • x)))) =
  0
    rw [smul_comm, ← Tensor.actionT_neg, ← Tensor.actionT_add]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ JΛ:↑(LorentzGroup d)x:SpaceTime d| Λ •
    (𝓕.μ₀⁻¹ • (contrT 1 0 1 ⋯) (Tensorial.toTensor (tensorDeriv A.toFieldStrength (Λ⁻¹ • x))) +
      -(permT ![0] ⋯) (Tensorial.toTensor (J (Λ⁻¹ • x)))) =
  0
  apply Iff.intrompd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ JΛ:↑(LorentzGroup d)⊢ (∀ (x : SpaceTime d),
    Λ •
        (𝓕.μ₀⁻¹ • (contrT 1 0 1 ⋯) (Tensorial.toTensor (tensorDeriv A.toFieldStrength (Λ⁻¹ • x))) +
          -(permT ![0] ⋯) (Tensorial.toTensor (J (Λ⁻¹ • x)))) =
      0) →
  IsExtrema 𝓕 A Jmprd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ JΛ:↑(LorentzGroup d)⊢ IsExtrema 𝓕 A J →
  ∀ (x : SpaceTime d),
    Λ •
        (𝓕.μ₀⁻¹ • (contrT 1 0 1 ⋯) (Tensorial.toTensor (tensorDeriv A.toFieldStrength (Λ⁻¹ • x))) +
          -(permT ![0] ⋯) (Tensorial.toTensor (J (Λ⁻¹ • x)))) =
      0hAd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ JΛ:↑(LorentzGroup d)⊢ ContDiff ℝ ∞ (Λ • A).valhJd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ JΛ:↑(LorentzGroup d)⊢ ContDiff ℝ ∞ fun x => Λ • J (Λ⁻¹ • x)
  ·mpd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ JΛ:↑(LorentzGroup d)⊢ (∀ (x : SpaceTime d),
    Λ •
        (𝓕.μ₀⁻¹ • (contrT 1 0 1 ⋯) (Tensorial.toTensor (tensorDeriv A.toFieldStrength (Λ⁻¹ • x))) +
          -(permT ![0] ⋯) (Tensorial.toTensor (J (Λ⁻¹ • x)))) =
      0) →
  IsExtrema 𝓕 A J intro hmpd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ JΛ:↑(LorentzGroup d)h:∀ (x : SpaceTime d),
  Λ •
      (𝓕.μ₀⁻¹ • (contrT 1 0 1 ⋯) (Tensorial.toTensor (tensorDeriv A.toFieldStrength (Λ⁻¹ • x))) +
        -(permT ![0] ⋯) (Tensorial.toTensor (J (Λ⁻¹ • x)))) =
    0⊢ IsExtrema 𝓕 A J
    rw [isExtrema_iff_tensors A hA J hJmpd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ JΛ:↑(LorentzGroup d)h:∀ (x : SpaceTime d),
  Λ •
      (𝓕.μ₀⁻¹ • (contrT 1 0 1 ⋯) (Tensorial.toTensor (tensorDeriv A.toFieldStrength (Λ⁻¹ • x))) +
        -(permT ![0] ⋯) (Tensorial.toTensor (J (Λ⁻¹ • x)))) =
    0⊢ ∀ (x : SpaceTime d),
  (contrT 1 0 1 ⋯) (Tensorial.toTensor ((1 / 𝓕.μ₀) • tensorDeriv A.toFieldStrength x)) +
      (permT ![0] ⋯) (-Tensorial.toTensor (J x)) =
    0mpd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ JΛ:↑(LorentzGroup d)h:∀ (x : SpaceTime d),
  Λ •
      (𝓕.μ₀⁻¹ • (contrT 1 0 1 ⋯) (Tensorial.toTensor (tensorDeriv A.toFieldStrength (Λ⁻¹ • x))) +
        -(permT ![0] ⋯) (Tensorial.toTensor (J (Λ⁻¹ • x)))) =
    0⊢ ∀ (x : SpaceTime d),
  (contrT 1 0 1 ⋯) (Tensorial.toTensor ((1 / 𝓕.μ₀) • tensorDeriv A.toFieldStrength x)) +
      (permT ![0] ⋯) (-Tensorial.toTensor (J x)) =
    0]mpd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ JΛ:↑(LorentzGroup d)h:∀ (x : SpaceTime d),
  Λ •
      (𝓕.μ₀⁻¹ • (contrT 1 0 1 ⋯) (Tensorial.toTensor (tensorDeriv A.toFieldStrength (Λ⁻¹ • x))) +
        -(permT ![0] ⋯) (Tensorial.toTensor (J (Λ⁻¹ • x)))) =
    0⊢ ∀ (x : SpaceTime d),
  (contrT 1 0 1 ⋯) (Tensorial.toTensor ((1 / 𝓕.μ₀) • tensorDeriv A.toFieldStrength x)) +
      (permT ![0] ⋯) (-Tensorial.toTensor (J x)) =
    0
    intro xmpd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ JΛ:↑(LorentzGroup d)h:∀ (x : SpaceTime d),
  Λ •
      (𝓕.μ₀⁻¹ • (contrT 1 0 1 ⋯) (Tensorial.toTensor (tensorDeriv A.toFieldStrength (Λ⁻¹ • x))) +
        -(permT ![0] ⋯) (Tensorial.toTensor (J (Λ⁻¹ • x)))) =
    0x:SpaceTime d⊢ (contrT 1 0 1 ⋯) (Tensorial.toTensor ((1 / 𝓕.μ₀) • tensorDeriv A.toFieldStrength x)) +
    (permT ![0] ⋯) (-Tensorial.toTensor (J x)) =
  0
    apply MulAction.injective Λmpd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ JΛ:↑(LorentzGroup d)h:∀ (x : SpaceTime d),
  Λ •
      (𝓕.μ₀⁻¹ • (contrT 1 0 1 ⋯) (Tensorial.toTensor (tensorDeriv A.toFieldStrength (Λ⁻¹ • x))) +
        -(permT ![0] ⋯) (Tensorial.toTensor (J (Λ⁻¹ • x)))) =
    0x:SpaceTime d⊢ (fun x => Λ • x)
    ((contrT 1 0 1 ⋯) (Tensorial.toTensor ((1 / 𝓕.μ₀) • tensorDeriv A.toFieldStrength x)) +
      (permT ![0] ⋯) (-Tensorial.toTensor (J x))) =
  (fun x => Λ • x) 0
    simp only [one_div, map_smul, map_neg,
      _root_.smul_add, actionT_smul, _root_.smul_neg, _root_.smul_zero]mpd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ JΛ:↑(LorentzGroup d)h:∀ (x : SpaceTime d),
  Λ •
      (𝓕.μ₀⁻¹ • (contrT 1 0 1 ⋯) (Tensorial.toTensor (tensorDeriv A.toFieldStrength (Λ⁻¹ • x))) +
        -(permT ![0] ⋯) (Tensorial.toTensor (J (Λ⁻¹ • x)))) =
    0x:SpaceTime d⊢ 𝓕.μ₀⁻¹ • Λ • (contrT 1 0 1 ⋯) (Tensorial.toTensor (tensorDeriv A.toFieldStrength x)) +
    -(Λ • (permT ![0] ⋯) (Tensorial.toTensor (J x))) =
  0
    simpa using h (Λ • x)All goals completed! 🐙
  ·mprd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ JΛ:↑(LorentzGroup d)⊢ IsExtrema 𝓕 A J →
  ∀ (x : SpaceTime d),
    Λ •
        (𝓕.μ₀⁻¹ • (contrT 1 0 1 ⋯) (Tensorial.toTensor (tensorDeriv A.toFieldStrength (Λ⁻¹ • x))) +
          -(permT ![0] ⋯) (Tensorial.toTensor (J (Λ⁻¹ • x)))) =
      0 intro h xmprd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ JΛ:↑(LorentzGroup d)h:IsExtrema 𝓕 A Jx:SpaceTime d⊢ Λ •
    (𝓕.μ₀⁻¹ • (contrT 1 0 1 ⋯) (Tensorial.toTensor (tensorDeriv A.toFieldStrength (Λ⁻¹ • x))) +
      -(permT ![0] ⋯) (Tensorial.toTensor (J (Λ⁻¹ • x)))) =
  0
    rw [isExtrema_iff_tensors A hA J hJmprd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ JΛ:↑(LorentzGroup d)h:∀ (x : SpaceTime d),
  (contrT 1 0 1 ⋯) (Tensorial.toTensor ((1 / 𝓕.μ₀) • tensorDeriv A.toFieldStrength x)) +
      (permT ![0] ⋯) (-Tensorial.toTensor (J x)) =
    0x:SpaceTime d⊢ Λ •
    (𝓕.μ₀⁻¹ • (contrT 1 0 1 ⋯) (Tensorial.toTensor (tensorDeriv A.toFieldStrength (Λ⁻¹ • x))) +
      -(permT ![0] ⋯) (Tensorial.toTensor (J (Λ⁻¹ • x)))) =
  0mprd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ JΛ:↑(LorentzGroup d)h:∀ (x : SpaceTime d),
  (contrT 1 0 1 ⋯) (Tensorial.toTensor ((1 / 𝓕.μ₀) • tensorDeriv A.toFieldStrength x)) +
      (permT ![0] ⋯) (-Tensorial.toTensor (J x)) =
    0x:SpaceTime d⊢ Λ •
    (𝓕.μ₀⁻¹ • (contrT 1 0 1 ⋯) (Tensorial.toTensor (tensorDeriv A.toFieldStrength (Λ⁻¹ • x))) +
      -(permT ![0] ⋯) (Tensorial.toTensor (J (Λ⁻¹ • x)))) =
  0] at hmprd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ JΛ:↑(LorentzGroup d)h:∀ (x : SpaceTime d),
  (contrT 1 0 1 ⋯) (Tensorial.toTensor ((1 / 𝓕.μ₀) • tensorDeriv A.toFieldStrength x)) +
      (permT ![0] ⋯) (-Tensorial.toTensor (J x)) =
    0x:SpaceTime d⊢ Λ •
    (𝓕.μ₀⁻¹ • (contrT 1 0 1 ⋯) (Tensorial.toTensor (tensorDeriv A.toFieldStrength (Λ⁻¹ • x))) +
      -(permT ![0] ⋯) (Tensorial.toTensor (J (Λ⁻¹ • x)))) =
  0
    specialize h (Λ⁻¹ • x)mprd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ JΛ:↑(LorentzGroup d)x:SpaceTime dh:(contrT 1 0 1 ⋯) (Tensorial.toTensor ((1 / 𝓕.μ₀) • tensorDeriv A.toFieldStrength (Λ⁻¹ • x))) +
    (permT ![0] ⋯) (-Tensorial.toTensor (J (Λ⁻¹ • x))) =
  0⊢ Λ •
    (𝓕.μ₀⁻¹ • (contrT 1 0 1 ⋯) (Tensorial.toTensor (tensorDeriv A.toFieldStrength (Λ⁻¹ • x))) +
      -(permT ![0] ⋯) (Tensorial.toTensor (J (Λ⁻¹ • x)))) =
  0
    simp at hmprd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ JΛ:↑(LorentzGroup d)x:SpaceTime dh:𝓕.μ₀⁻¹ • (contrT 1 0 1 ⋯) (Tensorial.toTensor (tensorDeriv A.toFieldStrength (Λ⁻¹ • x))) +
    -(permT ![0] ⋯) (Tensorial.toTensor (J (Λ⁻¹ • x))) =
  0⊢ Λ •
    (𝓕.μ₀⁻¹ • (contrT 1 0 1 ⋯) (Tensorial.toTensor (tensorDeriv A.toFieldStrength (Λ⁻¹ • x))) +
      -(permT ![0] ⋯) (Tensorial.toTensor (J (Λ⁻¹ • x)))) =
  0
    rw [hmprd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ JΛ:↑(LorentzGroup d)x:SpaceTime dh:𝓕.μ₀⁻¹ • (contrT 1 0 1 ⋯) (Tensorial.toTensor (tensorDeriv A.toFieldStrength (Λ⁻¹ • x))) +
    -(permT ![0] ⋯) (Tensorial.toTensor (J (Λ⁻¹ • x))) =
  0⊢ Λ • 0 = 0mprd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ JΛ:↑(LorentzGroup d)x:SpaceTime dh:𝓕.μ₀⁻¹ • (contrT 1 0 1 ⋯) (Tensorial.toTensor (tensorDeriv A.toFieldStrength (Λ⁻¹ • x))) +
    -(permT ![0] ⋯) (Tensorial.toTensor (J (Λ⁻¹ • x))) =
  0⊢ Λ • 0 = 0]mprd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ JΛ:↑(LorentzGroup d)x:SpaceTime dh:𝓕.μ₀⁻¹ • (contrT 1 0 1 ⋯) (Tensorial.toTensor (tensorDeriv A.toFieldStrength (Λ⁻¹ • x))) +
    -(permT ![0] ⋯) (Tensorial.toTensor (J (Λ⁻¹ • x))) =
  0⊢ Λ • 0 = 0
    simpAll goals completed! 🐙
  ·hAd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ JΛ:↑(LorentzGroup d)⊢ ContDiff ℝ ∞ (Λ • A).val change ContDiff ℝ ∞ (actionCLM Λ ∘ A ∘ actionCLM Λ⁻¹)hAd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ JΛ:↑(LorentzGroup d)⊢ ContDiff ℝ ∞ (⇑(actionCLM Λ) ∘ A.val ∘ ⇑(actionCLM Λ⁻¹))
    apply ContDiff.comphA.hgd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ JΛ:↑(LorentzGroup d)⊢ ContDiff ℝ ∞ ⇑(actionCLM Λ)hA.hfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ JΛ:↑(LorentzGroup d)⊢ ContDiff ℝ ∞ (A.val ∘ ⇑(actionCLM Λ⁻¹))
    ·hA.hgd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ JΛ:↑(LorentzGroup d)⊢ ContDiff ℝ ∞ ⇑(actionCLM Λ) exact ContinuousLinearMap.contDiff (actionCLM Λ)All goals completed! 🐙
    ·hA.hfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ JΛ:↑(LorentzGroup d)⊢ ContDiff ℝ ∞ (A.val ∘ ⇑(actionCLM Λ⁻¹)) apply ContDiff.comphA.hf.hgd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ JΛ:↑(LorentzGroup d)⊢ ContDiff ℝ ∞ A.valhA.hf.hfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ JΛ:↑(LorentzGroup d)⊢ ContDiff ℝ ∞ ⇑(actionCLM Λ⁻¹)
      ·hA.hf.hgd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ JΛ:↑(LorentzGroup d)⊢ ContDiff ℝ ∞ A.val exact hAAll goals completed! 🐙
      ·hA.hf.hfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ JΛ:↑(LorentzGroup d)⊢ ContDiff ℝ ∞ ⇑(actionCLM Λ⁻¹) exact ContinuousLinearMap.contDiff (actionCLM Λ⁻¹)All goals completed! 🐙
  ·hJd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ JΛ:↑(LorentzGroup d)⊢ ContDiff ℝ ∞ fun x => Λ • J (Λ⁻¹ • x) change ContDiff ℝ ∞ (actionCLM Λ ∘ J ∘ actionCLM Λ⁻¹)hJd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ JΛ:↑(LorentzGroup d)⊢ ContDiff ℝ ∞ (⇑(actionCLM Λ) ∘ J ∘ ⇑(actionCLM Λ⁻¹))
    apply ContDiff.comphJ.hgd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ JΛ:↑(LorentzGroup d)⊢ ContDiff ℝ ∞ ⇑(actionCLM Λ)hJ.hfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ JΛ:↑(LorentzGroup d)⊢ ContDiff ℝ ∞ (J ∘ ⇑(actionCLM Λ⁻¹))
    ·hJ.hgd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ JΛ:↑(LorentzGroup d)⊢ ContDiff ℝ ∞ ⇑(actionCLM Λ) exact ContinuousLinearMap.contDiff (actionCLM Λ)All goals completed! 🐙
    ·hJ.hfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ JΛ:↑(LorentzGroup d)⊢ ContDiff ℝ ∞ (J ∘ ⇑(actionCLM Λ⁻¹)) apply ContDiff.comphJ.hf.hgd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ JΛ:↑(LorentzGroup d)⊢ ContDiff ℝ ∞ JhJ.hf.hfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ JΛ:↑(LorentzGroup d)⊢ ContDiff ℝ ∞ ⇑(actionCLM Λ⁻¹)
      ·hJ.hf.hgd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ JΛ:↑(LorentzGroup d)⊢ ContDiff ℝ ∞ J exact hJAll goals completed! 🐙
      ·hJ.hf.hfd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ JΛ:↑(LorentzGroup d)⊢ ContDiff ℝ ∞ ⇑(actionCLM Λ⁻¹) exact ContinuousLinearMap.contDiff (actionCLM Λ⁻¹)All goals completed! 🐙

