Physlib.Particles.FlavorPhysics.CKMMatrix.StandardParameterization.StandardParameters

Imports

/-
Copyright (c) 2024 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.Particles.FlavorPhysics.CKMMatrix.PhaseFreedom
public import Physlib.Particles.FlavorPhysics.CKMMatrix.StandardParameterization.Basic

Standard parameters for the CKM Matrix

Given a CKM matrix V we can extract four real numbers θ₁₂, θ₁₃, θ₂₃ and δ₁₃. These, when used in the standard parameterization return V up to equivalence.

This leads to the theorem standParam.exists_for_CKMatrix which says that up to equivalence every CKM matrix can be written using the standard parameterization.

@[expose] public section

Given a CKM matrix V the real number corresponding to sin θ₁₂ in the standard parameterization. -

def S₁₂ (V : Quotient CKMMatrixSetoid) : ℝ := VusAbs V / (√ (VudAbs V ^ 2 + VusAbs V ^ 2))

Given a CKM matrix V the real number corresponding to sin θ₁₃ in the standard parameterization. -

def S₁₃ (V : Quotient CKMMatrixSetoid) : ℝ := VubAbs V

Given a CKM matrix V the real number corresponding to sin θ₂₃ in the standard parameterization. -

def S₂₃ (V : Quotient CKMMatrixSetoid) : ℝ :=
  if VubAbs V = 1 then VcdAbs V
  else VcbAbs V / √ (VudAbs V ^ 2 + VusAbs V ^ 2)

Given a CKM matrix V the real number corresponding to θ₁₂ in the standard parameterization. -

def θ₁₂ (V : Quotient CKMMatrixSetoid) : ℝ := Real.arcsin (S₁₂ V)

Given a CKM matrix V the real number corresponding to θ₁₃ in the standard parameterization. -

def θ₁₃ (V : Quotient CKMMatrixSetoid) : ℝ := Real.arcsin (S₁₃ V)

Given a CKM matrix V the real number corresponding to θ₂₃ in the standard parameterization. -

def θ₂₃ (V : Quotient CKMMatrixSetoid) : ℝ := Real.arcsin (S₂₃ V)

Given a CKM matrix V the real number corresponding to cos θ₁₂ in the standard parameterization. -

def C₁₂ (V : Quotient CKMMatrixSetoid) : ℝ := Real.cos (θ₁₂ V)

Given a CKM matrix V the real number corresponding to cos θ₁₃ in the standard parameterization. -

def C₁₃ (V : Quotient CKMMatrixSetoid) : ℝ := Real.cos (θ₁₃ V)

Given a CKM matrix V the real number corresponding to sin θ₂₃ in the standard parameterization. -

def C₂₃ (V : Quotient CKMMatrixSetoid) : ℝ := Real.cos (θ₂₃ V)

Given a CKM matrix V the real number corresponding to the phase δ₁₃ in the standard parameterization. -

def δ₁₃ (V : Quotient CKMMatrixSetoid) : ℝ :=
  arg (Invariant.mulExpδ₁₃ V)

For a CKM matrix sin θ₁₂ is non-negative.

lemma S₁₂_nonneg (V : Quotient CKMMatrixSetoid) : 0 ≤ S₁₂ V :=
  div_nonneg (VAbs_ge_zero 0 1 V) (Real.sqrt_nonneg _)

For a CKM matrix sin θ₁₃ is non-negative.

lemma S₁₃_nonneg (V : Quotient CKMMatrixSetoid) : 0 ≤ S₁₃ V :=
  VAbs_ge_zero 0 2 V

For a CKM matrix sin θ₂₃ is non-negative.


lemma S₂₃_nonneg (V : Quotient CKMMatrixSetoid) : 0 ≤ S₂₃ V := byV:Quotient CKMMatrixSetoid⊢ 0 ≤ S₂₃ V
  rw [S₂₃V:Quotient CKMMatrixSetoid⊢ 0 ≤ if VubAbs V = 1 then VcdAbs V else VcbAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2)V:Quotient CKMMatrixSetoid⊢ 0 ≤ if VubAbs V = 1 then VcdAbs V else VcbAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2)]V:Quotient CKMMatrixSetoid⊢ 0 ≤ if VubAbs V = 1 then VcdAbs V else VcbAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2)
  splitisTrueV:Quotient CKMMatrixSetoidh✝:VubAbs V = 1⊢ 0 ≤ VcdAbs VisFalseV:Quotient CKMMatrixSetoidh✝:¬VubAbs V = 1⊢ 0 ≤ VcbAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2)
  ·isTrueV:Quotient CKMMatrixSetoidh✝:VubAbs V = 1⊢ 0 ≤ VcdAbs V exact VAbs_ge_zero 1 0 VAll goals completed! 🐙
  ·isFalseV:Quotient CKMMatrixSetoidh✝:¬VubAbs V = 1⊢ 0 ≤ VcbAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2) exact div_nonneg (VAbs_ge_zero 1 2 V) (Real.sqrt_nonneg _)All goals completed! 🐙

For a CKM matrix sin θ₁₂ is less than or equal to 1.


lemma S₁₂_leq_one (V : Quotient CKMMatrixSetoid) : S₁₂ V ≤ 1 := byV:Quotient CKMMatrixSetoid⊢ S₁₂ V ≤ 1
  rw [S₁₂V:Quotient CKMMatrixSetoid⊢ VusAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2) ≤ 1V:Quotient CKMMatrixSetoid⊢ VusAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2) ≤ 1]V:Quotient CKMMatrixSetoid⊢ VusAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2) ≤ 1
  exact div_le_one_of_le₀ (Real.le_sqrt_of_sq_le (le_add_of_nonneg_left (sq_nonneg _)))
    (Real.sqrt_nonneg _)All goals completed! 🐙

For a CKM matrix sin θ₁₃ is less than or equal to 1.

lemma S₁₃_leq_one (V : Quotient CKMMatrixSetoid) : S₁₃ V ≤ 1 :=
  VAbs_leq_one 0 2 V

For a CKM matrix sin θ₂₃ is less than or equal to 1.


lemma S₂₃_leq_one (V : Quotient CKMMatrixSetoid) : S₂₃ V ≤ 1 := byV:Quotient CKMMatrixSetoid⊢ S₂₃ V ≤ 1
  rw [S₂₃V:Quotient CKMMatrixSetoid⊢ (if VubAbs V = 1 then VcdAbs V else VcbAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2)) ≤ 1V:Quotient CKMMatrixSetoid⊢ (if VubAbs V = 1 then VcdAbs V else VcbAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2)) ≤ 1]V:Quotient CKMMatrixSetoid⊢ (if VubAbs V = 1 then VcdAbs V else VcbAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2)) ≤ 1
  splitisTrueV:Quotient CKMMatrixSetoidh✝:VubAbs V = 1⊢ VcdAbs V ≤ 1isFalseV:Quotient CKMMatrixSetoidh✝:¬VubAbs V = 1⊢ VcbAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2) ≤ 1
  ·isTrueV:Quotient CKMMatrixSetoidh✝:VubAbs V = 1⊢ VcdAbs V ≤ 1 exact VAbs_leq_one 1 0 VAll goals completed! 🐙
  ·isFalseV:Quotient CKMMatrixSetoidh✝:¬VubAbs V = 1⊢ VcbAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2) ≤ 1 apply div_le_one_of_le₀ _ (Real.sqrt_nonneg _)V:Quotient CKMMatrixSetoidh✝:¬VubAbs V = 1⊢ VcbAbs V ≤ √(VudAbs V ^ 2 + VusAbs V ^ 2)
    rw [VudAbs_sq_add_VusAbs_sq,V:Quotient CKMMatrixSetoidh✝:¬VubAbs V = 1⊢ VcbAbs V ≤ √(1 - VubAbs V ^ 2)V:Quotient CKMMatrixSetoidh✝:¬VubAbs V = 1⊢ VcbAbs V ≤ √(VcbAbs V ^ 2 + VtbAbs V ^ 2) ← VcbAbs_sq_add_VtbAbs_sqV:Quotient CKMMatrixSetoidh✝:¬VubAbs V = 1⊢ VcbAbs V ≤ √(VcbAbs V ^ 2 + VtbAbs V ^ 2)V:Quotient CKMMatrixSetoidh✝:¬VubAbs V = 1⊢ VcbAbs V ≤ √(VcbAbs V ^ 2 + VtbAbs V ^ 2)]V:Quotient CKMMatrixSetoidh✝:¬VubAbs V = 1⊢ VcbAbs V ≤ √(VcbAbs V ^ 2 + VtbAbs V ^ 2)
    exact Real.le_sqrt_of_sq_le (le_add_of_nonneg_right (sq_nonneg _))All goals completed! 🐙