B. Gauss's law and Ampère's law and the extrema condition



lemma isExtrema_iff_gauss_ampere_magneticFieldMatrix {d} {𝓕 : FreeSpace}
    {A : ElectromagneticPotential d}
    (hA : ContDiff ℝ ∞ A) (J : LorentzCurrentDensity d)
    (hJ : ContDiff ℝ ∞ J) :
    IsExtrema 𝓕 A J ↔ ∀ t, ∀ x, (∇ ⬝ (A.electricField 𝓕.c t)) x = J.chargeDensity 𝓕.c t x / 𝓕.ε₀
    ∧ ∀ i, 𝓕.μ₀ * 𝓕.ε₀ * ∂ₜ (fun t => A.electricField 𝓕.c t x) t i =
    ∑ j, ∂[j] (A.magneticFieldMatrix 𝓕.c t · (j, i)) x - 𝓕.μ₀ * J.currentDensity 𝓕.c t x i := byd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ J⊢ IsExtrema 𝓕 A J ↔
  ∀ (t : Time) (x : Space d),
    Space.div (electricField 𝓕.c A t) x = LorentzCurrentDensity.chargeDensity 𝓕.c J t x / 𝓕.ε₀ ∧
      ∀ (i : Fin d),
        𝓕.μ₀ * 𝓕.ε₀ * (∂ₜ (fun t => electricField 𝓕.c A t x) t).ofLp i =
          ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x -
            𝓕.μ₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i
  rw [isExtrema_iff_gradLagrangiand:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ J⊢ gradLagrangian 𝓕 A J = 0 ↔
  ∀ (t : Time) (x : Space d),
    Space.div (electricField 𝓕.c A t) x = LorentzCurrentDensity.chargeDensity 𝓕.c J t x / 𝓕.ε₀ ∧
      ∀ (i : Fin d),
        𝓕.μ₀ * 𝓕.ε₀ * (∂ₜ (fun t => electricField 𝓕.c A t x) t).ofLp i =
          ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x -
            𝓕.μ₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp id:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ J⊢ gradLagrangian 𝓕 A J = 0 ↔
  ∀ (t : Time) (x : Space d),
    Space.div (electricField 𝓕.c A t) x = LorentzCurrentDensity.chargeDensity 𝓕.c J t x / 𝓕.ε₀ ∧
      ∀ (i : Fin d),
        𝓕.μ₀ * 𝓕.ε₀ * (∂ₜ (fun t => electricField 𝓕.c A t x) t).ofLp i =
          ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x -
            𝓕.μ₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ J⊢ gradLagrangian 𝓕 A J = 0 ↔
  ∀ (t : Time) (x : Space d),
    Space.div (electricField 𝓕.c A t) x = LorentzCurrentDensity.chargeDensity 𝓕.c J t x / 𝓕.ε₀ ∧
      ∀ (i : Fin d),
        𝓕.μ₀ * 𝓕.ε₀ * (∂ₜ (fun t => electricField 𝓕.c A t x) t).ofLp i =
          ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x -
            𝓕.μ₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i
  rw [funext_iffd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ J⊢ (∀ (x : SpaceTime d), gradLagrangian 𝓕 A J x = 0 x) ↔
  ∀ (t : Time) (x : Space d),
    Space.div (electricField 𝓕.c A t) x = LorentzCurrentDensity.chargeDensity 𝓕.c J t x / 𝓕.ε₀ ∧
      ∀ (i : Fin d),
        𝓕.μ₀ * 𝓕.ε₀ * (∂ₜ (fun t => electricField 𝓕.c A t x) t).ofLp i =
          ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x -
            𝓕.μ₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp id:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ J⊢ (∀ (x : SpaceTime d), gradLagrangian 𝓕 A J x = 0 x) ↔
  ∀ (t : Time) (x : Space d),
    Space.div (electricField 𝓕.c A t) x = LorentzCurrentDensity.chargeDensity 𝓕.c J t x / 𝓕.ε₀ ∧
      ∀ (i : Fin d),
        𝓕.μ₀ * 𝓕.ε₀ * (∂ₜ (fun t => electricField 𝓕.c A t x) t).ofLp i =
          ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x -
            𝓕.μ₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ J⊢ (∀ (x : SpaceTime d), gradLagrangian 𝓕 A J x = 0 x) ↔
  ∀ (t : Time) (x : Space d),
    Space.div (electricField 𝓕.c A t) x = LorentzCurrentDensity.chargeDensity 𝓕.c J t x / 𝓕.ε₀ ∧
      ∀ (i : Fin d),
        𝓕.μ₀ * 𝓕.ε₀ * (∂ₜ (fun t => electricField 𝓕.c A t x) t).ofLp i =
          ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x -
            𝓕.μ₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i
  conv_lhs =>
    enter [x]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jx:SpaceTime d| gradLagrangian 𝓕 A J x = 0 x
    rw [gradLagrangian_eq_electricField_magneticField (𝓕 := 𝓕) A hA J hJ]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jx:SpaceTime d| (1 / (𝓕.μ₀ * 𝓕.c.val) * Space.div (electricField 𝓕.c A ((time 𝓕.c) x)) (space x) +
        -𝓕.c.val * LorentzCurrentDensity.chargeDensity 𝓕.c J ((time 𝓕.c) x) (space x)) •
      Lorentz.Vector.basis (Sum.inl 0) +
    ∑ i,
      (𝓕.μ₀⁻¹ *
            (𝓕.ε₀ * 𝓕.μ₀ * (∂ₜ (fun x_1 => electricField 𝓕.c A x_1 (space x)) ((time 𝓕.c) x)).ofLp i -
              ∑ j, Space.deriv j (fun x_1 => magneticFieldMatrix 𝓕.c A ((time 𝓕.c) x) x_1 (j, i)) (space x)) +
          (LorentzCurrentDensity.currentDensity 𝓕.c J ((time 𝓕.c) x) (space x)).ofLp i) •
        Lorentz.Vector.basis (Sum.inr i) =
  0 x
    simp only [Pi.zero_apply]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jx:SpaceTime d| (1 / (𝓕.μ₀ * 𝓕.c.val) * Space.div (electricField 𝓕.c A ((time 𝓕.c) x)) (space x) +
        -𝓕.c.val * LorentzCurrentDensity.chargeDensity 𝓕.c J ((time 𝓕.c) x) (space x)) •
      Lorentz.Vector.basis (Sum.inl 0) +
    ∑ x_1,
      (𝓕.μ₀⁻¹ *
            (𝓕.ε₀ * 𝓕.μ₀ * (∂ₜ (fun x_2 => electricField 𝓕.c A x_2 (space x)) ((time 𝓕.c) x)).ofLp x_1 -
              ∑ j, Space.deriv j (fun x_2 => magneticFieldMatrix 𝓕.c A ((time 𝓕.c) x) x_2 (j, x_1)) (space x)) +
          (LorentzCurrentDensity.currentDensity 𝓕.c J ((time 𝓕.c) x) (space x)).ofLp x_1) •
        Lorentz.Vector.basis (Sum.inr x_1) =
  0
    rw [Lorentz.Vector.sum_inl_inr_basis_eq_zero_iff]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jx:SpaceTime d| 1 / (𝓕.μ₀ * 𝓕.c.val) * Space.div (electricField 𝓕.c A ((time 𝓕.c) x)) (space x) +
      -𝓕.c.val * LorentzCurrentDensity.chargeDensity 𝓕.c J ((time 𝓕.c) x) (space x) =
    0 ∧
  ∀ (i : Fin d),
    𝓕.μ₀⁻¹ *
          (𝓕.ε₀ * 𝓕.μ₀ * (∂ₜ (fun x_1 => electricField 𝓕.c A x_1 (space x)) ((time 𝓕.c) x)).ofLp i -
            ∑ j, Space.deriv j (fun x_1 => magneticFieldMatrix 𝓕.c A ((time 𝓕.c) x) x_1 (j, i)) (space x)) +
        (LorentzCurrentDensity.currentDensity 𝓕.c J ((time 𝓕.c) x) (space x)).ofLp i =
      0
  simp only [forall_and]d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ J⊢ ((∀ (x : SpaceTime d),
      1 / (𝓕.μ₀ * 𝓕.c.val) * Space.div (electricField 𝓕.c A ((time 𝓕.c) x)) (space x) +
          -𝓕.c.val * LorentzCurrentDensity.chargeDensity 𝓕.c J ((time 𝓕.c) x) (space x) =
        0) ∧
    ∀ (x : SpaceTime d) (i : Fin d),
      𝓕.μ₀⁻¹ *
            (𝓕.ε₀ * 𝓕.μ₀ * (∂ₜ (fun x_1 => electricField 𝓕.c A x_1 (space x)) ((time 𝓕.c) x)).ofLp i -
              ∑ j, Space.deriv j (fun x_1 => magneticFieldMatrix 𝓕.c A ((time 𝓕.c) x) x_1 (j, i)) (space x)) +
          (LorentzCurrentDensity.currentDensity 𝓕.c J ((time 𝓕.c) x) (space x)).ofLp i =
        0) ↔
  (∀ (x : Time) (x_1 : Space d),
      Space.div (electricField 𝓕.c A x) x_1 = LorentzCurrentDensity.chargeDensity 𝓕.c J x x_1 / 𝓕.ε₀) ∧
    ∀ (x : Time) (x_1 : Space d) (i : Fin d),
      𝓕.μ₀ * 𝓕.ε₀ * (∂ₜ (fun t => electricField 𝓕.c A t x_1) x).ofLp i =
        ∑ j, Space.deriv j (fun x_2 => magneticFieldMatrix 𝓕.c A x x_2 (j, i)) x_1 -
          𝓕.μ₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J x x_1).ofLp i
  apply and_congrh₁d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ J⊢ (∀ (x : SpaceTime d),
    1 / (𝓕.μ₀ * 𝓕.c.val) * Space.div (electricField 𝓕.c A ((time 𝓕.c) x)) (space x) +
        -𝓕.c.val * LorentzCurrentDensity.chargeDensity 𝓕.c J ((time 𝓕.c) x) (space x) =
      0) ↔
  ∀ (x : Time) (x_1 : Space d),
    Space.div (electricField 𝓕.c A x) x_1 = LorentzCurrentDensity.chargeDensity 𝓕.c J x x_1 / 𝓕.ε₀h₂d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ J⊢ (∀ (x : SpaceTime d) (i : Fin d),
    𝓕.μ₀⁻¹ *
          (𝓕.ε₀ * 𝓕.μ₀ * (∂ₜ (fun x_1 => electricField 𝓕.c A x_1 (space x)) ((time 𝓕.c) x)).ofLp i -
            ∑ j, Space.deriv j (fun x_1 => magneticFieldMatrix 𝓕.c A ((time 𝓕.c) x) x_1 (j, i)) (space x)) +
        (LorentzCurrentDensity.currentDensity 𝓕.c J ((time 𝓕.c) x) (space x)).ofLp i =
      0) ↔
  ∀ (x : Time) (x_1 : Space d) (i : Fin d),
    𝓕.μ₀ * 𝓕.ε₀ * (∂ₜ (fun t => electricField 𝓕.c A t x_1) x).ofLp i =
      ∑ j, Space.deriv j (fun x_2 => magneticFieldMatrix 𝓕.c A x x_2 (j, i)) x_1 -
        𝓕.μ₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J x x_1).ofLp i
  ·h₁d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ J⊢ (∀ (x : SpaceTime d),
    1 / (𝓕.μ₀ * 𝓕.c.val) * Space.div (electricField 𝓕.c A ((time 𝓕.c) x)) (space x) +
        -𝓕.c.val * LorentzCurrentDensity.chargeDensity 𝓕.c J ((time 𝓕.c) x) (space x) =
      0) ↔
  ∀ (x : Time) (x_1 : Space d),
    Space.div (electricField 𝓕.c A x) x_1 = LorentzCurrentDensity.chargeDensity 𝓕.c J x x_1 / 𝓕.ε₀ apply Iff.introh₁.mpd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ J⊢ (∀ (x : SpaceTime d),
    1 / (𝓕.μ₀ * 𝓕.c.val) * Space.div (electricField 𝓕.c A ((time 𝓕.c) x)) (space x) +
        -𝓕.c.val * LorentzCurrentDensity.chargeDensity 𝓕.c J ((time 𝓕.c) x) (space x) =
      0) →
  ∀ (x : Time) (x_1 : Space d),
    Space.div (electricField 𝓕.c A x) x_1 = LorentzCurrentDensity.chargeDensity 𝓕.c J x x_1 / 𝓕.ε₀h₁.mprd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ J⊢ (∀ (x : Time) (x_1 : Space d),
    Space.div (electricField 𝓕.c A x) x_1 = LorentzCurrentDensity.chargeDensity 𝓕.c J x x_1 / 𝓕.ε₀) →
  ∀ (x : SpaceTime d),
    1 / (𝓕.μ₀ * 𝓕.c.val) * Space.div (electricField 𝓕.c A ((time 𝓕.c) x)) (space x) +
        -𝓕.c.val * LorentzCurrentDensity.chargeDensity 𝓕.c J ((time 𝓕.c) x) (space x) =
      0
    ·h₁.mpd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ J⊢ (∀ (x : SpaceTime d),
    1 / (𝓕.μ₀ * 𝓕.c.val) * Space.div (electricField 𝓕.c A ((time 𝓕.c) x)) (space x) +
        -𝓕.c.val * LorentzCurrentDensity.chargeDensity 𝓕.c J ((time 𝓕.c) x) (space x) =
      0) →
  ∀ (x : Time) (x_1 : Space d),
    Space.div (electricField 𝓕.c A x) x_1 = LorentzCurrentDensity.chargeDensity 𝓕.c J x x_1 / 𝓕.ε₀ intro h t xh₁.mpd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:∀ (x : SpaceTime d),
  1 / (𝓕.μ₀ * 𝓕.c.val) * Space.div (electricField 𝓕.c A ((time 𝓕.c) x)) (space x) +
      -𝓕.c.val * LorentzCurrentDensity.chargeDensity 𝓕.c J ((time 𝓕.c) x) (space x) =
    0t:Timex:Space d⊢ Space.div (electricField 𝓕.c A t) x = LorentzCurrentDensity.chargeDensity 𝓕.c J t x / 𝓕.ε₀
      specialize h ((toTimeAndSpace 𝓕.c).symm (t, x))h₁.mpd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jt:Timex:Space dh:1 / (𝓕.μ₀ * 𝓕.c.val) *
      Space.div (electricField 𝓕.c A ((time 𝓕.c) ((toTimeAndSpace 𝓕.c).symm (t, x))))
        (space ((toTimeAndSpace 𝓕.c).symm (t, x))) +
    -𝓕.c.val *
      LorentzCurrentDensity.chargeDensity 𝓕.c J ((time 𝓕.c) ((toTimeAndSpace 𝓕.c).symm (t, x)))
        (space ((toTimeAndSpace 𝓕.c).symm (t, x))) =
  0⊢ Space.div (electricField 𝓕.c A t) x = LorentzCurrentDensity.chargeDensity 𝓕.c J t x / 𝓕.ε₀
      simp at hh₁.mpd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jt:Timex:Space dh:𝓕.c.val⁻¹ * 𝓕.μ₀⁻¹ * Space.div (electricField 𝓕.c A t) x + -(𝓕.c.val * LorentzCurrentDensity.chargeDensity 𝓕.c J t x) =
  0⊢ Space.div (electricField 𝓕.c A t) x = LorentzCurrentDensity.chargeDensity 𝓕.c J t x / 𝓕.ε₀
      linear_combination (norm := simpad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jt:Timex:Space dh:𝓕.c.val⁻¹ * 𝓕.μ₀⁻¹ * Space.div (electricField 𝓕.c A t) x + -(𝓕.c.val * LorentzCurrentDensity.chargeDensity 𝓕.c J t x) =
  0⊢ Space.div (electricField 𝓕.c A t) x -
    (LorentzCurrentDensity.chargeDensity 𝓕.c J t x / 𝓕.ε₀ +
      𝓕.μ₀ * 𝓕.c.val *
        (𝓕.c.val⁻¹ * 𝓕.μ₀⁻¹ * Space.div (electricField 𝓕.c A t) x +
          -(𝓕.c.val * LorentzCurrentDensity.chargeDensity 𝓕.c J t x))) =
  0) (𝓕.μ₀ * 𝓕.c) * h
      field_simpad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jt:Timex:Space dh:𝓕.c.val⁻¹ * 𝓕.μ₀⁻¹ * Space.div (electricField 𝓕.c A t) x + -(𝓕.c.val * LorentzCurrentDensity.chargeDensity 𝓕.c J t x) =
  0⊢ Space.div (electricField 𝓕.c A t) x * 𝓕.ε₀ -
    (LorentzCurrentDensity.chargeDensity 𝓕.c J t x +
      𝓕.ε₀ *
        (Space.div (electricField 𝓕.c A t) x + -(LorentzCurrentDensity.chargeDensity 𝓕.c J t x * 𝓕.μ₀ * 𝓕.c.val ^ 2))) =
  𝓕.ε₀ * 0
      simp only [FreeSpace.c_sq, one_div, mul_inv_rev, mul_zero]ad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jt:Timex:Space dh:𝓕.c.val⁻¹ * 𝓕.μ₀⁻¹ * Space.div (electricField 𝓕.c A t) x + -(𝓕.c.val * LorentzCurrentDensity.chargeDensity 𝓕.c J t x) =
  0⊢ Space.div (electricField 𝓕.c A t) x * 𝓕.ε₀ -
    (LorentzCurrentDensity.chargeDensity 𝓕.c J t x +
      𝓕.ε₀ *
        (Space.div (electricField 𝓕.c A t) x +
          -(LorentzCurrentDensity.chargeDensity 𝓕.c J t x * 𝓕.μ₀ * (𝓕.μ₀⁻¹ * 𝓕.ε₀⁻¹)))) =
  0
      field_simpad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jt:Timex:Space dh:𝓕.c.val⁻¹ * 𝓕.μ₀⁻¹ * Space.div (electricField 𝓕.c A t) x + -(𝓕.c.val * LorentzCurrentDensity.chargeDensity 𝓕.c J t x) =
  0⊢ Space.div (electricField 𝓕.c A t) x * 𝓕.ε₀ -
    (LorentzCurrentDensity.chargeDensity 𝓕.c J t x +
      (Space.div (electricField 𝓕.c A t) x * 𝓕.ε₀ + -LorentzCurrentDensity.chargeDensity 𝓕.c J t x)) =
  0
      ringAll goals completed! 🐙
    ·h₁.mprd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ J⊢ (∀ (x : Time) (x_1 : Space d),
    Space.div (electricField 𝓕.c A x) x_1 = LorentzCurrentDensity.chargeDensity 𝓕.c J x x_1 / 𝓕.ε₀) →
  ∀ (x : SpaceTime d),
    1 / (𝓕.μ₀ * 𝓕.c.val) * Space.div (electricField 𝓕.c A ((time 𝓕.c) x)) (space x) +
        -𝓕.c.val * LorentzCurrentDensity.chargeDensity 𝓕.c J ((time 𝓕.c) x) (space x) =
      0 intro h xh₁.mprd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:∀ (x : Time) (x_1 : Space d),
  Space.div (electricField 𝓕.c A x) x_1 = LorentzCurrentDensity.chargeDensity 𝓕.c J x x_1 / 𝓕.ε₀x:SpaceTime d⊢ 1 / (𝓕.μ₀ * 𝓕.c.val) * Space.div (electricField 𝓕.c A ((time 𝓕.c) x)) (space x) +
    -𝓕.c.val * LorentzCurrentDensity.chargeDensity 𝓕.c J ((time 𝓕.c) x) (space x) =
  0
      specialize h (x.time 𝓕.c) x.spaceh₁.mprd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jx:SpaceTime dh:Space.div (electricField 𝓕.c A ((time 𝓕.c) x)) (space x) =
  LorentzCurrentDensity.chargeDensity 𝓕.c J ((time 𝓕.c) x) (space x) / 𝓕.ε₀⊢ 1 / (𝓕.μ₀ * 𝓕.c.val) * Space.div (electricField 𝓕.c A ((time 𝓕.c) x)) (space x) +
    -𝓕.c.val * LorentzCurrentDensity.chargeDensity 𝓕.c J ((time 𝓕.c) x) (space x) =
  0
      linear_combination (norm := simpad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jx:SpaceTime dh:Space.div (electricField 𝓕.c A ((time 𝓕.c) x)) (space x) =
  LorentzCurrentDensity.chargeDensity 𝓕.c J ((time 𝓕.c) x) (space x) / 𝓕.ε₀⊢ 𝓕.c.val⁻¹ * 𝓕.μ₀⁻¹ * Space.div (electricField 𝓕.c A ((time 𝓕.c) x)) (space x) +
        -(𝓕.c.val * LorentzCurrentDensity.chargeDensity 𝓕.c J ((time 𝓕.c) x) (space x)) +
      𝓕.μ₀⁻¹ * 𝓕.c.val⁻¹ * (LorentzCurrentDensity.chargeDensity 𝓕.c J ((time 𝓕.c) x) (space x) / 𝓕.ε₀) -
    𝓕.μ₀⁻¹ * 𝓕.c.val⁻¹ * Space.div (electricField 𝓕.c A ((time 𝓕.c) x)) (space x) =
  0) (𝓕.μ₀⁻¹ * 𝓕.c⁻¹) * h
      field_simpad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jx:SpaceTime dh:Space.div (electricField 𝓕.c A ((time 𝓕.c) x)) (space x) =
  LorentzCurrentDensity.chargeDensity 𝓕.c J ((time 𝓕.c) x) (space x) / 𝓕.ε₀⊢ (Space.div (electricField 𝓕.c A ((time 𝓕.c) x)) (space x) +
          -(𝓕.c.val ^ 2 * 𝓕.μ₀ * LorentzCurrentDensity.chargeDensity 𝓕.c J ((time 𝓕.c) x) (space x))) *
        𝓕.ε₀ +
      LorentzCurrentDensity.chargeDensity 𝓕.c J ((time 𝓕.c) x) (space x) -
    Space.div (electricField 𝓕.c A ((time 𝓕.c) x)) (space x) * 𝓕.ε₀ =
  𝓕.c.val * 𝓕.μ₀ * 𝓕.ε₀ * 0
      simp only [FreeSpace.c_sq, one_div, mul_inv_rev, mul_zero]ad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jx:SpaceTime dh:Space.div (electricField 𝓕.c A ((time 𝓕.c) x)) (space x) =
  LorentzCurrentDensity.chargeDensity 𝓕.c J ((time 𝓕.c) x) (space x) / 𝓕.ε₀⊢ (Space.div (electricField 𝓕.c A ((time 𝓕.c) x)) (space x) +
          -(𝓕.μ₀⁻¹ * 𝓕.ε₀⁻¹ * 𝓕.μ₀ * LorentzCurrentDensity.chargeDensity 𝓕.c J ((time 𝓕.c) x) (space x))) *
        𝓕.ε₀ +
      LorentzCurrentDensity.chargeDensity 𝓕.c J ((time 𝓕.c) x) (space x) -
    Space.div (electricField 𝓕.c A ((time 𝓕.c) x)) (space x) * 𝓕.ε₀ =
  0
      field_simpad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jx:SpaceTime dh:Space.div (electricField 𝓕.c A ((time 𝓕.c) x)) (space x) =
  LorentzCurrentDensity.chargeDensity 𝓕.c J ((time 𝓕.c) x) (space x) / 𝓕.ε₀⊢ Space.div (electricField 𝓕.c A ((time 𝓕.c) x)) (space x) * 𝓕.ε₀ +
        -LorentzCurrentDensity.chargeDensity 𝓕.c J ((time 𝓕.c) x) (space x) +
      LorentzCurrentDensity.chargeDensity 𝓕.c J ((time 𝓕.c) x) (space x) -
    Space.div (electricField 𝓕.c A ((time 𝓕.c) x)) (space x) * 𝓕.ε₀ =
  0
      ringAll goals completed! 🐙
  ·h₂d:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ J⊢ (∀ (x : SpaceTime d) (i : Fin d),
    𝓕.μ₀⁻¹ *
          (𝓕.ε₀ * 𝓕.μ₀ * (∂ₜ (fun x_1 => electricField 𝓕.c A x_1 (space x)) ((time 𝓕.c) x)).ofLp i -
            ∑ j, Space.deriv j (fun x_1 => magneticFieldMatrix 𝓕.c A ((time 𝓕.c) x) x_1 (j, i)) (space x)) +
        (LorentzCurrentDensity.currentDensity 𝓕.c J ((time 𝓕.c) x) (space x)).ofLp i =
      0) ↔
  ∀ (x : Time) (x_1 : Space d) (i : Fin d),
    𝓕.μ₀ * 𝓕.ε₀ * (∂ₜ (fun t => electricField 𝓕.c A t x_1) x).ofLp i =
      ∑ j, Space.deriv j (fun x_2 => magneticFieldMatrix 𝓕.c A x x_2 (j, i)) x_1 -
        𝓕.μ₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J x x_1).ofLp i apply Iff.introh₂.mpd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ J⊢ (∀ (x : SpaceTime d) (i : Fin d),
    𝓕.μ₀⁻¹ *
          (𝓕.ε₀ * 𝓕.μ₀ * (∂ₜ (fun x_1 => electricField 𝓕.c A x_1 (space x)) ((time 𝓕.c) x)).ofLp i -
            ∑ j, Space.deriv j (fun x_1 => magneticFieldMatrix 𝓕.c A ((time 𝓕.c) x) x_1 (j, i)) (space x)) +
        (LorentzCurrentDensity.currentDensity 𝓕.c J ((time 𝓕.c) x) (space x)).ofLp i =
      0) →
  ∀ (x : Time) (x_1 : Space d) (i : Fin d),
    𝓕.μ₀ * 𝓕.ε₀ * (∂ₜ (fun t => electricField 𝓕.c A t x_1) x).ofLp i =
      ∑ j, Space.deriv j (fun x_2 => magneticFieldMatrix 𝓕.c A x x_2 (j, i)) x_1 -
        𝓕.μ₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J x x_1).ofLp ih₂.mprd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ J⊢ (∀ (x : Time) (x_1 : Space d) (i : Fin d),
    𝓕.μ₀ * 𝓕.ε₀ * (∂ₜ (fun t => electricField 𝓕.c A t x_1) x).ofLp i =
      ∑ j, Space.deriv j (fun x_2 => magneticFieldMatrix 𝓕.c A x x_2 (j, i)) x_1 -
        𝓕.μ₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J x x_1).ofLp i) →
  ∀ (x : SpaceTime d) (i : Fin d),
    𝓕.μ₀⁻¹ *
          (𝓕.ε₀ * 𝓕.μ₀ * (∂ₜ (fun x_1 => electricField 𝓕.c A x_1 (space x)) ((time 𝓕.c) x)).ofLp i -
            ∑ j, Space.deriv j (fun x_1 => magneticFieldMatrix 𝓕.c A ((time 𝓕.c) x) x_1 (j, i)) (space x)) +
        (LorentzCurrentDensity.currentDensity 𝓕.c J ((time 𝓕.c) x) (space x)).ofLp i =
      0
    ·h₂.mpd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ J⊢ (∀ (x : SpaceTime d) (i : Fin d),
    𝓕.μ₀⁻¹ *
          (𝓕.ε₀ * 𝓕.μ₀ * (∂ₜ (fun x_1 => electricField 𝓕.c A x_1 (space x)) ((time 𝓕.c) x)).ofLp i -
            ∑ j, Space.deriv j (fun x_1 => magneticFieldMatrix 𝓕.c A ((time 𝓕.c) x) x_1 (j, i)) (space x)) +
        (LorentzCurrentDensity.currentDensity 𝓕.c J ((time 𝓕.c) x) (space x)).ofLp i =
      0) →
  ∀ (x : Time) (x_1 : Space d) (i : Fin d),
    𝓕.μ₀ * 𝓕.ε₀ * (∂ₜ (fun t => electricField 𝓕.c A t x_1) x).ofLp i =
      ∑ j, Space.deriv j (fun x_2 => magneticFieldMatrix 𝓕.c A x x_2 (j, i)) x_1 -
        𝓕.μ₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J x x_1).ofLp i intro h t x ih₂.mpd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:∀ (x : SpaceTime d) (i : Fin d),
  𝓕.μ₀⁻¹ *
        (𝓕.ε₀ * 𝓕.μ₀ * (∂ₜ (fun x_1 => electricField 𝓕.c A x_1 (space x)) ((time 𝓕.c) x)).ofLp i -
          ∑ j, Space.deriv j (fun x_1 => magneticFieldMatrix 𝓕.c A ((time 𝓕.c) x) x_1 (j, i)) (space x)) +
      (LorentzCurrentDensity.currentDensity 𝓕.c J ((time 𝓕.c) x) (space x)).ofLp i =
    0t:Timex:Space di:Fin d⊢ 𝓕.μ₀ * 𝓕.ε₀ * (∂ₜ (fun t => electricField 𝓕.c A t x) t).ofLp i =
  ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x -
    𝓕.μ₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i
      specialize h ((toTimeAndSpace 𝓕.c).symm (t, x)) ih₂.mpd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jt:Timex:Space di:Fin dh:𝓕.μ₀⁻¹ *
      (𝓕.ε₀ * 𝓕.μ₀ *
          (∂ₜ (fun x_1 => electricField 𝓕.c A x_1 (space ((toTimeAndSpace 𝓕.c).symm (t, x))))
                ((time 𝓕.c) ((toTimeAndSpace 𝓕.c).symm (t, x)))).ofLp
            i -
        ∑ j,
          Space.deriv j
            (fun x_1 => magneticFieldMatrix 𝓕.c A ((time 𝓕.c) ((toTimeAndSpace 𝓕.c).symm (t, x))) x_1 (j, i))
            (space ((toTimeAndSpace 𝓕.c).symm (t, x)))) +
    (LorentzCurrentDensity.currentDensity 𝓕.c J ((time 𝓕.c) ((toTimeAndSpace 𝓕.c).symm (t, x)))
          (space ((toTimeAndSpace 𝓕.c).symm (t, x)))).ofLp
      i =
  0⊢ 𝓕.μ₀ * 𝓕.ε₀ * (∂ₜ (fun t => electricField 𝓕.c A t x) t).ofLp i =
  ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x -
    𝓕.μ₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i
      simp at hh₂.mpd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jt:Timex:Space di:Fin dh:𝓕.μ₀⁻¹ *
      (𝓕.ε₀ * 𝓕.μ₀ * (∂ₜ (fun x_1 => electricField 𝓕.c A x_1 x) t).ofLp i -
        ∑ x_1, Space.deriv x_1 (fun x => magneticFieldMatrix 𝓕.c A t x (x_1, i)) x) +
    (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i =
  0⊢ 𝓕.μ₀ * 𝓕.ε₀ * (∂ₜ (fun t => electricField 𝓕.c A t x) t).ofLp i =
  ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x -
    𝓕.μ₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i
      linear_combination (norm := simpad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jt:Timex:Space di:Fin dh:𝓕.μ₀⁻¹ *
      (𝓕.ε₀ * 𝓕.μ₀ * (∂ₜ (fun x_1 => electricField 𝓕.c A x_1 x) t).ofLp i -
        ∑ x_1, Space.deriv x_1 (fun x => magneticFieldMatrix 𝓕.c A t x (x_1, i)) x) +
    (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i =
  0⊢ 𝓕.μ₀ * 𝓕.ε₀ * (∂ₜ (fun t => electricField 𝓕.c A t x) t).ofLp i -
    (∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x -
        𝓕.μ₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i +
      𝓕.μ₀ *
        (𝓕.μ₀⁻¹ *
            (𝓕.ε₀ * 𝓕.μ₀ * (∂ₜ (fun t => electricField 𝓕.c A t x) t).ofLp i -
              ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x) +
          (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i)) =
  0) (𝓕.μ₀) * h
      field_simpad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jt:Timex:Space di:Fin dh:𝓕.μ₀⁻¹ *
      (𝓕.ε₀ * 𝓕.μ₀ * (∂ₜ (fun x_1 => electricField 𝓕.c A x_1 x) t).ofLp i -
        ∑ x_1, Space.deriv x_1 (fun x => magneticFieldMatrix 𝓕.c A t x (x_1, i)) x) +
    (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i =
  0⊢ 𝓕.μ₀ * 𝓕.ε₀ * (∂ₜ (fun t => electricField 𝓕.c A t x) t).ofLp i -
    (∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x -
        𝓕.μ₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i +
      (𝓕.μ₀ * 𝓕.ε₀ * (∂ₜ (fun t => electricField 𝓕.c A t x) t).ofLp i -
          ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x +
        𝓕.μ₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i)) =
  0
      simpAll goals completed! 🐙
    ·h₂.mprd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ J⊢ (∀ (x : Time) (x_1 : Space d) (i : Fin d),
    𝓕.μ₀ * 𝓕.ε₀ * (∂ₜ (fun t => electricField 𝓕.c A t x_1) x).ofLp i =
      ∑ j, Space.deriv j (fun x_2 => magneticFieldMatrix 𝓕.c A x x_2 (j, i)) x_1 -
        𝓕.μ₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J x x_1).ofLp i) →
  ∀ (x : SpaceTime d) (i : Fin d),
    𝓕.μ₀⁻¹ *
          (𝓕.ε₀ * 𝓕.μ₀ * (∂ₜ (fun x_1 => electricField 𝓕.c A x_1 (space x)) ((time 𝓕.c) x)).ofLp i -
            ∑ j, Space.deriv j (fun x_1 => magneticFieldMatrix 𝓕.c A ((time 𝓕.c) x) x_1 (j, i)) (space x)) +
        (LorentzCurrentDensity.currentDensity 𝓕.c J ((time 𝓕.c) x) (space x)).ofLp i =
      0 intro h x ih₂.mprd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:∀ (x : Time) (x_1 : Space d) (i : Fin d),
  𝓕.μ₀ * 𝓕.ε₀ * (∂ₜ (fun t => electricField 𝓕.c A t x_1) x).ofLp i =
    ∑ j, Space.deriv j (fun x_2 => magneticFieldMatrix 𝓕.c A x x_2 (j, i)) x_1 -
      𝓕.μ₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J x x_1).ofLp ix:SpaceTime di:Fin d⊢ 𝓕.μ₀⁻¹ *
      (𝓕.ε₀ * 𝓕.μ₀ * (∂ₜ (fun x_1 => electricField 𝓕.c A x_1 (space x)) ((time 𝓕.c) x)).ofLp i -
        ∑ j, Space.deriv j (fun x_1 => magneticFieldMatrix 𝓕.c A ((time 𝓕.c) x) x_1 (j, i)) (space x)) +
    (LorentzCurrentDensity.currentDensity 𝓕.c J ((time 𝓕.c) x) (space x)).ofLp i =
  0
      specialize h (x.time 𝓕.c) x.space ih₂.mprd:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jx:SpaceTime di:Fin dh:𝓕.μ₀ * 𝓕.ε₀ * (∂ₜ (fun t => electricField 𝓕.c A t (space x)) ((time 𝓕.c) x)).ofLp i =
  ∑ j, Space.deriv j (fun x_1 => magneticFieldMatrix 𝓕.c A ((time 𝓕.c) x) x_1 (j, i)) (space x) -
    𝓕.μ₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J ((time 𝓕.c) x) (space x)).ofLp i⊢ 𝓕.μ₀⁻¹ *
      (𝓕.ε₀ * 𝓕.μ₀ * (∂ₜ (fun x_1 => electricField 𝓕.c A x_1 (space x)) ((time 𝓕.c) x)).ofLp i -
        ∑ j, Space.deriv j (fun x_1 => magneticFieldMatrix 𝓕.c A ((time 𝓕.c) x) x_1 (j, i)) (space x)) +
    (LorentzCurrentDensity.currentDensity 𝓕.c J ((time 𝓕.c) x) (space x)).ofLp i =
  0
      linear_combination (norm := simpad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jx:SpaceTime di:Fin dh:𝓕.μ₀ * 𝓕.ε₀ * (∂ₜ (fun t => electricField 𝓕.c A t (space x)) ((time 𝓕.c) x)).ofLp i =
  ∑ j, Space.deriv j (fun x_1 => magneticFieldMatrix 𝓕.c A ((time 𝓕.c) x) x_1 (j, i)) (space x) -
    𝓕.μ₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J ((time 𝓕.c) x) (space x)).ofLp i⊢ 𝓕.μ₀⁻¹ *
          (𝓕.ε₀ * 𝓕.μ₀ * (∂ₜ (fun x_1 => electricField 𝓕.c A x_1 (space x)) ((time 𝓕.c) x)).ofLp i -
            ∑ j, Space.deriv j (fun x_1 => magneticFieldMatrix 𝓕.c A ((time 𝓕.c) x) x_1 (j, i)) (space x)) +
        (LorentzCurrentDensity.currentDensity 𝓕.c J ((time 𝓕.c) x) (space x)).ofLp i +
      𝓕.μ₀⁻¹ *
        (∑ j, Space.deriv j (fun x_1 => magneticFieldMatrix 𝓕.c A ((time 𝓕.c) x) x_1 (j, i)) (space x) -
          𝓕.μ₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J ((time 𝓕.c) x) (space x)).ofLp i) -
    𝓕.μ₀⁻¹ * (𝓕.μ₀ * 𝓕.ε₀ * (∂ₜ (fun x_1 => electricField 𝓕.c A x_1 (space x)) ((time 𝓕.c) x)).ofLp i) =
  0) (𝓕.μ₀⁻¹) * h
      field_simpad:ℕ𝓕:FreeSpaceA:ElectromagneticPotential dhA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jx:SpaceTime di:Fin dh:𝓕.μ₀ * 𝓕.ε₀ * (∂ₜ (fun t => electricField 𝓕.c A t (space x)) ((time 𝓕.c) x)).ofLp i =
  ∑ j, Space.deriv j (fun x_1 => magneticFieldMatrix 𝓕.c A ((time 𝓕.c) x) x_1 (j, i)) (space x) -
    𝓕.μ₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J ((time 𝓕.c) x) (space x)).ofLp i⊢ 𝓕.μ₀ * 𝓕.ε₀ * (∂ₜ (fun x_1 => electricField 𝓕.c A x_1 (space x)) ((time 𝓕.c) x)).ofLp i -
          ∑ j, Space.deriv j (fun x_1 => magneticFieldMatrix 𝓕.c A ((time 𝓕.c) x) x_1 (j, i)) (space x) +
        𝓕.μ₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J ((time 𝓕.c) x) (space x)).ofLp i +
      (∑ j, Space.deriv j (fun x_1 => magneticFieldMatrix 𝓕.c A ((time 𝓕.c) x) x_1 (j, i)) (space x) -
        𝓕.μ₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J ((time 𝓕.c) x) (space x)).ofLp i) -
    𝓕.μ₀ * 𝓕.ε₀ * (∂ₜ (fun x_1 => electricField 𝓕.c A x_1 (space x)) ((time 𝓕.c) x)).ofLp i =
  𝓕.μ₀ * 0
      simpAll goals completed! 🐙