              lemma S₁₂_eq_sin_θ₁₂ (V : Quotient CKMMatrixSetoid) : Real.sin (θ₁₂ V) = S₁₂ V :=
  Real.sin_arcsin (le_trans (byV:Quotient CKMMatrixSetoid⊢ -1 ≤ 0 simpAll goals completed! 🐙) (S₁₂_nonneg V)) (S₁₂_leq_one V)lemma S₁₃_eq_sin_θ₁₃ (V : Quotient CKMMatrixSetoid) : Real.sin (θ₁₃ V) = S₁₃ V :=
  Real.sin_arcsin (le_trans (byV:Quotient CKMMatrixSetoid⊢ -1 ≤ 0 simpAll goals completed! 🐙) (S₁₃_nonneg V)) (S₁₃_leq_one V)lemma S₂₃_eq_sin_θ₂₃ (V : Quotient CKMMatrixSetoid) : Real.sin (θ₂₃ V) = S₂₃ V :=
  Real.sin_arcsin (le_trans (byV:Quotient CKMMatrixSetoid⊢ -1 ≤ 0 simpAll goals completed! 🐙) (S₂₃_nonneg V)) (S₂₃_leq_one V)lemma S₁₂_eq_ℂsin_θ₁₂ (V : Quotient CKMMatrixSetoid) : Complex.sin (θ₁₂ V) = S₁₂ V :=
  (ofReal_sin _).symm.trans (congrArg ofReal (S₁₂_eq_sin_θ₁₂ V))lemma S₁₃_eq_ℂsin_θ₁₃ (V : Quotient CKMMatrixSetoid) : Complex.sin (θ₁₃ V) = S₁₃ V :=
  (ofReal_sin _).symm.trans (congrArg ofReal (S₁₃_eq_sin_θ₁₃ V))lemma S₂₃_eq_ℂsin_θ₂₃ (V : Quotient CKMMatrixSetoid) : Complex.sin (θ₂₃ V) = S₂₃ V :=
  (ofReal_sin _).symm.trans (congrArg ofReal (S₂₃_eq_sin_θ₂₃ V))

lemma complexAbs_sin_θ₁₂ (V : Quotient CKMMatrixSetoid) :
    norm (Complex.sin (θ₁₂ V)) = sin (θ₁₂ V) := byV:Quotient CKMMatrixSetoid⊢ ↑‖sin ↑(θ₁₂ V)‖ = sin ↑(θ₁₂ V)
  rw [S₁₂_eq_ℂsin_θ₁₂,V:Quotient CKMMatrixSetoid⊢ ↑‖↑(S₁₂ V)‖ = ↑(S₁₂ V)V:Quotient CKMMatrixSetoid⊢ ‖S₁₂ V‖ = S₁₂ V Complex.norm_real,V:Quotient CKMMatrixSetoid⊢ ↑‖S₁₂ V‖ = ↑(S₁₂ V)V:Quotient CKMMatrixSetoid⊢ ‖S₁₂ V‖ = S₁₂ V ofReal_injV:Quotient CKMMatrixSetoid⊢ ‖S₁₂ V‖ = S₁₂ VV:Quotient CKMMatrixSetoid⊢ ‖S₁₂ V‖ = S₁₂ V]V:Quotient CKMMatrixSetoid⊢ ‖S₁₂ V‖ = S₁₂ V
  exact Real.norm_of_nonneg (S₁₂_nonneg _)All goals completed! 🐙

lemma complexAbs_sin_θ₁₃ (V : Quotient CKMMatrixSetoid) :
    norm (Complex.sin (θ₁₃ V)) = sin (θ₁₃ V) := byV:Quotient CKMMatrixSetoid⊢ ↑‖sin ↑(θ₁₃ V)‖ = sin ↑(θ₁₃ V)
  rw [S₁₃_eq_ℂsin_θ₁₃,V:Quotient CKMMatrixSetoid⊢ ↑‖↑(S₁₃ V)‖ = ↑(S₁₃ V)V:Quotient CKMMatrixSetoid⊢ ‖S₁₃ V‖ = S₁₃ V Complex.norm_real,V:Quotient CKMMatrixSetoid⊢ ↑‖S₁₃ V‖ = ↑(S₁₃ V)V:Quotient CKMMatrixSetoid⊢ ‖S₁₃ V‖ = S₁₃ V ofReal_injV:Quotient CKMMatrixSetoid⊢ ‖S₁₃ V‖ = S₁₃ VV:Quotient CKMMatrixSetoid⊢ ‖S₁₃ V‖ = S₁₃ V]V:Quotient CKMMatrixSetoid⊢ ‖S₁₃ V‖ = S₁₃ V
  exact Real.norm_of_nonneg (S₁₃_nonneg _)All goals completed! 🐙

lemma complexAbs_sin_θ₂₃ (V : Quotient CKMMatrixSetoid) :
    norm (Complex.sin (θ₂₃ V)) = sin (θ₂₃ V) := byV:Quotient CKMMatrixSetoid⊢ ↑‖sin ↑(θ₂₃ V)‖ = sin ↑(θ₂₃ V)
  rw [S₂₃_eq_ℂsin_θ₂₃,V:Quotient CKMMatrixSetoid⊢ ↑‖↑(S₂₃ V)‖ = ↑(S₂₃ V)V:Quotient CKMMatrixSetoid⊢ ‖S₂₃ V‖ = S₂₃ V Complex.norm_real,V:Quotient CKMMatrixSetoid⊢ ↑‖S₂₃ V‖ = ↑(S₂₃ V)V:Quotient CKMMatrixSetoid⊢ ‖S₂₃ V‖ = S₂₃ V ofReal_injV:Quotient CKMMatrixSetoid⊢ ‖S₂₃ V‖ = S₂₃ VV:Quotient CKMMatrixSetoid⊢ ‖S₂₃ V‖ = S₂₃ V]V:Quotient CKMMatrixSetoid⊢ ‖S₂₃ V‖ = S₂₃ V
  exact Real.norm_of_nonneg (S₂₃_nonneg _)All goals completed! 🐙lemma S₁₂_of_Vub_one {V : Quotient CKMMatrixSetoid} (ha : VubAbs V = 1) : S₁₂ V = 0 := byV:Quotient CKMMatrixSetoidha:VubAbs V = 1⊢ S₁₂ V = 0
  simp [S₁₂, VudAbs_sq_add_VusAbs_sq, ha]All goals completed! 🐙lemma S₁₃_of_Vub_one {V : Quotient CKMMatrixSetoid} (ha : VubAbs V = 1) : S₁₃ V = 1 := ha

lemma S₂₃_of_Vub_eq_one {V : Quotient CKMMatrixSetoid} (ha : VubAbs V = 1) : S₂₃ V = VcdAbs V := byV:Quotient CKMMatrixSetoidha:VubAbs V = 1⊢ S₂₃ V = VcdAbs V
  rw [S₂₃,V:Quotient CKMMatrixSetoidha:VubAbs V = 1⊢ (if VubAbs V = 1 then VcdAbs V else VcbAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2)) = VcdAbs VAll goals completed! 🐙 if_pos haV:Quotient CKMMatrixSetoidha:VubAbs V = 1⊢ VcdAbs V = VcdAbs VAll goals completed! 🐙]All goals completed! 🐙

lemma S₂₃_of_Vub_ne_one {V : Quotient CKMMatrixSetoid} (ha : VubAbs V ≠ 1) :
    S₂₃ V = VcbAbs V / √ (VudAbs V ^ 2 + VusAbs V ^ 2) := byV:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ S₂₃ V = VcbAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2)
  rw [S₂₃,V:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ (if VubAbs V = 1 then VcdAbs V else VcbAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2)) =
  VcbAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2)All goals completed! 🐙 if_neg haV:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ VcbAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2) = VcbAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2)All goals completed! 🐙]All goals completed! 🐙lemma C₁₂_eq_ℂcos_θ₁₂ (V : Quotient CKMMatrixSetoid) : Complex.cos (θ₁₂ V) = C₁₂ V := byV:Quotient CKMMatrixSetoid⊢ cos ↑(θ₁₂ V) = ↑(C₁₂ V)
  simp [C₁₂]All goals completed! 🐙lemma C₁₃_eq_ℂcos_θ₁₃ (V : Quotient CKMMatrixSetoid) : Complex.cos (θ₁₃ V) = C₁₃ V := byV:Quotient CKMMatrixSetoid⊢ cos ↑(θ₁₃ V) = ↑(C₁₃ V)
  simp [C₁₃]All goals completed! 🐙lemma C₂₃_eq_ℂcos_θ₂₃ (V : Quotient CKMMatrixSetoid) : Complex.cos (θ₂₃ V) = C₂₃ V := byV:Quotient CKMMatrixSetoid⊢ cos ↑(θ₂₃ V) = ↑(C₂₃ V)
  simp [C₂₃]All goals completed! 🐙

lemma complexAbs_cos_θ₁₂ (V : Quotient CKMMatrixSetoid) : norm (Complex.cos (θ₁₂ V)) =
    cos (θ₁₂ V) := byV:Quotient CKMMatrixSetoid⊢ ↑‖cos ↑(θ₁₂ V)‖ = cos ↑(θ₁₂ V)
  rw [C₁₂_eq_ℂcos_θ₁₂,V:Quotient CKMMatrixSetoid⊢ ↑‖↑(C₁₂ V)‖ = ↑(C₁₂ V)V:Quotient CKMMatrixSetoid⊢ ↑‖C₁₂ V‖ = ↑(C₁₂ V) Complex.norm_realV:Quotient CKMMatrixSetoid⊢ ↑‖C₁₂ V‖ = ↑(C₁₂ V)V:Quotient CKMMatrixSetoid⊢ ↑‖C₁₂ V‖ = ↑(C₁₂ V)]V:Quotient CKMMatrixSetoid⊢ ↑‖C₁₂ V‖ = ↑(C₁₂ V)
  exact_mod_cast Real.norm_of_nonneg (Real.cos_arcsin_nonneg _)All goals completed! 🐙

lemma complexAbs_cos_θ₁₃ (V : Quotient CKMMatrixSetoid) : norm (Complex.cos (θ₁₃ V)) =
    cos (θ₁₃ V) := byV:Quotient CKMMatrixSetoid⊢ ↑‖cos ↑(θ₁₃ V)‖ = cos ↑(θ₁₃ V)
  rw [C₁₃_eq_ℂcos_θ₁₃,V:Quotient CKMMatrixSetoid⊢ ↑‖↑(C₁₃ V)‖ = ↑(C₁₃ V)V:Quotient CKMMatrixSetoid⊢ ↑‖C₁₃ V‖ = ↑(C₁₃ V) Complex.norm_realV:Quotient CKMMatrixSetoid⊢ ↑‖C₁₃ V‖ = ↑(C₁₃ V)V:Quotient CKMMatrixSetoid⊢ ↑‖C₁₃ V‖ = ↑(C₁₃ V)]V:Quotient CKMMatrixSetoid⊢ ↑‖C₁₃ V‖ = ↑(C₁₃ V)
  exact_mod_cast Real.norm_of_nonneg (Real.cos_arcsin_nonneg _)All goals completed! 🐙

lemma complexAbs_cos_θ₂₃ (V : Quotient CKMMatrixSetoid) : norm (Complex.cos (θ₂₃ V)) =
    cos (θ₂₃ V) := byV:Quotient CKMMatrixSetoid⊢ ↑‖cos ↑(θ₂₃ V)‖ = cos ↑(θ₂₃ V)
  rw [C₂₃_eq_ℂcos_θ₂₃,V:Quotient CKMMatrixSetoid⊢ ↑‖↑(C₂₃ V)‖ = ↑(C₂₃ V)V:Quotient CKMMatrixSetoid⊢ ↑‖C₂₃ V‖ = ↑(C₂₃ V) Complex.norm_realV:Quotient CKMMatrixSetoid⊢ ↑‖C₂₃ V‖ = ↑(C₂₃ V)V:Quotient CKMMatrixSetoid⊢ ↑‖C₂₃ V‖ = ↑(C₂₃ V)]V:Quotient CKMMatrixSetoid⊢ ↑‖C₂₃ V‖ = ↑(C₂₃ V)
  exact_mod_cast Real.norm_of_nonneg (Real.cos_arcsin_nonneg _)All goals completed! 🐙lemma S₁₂_sq_add_C₁₂_sq (V : Quotient CKMMatrixSetoid) : S₁₂ V ^ 2 + C₁₂ V ^ 2 = 1 := byV:Quotient CKMMatrixSetoid⊢ S₁₂ V ^ 2 + C₁₂ V ^ 2 = 1
  simp only [← S₁₂_eq_sin_θ₁₂ V, C₁₂, Real.sin_sq_add_cos_sq]All goals completed! 🐙lemma S₁₃_sq_add_C₁₃_sq (V : Quotient CKMMatrixSetoid) : S₁₃ V ^ 2 + C₁₃ V ^ 2 = 1 := byV:Quotient CKMMatrixSetoid⊢ S₁₃ V ^ 2 + C₁₃ V ^ 2 = 1
  simp only [← S₁₃_eq_sin_θ₁₃ V, C₁₃, Real.sin_sq_add_cos_sq]All goals completed! 🐙lemma S₂₃_sq_add_C₂₃_sq (V : Quotient CKMMatrixSetoid) : S₂₃ V ^ 2 + C₂₃ V ^ 2 = 1 := byV:Quotient CKMMatrixSetoid⊢ S₂₃ V ^ 2 + C₂₃ V ^ 2 = 1
  simp only [← S₂₃_eq_sin_θ₂₃ V, C₂₃, Real.sin_sq_add_cos_sq]All goals completed! 🐙lemma C₁₂_of_Vub_one {V : Quotient CKMMatrixSetoid} (ha : VubAbs V = 1) : C₁₂ V = 1 := byV:Quotient CKMMatrixSetoidha:VubAbs V = 1⊢ C₁₂ V = 1
  simp [C₁₂, θ₁₂, S₁₂_of_Vub_one ha]All goals completed! 🐙lemma C₁₃_of_Vub_eq_one {V : Quotient CKMMatrixSetoid} (ha : VubAbs V = 1) : C₁₃ V = 0 := byV:Quotient CKMMatrixSetoidha:VubAbs V = 1⊢ C₁₃ V = 0
  simp [C₁₃, θ₁₃, S₁₃, ha]All goals completed! 🐙

--rename
lemma C₁₂_eq_Vud_div_sqrt {V : Quotient CKMMatrixSetoid} (ha : VubAbs V ≠ 1) :
    C₁₂ V = VudAbs V / √ (VudAbs V ^ 2 + VusAbs V ^ 2) := byV:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ C₁₂ V = VudAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2)
  rw [C₁₂,V:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ Real.cos (θ₁₂ V) = VudAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2)V:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ √(1 - VusAbs V ^ 2 / (VudAbs V ^ 2 + VusAbs V ^ 2)) = VudAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2)V:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ 0 ≤ VudAbs V ^ 2 + VusAbs V ^ 2 θ₁₂,V:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ Real.cos (Real.arcsin (S₁₂ V)) = VudAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2)V:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ √(1 - VusAbs V ^ 2 / (VudAbs V ^ 2 + VusAbs V ^ 2)) = VudAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2)V:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ 0 ≤ VudAbs V ^ 2 + VusAbs V ^ 2 Real.cos_arcsin,V:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ √(1 - S₁₂ V ^ 2) = VudAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2)V:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ √(1 - VusAbs V ^ 2 / (VudAbs V ^ 2 + VusAbs V ^ 2)) = VudAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2)V:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ 0 ≤ VudAbs V ^ 2 + VusAbs V ^ 2 S₁₂,V:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ √(1 - (VusAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2)) ^ 2) = VudAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2)V:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ √(1 - VusAbs V ^ 2 / (VudAbs V ^ 2 + VusAbs V ^ 2)) = VudAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2)V:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ 0 ≤ VudAbs V ^ 2 + VusAbs V ^ 2 div_pow,V:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ √(1 - VusAbs V ^ 2 / √(VudAbs V ^ 2 + VusAbs V ^ 2) ^ 2) = VudAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2)V:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ √(1 - VusAbs V ^ 2 / (VudAbs V ^ 2 + VusAbs V ^ 2)) = VudAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2)V:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ 0 ≤ VudAbs V ^ 2 + VusAbs V ^ 2 Real.sq_sqrtV:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ √(1 - VusAbs V ^ 2 / (VudAbs V ^ 2 + VusAbs V ^ 2)) = VudAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2)V:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ 0 ≤ VudAbs V ^ 2 + VusAbs V ^ 2V:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ √(1 - VusAbs V ^ 2 / (VudAbs V ^ 2 + VusAbs V ^ 2)) = VudAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2)V:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ 0 ≤ VudAbs V ^ 2 + VusAbs V ^ 2]V:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ √(1 - VusAbs V ^ 2 / (VudAbs V ^ 2 + VusAbs V ^ 2)) = VudAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2)V:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ 0 ≤ VudAbs V ^ 2 + VusAbs V ^ 2
  ·V:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ √(1 - VusAbs V ^ 2 / (VudAbs V ^ 2 + VusAbs V ^ 2)) = VudAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2) rw [one_sub_divV:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ √((VudAbs V ^ 2 + VusAbs V ^ 2 - VusAbs V ^ 2) / (VudAbs V ^ 2 + VusAbs V ^ 2)) =
  VudAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2)V:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ VudAbs V ^ 2 + VusAbs V ^ 2 ≠ 0V:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ √((VudAbs V ^ 2 + VusAbs V ^ 2 - VusAbs V ^ 2) / (VudAbs V ^ 2 + VusAbs V ^ 2)) =
  VudAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2)V:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ VudAbs V ^ 2 + VusAbs V ^ 2 ≠ 0]V:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ √((VudAbs V ^ 2 + VusAbs V ^ 2 - VusAbs V ^ 2) / (VudAbs V ^ 2 + VusAbs V ^ 2)) =
  VudAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2)V:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ VudAbs V ^ 2 + VusAbs V ^ 2 ≠ 0
    ·V:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ √((VudAbs V ^ 2 + VusAbs V ^ 2 - VusAbs V ^ 2) / (VudAbs V ^ 2 + VusAbs V ^ 2)) =
  VudAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2) simp only [VudAbs, Fin.isValue, VusAbs, add_sub_cancel_right]V:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ √(VAbs 0 0 V ^ 2 / (VAbs 0 0 V ^ 2 + VAbs 0 1 V ^ 2)) = VAbs 0 0 V / √(VAbs 0 0 V ^ 2 + VAbs 0 1 V ^ 2)
      rw [Real.sqrt_div (sq_nonneg (VAbs 0 0 V)),V:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ √(VAbs 0 0 V ^ 2) / √(VAbs 0 0 V ^ 2 + VAbs 0 1 V ^ 2) = VAbs 0 0 V / √(VAbs 0 0 V ^ 2 + VAbs 0 1 V ^ 2)All goals completed! 🐙 Real.sqrt_sq (VAbs_ge_zero 0 0 V)V:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ VAbs 0 0 V / √(VAbs 0 0 V ^ 2 + VAbs 0 1 V ^ 2) = VAbs 0 0 V / √(VAbs 0 0 V ^ 2 + VAbs 0 1 V ^ 2)All goals completed! 🐙]All goals completed! 🐙
    exact VAbsub_ne_zero_Vud_Vus_ne_zero haAll goals completed! 🐙
  exact (Left.add_nonneg (sq_nonneg (VAbs 0 0 V)) (sq_nonneg (VAbs 0 1 V)))All goals completed! 🐙