C. Time derivatives from the extrema condition



lemma time_deriv_electricField_of_isExtrema {A : ElectromagneticPotential d}
    {𝓕 : FreeSpace}
    (hA : ContDiff ℝ ∞ A) (J : LorentzCurrentDensity d) (hJ : ContDiff ℝ ∞ J)
    (h : IsExtrema 𝓕 A J) (t : Time) (x : Space d) (i : Fin d) :
    ∂ₜ (A.electricField 𝓕.c · x) t i =
      1 / (𝓕.μ₀ * 𝓕.ε₀) * ∑ j, ∂[j] (A.magneticFieldMatrix 𝓕.c t · (j, i)) x -
      (1/ 𝓕.ε₀) * J.currentDensity 𝓕.c t x i := byd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin d⊢ (∂ₜ (fun x_1 => electricField 𝓕.c A x_1 x) t).ofLp i =
  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 J t x).ofLp i
  rw [isExtrema_iff_gauss_ampere_magneticFieldMatrix hA J hJd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:∀ (t : Time) (x : Space d),
  Space.div (electricField 𝓕.c A t) x = LorentzCurrentDensity.chargeDensity 𝓕.c J t x / 𝓕.ε₀ ∧
    ∀ (i : Fin d),
      𝓕.μ₀ * 𝓕.ε₀ * (∂ₜ (fun t => electricField 𝓕.c A t x) t).ofLp i =
        ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x -
          𝓕.μ₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp it:Timex:Space di:Fin d⊢ (∂ₜ (fun x_1 => electricField 𝓕.c A x_1 x) t).ofLp i =
  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 J t x).ofLp id:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:∀ (t : Time) (x : Space d),
  Space.div (electricField 𝓕.c A t) x = LorentzCurrentDensity.chargeDensity 𝓕.c J t x / 𝓕.ε₀ ∧
    ∀ (i : Fin d),
      𝓕.μ₀ * 𝓕.ε₀ * (∂ₜ (fun t => electricField 𝓕.c A t x) t).ofLp i =
        ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x -
          𝓕.μ₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp it:Timex:Space di:Fin d⊢ (∂ₜ (fun x_1 => electricField 𝓕.c A x_1 x) t).ofLp i =
  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 J t x).ofLp i] at hd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:∀ (t : Time) (x : Space d),
  Space.div (electricField 𝓕.c A t) x = LorentzCurrentDensity.chargeDensity 𝓕.c J t x / 𝓕.ε₀ ∧
    ∀ (i : Fin d),
      𝓕.μ₀ * 𝓕.ε₀ * (∂ₜ (fun t => electricField 𝓕.c A t x) t).ofLp i =
        ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x -
          𝓕.μ₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp it:Timex:Space di:Fin d⊢ (∂ₜ (fun x_1 => electricField 𝓕.c A x_1 x) t).ofLp i =
  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 J t x).ofLp i
  linear_combination (norm := simpad:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:∀ (t : Time) (x : Space d),
  Space.div (electricField 𝓕.c A t) x = LorentzCurrentDensity.chargeDensity 𝓕.c J t x / 𝓕.ε₀ ∧
    ∀ (i : Fin d),
      𝓕.μ₀ * 𝓕.ε₀ * (∂ₜ (fun t => electricField 𝓕.c A t x) t).ofLp i =
        ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x -
          𝓕.μ₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp it:Timex:Space di:Fin d⊢ (∂ₜ (fun x_1 => electricField 𝓕.c A x_1 x) t).ofLp i +
      𝓕.ε₀⁻¹ * 𝓕.μ₀⁻¹ *
        (∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x -
          𝓕.μ₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i) -
    (𝓕.ε₀⁻¹ * 𝓕.μ₀⁻¹ * ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x -
        𝓕.ε₀⁻¹ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i +
      𝓕.ε₀⁻¹ * 𝓕.μ₀⁻¹ * (𝓕.μ₀ * 𝓕.ε₀ * (∂ₜ (fun x_1 => electricField 𝓕.c A x_1 x) t).ofLp i)) =
  0) (𝓕.μ₀ * 𝓕.ε₀)⁻¹ * ((h t x).2 i)
  field_simpad:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:∀ (t : Time) (x : Space d),
  Space.div (electricField 𝓕.c A t) x = LorentzCurrentDensity.chargeDensity 𝓕.c J t x / 𝓕.ε₀ ∧
    ∀ (i : Fin d),
      𝓕.μ₀ * 𝓕.ε₀ * (∂ₜ (fun t => electricField 𝓕.c A t x) t).ofLp i =
        ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x -
          𝓕.μ₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp it:Timex:Space di:Fin d⊢ (∂ₜ (fun x_1 => electricField 𝓕.c A x_1 x) t).ofLp i * 𝓕.ε₀ * 𝓕.μ₀ +
      (∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x -
        𝓕.μ₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i) -
    (∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x -
        𝓕.μ₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i +
      (∂ₜ (fun x_1 => electricField 𝓕.c A x_1 x) t).ofLp i * 𝓕.ε₀ * 𝓕.μ₀) =
  𝓕.ε₀ * 𝓕.μ₀ * 0
  ringAll goals completed! 🐙

D. Second time derivatives from the extrema condition

D.1. Second time derivatives of the magnetic field from the extrema condition

We show that the magnetic field matrix $B_{ij}$ satisfies the following wave-like equation

$$\frac{\partial^2 B_{ij}}{\partial t^2} = c^2 \sum_k \frac{\partial^2 B_{ij}}{\partial x_k^2} + \frac{1}{\epsilon_0} \left(\frac{\partial J_i}{\partial x_j} - \frac{\partial J_j}{\partial x_i} \right).$$ When the free current density is zero, this reduces to the wave equation.