--rename
lemma C₁₃_eq_add_sq (V : Quotient CKMMatrixSetoid) : C₁₃ V = √ (VudAbs V ^ 2 + VusAbs V ^ 2) := byV:Quotient CKMMatrixSetoid⊢ C₁₃ V = √(VudAbs V ^ 2 + VusAbs V ^ 2)
  rw [C₁₃,V:Quotient CKMMatrixSetoid⊢ Real.cos (θ₁₃ V) = √(VudAbs V ^ 2 + VusAbs V ^ 2)All goals completed! 🐙 θ₁₃,V:Quotient CKMMatrixSetoid⊢ Real.cos (Real.arcsin (S₁₃ V)) = √(VudAbs V ^ 2 + VusAbs V ^ 2)All goals completed! 🐙 Real.cos_arcsin,V:Quotient CKMMatrixSetoid⊢ √(1 - S₁₃ V ^ 2) = √(VudAbs V ^ 2 + VusAbs V ^ 2)All goals completed! 🐙 S₁₃,V:Quotient CKMMatrixSetoid⊢ √(1 - VubAbs V ^ 2) = √(VudAbs V ^ 2 + VusAbs V ^ 2)All goals completed! 🐙 ← VudAbs_sq_add_VusAbs_sqV:Quotient CKMMatrixSetoid⊢ √(VudAbs V ^ 2 + VusAbs V ^ 2) = √(VudAbs V ^ 2 + VusAbs V ^ 2)All goals completed! 🐙]All goals completed! 🐙

lemma C₂₃_of_Vub_ne_one {V : Quotient CKMMatrixSetoid} (ha : VubAbs V ≠ 1) :
    C₂₃ V = VtbAbs V / √ (VudAbs V ^ 2 + VusAbs V ^ 2) := byV:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ C₂₃ V = VtbAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2)
  rw [C₂₃,V:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ Real.cos (θ₂₃ V) = VtbAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2)V:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ √(1 - VcbAbs V ^ 2 / (VudAbs V ^ 2 + VusAbs V ^ 2)) = VtbAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2)V:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ 0 ≤ VudAbs V ^ 2 + VusAbs V ^ 2 θ₂₃,V:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ Real.cos (Real.arcsin (S₂₃ V)) = VtbAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2)V:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ √(1 - VcbAbs V ^ 2 / (VudAbs V ^ 2 + VusAbs V ^ 2)) = VtbAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2)V:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ 0 ≤ VudAbs V ^ 2 + VusAbs V ^ 2 Real.cos_arcsin,V:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ √(1 - S₂₃ V ^ 2) = VtbAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2)V:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ √(1 - VcbAbs V ^ 2 / (VudAbs V ^ 2 + VusAbs V ^ 2)) = VtbAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2)V:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ 0 ≤ VudAbs V ^ 2 + VusAbs V ^ 2 S₂₃_of_Vub_ne_one ha,V:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ √(1 - (VcbAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2)) ^ 2) = VtbAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2)V:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ √(1 - VcbAbs V ^ 2 / (VudAbs V ^ 2 + VusAbs V ^ 2)) = VtbAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2)V:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ 0 ≤ VudAbs V ^ 2 + VusAbs V ^ 2 div_pow,V:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ √(1 - VcbAbs V ^ 2 / √(VudAbs V ^ 2 + VusAbs V ^ 2) ^ 2) = VtbAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2)V:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ √(1 - VcbAbs V ^ 2 / (VudAbs V ^ 2 + VusAbs V ^ 2)) = VtbAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2)V:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ 0 ≤ VudAbs V ^ 2 + VusAbs V ^ 2 Real.sq_sqrtV:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ √(1 - VcbAbs V ^ 2 / (VudAbs V ^ 2 + VusAbs V ^ 2)) = VtbAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2)V:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ 0 ≤ VudAbs V ^ 2 + VusAbs V ^ 2V:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ √(1 - VcbAbs V ^ 2 / (VudAbs V ^ 2 + VusAbs V ^ 2)) = VtbAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2)V:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ 0 ≤ VudAbs V ^ 2 + VusAbs V ^ 2]V:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ √(1 - VcbAbs V ^ 2 / (VudAbs V ^ 2 + VusAbs V ^ 2)) = VtbAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2)V:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ 0 ≤ VudAbs V ^ 2 + VusAbs V ^ 2
  ·V:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ √(1 - VcbAbs V ^ 2 / (VudAbs V ^ 2 + VusAbs V ^ 2)) = VtbAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2) rw [VudAbs_sq_add_VusAbs_sq,V:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ √(1 - VcbAbs V ^ 2 / (1 - VubAbs V ^ 2)) = VtbAbs V / √(1 - VubAbs V ^ 2)V:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ √(1 - VcbAbs V ^ 2 / (VcbAbs V ^ 2 + VtbAbs V ^ 2)) = VtbAbs V / √(VcbAbs V ^ 2 + VtbAbs V ^ 2) ← VcbAbs_sq_add_VtbAbs_sqV:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ √(1 - VcbAbs V ^ 2 / (VcbAbs V ^ 2 + VtbAbs V ^ 2)) = VtbAbs V / √(VcbAbs V ^ 2 + VtbAbs V ^ 2)V:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ √(1 - VcbAbs V ^ 2 / (VcbAbs V ^ 2 + VtbAbs V ^ 2)) = VtbAbs V / √(VcbAbs V ^ 2 + VtbAbs V ^ 2)]V:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ √(1 - VcbAbs V ^ 2 / (VcbAbs V ^ 2 + VtbAbs V ^ 2)) = VtbAbs V / √(VcbAbs V ^ 2 + VtbAbs V ^ 2)
    rw [one_sub_divV:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ √((VcbAbs V ^ 2 + VtbAbs V ^ 2 - VcbAbs V ^ 2) / (VcbAbs V ^ 2 + VtbAbs V ^ 2)) =
  VtbAbs V / √(VcbAbs V ^ 2 + VtbAbs V ^ 2)V:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ VcbAbs V ^ 2 + VtbAbs V ^ 2 ≠ 0V:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ √((VcbAbs V ^ 2 + VtbAbs V ^ 2 - VcbAbs V ^ 2) / (VcbAbs V ^ 2 + VtbAbs V ^ 2)) =
  VtbAbs V / √(VcbAbs V ^ 2 + VtbAbs V ^ 2)V:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ VcbAbs V ^ 2 + VtbAbs V ^ 2 ≠ 0]V:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ √((VcbAbs V ^ 2 + VtbAbs V ^ 2 - VcbAbs V ^ 2) / (VcbAbs V ^ 2 + VtbAbs V ^ 2)) =
  VtbAbs V / √(VcbAbs V ^ 2 + VtbAbs V ^ 2)V:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ VcbAbs V ^ 2 + VtbAbs V ^ 2 ≠ 0
    ·V:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ √((VcbAbs V ^ 2 + VtbAbs V ^ 2 - VcbAbs V ^ 2) / (VcbAbs V ^ 2 + VtbAbs V ^ 2)) =
  VtbAbs V / √(VcbAbs V ^ 2 + VtbAbs V ^ 2) simp only [VcbAbs, Fin.isValue, VtbAbs, add_sub_cancel_left]V:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ √(VAbs 2 2 V ^ 2 / (VAbs 1 2 V ^ 2 + VAbs 2 2 V ^ 2)) = VAbs 2 2 V / √(VAbs 1 2 V ^ 2 + VAbs 2 2 V ^ 2)
      rw [Real.sqrt_div (sq_nonneg (VAbs 2 2 V)),V:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ √(VAbs 2 2 V ^ 2) / √(VAbs 1 2 V ^ 2 + VAbs 2 2 V ^ 2) = VAbs 2 2 V / √(VAbs 1 2 V ^ 2 + VAbs 2 2 V ^ 2)All goals completed! 🐙 Real.sqrt_sq (VAbs_ge_zero 2 2 V)V:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ VAbs 2 2 V / √(VAbs 1 2 V ^ 2 + VAbs 2 2 V ^ 2) = VAbs 2 2 V / √(VAbs 1 2 V ^ 2 + VAbs 2 2 V ^ 2)All goals completed! 🐙]All goals completed! 🐙
    ·V:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ VcbAbs V ^ 2 + VtbAbs V ^ 2 ≠ 0 rw [VcbAbs_sq_add_VtbAbs_sq,V:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ 1 - VubAbs V ^ 2 ≠ 0V:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ VudAbs V ^ 2 + VusAbs V ^ 2 ≠ 0 ← VudAbs_sq_add_VusAbs_sqV:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ VudAbs V ^ 2 + VusAbs V ^ 2 ≠ 0V:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ VudAbs V ^ 2 + VusAbs V ^ 2 ≠ 0]V:Quotient CKMMatrixSetoidha:VubAbs V ≠ 1⊢ VudAbs V ^ 2 + VusAbs V ^ 2 ≠ 0
      exact VAbsub_ne_zero_Vud_Vus_ne_zero haAll goals completed! 🐙
  exact (Left.add_nonneg (sq_nonneg (VAbs 0 0 V)) (sq_nonneg (VAbs 0 1 V)))All goals completed! 🐙