lemma time_deriv_time_deriv_magneticFieldMatrix_of_isExtrema {A : ElectromagneticPotential d}
    {𝓕 : FreeSpace}
    (hA : ContDiff ℝ ∞ A) (J : LorentzCurrentDensity d)
    (hJ : ContDiff ℝ ∞ J) (h : IsExtrema 𝓕 A J)
    (t : Time) (x : Space d) (i j : Fin d) :
    ∂ₜ (∂ₜ (A.magneticFieldMatrix 𝓕.c · x (i, j))) t =
    𝓕.c ^ 2 * ∑ k, ∂[k] (∂[k] (A.magneticFieldMatrix 𝓕.c t · (i, j))) x +
    𝓕.ε₀⁻¹ * (∂[j] (J.currentDensity 𝓕.c t · i) x - ∂[i] (J.currentDensity 𝓕.c t · j) x) := byd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dj:Fin d⊢ ∂ₜ (∂ₜ fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, j)) t =
  𝓕.c.val ^ 2 * ∑ k, Space.deriv k (Space.deriv k fun x => magneticFieldMatrix 𝓕.c A t x (i, j)) x +
    𝓕.ε₀⁻¹ *
      (Space.deriv j (fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i) x -
        Space.deriv i (fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp j) x)
  have hcd : ∀ ij, ContDiff ℝ 2 (fun y => A.magneticFieldMatrix 𝓕.c t y ij) :=
    fun ij => magneticFieldMatrix_space_contDiff _ (hA.of_le (right_eq_inf.mp rfl)) t ijd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dj:Fin dhcd:∀ (ij : Fin d × Fin d), ContDiff ℝ 2 fun y => magneticFieldMatrix 𝓕.c A t y ij⊢ ∂ₜ (∂ₜ fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, j)) t =
  𝓕.c.val ^ 2 * ∑ k, Space.deriv k (Space.deriv k fun x => magneticFieldMatrix 𝓕.c A t x (i, j)) x +
    𝓕.ε₀⁻¹ *
      (Space.deriv j (fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i) x -
        Space.deriv i (fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp j) x)
  have hsd : ∀ ij k, Differentiable ℝ (∂[k] (fun y => A.magneticFieldMatrix 𝓕.c t y ij)) :=
    fun ij k => Space.deriv_differentiable (hcd ij) kd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dj:Fin dhcd:∀ (ij : Fin d × Fin d), ContDiff ℝ 2 fun y => magneticFieldMatrix 𝓕.c A t y ijhsd:∀ (ij : Fin d × Fin d) (k : Fin d), Differentiable ℝ (Space.deriv k fun y => magneticFieldMatrix 𝓕.c A t y ij)⊢ ∂ₜ (∂ₜ fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, j)) t =
  𝓕.c.val ^ 2 * ∑ k, Space.deriv k (Space.deriv k fun x => magneticFieldMatrix 𝓕.c A t x (i, j)) x +
    𝓕.ε₀⁻¹ *
      (Space.deriv j (fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i) x -
        Space.deriv i (fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp j) x)
  have hJd : ∀ i, Differentiable ℝ (fun x => J.currentDensity 𝓕.c t x i) :=
    fun i => LorentzCurrentDensity.currentDensity_apply_differentiable_space
      (hJ.differentiable (byd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di✝:Fin dj:Fin dhcd:∀ (ij : Fin d × Fin d), ContDiff ℝ 2 fun y => magneticFieldMatrix 𝓕.c A t y ijhsd:∀ (ij : Fin d × Fin d) (k : Fin d), Differentiable ℝ (Space.deriv k fun y => magneticFieldMatrix 𝓕.c A t y ij)i:Fin d⊢ ∞ ≠ 0d:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dj:Fin dhcd:∀ (ij : Fin d × Fin d), ContDiff ℝ 2 fun y => magneticFieldMatrix 𝓕.c A t y ijhsd:∀ (ij : Fin d × Fin d) (k : Fin d), Differentiable ℝ (Space.deriv k fun y => magneticFieldMatrix 𝓕.c A t y ij)hJd:∀ (i : Fin d), Differentiable ℝ fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i⊢ ∂ₜ (∂ₜ fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, j)) t =
  𝓕.c.val ^ 2 * ∑ k, Space.deriv k (Space.deriv k fun x => magneticFieldMatrix 𝓕.c A t x (i, j)) x +
    𝓕.ε₀⁻¹ *
      (Space.deriv j (fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i) x -
        Space.deriv i (fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp j) x) simpAll goals completed! 🐙d:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dj:Fin dhcd:∀ (ij : Fin d × Fin d), ContDiff ℝ 2 fun y => magneticFieldMatrix 𝓕.c A t y ijhsd:∀ (ij : Fin d × Fin d) (k : Fin d), Differentiable ℝ (Space.deriv k fun y => magneticFieldMatrix 𝓕.c A t y ij)hJd:∀ (i : Fin d), Differentiable ℝ fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i⊢ ∂ₜ (∂ₜ fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, j)) t =
  𝓕.c.val ^ 2 * ∑ k, Space.deriv k (Space.deriv k fun x => magneticFieldMatrix 𝓕.c A t x (i, j)) x +
    𝓕.ε₀⁻¹ *
      (Space.deriv j (fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i) x -
        Space.deriv i (fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp j) x))) t id:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dj:Fin dhcd:∀ (ij : Fin d × Fin d), ContDiff ℝ 2 fun y => magneticFieldMatrix 𝓕.c A t y ijhsd:∀ (ij : Fin d × Fin d) (k : Fin d), Differentiable ℝ (Space.deriv k fun y => magneticFieldMatrix 𝓕.c A t y ij)hJd:∀ (i : Fin d), Differentiable ℝ fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i⊢ ∂ₜ (∂ₜ fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (i, j)) t =
  𝓕.c.val ^ 2 * ∑ k, Space.deriv k (Space.deriv k fun x => magneticFieldMatrix 𝓕.c A t x (i, j)) x +
    𝓕.ε₀⁻¹ *
      (Space.deriv j (fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i) x -
        Space.deriv i (fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp j) x)
  rw [time_deriv_time_deriv_magneticFieldMatrix A (hA.of_le (ENat.LEInfty.out))d:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dj:Fin dhcd:∀ (ij : Fin d × Fin d), ContDiff ℝ 2 fun y => magneticFieldMatrix 𝓕.c A t y ijhsd:∀ (ij : Fin d × Fin d) (k : Fin d), Differentiable ℝ (Space.deriv k fun y => magneticFieldMatrix 𝓕.c A t y ij)hJd:∀ (i : Fin d), Differentiable ℝ fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i⊢ Space.deriv i (fun x => (∂ₜ (fun t => electricField 𝓕.c A t x) t).ofLp j) x -
    Space.deriv j (fun x => (∂ₜ (fun t => electricField 𝓕.c A t x) t).ofLp i) x =
  𝓕.c.val ^ 2 * ∑ k, Space.deriv k (Space.deriv k fun x => magneticFieldMatrix 𝓕.c A t x (i, j)) x +
    𝓕.ε₀⁻¹ *
      (Space.deriv j (fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i) x -
        Space.deriv i (fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp j) x)d:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dj:Fin dhcd:∀ (ij : Fin d × Fin d), ContDiff ℝ 2 fun y => magneticFieldMatrix 𝓕.c A t y ijhsd:∀ (ij : Fin d × Fin d) (k : Fin d), Differentiable ℝ (Space.deriv k fun y => magneticFieldMatrix 𝓕.c A t y ij)hJd:∀ (i : Fin d), Differentiable ℝ fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i⊢ Space.deriv i (fun x => (∂ₜ (fun t => electricField 𝓕.c A t x) t).ofLp j) x -
    Space.deriv j (fun x => (∂ₜ (fun t => electricField 𝓕.c A t x) t).ofLp i) x =
  𝓕.c.val ^ 2 * ∑ k, Space.deriv k (Space.deriv k fun x => magneticFieldMatrix 𝓕.c A t x (i, j)) x +
    𝓕.ε₀⁻¹ *
      (Space.deriv j (fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i) x -
        Space.deriv i (fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp j) x)]d:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dj:Fin dhcd:∀ (ij : Fin d × Fin d), ContDiff ℝ 2 fun y => magneticFieldMatrix 𝓕.c A t y ijhsd:∀ (ij : Fin d × Fin d) (k : Fin d), Differentiable ℝ (Space.deriv k fun y => magneticFieldMatrix 𝓕.c A t y ij)hJd:∀ (i : Fin d), Differentiable ℝ fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i⊢ Space.deriv i (fun x => (∂ₜ (fun t => electricField 𝓕.c A t x) t).ofLp j) x -
    Space.deriv j (fun x => (∂ₜ (fun t => electricField 𝓕.c A t x) t).ofLp i) x =
  𝓕.c.val ^ 2 * ∑ k, Space.deriv k (Space.deriv k fun x => magneticFieldMatrix 𝓕.c A t x (i, j)) x +
    𝓕.ε₀⁻¹ *
      (Space.deriv j (fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i) x -
        Space.deriv i (fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp j) x)
  conv_lhs =>
    enter [2, 2, x]d:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex✝:Space di:Fin dj:Fin dhcd:∀ (ij : Fin d × Fin d), ContDiff ℝ 2 fun y => magneticFieldMatrix 𝓕.c A t y ijhsd:∀ (ij : Fin d × Fin d) (k : Fin d), Differentiable ℝ (Space.deriv k fun y => magneticFieldMatrix 𝓕.c A t y ij)hJd:∀ (i : Fin d), Differentiable ℝ fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp ix:Space d| (∂ₜ (fun t => electricField 𝓕.c A t x) t).ofLp i
    rw [time_deriv_electricField_of_isExtrema hA J hJ h]d:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex✝:Space di:Fin dj:Fin dhcd:∀ (ij : Fin d × Fin d), ContDiff ℝ 2 fun y => magneticFieldMatrix 𝓕.c A t y ijhsd:∀ (ij : Fin d × Fin d) (k : Fin d), Differentiable ℝ (Space.deriv k fun y => magneticFieldMatrix 𝓕.c A t y ij)hJd:∀ (i : Fin d), Differentiable ℝ fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp ix:Space d| 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 J t x).ofLp i
  conv_lhs =>
    enter [1, 2, x]d:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex✝:Space di:Fin dj:Fin dhcd:∀ (ij : Fin d × Fin d), ContDiff ℝ 2 fun y => magneticFieldMatrix 𝓕.c A t y ijhsd:∀ (ij : Fin d × Fin d) (k : Fin d), Differentiable ℝ (Space.deriv k fun y => magneticFieldMatrix 𝓕.c A t y ij)hJd:∀ (i : Fin d), Differentiable ℝ fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp ix:Space d| (∂ₜ (fun t => electricField 𝓕.c A t x) t).ofLp j
    rw [time_deriv_electricField_of_isExtrema hA J hJ h]d:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex✝:Space di:Fin dj:Fin dhcd:∀ (ij : Fin d × Fin d), ContDiff ℝ 2 fun y => magneticFieldMatrix 𝓕.c A t y ijhsd:∀ (ij : Fin d × Fin d) (k : Fin d), Differentiable ℝ (Space.deriv k fun y => magneticFieldMatrix 𝓕.c A t y ij)hJd:∀ (i : Fin d), Differentiable ℝ fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp ix:Space d| 1 / (𝓕.μ₀ * 𝓕.ε₀) * ∑ j_1, Space.deriv j_1 (fun x => magneticFieldMatrix 𝓕.c A t x (j_1, j)) x -
  1 / 𝓕.ε₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp j
  rw [Space.deriv_eq_fderiv_basisd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dj:Fin dhcd:∀ (ij : Fin d × Fin d), ContDiff ℝ 2 fun y => magneticFieldMatrix 𝓕.c A t y ijhsd:∀ (ij : Fin d × Fin d) (k : Fin d), Differentiable ℝ (Space.deriv k fun y => magneticFieldMatrix 𝓕.c A t y ij)hJd:∀ (i : Fin d), Differentiable ℝ fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i⊢ (fderiv ℝ
        (fun x =>
          1 / (𝓕.μ₀ * 𝓕.ε₀) * ∑ j_1, Space.deriv j_1 (fun x => magneticFieldMatrix 𝓕.c A t x (j_1, j)) x -
            1 / 𝓕.ε₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp j)
        x)
      (Space.basis i) -
    Space.deriv j
      (fun x =>
        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 J t x).ofLp i)
      x =
  𝓕.c.val ^ 2 * ∑ k, Space.deriv k (Space.deriv k fun x => magneticFieldMatrix 𝓕.c A t x (i, j)) x +
    𝓕.ε₀⁻¹ *
      (Space.deriv j (fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i) x -
        Space.deriv i (fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp j) x)d:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dj:Fin dhcd:∀ (ij : Fin d × Fin d), ContDiff ℝ 2 fun y => magneticFieldMatrix 𝓕.c A t y ijhsd:∀ (ij : Fin d × Fin d) (k : Fin d), Differentiable ℝ (Space.deriv k fun y => magneticFieldMatrix 𝓕.c A t y ij)hJd:∀ (i : Fin d), Differentiable ℝ fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i⊢ (fderiv ℝ
        (fun x =>
          1 / (𝓕.μ₀ * 𝓕.ε₀) * ∑ j_1, Space.deriv j_1 (fun x => magneticFieldMatrix 𝓕.c A t x (j_1, j)) x -
            1 / 𝓕.ε₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp j)
        x)
      (Space.basis i) -
    Space.deriv j
      (fun x =>
        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 J t x).ofLp i)
      x =
  𝓕.c.val ^ 2 * ∑ k, Space.deriv k (Space.deriv k fun x => magneticFieldMatrix 𝓕.c A t x (i, j)) x +
    𝓕.ε₀⁻¹ *
      (Space.deriv j (fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i) x -
        Space.deriv i (fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp j) x)]d:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dj:Fin dhcd:∀ (ij : Fin d × Fin d), ContDiff ℝ 2 fun y => magneticFieldMatrix 𝓕.c A t y ijhsd:∀ (ij : Fin d × Fin d) (k : Fin d), Differentiable ℝ (Space.deriv k fun y => magneticFieldMatrix 𝓕.c A t y ij)hJd:∀ (i : Fin d), Differentiable ℝ fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i⊢ (fderiv ℝ
        (fun x =>
          1 / (𝓕.μ₀ * 𝓕.ε₀) * ∑ j_1, Space.deriv j_1 (fun x => magneticFieldMatrix 𝓕.c A t x (j_1, j)) x -
            1 / 𝓕.ε₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp j)
        x)
      (Space.basis i) -
    Space.deriv j
      (fun x =>
        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 J t x).ofLp i)
      x =
  𝓕.c.val ^ 2 * ∑ k, Space.deriv k (Space.deriv k fun x => magneticFieldMatrix 𝓕.c A t x (i, j)) x +
    𝓕.ε₀⁻¹ *
      (Space.deriv j (fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i) x -
        Space.deriv i (fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp j) x)
  rw [fderiv_fun_sub ((Differentiable.fun_sum fun i _ => hsd _ i).const_mul _).differentiableAt
      ((hJd _).const_mul _).differentiableAt,d:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dj:Fin dhcd:∀ (ij : Fin d × Fin d), ContDiff ℝ 2 fun y => magneticFieldMatrix 𝓕.c A t y ijhsd:∀ (ij : Fin d × Fin d) (k : Fin d), Differentiable ℝ (Space.deriv k fun y => magneticFieldMatrix 𝓕.c A t y ij)hJd:∀ (i : Fin d), Differentiable ℝ fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i⊢ (fderiv ℝ (fun y => 1 / (𝓕.μ₀ * 𝓕.ε₀) * ∑ i, Space.deriv i (fun y => magneticFieldMatrix 𝓕.c A t y (i, j)) y) x -
        fderiv ℝ (fun y => 1 / 𝓕.ε₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J t y).ofLp j) x)
      (Space.basis i) -
    Space.deriv j
      (fun x =>
        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 J t x).ofLp i)
      x =
  𝓕.c.val ^ 2 * ∑ k, Space.deriv k (Space.deriv k fun x => magneticFieldMatrix 𝓕.c A t x (i, j)) x +
    𝓕.ε₀⁻¹ *
      (Space.deriv j (fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i) x -
        Space.deriv i (fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp j) x)d:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dj:Fin dhcd:∀ (ij : Fin d × Fin d), ContDiff ℝ 2 fun y => magneticFieldMatrix 𝓕.c A t y ijhsd:∀ (ij : Fin d × Fin d) (k : Fin d), Differentiable ℝ (Space.deriv k fun y => magneticFieldMatrix 𝓕.c A t y ij)hJd:∀ (i : Fin d), Differentiable ℝ fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i⊢ ((1 / (𝓕.μ₀ * 𝓕.ε₀)) • ∑ i, fderiv ℝ (Space.deriv i fun y => magneticFieldMatrix 𝓕.c A t y (i, j)) x -
        (1 / 𝓕.ε₀) • fderiv ℝ (fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp j) x)
      (Space.basis i) -
    Space.deriv j
      (fun x =>
        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 J t x).ofLp i)
      x =
  𝓕.c.val ^ 2 * ∑ k, Space.deriv k (Space.deriv k fun x => magneticFieldMatrix 𝓕.c A t x (i, j)) x +
    𝓕.ε₀⁻¹ *
      (Space.deriv j (fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i) x -
        Space.deriv i (fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp j) x)
    fderiv_const_mul (Differentiable.fun_sum fun i _ => hsd _ i).differentiableAt,d:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dj:Fin dhcd:∀ (ij : Fin d × Fin d), ContDiff ℝ 2 fun y => magneticFieldMatrix 𝓕.c A t y ijhsd:∀ (ij : Fin d × Fin d) (k : Fin d), Differentiable ℝ (Space.deriv k fun y => magneticFieldMatrix 𝓕.c A t y ij)hJd:∀ (i : Fin d), Differentiable ℝ fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i⊢ ((1 / (𝓕.μ₀ * 𝓕.ε₀)) • fderiv ℝ (fun y => ∑ i, Space.deriv i (fun y => magneticFieldMatrix 𝓕.c A t y (i, j)) y) x -
        fderiv ℝ (fun y => 1 / 𝓕.ε₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J t y).ofLp j) x)
      (Space.basis i) -
    Space.deriv j
      (fun x =>
        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 J t x).ofLp i)
      x =
  𝓕.c.val ^ 2 * ∑ k, Space.deriv k (Space.deriv k fun x => magneticFieldMatrix 𝓕.c A t x (i, j)) x +
    𝓕.ε₀⁻¹ *
      (Space.deriv j (fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i) x -
        Space.deriv i (fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp j) x)d:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dj:Fin dhcd:∀ (ij : Fin d × Fin d), ContDiff ℝ 2 fun y => magneticFieldMatrix 𝓕.c A t y ijhsd:∀ (ij : Fin d × Fin d) (k : Fin d), Differentiable ℝ (Space.deriv k fun y => magneticFieldMatrix 𝓕.c A t y ij)hJd:∀ (i : Fin d), Differentiable ℝ fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i⊢ ((1 / (𝓕.μ₀ * 𝓕.ε₀)) • ∑ i, fderiv ℝ (Space.deriv i fun y => magneticFieldMatrix 𝓕.c A t y (i, j)) x -
        (1 / 𝓕.ε₀) • fderiv ℝ (fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp j) x)
      (Space.basis i) -
    Space.deriv j
      (fun x =>
        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 J t x).ofLp i)
      x =
  𝓕.c.val ^ 2 * ∑ k, Space.deriv k (Space.deriv k fun x => magneticFieldMatrix 𝓕.c A t x (i, j)) x +
    𝓕.ε₀⁻¹ *
      (Space.deriv j (fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i) x -
        Space.deriv i (fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp j) x)
    fderiv_const_mul (hJd _).differentiableAt,d:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dj:Fin dhcd:∀ (ij : Fin d × Fin d), ContDiff ℝ 2 fun y => magneticFieldMatrix 𝓕.c A t y ijhsd:∀ (ij : Fin d × Fin d) (k : Fin d), Differentiable ℝ (Space.deriv k fun y => magneticFieldMatrix 𝓕.c A t y ij)hJd:∀ (i : Fin d), Differentiable ℝ fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i⊢ ((1 / (𝓕.μ₀ * 𝓕.ε₀)) • fderiv ℝ (fun y => ∑ i, Space.deriv i (fun y => magneticFieldMatrix 𝓕.c A t y (i, j)) y) x -
        (1 / 𝓕.ε₀) • fderiv ℝ (fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp j) x)
      (Space.basis i) -
    Space.deriv j
      (fun x =>
        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 J t x).ofLp i)
      x =
  𝓕.c.val ^ 2 * ∑ k, Space.deriv k (Space.deriv k fun x => magneticFieldMatrix 𝓕.c A t x (i, j)) x +
    𝓕.ε₀⁻¹ *
      (Space.deriv j (fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i) x -
        Space.deriv i (fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp j) x)d:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dj:Fin dhcd:∀ (ij : Fin d × Fin d), ContDiff ℝ 2 fun y => magneticFieldMatrix 𝓕.c A t y ijhsd:∀ (ij : Fin d × Fin d) (k : Fin d), Differentiable ℝ (Space.deriv k fun y => magneticFieldMatrix 𝓕.c A t y ij)hJd:∀ (i : Fin d), Differentiable ℝ fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i⊢ ((1 / (𝓕.μ₀ * 𝓕.ε₀)) • ∑ i, fderiv ℝ (Space.deriv i fun y => magneticFieldMatrix 𝓕.c A t y (i, j)) x -
        (1 / 𝓕.ε₀) • fderiv ℝ (fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp j) x)
      (Space.basis i) -
    Space.deriv j
      (fun x =>
        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 J t x).ofLp i)
      x =
  𝓕.c.val ^ 2 * ∑ k, Space.deriv k (Space.deriv k fun x => magneticFieldMatrix 𝓕.c A t x (i, j)) x +
    𝓕.ε₀⁻¹ *
      (Space.deriv j (fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i) x -
        Space.deriv i (fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp j) x)
    fderiv_fun_sum fun i _ => (hsd _ i).differentiableAtd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dj:Fin dhcd:∀ (ij : Fin d × Fin d), ContDiff ℝ 2 fun y => magneticFieldMatrix 𝓕.c A t y ijhsd:∀ (ij : Fin d × Fin d) (k : Fin d), Differentiable ℝ (Space.deriv k fun y => magneticFieldMatrix 𝓕.c A t y ij)hJd:∀ (i : Fin d), Differentiable ℝ fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i⊢ ((1 / (𝓕.μ₀ * 𝓕.ε₀)) • ∑ i, fderiv ℝ (Space.deriv i fun y => magneticFieldMatrix 𝓕.c A t y (i, j)) x -
        (1 / 𝓕.ε₀) • fderiv ℝ (fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp j) x)
      (Space.basis i) -
    Space.deriv j
      (fun x =>
        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 J t x).ofLp i)
      x =
  𝓕.c.val ^ 2 * ∑ k, Space.deriv k (Space.deriv k fun x => magneticFieldMatrix 𝓕.c A t x (i, j)) x +
    𝓕.ε₀⁻¹ *
      (Space.deriv j (fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i) x -
        Space.deriv i (fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp j) x)d:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dj:Fin dhcd:∀ (ij : Fin d × Fin d), ContDiff ℝ 2 fun y => magneticFieldMatrix 𝓕.c A t y ijhsd:∀ (ij : Fin d × Fin d) (k : Fin d), Differentiable ℝ (Space.deriv k fun y => magneticFieldMatrix 𝓕.c A t y ij)hJd:∀ (i : Fin d), Differentiable ℝ fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i⊢ ((1 / (𝓕.μ₀ * 𝓕.ε₀)) • ∑ i, fderiv ℝ (Space.deriv i fun y => magneticFieldMatrix 𝓕.c A t y (i, j)) x -
        (1 / 𝓕.ε₀) • fderiv ℝ (fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp j) x)
      (Space.basis i) -
    Space.deriv j
      (fun x =>
        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 J t x).ofLp i)
      x =
  𝓕.c.val ^ 2 * ∑ k, Space.deriv k (Space.deriv k fun x => magneticFieldMatrix 𝓕.c A t x (i, j)) x +
    𝓕.ε₀⁻¹ *
      (Space.deriv j (fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i) x -
        Space.deriv i (fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp j) x)]d:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dj:Fin dhcd:∀ (ij : Fin d × Fin d), ContDiff ℝ 2 fun y => magneticFieldMatrix 𝓕.c A t y ijhsd:∀ (ij : Fin d × Fin d) (k : Fin d), Differentiable ℝ (Space.deriv k fun y => magneticFieldMatrix 𝓕.c A t y ij)hJd:∀ (i : Fin d), Differentiable ℝ fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i⊢ ((1 / (𝓕.μ₀ * 𝓕.ε₀)) • ∑ i, fderiv ℝ (Space.deriv i fun y => magneticFieldMatrix 𝓕.c A t y (i, j)) x -
        (1 / 𝓕.ε₀) • fderiv ℝ (fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp j) x)
      (Space.basis i) -
    Space.deriv j
      (fun x =>
        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 J t x).ofLp i)
      x =
  𝓕.c.val ^ 2 * ∑ k, Space.deriv k (Space.deriv k fun x => magneticFieldMatrix 𝓕.c A t x (i, j)) x +
    𝓕.ε₀⁻¹ *
      (Space.deriv j (fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i) x -
        Space.deriv i (fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp j) x)
  conv_lhs =>
    enter [2]d:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dj:Fin dhcd:∀ (ij : Fin d × Fin d), ContDiff ℝ 2 fun y => magneticFieldMatrix 𝓕.c A t y ijhsd:∀ (ij : Fin d × Fin d) (k : Fin d), Differentiable ℝ (Space.deriv k fun y => magneticFieldMatrix 𝓕.c A t y ij)hJd:∀ (i : Fin d), Differentiable ℝ fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i| Space.deriv j
  (fun x =>
    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 J t x).ofLp i)
  x
    rw [Space.deriv_eq_fderiv_basis]d:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dj:Fin dhcd:∀ (ij : Fin d × Fin d), ContDiff ℝ 2 fun y => magneticFieldMatrix 𝓕.c A t y ijhsd:∀ (ij : Fin d × Fin d) (k : Fin d), Differentiable ℝ (Space.deriv k fun y => magneticFieldMatrix 𝓕.c A t y ij)hJd:∀ (i : Fin d), Differentiable ℝ fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i| (fderiv ℝ
    (fun x =>
      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 J t x).ofLp i)
    x)
  (Space.basis j)
    rw [fderiv_fun_sub ((Differentiable.fun_sum fun i _ => hsd _ i).const_mul _).differentiableAt
        ((hJd _).const_mul _).differentiableAt,
    fderiv_const_mul (Differentiable.fun_sum fun i _ => hsd _ i).differentiableAt,
    fderiv_const_mul (hJd _).differentiableAt,
    fderiv_fun_sum fun i _ => (hsd _ i).differentiableAt]d:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dj:Fin dhcd:∀ (ij : Fin d × Fin d), ContDiff ℝ 2 fun y => magneticFieldMatrix 𝓕.c A t y ijhsd:∀ (ij : Fin d × Fin d) (k : Fin d), Differentiable ℝ (Space.deriv k fun y => magneticFieldMatrix 𝓕.c A t y ij)hJd:∀ (i : Fin d), Differentiable ℝ fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i| ((1 / (𝓕.μ₀ * 𝓕.ε₀)) • ∑ i_1, fderiv ℝ (Space.deriv i_1 fun y => magneticFieldMatrix 𝓕.c A t y (i_1, i)) x -
    (1 / 𝓕.ε₀) • fderiv ℝ (fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i) x)
  (Space.basis j)
  simp [← Space.deriv_eq_fderiv_basis, FreeSpace.c_sq]d:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dj:Fin dhcd:∀ (ij : Fin d × Fin d), ContDiff ℝ 2 fun y => magneticFieldMatrix 𝓕.c A t y ijhsd:∀ (ij : Fin d × Fin d) (k : Fin d), Differentiable ℝ (Space.deriv k fun y => magneticFieldMatrix 𝓕.c A t y ij)hJd:∀ (i : Fin d), Differentiable ℝ fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i⊢ 𝓕.ε₀⁻¹ * 𝓕.μ₀⁻¹ * ∑ x_1, Space.deriv i (Space.deriv x_1 fun y => magneticFieldMatrix 𝓕.c A t y (x_1, j)) x -
      𝓕.ε₀⁻¹ * Space.deriv i (fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp j) x -
    (𝓕.ε₀⁻¹ * 𝓕.μ₀⁻¹ * ∑ x_1, Space.deriv j (Space.deriv x_1 fun y => magneticFieldMatrix 𝓕.c A t y (x_1, i)) x -
      𝓕.ε₀⁻¹ * Space.deriv j (fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i) x) =
  𝓕.μ₀⁻¹ * 𝓕.ε₀⁻¹ * ∑ k, Space.deriv k (Space.deriv k fun x => magneticFieldMatrix 𝓕.c A t x (i, j)) x +
    𝓕.ε₀⁻¹ *
      (Space.deriv j (fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i) x -
        Space.deriv i (fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp j) x)
  field_simpd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dj:Fin dhcd:∀ (ij : Fin d × Fin d), ContDiff ℝ 2 fun y => magneticFieldMatrix 𝓕.c A t y ijhsd:∀ (ij : Fin d × Fin d) (k : Fin d), Differentiable ℝ (Space.deriv k fun y => magneticFieldMatrix 𝓕.c A t y ij)hJd:∀ (i : Fin d), Differentiable ℝ fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i⊢ ∑ x_1, Space.deriv i (Space.deriv x_1 fun y => magneticFieldMatrix 𝓕.c A t y (x_1, j)) x -
      𝓕.μ₀ * Space.deriv i (fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp j) x -
    (∑ x_1, Space.deriv j (Space.deriv x_1 fun y => magneticFieldMatrix 𝓕.c A t y (x_1, i)) x -
      𝓕.μ₀ * Space.deriv j (fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i) x) =
  ∑ k, Space.deriv k (Space.deriv k fun x => magneticFieldMatrix 𝓕.c A t x (i, j)) x +
    𝓕.μ₀ *
      (Space.deriv j (fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i) x -
        Space.deriv i (fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp j) x)
  conv_rhs =>
    enter [1, 2, k, 2, x]d:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex✝:Space di:Fin dj:Fin dhcd:∀ (ij : Fin d × Fin d), ContDiff ℝ 2 fun y => magneticFieldMatrix 𝓕.c A t y ijhsd:∀ (ij : Fin d × Fin d) (k : Fin d), Differentiable ℝ (Space.deriv k fun y => magneticFieldMatrix 𝓕.c A t y ij)hJd:∀ (i : Fin d), Differentiable ℝ fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp ik:Fin dx:Space d| Space.deriv k (fun x => magneticFieldMatrix 𝓕.c A t x (i, j)) x
    rw [magneticFieldMatrix_space_deriv_eq A (hA.of_le (right_eq_inf.mp rfl))]d:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex✝:Space di:Fin dj:Fin dhcd:∀ (ij : Fin d × Fin d), ContDiff ℝ 2 fun y => magneticFieldMatrix 𝓕.c A t y ijhsd:∀ (ij : Fin d × Fin d) (k : Fin d), Differentiable ℝ (Space.deriv k fun y => magneticFieldMatrix 𝓕.c A t y ij)hJd:∀ (i : Fin d), Differentiable ℝ fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp ik:Fin dx:Space d| Space.deriv i (fun x => magneticFieldMatrix 𝓕.c A t x (k, j)) x -
  Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (k, i)) x
  conv_rhs =>
    enter [1, 2, k]d:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dj:Fin dhcd:∀ (ij : Fin d × Fin d), ContDiff ℝ 2 fun y => magneticFieldMatrix 𝓕.c A t y ijhsd:∀ (ij : Fin d × Fin d) (k : Fin d), Differentiable ℝ (Space.deriv k fun y => magneticFieldMatrix 𝓕.c A t y ij)hJd:∀ (i : Fin d), Differentiable ℝ fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp ik:Fin d| Space.deriv k
  (fun x =>
    Space.deriv i (fun x => magneticFieldMatrix 𝓕.c A t x (k, j)) x -
      Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (k, i)) x)
  x
    rw [Space.deriv_eq_fderiv_basis]d:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dj:Fin dhcd:∀ (ij : Fin d × Fin d), ContDiff ℝ 2 fun y => magneticFieldMatrix 𝓕.c A t y ijhsd:∀ (ij : Fin d × Fin d) (k : Fin d), Differentiable ℝ (Space.deriv k fun y => magneticFieldMatrix 𝓕.c A t y ij)hJd:∀ (i : Fin d), Differentiable ℝ fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp ik:Fin d| (fderiv ℝ
    (fun x =>
      Space.deriv i (fun x => magneticFieldMatrix 𝓕.c A t x (k, j)) x -
        Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (k, i)) x)
    x)
  (Space.basis k)
    rw [fderiv_fun_sub (hsd _ _).differentiableAt (hsd _ _).differentiableAt]d:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dj:Fin dhcd:∀ (ij : Fin d × Fin d), ContDiff ℝ 2 fun y => magneticFieldMatrix 𝓕.c A t y ijhsd:∀ (ij : Fin d × Fin d) (k : Fin d), Differentiable ℝ (Space.deriv k fun y => magneticFieldMatrix 𝓕.c A t y ij)hJd:∀ (i : Fin d), Differentiable ℝ fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp ik:Fin d| (fderiv ℝ (Space.deriv i fun y => magneticFieldMatrix 𝓕.c A t y (k, j)) x -
    fderiv ℝ (Space.deriv j fun y => magneticFieldMatrix 𝓕.c A t y (k, i)) x)
  (Space.basis k)
    simp [← Space.deriv_eq_fderiv_basis]d:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dj:Fin dhcd:∀ (ij : Fin d × Fin d), ContDiff ℝ 2 fun y => magneticFieldMatrix 𝓕.c A t y ijhsd:∀ (ij : Fin d × Fin d) (k : Fin d), Differentiable ℝ (Space.deriv k fun y => magneticFieldMatrix 𝓕.c A t y ij)hJd:∀ (i : Fin d), Differentiable ℝ fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp ik:Fin d| Space.deriv k (Space.deriv i fun y => magneticFieldMatrix 𝓕.c A t y (k, j)) x -
  Space.deriv k (Space.deriv j fun y => magneticFieldMatrix 𝓕.c A t y (k, i)) x
    rw [Space.deriv_commute _ (hcd _)]d:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dj:Fin dhcd:∀ (ij : Fin d × Fin d), ContDiff ℝ 2 fun y => magneticFieldMatrix 𝓕.c A t y ijhsd:∀ (ij : Fin d × Fin d) (k : Fin d), Differentiable ℝ (Space.deriv k fun y => magneticFieldMatrix 𝓕.c A t y ij)hJd:∀ (i : Fin d), Differentiable ℝ fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp ik:Fin d| Space.deriv i (Space.deriv k fun y => magneticFieldMatrix 𝓕.c A t y (k, j)) x -
  Space.deriv k (Space.deriv j fun y => magneticFieldMatrix 𝓕.c A t y (k, i)) x
    enter [2]d:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dj:Fin dhcd:∀ (ij : Fin d × Fin d), ContDiff ℝ 2 fun y => magneticFieldMatrix 𝓕.c A t y ijhsd:∀ (ij : Fin d × Fin d) (k : Fin d), Differentiable ℝ (Space.deriv k fun y => magneticFieldMatrix 𝓕.c A t y ij)hJd:∀ (i : Fin d), Differentiable ℝ fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp ik:Fin d| Space.deriv k (Space.deriv j fun y => magneticFieldMatrix 𝓕.c A t y (k, i)) x
    rw [Space.deriv_commute _ (hcd _)]d:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dj:Fin dhcd:∀ (ij : Fin d × Fin d), ContDiff ℝ 2 fun y => magneticFieldMatrix 𝓕.c A t y ijhsd:∀ (ij : Fin d × Fin d) (k : Fin d), Differentiable ℝ (Space.deriv k fun y => magneticFieldMatrix 𝓕.c A t y ij)hJd:∀ (i : Fin d), Differentiable ℝ fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp ik:Fin d| Space.deriv j (Space.deriv k fun y => magneticFieldMatrix 𝓕.c A t y (k, i)) x
  simp only [Finset.sum_sub_distrib]d:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dj:Fin dhcd:∀ (ij : Fin d × Fin d), ContDiff ℝ 2 fun y => magneticFieldMatrix 𝓕.c A t y ijhsd:∀ (ij : Fin d × Fin d) (k : Fin d), Differentiable ℝ (Space.deriv k fun y => magneticFieldMatrix 𝓕.c A t y ij)hJd:∀ (i : Fin d), Differentiable ℝ fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i⊢ ∑ x_1, Space.deriv i (Space.deriv x_1 fun y => magneticFieldMatrix 𝓕.c A t y (x_1, j)) x -
      𝓕.μ₀ * Space.deriv i (fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp j) x -
    (∑ x_1, Space.deriv j (Space.deriv x_1 fun y => magneticFieldMatrix 𝓕.c A t y (x_1, i)) x -
      𝓕.μ₀ * Space.deriv j (fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i) x) =
  ∑ x_1, Space.deriv i (Space.deriv x_1 fun y => magneticFieldMatrix 𝓕.c A t y (x_1, j)) x -
      ∑ x_1, Space.deriv j (Space.deriv x_1 fun y => magneticFieldMatrix 𝓕.c A t y (x_1, i)) x +
    𝓕.μ₀ *
      (Space.deriv j (fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i) x -
        Space.deriv i (fun x => (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp j) x)
  ringAll goals completed! 🐙