-- rename to VudAbs_standard_param
lemma VudAbs_eq_C₁₂_mul_C₁₃ (V : Quotient CKMMatrixSetoid) : VudAbs V = C₁₂ V * C₁₃ V := byV:Quotient CKMMatrixSetoid⊢ VudAbs V = C₁₂ V * C₁₃ V
  by_cases ha : VubAbs V = 1posV:Quotient CKMMatrixSetoidha:VubAbs V = 1⊢ VudAbs V = C₁₂ V * C₁₃ VnegV:Quotient CKMMatrixSetoidha:¬VubAbs V = 1⊢ VudAbs V = C₁₂ V * C₁₃ V
  ·posV:Quotient CKMMatrixSetoidha:VubAbs V = 1⊢ VudAbs V = C₁₂ V * C₁₃ V rw [C₁₃_of_Vub_eq_one ha,posV:Quotient CKMMatrixSetoidha:VubAbs V = 1⊢ VudAbs V = C₁₂ V * 0posV:Quotient CKMMatrixSetoidha:VubAbs V = 1⊢ VudAbs V = 0 mul_zeroposV:Quotient CKMMatrixSetoidha:VubAbs V = 1⊢ VudAbs V = 0posV:Quotient CKMMatrixSetoidha:VubAbs V = 1⊢ VudAbs V = 0]posV:Quotient CKMMatrixSetoidha:VubAbs V = 1⊢ VudAbs V = 0
    exact VAbs_thd_eq_one_fst_eq_zero haAll goals completed! 🐙
  ·negV:Quotient CKMMatrixSetoidha:¬VubAbs V = 1⊢ VudAbs V = C₁₂ V * C₁₃ V rw [C₁₂_eq_Vud_div_sqrt ha,negV:Quotient CKMMatrixSetoidha:¬VubAbs V = 1⊢ VudAbs V = VudAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2) * C₁₃ VnegV:Quotient CKMMatrixSetoidha:¬VubAbs V = 1⊢ VudAbs V = √(VudAbs V ^ 2 + VusAbs V ^ 2) * (VudAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2)) C₁₃,negV:Quotient CKMMatrixSetoidha:¬VubAbs V = 1⊢ VudAbs V = VudAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2) * Real.cos (θ₁₃ V)negV:Quotient CKMMatrixSetoidha:¬VubAbs V = 1⊢ VudAbs V = √(VudAbs V ^ 2 + VusAbs V ^ 2) * (VudAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2)) θ₁₃,negV:Quotient CKMMatrixSetoidha:¬VubAbs V = 1⊢ VudAbs V = VudAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2) * Real.cos (Real.arcsin (S₁₃ V))negV:Quotient CKMMatrixSetoidha:¬VubAbs V = 1⊢ VudAbs V = √(VudAbs V ^ 2 + VusAbs V ^ 2) * (VudAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2)) Real.cos_arcsin,negV:Quotient CKMMatrixSetoidha:¬VubAbs V = 1⊢ VudAbs V = VudAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2) * √(1 - S₁₃ V ^ 2)negV:Quotient CKMMatrixSetoidha:¬VubAbs V = 1⊢ VudAbs V = √(VudAbs V ^ 2 + VusAbs V ^ 2) * (VudAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2)) S₁₃,negV:Quotient CKMMatrixSetoidha:¬VubAbs V = 1⊢ VudAbs V = VudAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2) * √(1 - VubAbs V ^ 2)negV:Quotient CKMMatrixSetoidha:¬VubAbs V = 1⊢ VudAbs V = √(VudAbs V ^ 2 + VusAbs V ^ 2) * (VudAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2)) ← VudAbs_sq_add_VusAbs_sq,negV:Quotient CKMMatrixSetoidha:¬VubAbs V = 1⊢ VudAbs V = VudAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2) * √(VudAbs V ^ 2 + VusAbs V ^ 2)negV:Quotient CKMMatrixSetoidha:¬VubAbs V = 1⊢ VudAbs V = √(VudAbs V ^ 2 + VusAbs V ^ 2) * (VudAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2)) mul_commnegV:Quotient CKMMatrixSetoidha:¬VubAbs V = 1⊢ VudAbs V = √(VudAbs V ^ 2 + VusAbs V ^ 2) * (VudAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2))negV:Quotient CKMMatrixSetoidha:¬VubAbs V = 1⊢ VudAbs V = √(VudAbs V ^ 2 + VusAbs V ^ 2) * (VudAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2))]negV:Quotient CKMMatrixSetoidha:¬VubAbs V = 1⊢ VudAbs V = √(VudAbs V ^ 2 + VusAbs V ^ 2) * (VudAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2))
    exact (mul_div_cancel₀ (VudAbs V) (VAbsub_ne_zero_sqrt_Vud_Vus_ne_zero ha)).symmAll goals completed! 🐙

lemma VusAbs_eq_S₁₂_mul_C₁₃ (V : Quotient CKMMatrixSetoid) : VusAbs V = S₁₂ V * C₁₃ V := byV:Quotient CKMMatrixSetoid⊢ VusAbs V = S₁₂ V * C₁₃ V
  by_cases ha : VubAbs V = 1posV:Quotient CKMMatrixSetoidha:VubAbs V = 1⊢ VusAbs V = S₁₂ V * C₁₃ VnegV:Quotient CKMMatrixSetoidha:¬VubAbs V = 1⊢ VusAbs V = S₁₂ V * C₁₃ V
  ·posV:Quotient CKMMatrixSetoidha:VubAbs V = 1⊢ VusAbs V = S₁₂ V * C₁₃ V rw [C₁₃_of_Vub_eq_one ha,posV:Quotient CKMMatrixSetoidha:VubAbs V = 1⊢ VusAbs V = S₁₂ V * 0posV:Quotient CKMMatrixSetoidha:VubAbs V = 1⊢ VusAbs V = 0 mul_zeroposV:Quotient CKMMatrixSetoidha:VubAbs V = 1⊢ VusAbs V = 0posV:Quotient CKMMatrixSetoidha:VubAbs V = 1⊢ VusAbs V = 0]posV:Quotient CKMMatrixSetoidha:VubAbs V = 1⊢ VusAbs V = 0
    exact VAbs_thd_eq_one_snd_eq_zero haAll goals completed! 🐙
  ·negV:Quotient CKMMatrixSetoidha:¬VubAbs V = 1⊢ VusAbs V = S₁₂ V * C₁₃ V rw [C₁₃,negV:Quotient CKMMatrixSetoidha:¬VubAbs V = 1⊢ VusAbs V = S₁₂ V * Real.cos (θ₁₃ V)negV:Quotient CKMMatrixSetoidha:¬VubAbs V = 1⊢ VusAbs V = √(VudAbs V ^ 2 + VusAbs V ^ 2) * (VusAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2)) θ₁₃,negV:Quotient CKMMatrixSetoidha:¬VubAbs V = 1⊢ VusAbs V = S₁₂ V * Real.cos (Real.arcsin (S₁₃ V))negV:Quotient CKMMatrixSetoidha:¬VubAbs V = 1⊢ VusAbs V = √(VudAbs V ^ 2 + VusAbs V ^ 2) * (VusAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2)) Real.cos_arcsin,negV:Quotient CKMMatrixSetoidha:¬VubAbs V = 1⊢ VusAbs V = S₁₂ V * √(1 - S₁₃ V ^ 2)negV:Quotient CKMMatrixSetoidha:¬VubAbs V = 1⊢ VusAbs V = √(VudAbs V ^ 2 + VusAbs V ^ 2) * (VusAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2)) S₁₂,negV:Quotient CKMMatrixSetoidha:¬VubAbs V = 1⊢ VusAbs V = VusAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2) * √(1 - S₁₃ V ^ 2)negV:Quotient CKMMatrixSetoidha:¬VubAbs V = 1⊢ VusAbs V = √(VudAbs V ^ 2 + VusAbs V ^ 2) * (VusAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2)) S₁₃,negV:Quotient CKMMatrixSetoidha:¬VubAbs V = 1⊢ VusAbs V = VusAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2) * √(1 - VubAbs V ^ 2)negV:Quotient CKMMatrixSetoidha:¬VubAbs V = 1⊢ VusAbs V = √(VudAbs V ^ 2 + VusAbs V ^ 2) * (VusAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2)) ← VudAbs_sq_add_VusAbs_sq,negV:Quotient CKMMatrixSetoidha:¬VubAbs V = 1⊢ VusAbs V = VusAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2) * √(VudAbs V ^ 2 + VusAbs V ^ 2)negV:Quotient CKMMatrixSetoidha:¬VubAbs V = 1⊢ VusAbs V = √(VudAbs V ^ 2 + VusAbs V ^ 2) * (VusAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2)) mul_commnegV:Quotient CKMMatrixSetoidha:¬VubAbs V = 1⊢ VusAbs V = √(VudAbs V ^ 2 + VusAbs V ^ 2) * (VusAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2))negV:Quotient CKMMatrixSetoidha:¬VubAbs V = 1⊢ VusAbs V = √(VudAbs V ^ 2 + VusAbs V ^ 2) * (VusAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2))]negV:Quotient CKMMatrixSetoidha:¬VubAbs V = 1⊢ VusAbs V = √(VudAbs V ^ 2 + VusAbs V ^ 2) * (VusAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2))
    exact (mul_div_cancel₀ (VusAbs V) (VAbsub_ne_zero_sqrt_Vud_Vus_ne_zero ha)).symmAll goals completed! 🐙lemma VubAbs_eq_S₁₃ (V : Quotient CKMMatrixSetoid) : VubAbs V = S₁₃ V := rfl

lemma VcbAbs_eq_S₂₃_mul_C₁₃ (V : Quotient CKMMatrixSetoid) : VcbAbs V = S₂₃ V * C₁₃ V := byV:Quotient CKMMatrixSetoid⊢ VcbAbs V = S₂₃ V * C₁₃ V
  by_cases ha : VubAbs V = 1posV:Quotient CKMMatrixSetoidha:VubAbs V = 1⊢ VcbAbs V = S₂₃ V * C₁₃ VnegV:Quotient CKMMatrixSetoidha:¬VubAbs V = 1⊢ VcbAbs V = S₂₃ V * C₁₃ V
  ·posV:Quotient CKMMatrixSetoidha:VubAbs V = 1⊢ VcbAbs V = S₂₃ V * C₁₃ V rw [C₁₃_of_Vub_eq_one ha,posV:Quotient CKMMatrixSetoidha:VubAbs V = 1⊢ VcbAbs V = S₂₃ V * 0posV:Quotient CKMMatrixSetoidha:VubAbs V = 1⊢ VcbAbs V = 0 mul_zeroposV:Quotient CKMMatrixSetoidha:VubAbs V = 1⊢ VcbAbs V = 0posV:Quotient CKMMatrixSetoidha:VubAbs V = 1⊢ VcbAbs V = 0]posV:Quotient CKMMatrixSetoidha:VubAbs V = 1⊢ VcbAbs V = 0
    exact VAbs_fst_col_eq_one_snd_eq_zero haAll goals completed! 🐙
  ·negV:Quotient CKMMatrixSetoidha:¬VubAbs V = 1⊢ VcbAbs V = S₂₃ V * C₁₃ V rw [S₂₃_of_Vub_ne_one ha,negV:Quotient CKMMatrixSetoidha:¬VubAbs V = 1⊢ VcbAbs V = VcbAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2) * C₁₃ VnegV:Quotient CKMMatrixSetoidha:¬VubAbs V = 1⊢ VcbAbs V = √(VudAbs V ^ 2 + VusAbs V ^ 2) * (VcbAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2)) C₁₃_eq_add_sq,negV:Quotient CKMMatrixSetoidha:¬VubAbs V = 1⊢ VcbAbs V = VcbAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2) * √(VudAbs V ^ 2 + VusAbs V ^ 2)negV:Quotient CKMMatrixSetoidha:¬VubAbs V = 1⊢ VcbAbs V = √(VudAbs V ^ 2 + VusAbs V ^ 2) * (VcbAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2)) mul_commnegV:Quotient CKMMatrixSetoidha:¬VubAbs V = 1⊢ VcbAbs V = √(VudAbs V ^ 2 + VusAbs V ^ 2) * (VcbAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2))negV:Quotient CKMMatrixSetoidha:¬VubAbs V = 1⊢ VcbAbs V = √(VudAbs V ^ 2 + VusAbs V ^ 2) * (VcbAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2))]negV:Quotient CKMMatrixSetoidha:¬VubAbs V = 1⊢ VcbAbs V = √(VudAbs V ^ 2 + VusAbs V ^ 2) * (VcbAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2))
    exact (mul_div_cancel₀ (VcbAbs V) (VAbsub_ne_zero_sqrt_Vud_Vus_ne_zero ha)).symmAll goals completed! 🐙

lemma VtbAbs_eq_C₂₃_mul_C₁₃ (V : Quotient CKMMatrixSetoid) : VtbAbs V = C₂₃ V * C₁₃ V := byV:Quotient CKMMatrixSetoid⊢ VtbAbs V = C₂₃ V * C₁₃ V
  by_cases ha : VubAbs V = 1posV:Quotient CKMMatrixSetoidha:VubAbs V = 1⊢ VtbAbs V = C₂₃ V * C₁₃ VnegV:Quotient CKMMatrixSetoidha:¬VubAbs V = 1⊢ VtbAbs V = C₂₃ V * C₁₃ V
  ·posV:Quotient CKMMatrixSetoidha:VubAbs V = 1⊢ VtbAbs V = C₂₃ V * C₁₃ V rw [C₁₃_of_Vub_eq_one ha,posV:Quotient CKMMatrixSetoidha:VubAbs V = 1⊢ VtbAbs V = C₂₃ V * 0posV:Quotient CKMMatrixSetoidha:VubAbs V = 1⊢ VtbAbs V = 0 mul_zeroposV:Quotient CKMMatrixSetoidha:VubAbs V = 1⊢ VtbAbs V = 0posV:Quotient CKMMatrixSetoidha:VubAbs V = 1⊢ VtbAbs V = 0]posV:Quotient CKMMatrixSetoidha:VubAbs V = 1⊢ VtbAbs V = 0
    exact VAbs_fst_col_eq_one_thd_eq_zero haAll goals completed! 🐙
  ·negV:Quotient CKMMatrixSetoidha:¬VubAbs V = 1⊢ VtbAbs V = C₂₃ V * C₁₃ V rw [C₂₃_of_Vub_ne_one ha,negV:Quotient CKMMatrixSetoidha:¬VubAbs V = 1⊢ VtbAbs V = VtbAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2) * C₁₃ VnegV:Quotient CKMMatrixSetoidha:¬VubAbs V = 1⊢ VtbAbs V = √(VudAbs V ^ 2 + VusAbs V ^ 2) * (VtbAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2)) C₁₃_eq_add_sq,negV:Quotient CKMMatrixSetoidha:¬VubAbs V = 1⊢ VtbAbs V = VtbAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2) * √(VudAbs V ^ 2 + VusAbs V ^ 2)negV:Quotient CKMMatrixSetoidha:¬VubAbs V = 1⊢ VtbAbs V = √(VudAbs V ^ 2 + VusAbs V ^ 2) * (VtbAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2)) mul_commnegV:Quotient CKMMatrixSetoidha:¬VubAbs V = 1⊢ VtbAbs V = √(VudAbs V ^ 2 + VusAbs V ^ 2) * (VtbAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2))negV:Quotient CKMMatrixSetoidha:¬VubAbs V = 1⊢ VtbAbs V = √(VudAbs V ^ 2 + VusAbs V ^ 2) * (VtbAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2))]negV:Quotient CKMMatrixSetoidha:¬VubAbs V = 1⊢ VtbAbs V = √(VudAbs V ^ 2 + VusAbs V ^ 2) * (VtbAbs V / √(VudAbs V ^ 2 + VusAbs V ^ 2))
    exact (mul_div_cancel₀ (VtbAbs V) (VAbsub_ne_zero_sqrt_Vud_Vus_ne_zero ha)).symmAll goals completed! 🐙