D.2. Second time derivatives of the electric field from the extrema condition

We show that the electric field $E_i$ satisfies the following wave-like equation:

$$\frac{\partial^2 E_{i}}{\partial t^2} = c^2 \sum_k \frac{\partial^2 E_{i}}{\partial x_k^2} - \frac{c ^ 2}{\epsilon_0} \frac{\partial \rho}{\partial x_i} - c ^ 2 μ_0 \frac{\partial J_i}{\partial t}.$$

When the free current density and charge density are zero, this reduces to the wave equation.



lemma time_deriv_time_deriv_electricField_of_isExtrema {A : ElectromagneticPotential d}
    {𝓕 : FreeSpace}
    (hA : ContDiff ℝ ∞ A) (J : LorentzCurrentDensity d)
    (hJ : ContDiff ℝ ∞ J) (h : IsExtrema 𝓕 A J)
    (t : Time) (x : Space d) (i : Fin d) :
    ∂ₜ (∂ₜ (A.electricField 𝓕.c · x i)) t =
      𝓕.c ^ 2 * ∑ j, (∂[j] (∂[j] (A.electricField 𝓕.c t · i)) x) -
      𝓕.c ^ 2 / 𝓕.ε₀ * ∂[i] (J.chargeDensity 𝓕.c t ·) x -
      𝓕.c ^ 2 * 𝓕.μ₀ * ∂ₜ (J.currentDensity 𝓕.c · x i) t := byd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin d⊢ ∂ₜ (∂ₜ fun x_1 => (electricField 𝓕.c A x_1 x).ofLp i) t =
  𝓕.c.val ^ 2 * ∑ j, Space.deriv j (Space.deriv j fun x => (electricField 𝓕.c A t x).ofLp i) x -
      𝓕.c.val ^ 2 / 𝓕.ε₀ * Space.deriv i (fun x => LorentzCurrentDensity.chargeDensity 𝓕.c J t x) x -
    𝓕.c.val ^ 2 * 𝓕.μ₀ * ∂ₜ (fun x_1 => (LorentzCurrentDensity.currentDensity 𝓕.c J x_1 x).ofLp i) t
  have hEs : ∀ j, ContDiff ℝ 2 (fun y => A.electricField 𝓕.c t y j) :=
    fun j => electricField_apply_contDiff_space (i := j) (hA.of_le (right_eq_inf.mp rfl)) td:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp j⊢ ∂ₜ (∂ₜ fun x_1 => (electricField 𝓕.c A x_1 x).ofLp i) t =
  𝓕.c.val ^ 2 * ∑ j, Space.deriv j (Space.deriv j fun x => (electricField 𝓕.c A t x).ofLp i) x -
      𝓕.c.val ^ 2 / 𝓕.ε₀ * Space.deriv i (fun x => LorentzCurrentDensity.chargeDensity 𝓕.c J t x) x -
    𝓕.c.val ^ 2 * 𝓕.μ₀ * ∂ₜ (fun x_1 => (LorentzCurrentDensity.currentDensity 𝓕.c J x_1 x).ofLp i) t
  have hEd : ∀ j k, Differentiable ℝ (∂[k] (fun y => A.electricField 𝓕.c t y j)) :=
    fun j k => Space.deriv_differentiable (hEs j) kd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)⊢ ∂ₜ (∂ₜ fun x_1 => (electricField 𝓕.c A x_1 x).ofLp i) t =
  𝓕.c.val ^ 2 * ∑ j, Space.deriv j (Space.deriv j fun x => (electricField 𝓕.c A t x).ofLp i) x -
      𝓕.c.val ^ 2 / 𝓕.ε₀ * Space.deriv i (fun x => LorentzCurrentDensity.chargeDensity 𝓕.c J t x) x -
    𝓕.c.val ^ 2 * 𝓕.μ₀ * ∂ₜ (fun x_1 => (LorentzCurrentDensity.currentDensity 𝓕.c J x_1 x).ofLp i) t
  have hBt : ∀ j, Differentiable ℝ
      (fun s => ∂[j] (fun y => A.magneticFieldMatrix 𝓕.c s y (j, i)) x) :=
    fun j => Space.space_deriv_differentiable_time (i := j)
      (magneticFieldMatrix_contDiff _ (hA.of_le (right_eq_inf.mp rfl)) (j, i)) xd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) x⊢ ∂ₜ (∂ₜ fun x_1 => (electricField 𝓕.c A x_1 x).ofLp i) t =
  𝓕.c.val ^ 2 * ∑ j, Space.deriv j (Space.deriv j fun x => (electricField 𝓕.c A t x).ofLp i) x -
      𝓕.c.val ^ 2 / 𝓕.ε₀ * Space.deriv i (fun x => LorentzCurrentDensity.chargeDensity 𝓕.c J t x) x -
    𝓕.c.val ^ 2 * 𝓕.μ₀ * ∂ₜ (fun x_1 => (LorentzCurrentDensity.currentDensity 𝓕.c J x_1 x).ofLp i) t
  have hJt : Differentiable ℝ (fun s => J.currentDensity 𝓕.c s x i) :=
    LorentzCurrentDensity.currentDensity_apply_differentiable_time (hJ.differentiable (byd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) x⊢ ∞ ≠ 0d:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ ∂ₜ (∂ₜ fun x_1 => (electricField 𝓕.c A x_1 x).ofLp i) t =
  𝓕.c.val ^ 2 * ∑ j, Space.deriv j (Space.deriv j fun x => (electricField 𝓕.c A t x).ofLp i) x -
      𝓕.c.val ^ 2 / 𝓕.ε₀ * Space.deriv i (fun x => LorentzCurrentDensity.chargeDensity 𝓕.c J t x) x -
    𝓕.c.val ^ 2 * 𝓕.μ₀ * ∂ₜ (fun x_1 => (LorentzCurrentDensity.currentDensity 𝓕.c J x_1 x).ofLp i) t simpAll goals completed! 🐙d:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ ∂ₜ (∂ₜ fun x_1 => (electricField 𝓕.c A x_1 x).ofLp i) t =
  𝓕.c.val ^ 2 * ∑ j, Space.deriv j (Space.deriv j fun x => (electricField 𝓕.c A t x).ofLp i) x -
      𝓕.c.val ^ 2 / 𝓕.ε₀ * Space.deriv i (fun x => LorentzCurrentDensity.chargeDensity 𝓕.c J t x) x -
    𝓕.c.val ^ 2 * 𝓕.μ₀ * ∂ₜ (fun x_1 => (LorentzCurrentDensity.currentDensity 𝓕.c J x_1 x).ofLp i) t)) x id:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ ∂ₜ (∂ₜ fun x_1 => (electricField 𝓕.c A x_1 x).ofLp i) t =
  𝓕.c.val ^ 2 * ∑ j, Space.deriv j (Space.deriv j fun x => (electricField 𝓕.c A t x).ofLp i) x -
      𝓕.c.val ^ 2 / 𝓕.ε₀ * Space.deriv i (fun x => LorentzCurrentDensity.chargeDensity 𝓕.c J t x) x -
    𝓕.c.val ^ 2 * 𝓕.μ₀ * ∂ₜ (fun x_1 => (LorentzCurrentDensity.currentDensity 𝓕.c J x_1 x).ofLp i) t
  calc _
    _= ∂ₜ (fun t =>
      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 J t x i) t := byd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ ∂ₜ (∂ₜ fun x_1 => (electricField 𝓕.c A x_1 x).ofLp i) t =
  ∂ₜ
    (fun t =>
      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 J t x).ofLp i)
    t
        conv_lhs =>
          enter [1]d:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i| ∂ₜ fun x_1 => (electricField 𝓕.c A x_1 x).ofLp i
          change fun t => ∂ₜ (A.electricField 𝓕.c · x i) td:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i| fun t => ∂ₜ (fun x_1 => (electricField 𝓕.c A x_1 x).ofLp i) t
          enter [t]d:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt✝:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp it:Time| ∂ₜ (fun x_1 => (electricField 𝓕.c A x_1 x).ofLp i) t
          rw [Time.deriv_euclid (electricField_differentiable_time
            (hA.of_le (right_eq_inf.mp rfl)) _),
            time_deriv_electricField_of_isExtrema hA J hJ h]d:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt✝:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp it:Time| 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 J t x).ofLp i
    _ = 1 / (𝓕.μ₀ * 𝓕.ε₀) * ∂ₜ (fun t => ∑ j, ∂[j] (A.magneticFieldMatrix 𝓕.c t · (j, i)) x) t -
      1 / 𝓕.ε₀ * ∂ₜ (J.currentDensity 𝓕.c · x i) t := byd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ ∂ₜ
    (fun t =>
      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 J t x).ofLp i)
    t =
  1 / (𝓕.μ₀ * 𝓕.ε₀) * ∂ₜ (fun t => ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x) t -
    1 / 𝓕.ε₀ * ∂ₜ (fun x_1 => (LorentzCurrentDensity.currentDensity 𝓕.c J x_1 x).ofLp i) t
      rw [Time.deriv_eqd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ (fderiv ℝ
      (fun t =>
        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 J t x).ofLp i)
      t)
    1 =
  1 / (𝓕.μ₀ * 𝓕.ε₀) * ∂ₜ (fun t => ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x) t -
    1 / 𝓕.ε₀ * ∂ₜ (fun x_1 => (LorentzCurrentDensity.currentDensity 𝓕.c J x_1 x).ofLp i) td:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ (fderiv ℝ
      (fun t =>
        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 J t x).ofLp i)
      t)
    1 =
  1 / (𝓕.μ₀ * 𝓕.ε₀) * ∂ₜ (fun t => ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x) t -
    1 / 𝓕.ε₀ * ∂ₜ (fun x_1 => (LorentzCurrentDensity.currentDensity 𝓕.c J x_1 x).ofLp i) t]d:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ (fderiv ℝ
      (fun t =>
        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 J t x).ofLp i)
      t)
    1 =
  1 / (𝓕.μ₀ * 𝓕.ε₀) * ∂ₜ (fun t => ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x) t -
    1 / 𝓕.ε₀ * ∂ₜ (fun x_1 => (LorentzCurrentDensity.currentDensity 𝓕.c J x_1 x).ofLp i) t
      rw [fderiv_fun_subd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ (fderiv ℝ (fun t => 1 / (𝓕.μ₀ * 𝓕.ε₀) * ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x) t -
      fderiv ℝ (fun t => 1 / 𝓕.ε₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i) t)
    1 =
  1 / (𝓕.μ₀ * 𝓕.ε₀) * ∂ₜ (fun t => ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x) t -
    1 / 𝓕.ε₀ * ∂ₜ (fun x_1 => (LorentzCurrentDensity.currentDensity 𝓕.c J x_1 x).ofLp i) thfd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ DifferentiableAt ℝ (fun t => 1 / (𝓕.μ₀ * 𝓕.ε₀) * ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x) thgd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ DifferentiableAt ℝ (fun t => 1 / 𝓕.ε₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i) td:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ (fderiv ℝ (fun t => 1 / (𝓕.μ₀ * 𝓕.ε₀) * ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x) t -
      fderiv ℝ (fun t => 1 / 𝓕.ε₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i) t)
    1 =
  1 / (𝓕.μ₀ * 𝓕.ε₀) * ∂ₜ (fun t => ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x) t -
    1 / 𝓕.ε₀ * ∂ₜ (fun x_1 => (LorentzCurrentDensity.currentDensity 𝓕.c J x_1 x).ofLp i) thfd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ DifferentiableAt ℝ (fun t => 1 / (𝓕.μ₀ * 𝓕.ε₀) * ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x) thgd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ DifferentiableAt ℝ (fun t => 1 / 𝓕.ε₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i) t]d:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ (fderiv ℝ (fun t => 1 / (𝓕.μ₀ * 𝓕.ε₀) * ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x) t -
      fderiv ℝ (fun t => 1 / 𝓕.ε₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i) t)
    1 =
  1 / (𝓕.μ₀ * 𝓕.ε₀) * ∂ₜ (fun t => ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x) t -
    1 / 𝓕.ε₀ * ∂ₜ (fun x_1 => (LorentzCurrentDensity.currentDensity 𝓕.c J x_1 x).ofLp i) thfd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ DifferentiableAt ℝ (fun t => 1 / (𝓕.μ₀ * 𝓕.ε₀) * ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x) thgd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ DifferentiableAt ℝ (fun t => 1 / 𝓕.ε₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i) t
      simp only [one_div, mul_inv_rev, FunLike.coe_sub, Pi.sub_apply]d:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ (fderiv ℝ (fun t => 𝓕.ε₀⁻¹ * 𝓕.μ₀⁻¹ * ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x) t) 1 -
    (fderiv ℝ (fun t => 𝓕.ε₀⁻¹ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i) t) 1 =
  𝓕.ε₀⁻¹ * 𝓕.μ₀⁻¹ * ∂ₜ (fun t => ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x) t -
    𝓕.ε₀⁻¹ * ∂ₜ (fun x_1 => (LorentzCurrentDensity.currentDensity 𝓕.c J x_1 x).ofLp i) thfd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ DifferentiableAt ℝ (fun t => 1 / (𝓕.μ₀ * 𝓕.ε₀) * ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x) thgd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ DifferentiableAt ℝ (fun t => 1 / 𝓕.ε₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i) t
      rw [fderiv_const_mul (Differentiable.fun_sum fun j _ => hBt j).differentiableAtd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ ((𝓕.ε₀⁻¹ * 𝓕.μ₀⁻¹) •
        fderiv ℝ (fun y => ∑ i_1, Space.deriv i_1 (fun y_1 => magneticFieldMatrix 𝓕.c A y y_1 (i_1, i)) x) t)
      1 -
    (fderiv ℝ (fun t => 𝓕.ε₀⁻¹ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i) t) 1 =
  𝓕.ε₀⁻¹ * 𝓕.μ₀⁻¹ * ∂ₜ (fun t => ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x) t -
    𝓕.ε₀⁻¹ * ∂ₜ (fun x_1 => (LorentzCurrentDensity.currentDensity 𝓕.c J x_1 x).ofLp i) thfd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ DifferentiableAt ℝ (fun t => 1 / (𝓕.μ₀ * 𝓕.ε₀) * ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x) thgd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ DifferentiableAt ℝ (fun t => 1 / 𝓕.ε₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i) td:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ ((𝓕.ε₀⁻¹ * 𝓕.μ₀⁻¹) •
        fderiv ℝ (fun y => ∑ i_1, Space.deriv i_1 (fun y_1 => magneticFieldMatrix 𝓕.c A y y_1 (i_1, i)) x) t)
      1 -
    (fderiv ℝ (fun t => 𝓕.ε₀⁻¹ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i) t) 1 =
  𝓕.ε₀⁻¹ * 𝓕.μ₀⁻¹ * ∂ₜ (fun t => ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x) t -
    𝓕.ε₀⁻¹ * ∂ₜ (fun x_1 => (LorentzCurrentDensity.currentDensity 𝓕.c J x_1 x).ofLp i) thfd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ DifferentiableAt ℝ (fun t => 1 / (𝓕.μ₀ * 𝓕.ε₀) * ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x) thgd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ DifferentiableAt ℝ (fun t => 1 / 𝓕.ε₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i) t]d:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ ((𝓕.ε₀⁻¹ * 𝓕.μ₀⁻¹) •
        fderiv ℝ (fun y => ∑ i_1, Space.deriv i_1 (fun y_1 => magneticFieldMatrix 𝓕.c A y y_1 (i_1, i)) x) t)
      1 -
    (fderiv ℝ (fun t => 𝓕.ε₀⁻¹ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i) t) 1 =
  𝓕.ε₀⁻¹ * 𝓕.μ₀⁻¹ * ∂ₜ (fun t => ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x) t -
    𝓕.ε₀⁻¹ * ∂ₜ (fun x_1 => (LorentzCurrentDensity.currentDensity 𝓕.c J x_1 x).ofLp i) thfd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ DifferentiableAt ℝ (fun t => 1 / (𝓕.μ₀ * 𝓕.ε₀) * ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x) thgd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ DifferentiableAt ℝ (fun t => 1 / 𝓕.ε₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i) t
      rw [fderiv_const_mul hJt.differentiableAtd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ ((𝓕.ε₀⁻¹ * 𝓕.μ₀⁻¹) •
        fderiv ℝ (fun y => ∑ i_1, Space.deriv i_1 (fun y_1 => magneticFieldMatrix 𝓕.c A y y_1 (i_1, i)) x) t)
      1 -
    (𝓕.ε₀⁻¹ • fderiv ℝ (fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i) t) 1 =
  𝓕.ε₀⁻¹ * 𝓕.μ₀⁻¹ * ∂ₜ (fun t => ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x) t -
    𝓕.ε₀⁻¹ * ∂ₜ (fun x_1 => (LorentzCurrentDensity.currentDensity 𝓕.c J x_1 x).ofLp i) thfd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ DifferentiableAt ℝ (fun t => 1 / (𝓕.μ₀ * 𝓕.ε₀) * ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x) thgd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ DifferentiableAt ℝ (fun t => 1 / 𝓕.ε₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i) td:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ ((𝓕.ε₀⁻¹ * 𝓕.μ₀⁻¹) •
        fderiv ℝ (fun y => ∑ i_1, Space.deriv i_1 (fun y_1 => magneticFieldMatrix 𝓕.c A y y_1 (i_1, i)) x) t)
      1 -
    (𝓕.ε₀⁻¹ • fderiv ℝ (fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i) t) 1 =
  𝓕.ε₀⁻¹ * 𝓕.μ₀⁻¹ * ∂ₜ (fun t => ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x) t -
    𝓕.ε₀⁻¹ * ∂ₜ (fun x_1 => (LorentzCurrentDensity.currentDensity 𝓕.c J x_1 x).ofLp i) thfd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ DifferentiableAt ℝ (fun t => 1 / (𝓕.μ₀ * 𝓕.ε₀) * ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x) thgd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ DifferentiableAt ℝ (fun t => 1 / 𝓕.ε₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i) t]d:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ ((𝓕.ε₀⁻¹ * 𝓕.μ₀⁻¹) •
        fderiv ℝ (fun y => ∑ i_1, Space.deriv i_1 (fun y_1 => magneticFieldMatrix 𝓕.c A y y_1 (i_1, i)) x) t)
      1 -
    (𝓕.ε₀⁻¹ • fderiv ℝ (fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i) t) 1 =
  𝓕.ε₀⁻¹ * 𝓕.μ₀⁻¹ * ∂ₜ (fun t => ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x) t -
    𝓕.ε₀⁻¹ * ∂ₜ (fun x_1 => (LorentzCurrentDensity.currentDensity 𝓕.c J x_1 x).ofLp i) thfd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ DifferentiableAt ℝ (fun t => 1 / (𝓕.μ₀ * 𝓕.ε₀) * ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x) thgd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ DifferentiableAt ℝ (fun t => 1 / 𝓕.ε₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i) t
      simp [Time.deriv_eq]hfd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ DifferentiableAt ℝ (fun t => 1 / (𝓕.μ₀ * 𝓕.ε₀) * ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x) thgd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ DifferentiableAt ℝ (fun t => 1 / 𝓕.ε₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i) t
      ·hfd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ DifferentiableAt ℝ (fun t => 1 / (𝓕.μ₀ * 𝓕.ε₀) * ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x) t exact ((Differentiable.fun_sum fun j _ => hBt j).const_mul _).differentiableAtAll goals completed! 🐙
      ·hgd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ DifferentiableAt ℝ (fun t => 1 / 𝓕.ε₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp i) t exact hJt.differentiableAt.const_mul _All goals completed! 🐙
    _ = 1 / (𝓕.μ₀ * 𝓕.ε₀) * ((∑ j, ∂ₜ (fun t => ∂[j] (A.magneticFieldMatrix 𝓕.c t · (j, i)) x)) t) -
      1 / 𝓕.ε₀ * (∂ₜ (J.currentDensity 𝓕.c · x i) t) := byd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ 1 / (𝓕.μ₀ * 𝓕.ε₀) * ∂ₜ (fun t => ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x) t -
    1 / 𝓕.ε₀ * ∂ₜ (fun x_1 => (LorentzCurrentDensity.currentDensity 𝓕.c J x_1 x).ofLp i) t =
  1 / (𝓕.μ₀ * 𝓕.ε₀) * (∑ j, ∂ₜ fun t => Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x) t -
    1 / 𝓕.ε₀ * ∂ₜ (fun x_1 => (LorentzCurrentDensity.currentDensity 𝓕.c J x_1 x).ofLp i) t
      congre_a.e_ad:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ ∂ₜ (fun t => ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x) t =
  (∑ j, ∂ₜ fun t => Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x) t
      rw [Time.deriv_eqe_a.e_ad:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ (fderiv ℝ (fun t => ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x) t) 1 =
  (∑ j, ∂ₜ fun t => Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x) te_a.e_ad:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ (fderiv ℝ (fun t => ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x) t) 1 =
  (∑ j, ∂ₜ fun t => Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x) t]e_a.e_ad:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ (fderiv ℝ (fun t => ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x) t) 1 =
  (∑ j, ∂ₜ fun t => Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x) t
      rw [fderiv_fun_sum fun i _ => (hBt i).differentiableAte_a.e_ad:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ (∑ i_1, fderiv ℝ (fun s => Space.deriv i_1 (fun y => magneticFieldMatrix 𝓕.c A s y (i_1, i)) x) t) 1 =
  (∑ j, ∂ₜ fun t => Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x) te_a.e_ad:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ (∑ i_1, fderiv ℝ (fun s => Space.deriv i_1 (fun y => magneticFieldMatrix 𝓕.c A s y (i_1, i)) x) t) 1 =
  (∑ j, ∂ₜ fun t => Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x) t]e_a.e_ad:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ (∑ i_1, fderiv ℝ (fun s => Space.deriv i_1 (fun y => magneticFieldMatrix 𝓕.c A s y (i_1, i)) x) t) 1 =
  (∑ j, ∂ₜ fun t => Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x) t
      simp only [FunLike.coe_sum, Finset.sum_apply]e_a.e_ad:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ ∑ c, (fderiv ℝ (fun s => Space.deriv c (fun y => magneticFieldMatrix 𝓕.c A s y (c, i)) x) t) 1 =
  ∑ c, ∂ₜ (fun t => Space.deriv c (fun x => magneticFieldMatrix 𝓕.c A t x (c, i)) x) t
      rflAll goals completed! 🐙
    _ = 1 / (𝓕.μ₀ * 𝓕.ε₀) * (∑ j, ∂[j] (fun x => ∂ₜ (A.magneticFieldMatrix 𝓕.c · x (j, i)) t)) x -
        1 / 𝓕.ε₀ * ∂ₜ (J.currentDensity 𝓕.c · x i) t := byd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ 1 / (𝓕.μ₀ * 𝓕.ε₀) * (∑ j, ∂ₜ fun t => Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x) t -
    1 / 𝓕.ε₀ * ∂ₜ (fun x_1 => (LorentzCurrentDensity.currentDensity 𝓕.c J x_1 x).ofLp i) t =
  1 / (𝓕.μ₀ * 𝓕.ε₀) * (∑ j, Space.deriv j fun x => ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (j, i)) t) x -
    1 / 𝓕.ε₀ * ∂ₜ (fun x_1 => (LorentzCurrentDensity.currentDensity 𝓕.c J x_1 x).ofLp i) t
      congre_a.e_ad:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ (∑ j, ∂ₜ fun t => Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x) t =
  (∑ j, Space.deriv j fun x => ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (j, i)) t) x
      simp only [Finset.sum_apply]e_a.e_ad:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ ∑ c, ∂ₜ (fun t => Space.deriv c (fun x => magneticFieldMatrix 𝓕.c A t x (c, i)) x) t =
  ∑ c, Space.deriv c (fun x => ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (c, i)) t) x
      congre_a.e_a.e_fd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ (fun c => ∂ₜ (fun t => Space.deriv c (fun x => magneticFieldMatrix 𝓕.c A t x (c, i)) x) t) = fun c =>
  Space.deriv c (fun x => ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (c, i)) t) x
      funext ke_a.e_a.e_fd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp ik:Fin d⊢ ∂ₜ (fun t => Space.deriv k (fun x => magneticFieldMatrix 𝓕.c A t x (k, i)) x) t =
  Space.deriv k (fun x => ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (k, i)) t) x
      rw [Space.time_deriv_comm_space_derive_a.e_a.e_fd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp ik:Fin d⊢ Space.deriv k (fun x' => ∂ₜ (fun t' => magneticFieldMatrix 𝓕.c A t' x' (k, i)) t) x =
  Space.deriv k (fun x => ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (k, i)) t) xe_a.e_a.e_f.hfd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp ik:Fin d⊢ ContDiff ℝ 2 ↿fun t x => magneticFieldMatrix 𝓕.c A t x (k, i)e_a.e_a.e_f.hfd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp ik:Fin d⊢ ContDiff ℝ 2 ↿fun t x => magneticFieldMatrix 𝓕.c A t x (k, i)]e_a.e_a.e_f.hfd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp ik:Fin d⊢ ContDiff ℝ 2 ↿fun t x => magneticFieldMatrix 𝓕.c A t x (k, i)