lemma VubAbs_of_cos_θ₁₃_zero {V : Quotient CKMMatrixSetoid} (h1 : Real.cos (θ₁₃ V) = 0) :
    VubAbs V = 1 := byV:Quotient CKMMatrixSetoidh1:Real.cos (θ₁₃ V) = 0⊢ VubAbs V = 1
  rw [VubAbs_eq_S₁₃V:Quotient CKMMatrixSetoidh1:Real.cos (θ₁₃ V) = 0⊢ S₁₃ V = 1V:Quotient CKMMatrixSetoidh1:Real.cos (θ₁₃ V) = 0⊢ S₁₃ V = 1]V:Quotient CKMMatrixSetoidh1:Real.cos (θ₁₃ V) = 0⊢ S₁₃ V = 1
  rw [θ₁₃,V:Quotient CKMMatrixSetoidh1:Real.cos (Real.arcsin (S₁₃ V)) = 0⊢ S₁₃ V = 1V:Quotient CKMMatrixSetoidh1:1 - S₁₃ V ^ 2 ≤ 0⊢ S₁₃ V = 1 Real.cos_arcsin,V:Quotient CKMMatrixSetoidh1:√(1 - S₁₃ V ^ 2) = 0⊢ S₁₃ V = 1V:Quotient CKMMatrixSetoidh1:1 - S₁₃ V ^ 2 ≤ 0⊢ S₁₃ V = 1 Real.sqrt_eq_zero'V:Quotient CKMMatrixSetoidh1:1 - S₁₃ V ^ 2 ≤ 0⊢ S₁₃ V = 1V:Quotient CKMMatrixSetoidh1:1 - S₁₃ V ^ 2 ≤ 0⊢ S₁₃ V = 1] at h1V:Quotient CKMMatrixSetoidh1:1 - S₁₃ V ^ 2 ≤ 0⊢ S₁₃ V = 1
  refine le_antisymm (S₁₃_leq_one V) ?_V:Quotient CKMMatrixSetoidh1:1 - S₁₃ V ^ 2 ≤ 0⊢ 1 ≤ S₁₃ V
  nlinarith [S₁₃_nonneg V, S₁₃_leq_one V, h1]All goals completed! 🐙