      apply magneticFieldMatrix_contDiffe_a.e_a.e_f.hfd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp ik:Fin d⊢ ContDiff ℝ (2 + 1) A.val
      apply hA.of_le (right_eq_inf.mp rfl)All goals completed! 🐙
    _ = 1 / (𝓕.μ₀ * 𝓕.ε₀) *(∑ j, ∂[j] (fun x => ∂[j] (A.electricField 𝓕.c t · i) x -
        ∂[i] (A.electricField 𝓕.c t · j) x)) x -
        1 / 𝓕.ε₀ * ∂ₜ (J.currentDensity 𝓕.c · x i) t := byd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ 1 / (𝓕.μ₀ * 𝓕.ε₀) * (∑ j, Space.deriv j fun x => ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (j, i)) t) x -
    1 / 𝓕.ε₀ * ∂ₜ (fun x_1 => (LorentzCurrentDensity.currentDensity 𝓕.c J x_1 x).ofLp i) t =
  1 / (𝓕.μ₀ * 𝓕.ε₀) *
      (∑ j,
          Space.deriv j fun x =>
            Space.deriv j (fun x => (electricField 𝓕.c A t x).ofLp i) x -
              Space.deriv i (fun x => (electricField 𝓕.c A t x).ofLp j) x)
        x -
    1 / 𝓕.ε₀ * ∂ₜ (fun x_1 => (LorentzCurrentDensity.currentDensity 𝓕.c J x_1 x).ofLp i) t
        congre_a.e_ad:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ (∑ j, Space.deriv j fun x => ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (j, i)) t) x =
  (∑ j,
      Space.deriv j fun x =>
        Space.deriv j (fun x => (electricField 𝓕.c A t x).ofLp i) x -
          Space.deriv i (fun x => (electricField 𝓕.c A t x).ofLp j) x)
    x
        simp only [Finset.sum_apply]e_a.e_ad:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ ∑ c, Space.deriv c (fun x => ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (c, i)) t) x =
  ∑ c,
    Space.deriv c
      (fun x =>
        Space.deriv c (fun x => (electricField 𝓕.c A t x).ofLp i) x -
          Space.deriv i (fun x => (electricField 𝓕.c A t x).ofLp c) x)
      x
        congre_a.e_a.e_fd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ (fun c => Space.deriv c (fun x => ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (c, i)) t) x) = fun c =>
  Space.deriv c
    (fun x =>
      Space.deriv c (fun x => (electricField 𝓕.c A t x).ofLp i) x -
        Space.deriv i (fun x => (electricField 𝓕.c A t x).ofLp c) x)
    x
        funext ke_a.e_a.e_fd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp ik:Fin d⊢ Space.deriv k (fun x => ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (k, i)) t) x =
  Space.deriv k
    (fun x =>
      Space.deriv k (fun x => (electricField 𝓕.c A t x).ofLp i) x -
        Space.deriv i (fun x => (electricField 𝓕.c A t x).ofLp k) x)
    x
        congre_a.e_a.e_f.e_fd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp ik:Fin d⊢ (fun x => ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (k, i)) t) = fun x =>
  Space.deriv k (fun x => (electricField 𝓕.c A t x).ofLp i) x -
    Space.deriv i (fun x => (electricField 𝓕.c A t x).ofLp k) x
        funext xe_a.e_a.e_f.e_fd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex✝:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp ik:Fin dx:Space d⊢ ∂ₜ (fun x_1 => magneticFieldMatrix 𝓕.c A x_1 x (k, i)) t =
  Space.deriv k (fun x => (electricField 𝓕.c A t x).ofLp i) x -
    Space.deriv i (fun x => (electricField 𝓕.c A t x).ofLp k) x
        rw [time_deriv_magneticFieldMatrix _ (hA.of_le (ENat.LEInfty.out))e_a.e_a.e_f.e_fd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex✝:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp ik:Fin dx:Space d⊢ Space.deriv k (fun x => (electricField 𝓕.c A t x).ofLp i) x -
    Space.deriv i (fun x => (electricField 𝓕.c A t x).ofLp k) x =
  Space.deriv k (fun x => (electricField 𝓕.c A t x).ofLp i) x -
    Space.deriv i (fun x => (electricField 𝓕.c A t x).ofLp k) xAll goals completed! 🐙]All goals completed! 🐙
    _ = (1 / (𝓕.μ₀ * 𝓕.ε₀) * ∑ j, (∂[j] (fun x => ∂[j] (A.electricField 𝓕.c t · i) x) x -
          ∂[j] (fun x => ∂[i] (A.electricField 𝓕.c t · j) x) x)) -
          1 / 𝓕.ε₀ * ∂ₜ (J.currentDensity 𝓕.c · x i) t := byd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ 1 / (𝓕.μ₀ * 𝓕.ε₀) *
      (∑ j,
          Space.deriv j fun x =>
            Space.deriv j (fun x => (electricField 𝓕.c A t x).ofLp i) x -
              Space.deriv i (fun x => (electricField 𝓕.c A t x).ofLp j) x)
        x -
    1 / 𝓕.ε₀ * ∂ₜ (fun x_1 => (LorentzCurrentDensity.currentDensity 𝓕.c J x_1 x).ofLp i) t =
  1 / (𝓕.μ₀ * 𝓕.ε₀) *
      ∑ j,
        (Space.deriv j (fun x => Space.deriv j (fun x => (electricField 𝓕.c A t x).ofLp i) x) x -
          Space.deriv j (fun x => Space.deriv i (fun x => (electricField 𝓕.c A t x).ofLp j) x) x) -
    1 / 𝓕.ε₀ * ∂ₜ (fun x_1 => (LorentzCurrentDensity.currentDensity 𝓕.c J x_1 x).ofLp i) t
        congre_a.e_ad:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ (∑ j,
      Space.deriv j fun x =>
        Space.deriv j (fun x => (electricField 𝓕.c A t x).ofLp i) x -
          Space.deriv i (fun x => (electricField 𝓕.c A t x).ofLp j) x)
    x =
  ∑ j,
    (Space.deriv j (fun x => Space.deriv j (fun x => (electricField 𝓕.c A t x).ofLp i) x) x -
      Space.deriv j (fun x => Space.deriv i (fun x => (electricField 𝓕.c A t x).ofLp j) x) x)
        simp only [Finset.sum_apply]e_a.e_ad:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ ∑ c,
    Space.deriv c
      (fun x =>
        Space.deriv c (fun x => (electricField 𝓕.c A t x).ofLp i) x -
          Space.deriv i (fun x => (electricField 𝓕.c A t x).ofLp c) x)
      x =
  ∑ x_1,
    (Space.deriv x_1 (fun x => Space.deriv x_1 (fun x => (electricField 𝓕.c A t x).ofLp i) x) x -
      Space.deriv x_1 (fun x => Space.deriv i (fun x => (electricField 𝓕.c A t x).ofLp x_1) x) x)
        congre_a.e_a.e_fd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ (fun c =>
    Space.deriv c
      (fun x =>
        Space.deriv c (fun x => (electricField 𝓕.c A t x).ofLp i) x -
          Space.deriv i (fun x => (electricField 𝓕.c A t x).ofLp c) x)
      x) =
  fun x_1 =>
  Space.deriv x_1 (fun x => Space.deriv x_1 (fun x => (electricField 𝓕.c A t x).ofLp i) x) x -
    Space.deriv x_1 (fun x => Space.deriv i (fun x => (electricField 𝓕.c A t x).ofLp x_1) x) x
        funext je_a.e_a.e_fd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp ij:Fin d⊢ Space.deriv j
    (fun x =>
      Space.deriv j (fun x => (electricField 𝓕.c A t x).ofLp i) x -
        Space.deriv i (fun x => (electricField 𝓕.c A t x).ofLp j) x)
    x =
  Space.deriv j (fun x => Space.deriv j (fun x => (electricField 𝓕.c A t x).ofLp i) x) x -
    Space.deriv j (fun x => Space.deriv i (fun x => (electricField 𝓕.c A t x).ofLp j) x) x
        rw [Space.deriv_eq_fderiv_basise_a.e_a.e_fd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp ij:Fin d⊢ (fderiv ℝ
      (fun x =>
        Space.deriv j (fun x => (electricField 𝓕.c A t x).ofLp i) x -
          Space.deriv i (fun x => (electricField 𝓕.c A t x).ofLp j) x)
      x)
    (Space.basis j) =
  Space.deriv j (fun x => Space.deriv j (fun x => (electricField 𝓕.c A t x).ofLp i) x) x -
    Space.deriv j (fun x => Space.deriv i (fun x => (electricField 𝓕.c A t x).ofLp j) x) xe_a.e_a.e_fd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp ij:Fin d⊢ (fderiv ℝ
      (fun x =>
        Space.deriv j (fun x => (electricField 𝓕.c A t x).ofLp i) x -
          Space.deriv i (fun x => (electricField 𝓕.c A t x).ofLp j) x)
      x)
    (Space.basis j) =
  Space.deriv j (fun x => Space.deriv j (fun x => (electricField 𝓕.c A t x).ofLp i) x) x -
    Space.deriv j (fun x => Space.deriv i (fun x => (electricField 𝓕.c A t x).ofLp j) x) x]e_a.e_a.e_fd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp ij:Fin d⊢ (fderiv ℝ
      (fun x =>
        Space.deriv j (fun x => (electricField 𝓕.c A t x).ofLp i) x -
          Space.deriv i (fun x => (electricField 𝓕.c A t x).ofLp j) x)
      x)
    (Space.basis j) =
  Space.deriv j (fun x => Space.deriv j (fun x => (electricField 𝓕.c A t x).ofLp i) x) x -
    Space.deriv j (fun x => Space.deriv i (fun x => (electricField 𝓕.c A t x).ofLp j) x) x
        rw [fderiv_fun_sub (hEd _ _).differentiableAt (hEd _ _).differentiableAte_a.e_a.e_fd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp ij:Fin d⊢ (fderiv ℝ (Space.deriv j fun y => (electricField 𝓕.c A t y).ofLp i) x -
      fderiv ℝ (Space.deriv i fun y => (electricField 𝓕.c A t y).ofLp j) x)
    (Space.basis j) =
  Space.deriv j (fun x => Space.deriv j (fun x => (electricField 𝓕.c A t x).ofLp i) x) x -
    Space.deriv j (fun x => Space.deriv i (fun x => (electricField 𝓕.c A t x).ofLp j) x) xe_a.e_a.e_fd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp ij:Fin d⊢ (fderiv ℝ (Space.deriv j fun y => (electricField 𝓕.c A t y).ofLp i) x -
      fderiv ℝ (Space.deriv i fun y => (electricField 𝓕.c A t y).ofLp j) x)
    (Space.basis j) =
  Space.deriv j (fun x => Space.deriv j (fun x => (electricField 𝓕.c A t x).ofLp i) x) x -
    Space.deriv j (fun x => Space.deriv i (fun x => (electricField 𝓕.c A t x).ofLp j) x) x]e_a.e_a.e_fd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp ij:Fin d⊢ (fderiv ℝ (Space.deriv j fun y => (electricField 𝓕.c A t y).ofLp i) x -
      fderiv ℝ (Space.deriv i fun y => (electricField 𝓕.c A t y).ofLp j) x)
    (Space.basis j) =
  Space.deriv j (fun x => Space.deriv j (fun x => (electricField 𝓕.c A t x).ofLp i) x) x -
    Space.deriv j (fun x => Space.deriv i (fun x => (electricField 𝓕.c A t x).ofLp j) x) x
        simp [← Space.deriv_eq_fderiv_basis]All goals completed! 🐙
    _ = 1 / (𝓕.μ₀ * 𝓕.ε₀) * ∑ j, (∂[j] (fun x => ∂[j] (A.electricField 𝓕.c t · i) x) x) -
          1 / (𝓕.μ₀ * 𝓕.ε₀) * ∑ j, (∂[j] (fun x => ∂[i] (A.electricField 𝓕.c t · j) x) x) -
          1 / 𝓕.ε₀ * ∂ₜ (J.currentDensity 𝓕.c · x i) t := byd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ 1 / (𝓕.μ₀ * 𝓕.ε₀) *
      ∑ j,
        (Space.deriv j (fun x => Space.deriv j (fun x => (electricField 𝓕.c A t x).ofLp i) x) x -
          Space.deriv j (fun x => Space.deriv i (fun x => (electricField 𝓕.c A t x).ofLp j) x) x) -
    1 / 𝓕.ε₀ * ∂ₜ (fun x_1 => (LorentzCurrentDensity.currentDensity 𝓕.c J x_1 x).ofLp i) t =
  1 / (𝓕.μ₀ * 𝓕.ε₀) * ∑ j, Space.deriv j (fun x => Space.deriv j (fun x => (electricField 𝓕.c A t x).ofLp i) x) x -
      1 / (𝓕.μ₀ * 𝓕.ε₀) * ∑ j, Space.deriv j (fun x => Space.deriv i (fun x => (electricField 𝓕.c A t x).ofLp j) x) x -
    1 / 𝓕.ε₀ * ∂ₜ (fun x_1 => (LorentzCurrentDensity.currentDensity 𝓕.c J x_1 x).ofLp i) t simp [mul_sub]All goals completed! 🐙
    _ = 1 / (𝓕.μ₀ * 𝓕.ε₀) * ∑ j, (∂[j] (fun x => ∂[j] (A.electricField 𝓕.c t · i) x) x) -
        1 / (𝓕.μ₀ * 𝓕.ε₀) * ∑ j, (∂[i] (fun x => ∂[j] (A.electricField 𝓕.c t · j) x) x) -
        1 / 𝓕.ε₀ * ∂ₜ (J.currentDensity 𝓕.c · x i) t := byd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ 1 / (𝓕.μ₀ * 𝓕.ε₀) * ∑ j, Space.deriv j (fun x => Space.deriv j (fun x => (electricField 𝓕.c A t x).ofLp i) x) x -
      1 / (𝓕.μ₀ * 𝓕.ε₀) * ∑ j, Space.deriv j (fun x => Space.deriv i (fun x => (electricField 𝓕.c A t x).ofLp j) x) x -
    1 / 𝓕.ε₀ * ∂ₜ (fun x_1 => (LorentzCurrentDensity.currentDensity 𝓕.c J x_1 x).ofLp i) t =
  1 / (𝓕.μ₀ * 𝓕.ε₀) * ∑ j, Space.deriv j (fun x => Space.deriv j (fun x => (electricField 𝓕.c A t x).ofLp i) x) x -
      1 / (𝓕.μ₀ * 𝓕.ε₀) * ∑ j, Space.deriv i (fun x => Space.deriv j (fun x => (electricField 𝓕.c A t x).ofLp j) x) x -
    1 / 𝓕.ε₀ * ∂ₜ (fun x_1 => (LorentzCurrentDensity.currentDensity 𝓕.c J x_1 x).ofLp i) t
        congre_a.e_a.e_a.e_fd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ (fun j => Space.deriv j (fun x => Space.deriv i (fun x => (electricField 𝓕.c A t x).ofLp j) x) x) = fun j =>
  Space.deriv i (fun x => Space.deriv j (fun x => (electricField 𝓕.c A t x).ofLp j) x) x
        funext je_a.e_a.e_a.e_fd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp ij:Fin d⊢ Space.deriv j (fun x => Space.deriv i (fun x => (electricField 𝓕.c A t x).ofLp j) x) x =
  Space.deriv i (fun x => Space.deriv j (fun x => (electricField 𝓕.c A t x).ofLp j) x) x
        rw [Space.deriv_commute _ (hEs _),e_a.e_a.e_a.e_fd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp ij:Fin d⊢ Space.deriv i (Space.deriv j fun y => (electricField 𝓕.c A t y).ofLp j) x =
  Space.deriv i (fun x => Space.deriv j (fun x => (electricField 𝓕.c A t x).ofLp j) x) xAll goals completed! 🐙 Space.deriv_eq_fderiv_basise_a.e_a.e_a.e_fd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp ij:Fin d⊢ (fderiv ℝ (Space.deriv j fun y => (electricField 𝓕.c A t y).ofLp j) x) (Space.basis i) =
  (fderiv ℝ (Space.deriv j fun y => (electricField 𝓕.c A t y).ofLp j) x) (Space.basis i)All goals completed! 🐙]All goals completed! 🐙
      _ = 1 / (𝓕.μ₀ * 𝓕.ε₀) * ∑ j, (∂[j] (fun x => ∂[j] (A.electricField 𝓕.c t · i) x) x) -
          1 / (𝓕.μ₀ * 𝓕.ε₀) * (∂[i] (fun x => ∑ j, ∂[j] (A.electricField 𝓕.c t · j) x) x) -
          1 / 𝓕.ε₀ * ∂ₜ (J.currentDensity 𝓕.c · x i) t := byd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ 1 / (𝓕.μ₀ * 𝓕.ε₀) * ∑ j, Space.deriv j (fun x => Space.deriv j (fun x => (electricField 𝓕.c A t x).ofLp i) x) x -
      1 / (𝓕.μ₀ * 𝓕.ε₀) * ∑ j, Space.deriv i (fun x => Space.deriv j (fun x => (electricField 𝓕.c A t x).ofLp j) x) x -
    1 / 𝓕.ε₀ * ∂ₜ (fun x_1 => (LorentzCurrentDensity.currentDensity 𝓕.c J x_1 x).ofLp i) t =
  1 / (𝓕.μ₀ * 𝓕.ε₀) * ∑ j, Space.deriv j (fun x => Space.deriv j (fun x => (electricField 𝓕.c A t x).ofLp i) x) x -
      1 / (𝓕.μ₀ * 𝓕.ε₀) * Space.deriv i (fun x => ∑ j, Space.deriv j (fun x => (electricField 𝓕.c A t x).ofLp j) x) x -
    1 / 𝓕.ε₀ * ∂ₜ (fun x_1 => (LorentzCurrentDensity.currentDensity 𝓕.c J x_1 x).ofLp i) t
        congre_a.e_a.e_ad:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ ∑ j, Space.deriv i (fun x => Space.deriv j (fun x => (electricField 𝓕.c A t x).ofLp j) x) x =
  Space.deriv i (fun x => ∑ j, Space.deriv j (fun x => (electricField 𝓕.c A t x).ofLp j) x) x
        rw [Space.deriv_eq_fderiv_basise_a.e_a.e_ad:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ ∑ j, Space.deriv i (fun x => Space.deriv j (fun x => (electricField 𝓕.c A t x).ofLp j) x) x =
  (fderiv ℝ (fun x => ∑ j, Space.deriv j (fun x => (electricField 𝓕.c A t x).ofLp j) x) x) (Space.basis i)e_a.e_a.e_ad:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ ∑ j, Space.deriv i (fun x => Space.deriv j (fun x => (electricField 𝓕.c A t x).ofLp j) x) x =
  (fderiv ℝ (fun x => ∑ j, Space.deriv j (fun x => (electricField 𝓕.c A t x).ofLp j) x) x) (Space.basis i)]e_a.e_a.e_ad:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ ∑ j, Space.deriv i (fun x => Space.deriv j (fun x => (electricField 𝓕.c A t x).ofLp j) x) x =
  (fderiv ℝ (fun x => ∑ j, Space.deriv j (fun x => (electricField 𝓕.c A t x).ofLp j) x) x) (Space.basis i)
        rw [fderiv_fun_sume_a.e_a.e_ad:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ ∑ j, Space.deriv i (fun x => Space.deriv j (fun x => (electricField 𝓕.c A t x).ofLp j) x) x =
  (∑ i, fderiv ℝ (Space.deriv i fun x => (electricField 𝓕.c A t x).ofLp i) x) (Space.basis i)e_a.e_a.e_ad:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ ∀ i ∈ Finset.univ, DifferentiableAt ℝ (Space.deriv i fun x => (electricField 𝓕.c A t x).ofLp i) xe_a.e_a.e_ad:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ ∑ j, Space.deriv i (fun x => Space.deriv j (fun x => (electricField 𝓕.c A t x).ofLp j) x) x =
  (∑ i, fderiv ℝ (Space.deriv i fun x => (electricField 𝓕.c A t x).ofLp i) x) (Space.basis i)e_a.e_a.e_ad:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ ∀ i ∈ Finset.univ, DifferentiableAt ℝ (Space.deriv i fun x => (electricField 𝓕.c A t x).ofLp i) x]e_a.e_a.e_ad:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ ∑ j, Space.deriv i (fun x => Space.deriv j (fun x => (electricField 𝓕.c A t x).ofLp j) x) x =
  (∑ i, fderiv ℝ (Space.deriv i fun x => (electricField 𝓕.c A t x).ofLp i) x) (Space.basis i)e_a.e_a.e_ad:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ ∀ i ∈ Finset.univ, DifferentiableAt ℝ (Space.deriv i fun x => (electricField 𝓕.c A t x).ofLp i) x
        simp [← Space.deriv_eq_fderiv_basis]e_a.e_a.e_ad:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ ∀ i ∈ Finset.univ, DifferentiableAt ℝ (Space.deriv i fun x => (electricField 𝓕.c A t x).ofLp i) x
        intro j _e_a.e_a.e_ad:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp ij:Fin da✝:j ∈ Finset.univ⊢ DifferentiableAt ℝ (Space.deriv j fun x => (electricField 𝓕.c A t x).ofLp j) x
        exact (hEd j j).differentiableAtAll goals completed! 🐙
      _ = 1 / (𝓕.μ₀ * 𝓕.ε₀) * ∑ j, (∂[j] (fun x => ∂[j] (A.electricField 𝓕.c t · i) x) x) -
          1 / (𝓕.μ₀ * 𝓕.ε₀) * (∂[i] (fun x => (∇ ⬝ (A.electricField 𝓕.c t)) x) x) -
          1 / 𝓕.ε₀ * ∂ₜ (J.currentDensity 𝓕.c · x i) t := byd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ 1 / (𝓕.μ₀ * 𝓕.ε₀) * ∑ j, Space.deriv j (fun x => Space.deriv j (fun x => (electricField 𝓕.c A t x).ofLp i) x) x -
      1 / (𝓕.μ₀ * 𝓕.ε₀) * Space.deriv i (fun x => ∑ j, Space.deriv j (fun x => (electricField 𝓕.c A t x).ofLp j) x) x -
    1 / 𝓕.ε₀ * ∂ₜ (fun x_1 => (LorentzCurrentDensity.currentDensity 𝓕.c J x_1 x).ofLp i) t =
  1 / (𝓕.μ₀ * 𝓕.ε₀) * ∑ j, Space.deriv j (fun x => Space.deriv j (fun x => (electricField 𝓕.c A t x).ofLp i) x) x -
      1 / (𝓕.μ₀ * 𝓕.ε₀) * Space.deriv i (fun x => Space.div (electricField 𝓕.c A t) x) x -
    1 / 𝓕.ε₀ * ∂ₜ (fun x_1 => (LorentzCurrentDensity.currentDensity 𝓕.c J x_1 x).ofLp i) t
        rflAll goals completed! 🐙
      _ = 1 / (𝓕.μ₀ * 𝓕.ε₀) * ∑ j, (∂[j] (∂[j] (A.electricField 𝓕.c t · i)) x) -
          1 / (𝓕.μ₀ * 𝓕.ε₀ ^ 2) * ∂[i] (J.chargeDensity 𝓕.c t ·) x -
          1 / 𝓕.ε₀ * ∂ₜ (J.currentDensity 𝓕.c · x i) t := byd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ 1 / (𝓕.μ₀ * 𝓕.ε₀) * ∑ j, Space.deriv j (fun x => Space.deriv j (fun x => (electricField 𝓕.c A t x).ofLp i) x) x -
      1 / (𝓕.μ₀ * 𝓕.ε₀) * Space.deriv i (fun x => Space.div (electricField 𝓕.c A t) x) x -
    1 / 𝓕.ε₀ * ∂ₜ (fun x_1 => (LorentzCurrentDensity.currentDensity 𝓕.c J x_1 x).ofLp i) t =
  1 / (𝓕.μ₀ * 𝓕.ε₀) * ∑ j, Space.deriv j (Space.deriv j fun x => (electricField 𝓕.c A t x).ofLp i) x -
      1 / (𝓕.μ₀ * 𝓕.ε₀ ^ 2) * Space.deriv i (fun x => LorentzCurrentDensity.chargeDensity 𝓕.c J t x) x -
    1 / 𝓕.ε₀ * ∂ₜ (fun x_1 => (LorentzCurrentDensity.currentDensity 𝓕.c J x_1 x).ofLp i) t
        congr 2e_a.e_ad:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ 1 / (𝓕.μ₀ * 𝓕.ε₀) * Space.deriv i (fun x => Space.div (electricField 𝓕.c A t) x) x =
  1 / (𝓕.μ₀ * 𝓕.ε₀ ^ 2) * Space.deriv i (fun x => LorentzCurrentDensity.chargeDensity 𝓕.c J t x) x
        rw [isExtrema_iff_gauss_ampere_magneticFieldMatrixe_a.e_ad:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:∀ (t : Time) (x : Space d),
  Space.div (electricField 𝓕.c A t) x = LorentzCurrentDensity.chargeDensity 𝓕.c J t x / 𝓕.ε₀ ∧
    ∀ (i : Fin d),
      𝓕.μ₀ * 𝓕.ε₀ * (∂ₜ (fun t => electricField 𝓕.c A t x) t).ofLp i =
        ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x -
          𝓕.μ₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp it:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ 1 / (𝓕.μ₀ * 𝓕.ε₀) * Space.deriv i (fun x => Space.div (electricField 𝓕.c A t) x) x =
  1 / (𝓕.μ₀ * 𝓕.ε₀ ^ 2) * Space.deriv i (fun x => LorentzCurrentDensity.chargeDensity 𝓕.c J t x) xe_a.e_a.hAd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ ContDiff ℝ ∞ A.vale_a.e_a.hJd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ ContDiff ℝ ∞ Je_a.e_ad:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:∀ (t : Time) (x : Space d),
  Space.div (electricField 𝓕.c A t) x = LorentzCurrentDensity.chargeDensity 𝓕.c J t x / 𝓕.ε₀ ∧
    ∀ (i : Fin d),
      𝓕.μ₀ * 𝓕.ε₀ * (∂ₜ (fun t => electricField 𝓕.c A t x) t).ofLp i =
        ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x -
          𝓕.μ₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp it:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ 1 / (𝓕.μ₀ * 𝓕.ε₀) * Space.deriv i (fun x => Space.div (electricField 𝓕.c A t) x) x =
  1 / (𝓕.μ₀ * 𝓕.ε₀ ^ 2) * Space.deriv i (fun x => LorentzCurrentDensity.chargeDensity 𝓕.c J t x) xe_a.e_a.hAd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ ContDiff ℝ ∞ A.vale_a.e_a.hJd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ ContDiff ℝ ∞ J] at he_a.e_ad:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:∀ (t : Time) (x : Space d),
  Space.div (electricField 𝓕.c A t) x = LorentzCurrentDensity.chargeDensity 𝓕.c J t x / 𝓕.ε₀ ∧
    ∀ (i : Fin d),
      𝓕.μ₀ * 𝓕.ε₀ * (∂ₜ (fun t => electricField 𝓕.c A t x) t).ofLp i =
        ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x -
          𝓕.μ₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp it:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ 1 / (𝓕.μ₀ * 𝓕.ε₀) * Space.deriv i (fun x => Space.div (electricField 𝓕.c A t) x) x =
  1 / (𝓕.μ₀ * 𝓕.ε₀ ^ 2) * Space.deriv i (fun x => LorentzCurrentDensity.chargeDensity 𝓕.c J t x) xe_a.e_a.hAd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ ContDiff ℝ ∞ A.vale_a.e_a.hJd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ ContDiff ℝ ∞ J

        conv_lhs =>
          enter [2, 2, x]d:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:∀ (t : Time) (x : Space d),
  Space.div (electricField 𝓕.c A t) x = LorentzCurrentDensity.chargeDensity 𝓕.c J t x / 𝓕.ε₀ ∧
    ∀ (i : Fin d),
      𝓕.μ₀ * 𝓕.ε₀ * (∂ₜ (fun t => electricField 𝓕.c A t x) t).ofLp i =
        ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x -
          𝓕.μ₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp it:Timex✝:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp ix:Space d| Space.div (electricField 𝓕.c A t) x
          rw [(h t x).1]d:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:∀ (t : Time) (x : Space d),
  Space.div (electricField 𝓕.c A t) x = LorentzCurrentDensity.chargeDensity 𝓕.c J t x / 𝓕.ε₀ ∧
    ∀ (i : Fin d),
      𝓕.μ₀ * 𝓕.ε₀ * (∂ₜ (fun t => electricField 𝓕.c A t x) t).ofLp i =
        ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x -
          𝓕.μ₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp it:Timex✝:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp ix:Space d| LorentzCurrentDensity.chargeDensity 𝓕.c J t x / 𝓕.ε₀
        trans 1 / (𝓕.μ₀ * 𝓕.ε₀) * Space.deriv i
            (fun x => (1/ 𝓕.ε₀) * LorentzCurrentDensity.chargeDensity 𝓕.c J t x) xd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:∀ (t : Time) (x : Space d),
  Space.div (electricField 𝓕.c A t) x = LorentzCurrentDensity.chargeDensity 𝓕.c J t x / 𝓕.ε₀ ∧
    ∀ (i : Fin d),
      𝓕.μ₀ * 𝓕.ε₀ * (∂ₜ (fun t => electricField 𝓕.c A t x) t).ofLp i =
        ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x -
          𝓕.μ₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp it:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ 1 / (𝓕.μ₀ * 𝓕.ε₀) * Space.deriv i (fun x => LorentzCurrentDensity.chargeDensity 𝓕.c J t x / 𝓕.ε₀) x =
  1 / (𝓕.μ₀ * 𝓕.ε₀) * Space.deriv i (fun x => 1 / 𝓕.ε₀ * LorentzCurrentDensity.chargeDensity 𝓕.c J t x) xd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:∀ (t : Time) (x : Space d),
  Space.div (electricField 𝓕.c A t) x = LorentzCurrentDensity.chargeDensity 𝓕.c J t x / 𝓕.ε₀ ∧
    ∀ (i : Fin d),
      𝓕.μ₀ * 𝓕.ε₀ * (∂ₜ (fun t => electricField 𝓕.c A t x) t).ofLp i =
        ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x -
          𝓕.μ₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp it:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ 1 / (𝓕.μ₀ * 𝓕.ε₀) * Space.deriv i (fun x => 1 / 𝓕.ε₀ * LorentzCurrentDensity.chargeDensity 𝓕.c J t x) x =
  1 / (𝓕.μ₀ * 𝓕.ε₀ ^ 2) * Space.deriv i (fun x => LorentzCurrentDensity.chargeDensity 𝓕.c J t x) xe_a.e_a.hAd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ ContDiff ℝ ∞ A.vale_a.e_a.hJd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ ContDiff ℝ ∞ J
        ·d:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:∀ (t : Time) (x : Space d),
  Space.div (electricField 𝓕.c A t) x = LorentzCurrentDensity.chargeDensity 𝓕.c J t x / 𝓕.ε₀ ∧
    ∀ (i : Fin d),
      𝓕.μ₀ * 𝓕.ε₀ * (∂ₜ (fun t => electricField 𝓕.c A t x) t).ofLp i =
        ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x -
          𝓕.μ₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp it:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ 1 / (𝓕.μ₀ * 𝓕.ε₀) * Space.deriv i (fun x => LorentzCurrentDensity.chargeDensity 𝓕.c J t x / 𝓕.ε₀) x =
  1 / (𝓕.μ₀ * 𝓕.ε₀) * Space.deriv i (fun x => 1 / 𝓕.ε₀ * LorentzCurrentDensity.chargeDensity 𝓕.c J t x) x congre_a.e_fd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:∀ (t : Time) (x : Space d),
  Space.div (electricField 𝓕.c A t) x = LorentzCurrentDensity.chargeDensity 𝓕.c J t x / 𝓕.ε₀ ∧
    ∀ (i : Fin d),
      𝓕.μ₀ * 𝓕.ε₀ * (∂ₜ (fun t => electricField 𝓕.c A t x) t).ofLp i =
        ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x -
          𝓕.μ₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp it:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ (fun x => LorentzCurrentDensity.chargeDensity 𝓕.c J t x / 𝓕.ε₀) = fun x =>
  1 / 𝓕.ε₀ * LorentzCurrentDensity.chargeDensity 𝓕.c J t x
          funext xe_a.e_fd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:∀ (t : Time) (x : Space d),
  Space.div (electricField 𝓕.c A t) x = LorentzCurrentDensity.chargeDensity 𝓕.c J t x / 𝓕.ε₀ ∧
    ∀ (i : Fin d),
      𝓕.μ₀ * 𝓕.ε₀ * (∂ₜ (fun t => electricField 𝓕.c A t x) t).ofLp i =
        ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x -
          𝓕.μ₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp it:Timex✝:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp ix:Space d⊢ LorentzCurrentDensity.chargeDensity 𝓕.c J t x / 𝓕.ε₀ = 1 / 𝓕.ε₀ * LorentzCurrentDensity.chargeDensity 𝓕.c J t x
          ringAll goals completed! 🐙
        ·d:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:∀ (t : Time) (x : Space d),
  Space.div (electricField 𝓕.c A t) x = LorentzCurrentDensity.chargeDensity 𝓕.c J t x / 𝓕.ε₀ ∧
    ∀ (i : Fin d),
      𝓕.μ₀ * 𝓕.ε₀ * (∂ₜ (fun t => electricField 𝓕.c A t x) t).ofLp i =
        ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x -
          𝓕.μ₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp it:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ 1 / (𝓕.μ₀ * 𝓕.ε₀) * Space.deriv i (fun x => 1 / 𝓕.ε₀ * LorentzCurrentDensity.chargeDensity 𝓕.c J t x) x =
  1 / (𝓕.μ₀ * 𝓕.ε₀ ^ 2) * Space.deriv i (fun x => LorentzCurrentDensity.chargeDensity 𝓕.c J t x) x rw [Space.deriv_eq_fderiv_basisd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:∀ (t : Time) (x : Space d),
  Space.div (electricField 𝓕.c A t) x = LorentzCurrentDensity.chargeDensity 𝓕.c J t x / 𝓕.ε₀ ∧
    ∀ (i : Fin d),
      𝓕.μ₀ * 𝓕.ε₀ * (∂ₜ (fun t => electricField 𝓕.c A t x) t).ofLp i =
        ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x -
          𝓕.μ₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp it:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ 1 / (𝓕.μ₀ * 𝓕.ε₀) * (fderiv ℝ (fun x => 1 / 𝓕.ε₀ * LorentzCurrentDensity.chargeDensity 𝓕.c J t x) x) (Space.basis i) =
  1 / (𝓕.μ₀ * 𝓕.ε₀ ^ 2) * Space.deriv i (fun x => LorentzCurrentDensity.chargeDensity 𝓕.c J t x) xd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:∀ (t : Time) (x : Space d),
  Space.div (electricField 𝓕.c A t) x = LorentzCurrentDensity.chargeDensity 𝓕.c J t x / 𝓕.ε₀ ∧
    ∀ (i : Fin d),
      𝓕.μ₀ * 𝓕.ε₀ * (∂ₜ (fun t => electricField 𝓕.c A t x) t).ofLp i =
        ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x -
          𝓕.μ₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp it:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ 1 / (𝓕.μ₀ * 𝓕.ε₀) * (fderiv ℝ (fun x => 1 / 𝓕.ε₀ * LorentzCurrentDensity.chargeDensity 𝓕.c J t x) x) (Space.basis i) =
  1 / (𝓕.μ₀ * 𝓕.ε₀ ^ 2) * Space.deriv i (fun x => LorentzCurrentDensity.chargeDensity 𝓕.c J t x) x]d:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:∀ (t : Time) (x : Space d),
  Space.div (electricField 𝓕.c A t) x = LorentzCurrentDensity.chargeDensity 𝓕.c J t x / 𝓕.ε₀ ∧
    ∀ (i : Fin d),
      𝓕.μ₀ * 𝓕.ε₀ * (∂ₜ (fun t => electricField 𝓕.c A t x) t).ofLp i =
        ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x -
          𝓕.μ₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp it:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ 1 / (𝓕.μ₀ * 𝓕.ε₀) * (fderiv ℝ (fun x => 1 / 𝓕.ε₀ * LorentzCurrentDensity.chargeDensity 𝓕.c J t x) x) (Space.basis i) =
  1 / (𝓕.μ₀ * 𝓕.ε₀ ^ 2) * Space.deriv i (fun x => LorentzCurrentDensity.chargeDensity 𝓕.c J t x) x
          rw [fderiv_const_muld:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:∀ (t : Time) (x : Space d),
  Space.div (electricField 𝓕.c A t) x = LorentzCurrentDensity.chargeDensity 𝓕.c J t x / 𝓕.ε₀ ∧
    ∀ (i : Fin d),
      𝓕.μ₀ * 𝓕.ε₀ * (∂ₜ (fun t => electricField 𝓕.c A t x) t).ofLp i =
        ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x -
          𝓕.μ₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp it:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ 1 / (𝓕.μ₀ * 𝓕.ε₀) * ((1 / 𝓕.ε₀) • fderiv ℝ (LorentzCurrentDensity.chargeDensity 𝓕.c J t) x) (Space.basis i) =
  1 / (𝓕.μ₀ * 𝓕.ε₀ ^ 2) * Space.deriv i (fun x => LorentzCurrentDensity.chargeDensity 𝓕.c J t x) xhad:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:∀ (t : Time) (x : Space d),
  Space.div (electricField 𝓕.c A t) x = LorentzCurrentDensity.chargeDensity 𝓕.c J t x / 𝓕.ε₀ ∧
    ∀ (i : Fin d),
      𝓕.μ₀ * 𝓕.ε₀ * (∂ₜ (fun t => electricField 𝓕.c A t x) t).ofLp i =
        ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x -
          𝓕.μ₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp it:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ DifferentiableAt ℝ (LorentzCurrentDensity.chargeDensity 𝓕.c J t) xd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:∀ (t : Time) (x : Space d),
  Space.div (electricField 𝓕.c A t) x = LorentzCurrentDensity.chargeDensity 𝓕.c J t x / 𝓕.ε₀ ∧
    ∀ (i : Fin d),
      𝓕.μ₀ * 𝓕.ε₀ * (∂ₜ (fun t => electricField 𝓕.c A t x) t).ofLp i =
        ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x -
          𝓕.μ₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp it:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ 1 / (𝓕.μ₀ * 𝓕.ε₀) * ((1 / 𝓕.ε₀) • fderiv ℝ (LorentzCurrentDensity.chargeDensity 𝓕.c J t) x) (Space.basis i) =
  1 / (𝓕.μ₀ * 𝓕.ε₀ ^ 2) * Space.deriv i (fun x => LorentzCurrentDensity.chargeDensity 𝓕.c J t x) xhad:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:∀ (t : Time) (x : Space d),
  Space.div (electricField 𝓕.c A t) x = LorentzCurrentDensity.chargeDensity 𝓕.c J t x / 𝓕.ε₀ ∧
    ∀ (i : Fin d),
      𝓕.μ₀ * 𝓕.ε₀ * (∂ₜ (fun t => electricField 𝓕.c A t x) t).ofLp i =
        ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x -
          𝓕.μ₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp it:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ DifferentiableAt ℝ (LorentzCurrentDensity.chargeDensity 𝓕.c J t) x]d:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:∀ (t : Time) (x : Space d),
  Space.div (electricField 𝓕.c A t) x = LorentzCurrentDensity.chargeDensity 𝓕.c J t x / 𝓕.ε₀ ∧
    ∀ (i : Fin d),
      𝓕.μ₀ * 𝓕.ε₀ * (∂ₜ (fun t => electricField 𝓕.c A t x) t).ofLp i =
        ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x -
          𝓕.μ₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp it:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ 1 / (𝓕.μ₀ * 𝓕.ε₀) * ((1 / 𝓕.ε₀) • fderiv ℝ (LorentzCurrentDensity.chargeDensity 𝓕.c J t) x) (Space.basis i) =
  1 / (𝓕.μ₀ * 𝓕.ε₀ ^ 2) * Space.deriv i (fun x => LorentzCurrentDensity.chargeDensity 𝓕.c J t x) xhad:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:∀ (t : Time) (x : Space d),
  Space.div (electricField 𝓕.c A t) x = LorentzCurrentDensity.chargeDensity 𝓕.c J t x / 𝓕.ε₀ ∧
    ∀ (i : Fin d),
      𝓕.μ₀ * 𝓕.ε₀ * (∂ₜ (fun t => electricField 𝓕.c A t x) t).ofLp i =
        ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x -
          𝓕.μ₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp it:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ DifferentiableAt ℝ (LorentzCurrentDensity.chargeDensity 𝓕.c J t) x
          simp [← Space.deriv_eq_fderiv_basis]d:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:∀ (t : Time) (x : Space d),
  Space.div (electricField 𝓕.c A t) x = LorentzCurrentDensity.chargeDensity 𝓕.c J t x / 𝓕.ε₀ ∧
    ∀ (i : Fin d),
      𝓕.μ₀ * 𝓕.ε₀ * (∂ₜ (fun t => electricField 𝓕.c A t x) t).ofLp i =
        ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x -
          𝓕.μ₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp it:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ 𝓕.ε₀⁻¹ * 𝓕.μ₀⁻¹ * (𝓕.ε₀⁻¹ * Space.deriv i (LorentzCurrentDensity.chargeDensity 𝓕.c J t) x) =
  (𝓕.ε₀ ^ 2)⁻¹ * 𝓕.μ₀⁻¹ * Space.deriv i (fun x => LorentzCurrentDensity.chargeDensity 𝓕.c J t x) xhad:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:∀ (t : Time) (x : Space d),
  Space.div (electricField 𝓕.c A t) x = LorentzCurrentDensity.chargeDensity 𝓕.c J t x / 𝓕.ε₀ ∧
    ∀ (i : Fin d),
      𝓕.μ₀ * 𝓕.ε₀ * (∂ₜ (fun t => electricField 𝓕.c A t x) t).ofLp i =
        ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x -
          𝓕.μ₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp it:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ DifferentiableAt ℝ (LorentzCurrentDensity.chargeDensity 𝓕.c J t) x
          field_simphad:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:∀ (t : Time) (x : Space d),
  Space.div (electricField 𝓕.c A t) x = LorentzCurrentDensity.chargeDensity 𝓕.c J t x / 𝓕.ε₀ ∧
    ∀ (i : Fin d),
      𝓕.μ₀ * 𝓕.ε₀ * (∂ₜ (fun t => electricField 𝓕.c A t x) t).ofLp i =
        ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x -
          𝓕.μ₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp it:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ DifferentiableAt ℝ (LorentzCurrentDensity.chargeDensity 𝓕.c J t) x
          apply Differentiable.differentiableAthad:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:∀ (t : Time) (x : Space d),
  Space.div (electricField 𝓕.c A t) x = LorentzCurrentDensity.chargeDensity 𝓕.c J t x / 𝓕.ε₀ ∧
    ∀ (i : Fin d),
      𝓕.μ₀ * 𝓕.ε₀ * (∂ₜ (fun t => electricField 𝓕.c A t x) t).ofLp i =
        ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x -
          𝓕.μ₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp it:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ Differentiable ℝ (LorentzCurrentDensity.chargeDensity 𝓕.c J t)
          apply LorentzCurrentDensity.chargeDensity_differentiable_spacehad:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:∀ (t : Time) (x : Space d),
  Space.div (electricField 𝓕.c A t) x = LorentzCurrentDensity.chargeDensity 𝓕.c J t x / 𝓕.ε₀ ∧
    ∀ (i : Fin d),
      𝓕.μ₀ * 𝓕.ε₀ * (∂ₜ (fun t => electricField 𝓕.c A t x) t).ofLp i =
        ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x -
          𝓕.μ₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp it:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ Differentiable ℝ J
          exact hJ.differentiable (byd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:∀ (t : Time) (x : Space d),
  Space.div (electricField 𝓕.c A t) x = LorentzCurrentDensity.chargeDensity 𝓕.c J t x / 𝓕.ε₀ ∧
    ∀ (i : Fin d),
      𝓕.μ₀ * 𝓕.ε₀ * (∂ₜ (fun t => electricField 𝓕.c A t x) t).ofLp i =
        ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x -
          𝓕.μ₀ * (LorentzCurrentDensity.currentDensity 𝓕.c J t x).ofLp it:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ ∞ ≠ 0 simpAll goals completed! 🐙)
        ·e_a.e_a.hAd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ ContDiff ℝ ∞ A.val exact hAAll goals completed! 🐙
        ·e_a.e_a.hJd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ ContDiff ℝ ∞ J exact hJAll goals completed! 🐙
      _ = 𝓕.c ^ 2 * ∑ j, (∂[j] (∂[j] (A.electricField 𝓕.c t · i)) x) -
            𝓕.c ^ 2 / 𝓕.ε₀ * ∂[i] (J.chargeDensity 𝓕.c t ·) x -
            𝓕.c ^ 2 * 𝓕.μ₀ * ∂ₜ (J.currentDensity 𝓕.c · x i) t := byd:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ 1 / (𝓕.μ₀ * 𝓕.ε₀) * ∑ j, Space.deriv j (Space.deriv j fun x => (electricField 𝓕.c A t x).ofLp i) x -
      1 / (𝓕.μ₀ * 𝓕.ε₀ ^ 2) * Space.deriv i (fun x => LorentzCurrentDensity.chargeDensity 𝓕.c J t x) x -
    1 / 𝓕.ε₀ * ∂ₜ (fun x_1 => (LorentzCurrentDensity.currentDensity 𝓕.c J x_1 x).ofLp i) t =
  𝓕.c.val ^ 2 * ∑ j, Space.deriv j (Space.deriv j fun x => (electricField 𝓕.c A t x).ofLp i) x -
      𝓕.c.val ^ 2 / 𝓕.ε₀ * Space.deriv i (fun x => LorentzCurrentDensity.chargeDensity 𝓕.c J t x) x -
    𝓕.c.val ^ 2 * 𝓕.μ₀ * ∂ₜ (fun x_1 => (LorentzCurrentDensity.currentDensity 𝓕.c J x_1 x).ofLp i) t
          simp [FreeSpace.c_sq]d:ℕA:ElectromagneticPotential d𝓕:FreeSpacehA:ContDiff ℝ ∞ A.valJ:LorentzCurrentDensity dhJ:ContDiff ℝ ∞ Jh:IsExtrema 𝓕 A Jt:Timex:Space di:Fin dhEs:∀ (j : Fin d), ContDiff ℝ 2 fun y => (electricField 𝓕.c A t y).ofLp jhEd:∀ (j k : Fin d), Differentiable ℝ (Space.deriv k fun y => (electricField 𝓕.c A t y).ofLp j)hBt:∀ (j : Fin d), Differentiable ℝ fun s => Space.deriv j (fun y => magneticFieldMatrix 𝓕.c A s y (j, i)) xhJt:Differentiable ℝ fun s => (LorentzCurrentDensity.currentDensity 𝓕.c J s x).ofLp i⊢ 𝓕.ε₀⁻¹ * 𝓕.μ₀⁻¹ * ∑ x_1, Space.deriv x_1 (Space.deriv x_1 fun x => (electricField 𝓕.c A t x).ofLp i) x -
      (𝓕.ε₀ ^ 2)⁻¹ * 𝓕.μ₀⁻¹ * Space.deriv i (fun x => LorentzCurrentDensity.chargeDensity 𝓕.c J t x) x -
    𝓕.ε₀⁻¹ * ∂ₜ (fun x_1 => (LorentzCurrentDensity.currentDensity 𝓕.c J x_1 x).ofLp i) t =
  𝓕.μ₀⁻¹ * 𝓕.ε₀⁻¹ * ∑ x_1, Space.deriv x_1 (Space.deriv x_1 fun x => (electricField 𝓕.c A t x).ofLp i) x -
      𝓕.μ₀⁻¹ * 𝓕.ε₀⁻¹ / 𝓕.ε₀ * Space.deriv i (fun x => LorentzCurrentDensity.chargeDensity 𝓕.c J t x) x -
    𝓕.μ₀⁻¹ * 𝓕.ε₀⁻¹ * 𝓕.μ₀ * ∂ₜ (fun x_1 => (LorentzCurrentDensity.currentDensity 𝓕.c J x_1 x).ofLp i) t
          field_simpAll goals completed! 🐙