lemma Vs_zero_iff_cos_sin_zero (V : CKMMatrix) :
    VudAbs ⟦V⟧ = 0 ∨ VubAbs ⟦V⟧ = 0 ∨ VusAbs ⟦V⟧ = 0 ∨ VcbAbs ⟦V⟧ = 0 ∨ VtbAbs ⟦V⟧ = 0
    ↔ Real.cos (θ₁₂ ⟦V⟧) = 0 ∨ Real.cos (θ₁₃ ⟦V⟧) = 0 ∨ Real.cos (θ₂₃ ⟦V⟧) = 0 ∨
      Real.sin (θ₁₂ ⟦V⟧) = 0 ∨ Real.sin (θ₁₃ ⟦V⟧) = 0 ∨ Real.sin (θ₂₃ ⟦V⟧) = 0 := byV:CKMMatrix⊢ VudAbs ⟦V⟧ = 0 ∨ VubAbs ⟦V⟧ = 0 ∨ VusAbs ⟦V⟧ = 0 ∨ VcbAbs ⟦V⟧ = 0 ∨ VtbAbs ⟦V⟧ = 0 ↔
  Real.cos (θ₁₂ ⟦V⟧) = 0 ∨
    Real.cos (θ₁₃ ⟦V⟧) = 0 ∨
      Real.cos (θ₂₃ ⟦V⟧) = 0 ∨ Real.sin (θ₁₂ ⟦V⟧) = 0 ∨ Real.sin (θ₁₃ ⟦V⟧) = 0 ∨ Real.sin (θ₂₃ ⟦V⟧) = 0
  rw [VudAbs_eq_C₁₂_mul_C₁₃,V:CKMMatrix⊢ C₁₂ ⟦V⟧ * C₁₃ ⟦V⟧ = 0 ∨ VubAbs ⟦V⟧ = 0 ∨ VusAbs ⟦V⟧ = 0 ∨ VcbAbs ⟦V⟧ = 0 ∨ VtbAbs ⟦V⟧ = 0 ↔
  Real.cos (θ₁₂ ⟦V⟧) = 0 ∨
    Real.cos (θ₁₃ ⟦V⟧) = 0 ∨
      Real.cos (θ₂₃ ⟦V⟧) = 0 ∨ Real.sin (θ₁₂ ⟦V⟧) = 0 ∨ Real.sin (θ₁₃ ⟦V⟧) = 0 ∨ Real.sin (θ₂₃ ⟦V⟧) = 0V:CKMMatrix⊢ Real.cos (θ₁₂ ⟦V⟧) * Real.cos (θ₁₃ ⟦V⟧) = 0 ∨
    S₁₃ ⟦V⟧ = 0 ∨
      S₁₂ ⟦V⟧ * Real.cos (θ₁₃ ⟦V⟧) = 0 ∨
        S₂₃ ⟦V⟧ * Real.cos (θ₁₃ ⟦V⟧) = 0 ∨ Real.cos (θ₂₃ ⟦V⟧) * Real.cos (θ₁₃ ⟦V⟧) = 0 ↔
  Real.cos (θ₁₂ ⟦V⟧) = 0 ∨ Real.cos (θ₁₃ ⟦V⟧) = 0 ∨ Real.cos (θ₂₃ ⟦V⟧) = 0 ∨ S₁₂ ⟦V⟧ = 0 ∨ S₁₃ ⟦V⟧ = 0 ∨ S₂₃ ⟦V⟧ = 0 VubAbs_eq_S₁₃,V:CKMMatrix⊢ C₁₂ ⟦V⟧ * C₁₃ ⟦V⟧ = 0 ∨ S₁₃ ⟦V⟧ = 0 ∨ VusAbs ⟦V⟧ = 0 ∨ VcbAbs ⟦V⟧ = 0 ∨ VtbAbs ⟦V⟧ = 0 ↔
  Real.cos (θ₁₂ ⟦V⟧) = 0 ∨
    Real.cos (θ₁₃ ⟦V⟧) = 0 ∨
      Real.cos (θ₂₃ ⟦V⟧) = 0 ∨ Real.sin (θ₁₂ ⟦V⟧) = 0 ∨ Real.sin (θ₁₃ ⟦V⟧) = 0 ∨ Real.sin (θ₂₃ ⟦V⟧) = 0V:CKMMatrix⊢ Real.cos (θ₁₂ ⟦V⟧) * Real.cos (θ₁₃ ⟦V⟧) = 0 ∨
    S₁₃ ⟦V⟧ = 0 ∨
      S₁₂ ⟦V⟧ * Real.cos (θ₁₃ ⟦V⟧) = 0 ∨
        S₂₃ ⟦V⟧ * Real.cos (θ₁₃ ⟦V⟧) = 0 ∨ Real.cos (θ₂₃ ⟦V⟧) * Real.cos (θ₁₃ ⟦V⟧) = 0 ↔
  Real.cos (θ₁₂ ⟦V⟧) = 0 ∨ Real.cos (θ₁₃ ⟦V⟧) = 0 ∨ Real.cos (θ₂₃ ⟦V⟧) = 0 ∨ S₁₂ ⟦V⟧ = 0 ∨ S₁₃ ⟦V⟧ = 0 ∨ S₂₃ ⟦V⟧ = 0 VusAbs_eq_S₁₂_mul_C₁₃,V:CKMMatrix⊢ C₁₂ ⟦V⟧ * C₁₃ ⟦V⟧ = 0 ∨ S₁₃ ⟦V⟧ = 0 ∨ S₁₂ ⟦V⟧ * C₁₃ ⟦V⟧ = 0 ∨ VcbAbs ⟦V⟧ = 0 ∨ VtbAbs ⟦V⟧ = 0 ↔
  Real.cos (θ₁₂ ⟦V⟧) = 0 ∨
    Real.cos (θ₁₃ ⟦V⟧) = 0 ∨
      Real.cos (θ₂₃ ⟦V⟧) = 0 ∨ Real.sin (θ₁₂ ⟦V⟧) = 0 ∨ Real.sin (θ₁₃ ⟦V⟧) = 0 ∨ Real.sin (θ₂₃ ⟦V⟧) = 0V:CKMMatrix⊢ Real.cos (θ₁₂ ⟦V⟧) * Real.cos (θ₁₃ ⟦V⟧) = 0 ∨
    S₁₃ ⟦V⟧ = 0 ∨
      S₁₂ ⟦V⟧ * Real.cos (θ₁₃ ⟦V⟧) = 0 ∨
        S₂₃ ⟦V⟧ * Real.cos (θ₁₃ ⟦V⟧) = 0 ∨ Real.cos (θ₂₃ ⟦V⟧) * Real.cos (θ₁₃ ⟦V⟧) = 0 ↔
  Real.cos (θ₁₂ ⟦V⟧) = 0 ∨ Real.cos (θ₁₃ ⟦V⟧) = 0 ∨ Real.cos (θ₂₃ ⟦V⟧) = 0 ∨ S₁₂ ⟦V⟧ = 0 ∨ S₁₃ ⟦V⟧ = 0 ∨ S₂₃ ⟦V⟧ = 0 VcbAbs_eq_S₂₃_mul_C₁₃,V:CKMMatrix⊢ C₁₂ ⟦V⟧ * C₁₃ ⟦V⟧ = 0 ∨ S₁₃ ⟦V⟧ = 0 ∨ S₁₂ ⟦V⟧ * C₁₃ ⟦V⟧ = 0 ∨ S₂₃ ⟦V⟧ * C₁₃ ⟦V⟧ = 0 ∨ VtbAbs ⟦V⟧ = 0 ↔
  Real.cos (θ₁₂ ⟦V⟧) = 0 ∨
    Real.cos (θ₁₃ ⟦V⟧) = 0 ∨
      Real.cos (θ₂₃ ⟦V⟧) = 0 ∨ Real.sin (θ₁₂ ⟦V⟧) = 0 ∨ Real.sin (θ₁₃ ⟦V⟧) = 0 ∨ Real.sin (θ₂₃ ⟦V⟧) = 0V:CKMMatrix⊢ Real.cos (θ₁₂ ⟦V⟧) * Real.cos (θ₁₃ ⟦V⟧) = 0 ∨
    S₁₃ ⟦V⟧ = 0 ∨
      S₁₂ ⟦V⟧ * Real.cos (θ₁₃ ⟦V⟧) = 0 ∨
        S₂₃ ⟦V⟧ * Real.cos (θ₁₃ ⟦V⟧) = 0 ∨ Real.cos (θ₂₃ ⟦V⟧) * Real.cos (θ₁₃ ⟦V⟧) = 0 ↔
  Real.cos (θ₁₂ ⟦V⟧) = 0 ∨ Real.cos (θ₁₃ ⟦V⟧) = 0 ∨ Real.cos (θ₂₃ ⟦V⟧) = 0 ∨ S₁₂ ⟦V⟧ = 0 ∨ S₁₃ ⟦V⟧ = 0 ∨ S₂₃ ⟦V⟧ = 0
    VtbAbs_eq_C₂₃_mul_C₁₃,V:CKMMatrix⊢ C₁₂ ⟦V⟧ * C₁₃ ⟦V⟧ = 0 ∨ S₁₃ ⟦V⟧ = 0 ∨ S₁₂ ⟦V⟧ * C₁₃ ⟦V⟧ = 0 ∨ S₂₃ ⟦V⟧ * C₁₃ ⟦V⟧ = 0 ∨ C₂₃ ⟦V⟧ * C₁₃ ⟦V⟧ = 0 ↔
  Real.cos (θ₁₂ ⟦V⟧) = 0 ∨
    Real.cos (θ₁₃ ⟦V⟧) = 0 ∨
      Real.cos (θ₂₃ ⟦V⟧) = 0 ∨ Real.sin (θ₁₂ ⟦V⟧) = 0 ∨ Real.sin (θ₁₃ ⟦V⟧) = 0 ∨ Real.sin (θ₂₃ ⟦V⟧) = 0V:CKMMatrix⊢ Real.cos (θ₁₂ ⟦V⟧) * Real.cos (θ₁₃ ⟦V⟧) = 0 ∨
    S₁₃ ⟦V⟧ = 0 ∨
      S₁₂ ⟦V⟧ * Real.cos (θ₁₃ ⟦V⟧) = 0 ∨
        S₂₃ ⟦V⟧ * Real.cos (θ₁₃ ⟦V⟧) = 0 ∨ Real.cos (θ₂₃ ⟦V⟧) * Real.cos (θ₁₃ ⟦V⟧) = 0 ↔
  Real.cos (θ₁₂ ⟦V⟧) = 0 ∨ Real.cos (θ₁₃ ⟦V⟧) = 0 ∨ Real.cos (θ₂₃ ⟦V⟧) = 0 ∨ S₁₂ ⟦V⟧ = 0 ∨ S₁₃ ⟦V⟧ = 0 ∨ S₂₃ ⟦V⟧ = 0 C₁₂,V:CKMMatrix⊢ Real.cos (θ₁₂ ⟦V⟧) * C₁₃ ⟦V⟧ = 0 ∨ S₁₃ ⟦V⟧ = 0 ∨ S₁₂ ⟦V⟧ * C₁₃ ⟦V⟧ = 0 ∨ S₂₃ ⟦V⟧ * C₁₃ ⟦V⟧ = 0 ∨ C₂₃ ⟦V⟧ * C₁₃ ⟦V⟧ = 0 ↔
  Real.cos (θ₁₂ ⟦V⟧) = 0 ∨
    Real.cos (θ₁₃ ⟦V⟧) = 0 ∨
      Real.cos (θ₂₃ ⟦V⟧) = 0 ∨ Real.sin (θ₁₂ ⟦V⟧) = 0 ∨ Real.sin (θ₁₃ ⟦V⟧) = 0 ∨ Real.sin (θ₂₃ ⟦V⟧) = 0V:CKMMatrix⊢ Real.cos (θ₁₂ ⟦V⟧) * Real.cos (θ₁₃ ⟦V⟧) = 0 ∨
    S₁₃ ⟦V⟧ = 0 ∨
      S₁₂ ⟦V⟧ * Real.cos (θ₁₃ ⟦V⟧) = 0 ∨
        S₂₃ ⟦V⟧ * Real.cos (θ₁₃ ⟦V⟧) = 0 ∨ Real.cos (θ₂₃ ⟦V⟧) * Real.cos (θ₁₃ ⟦V⟧) = 0 ↔
  Real.cos (θ₁₂ ⟦V⟧) = 0 ∨ Real.cos (θ₁₃ ⟦V⟧) = 0 ∨ Real.cos (θ₂₃ ⟦V⟧) = 0 ∨ S₁₂ ⟦V⟧ = 0 ∨ S₁₃ ⟦V⟧ = 0 ∨ S₂₃ ⟦V⟧ = 0 C₁₃,V:CKMMatrix⊢ Real.cos (θ₁₂ ⟦V⟧) * Real.cos (θ₁₃ ⟦V⟧) = 0 ∨
    S₁₃ ⟦V⟧ = 0 ∨
      S₁₂ ⟦V⟧ * Real.cos (θ₁₃ ⟦V⟧) = 0 ∨ S₂₃ ⟦V⟧ * Real.cos (θ₁₃ ⟦V⟧) = 0 ∨ C₂₃ ⟦V⟧ * Real.cos (θ₁₃ ⟦V⟧) = 0 ↔
  Real.cos (θ₁₂ ⟦V⟧) = 0 ∨
    Real.cos (θ₁₃ ⟦V⟧) = 0 ∨
      Real.cos (θ₂₃ ⟦V⟧) = 0 ∨ Real.sin (θ₁₂ ⟦V⟧) = 0 ∨ Real.sin (θ₁₃ ⟦V⟧) = 0 ∨ Real.sin (θ₂₃ ⟦V⟧) = 0V:CKMMatrix⊢ Real.cos (θ₁₂ ⟦V⟧) * Real.cos (θ₁₃ ⟦V⟧) = 0 ∨
    S₁₃ ⟦V⟧ = 0 ∨
      S₁₂ ⟦V⟧ * Real.cos (θ₁₃ ⟦V⟧) = 0 ∨
        S₂₃ ⟦V⟧ * Real.cos (θ₁₃ ⟦V⟧) = 0 ∨ Real.cos (θ₂₃ ⟦V⟧) * Real.cos (θ₁₃ ⟦V⟧) = 0 ↔
  Real.cos (θ₁₂ ⟦V⟧) = 0 ∨ Real.cos (θ₁₃ ⟦V⟧) = 0 ∨ Real.cos (θ₂₃ ⟦V⟧) = 0 ∨ S₁₂ ⟦V⟧ = 0 ∨ S₁₃ ⟦V⟧ = 0 ∨ S₂₃ ⟦V⟧ = 0 C₂₃,V:CKMMatrix⊢ Real.cos (θ₁₂ ⟦V⟧) * Real.cos (θ₁₃ ⟦V⟧) = 0 ∨
    S₁₃ ⟦V⟧ = 0 ∨
      S₁₂ ⟦V⟧ * Real.cos (θ₁₃ ⟦V⟧) = 0 ∨
        S₂₃ ⟦V⟧ * Real.cos (θ₁₃ ⟦V⟧) = 0 ∨ Real.cos (θ₂₃ ⟦V⟧) * Real.cos (θ₁₃ ⟦V⟧) = 0 ↔
  Real.cos (θ₁₂ ⟦V⟧) = 0 ∨
    Real.cos (θ₁₃ ⟦V⟧) = 0 ∨
      Real.cos (θ₂₃ ⟦V⟧) = 0 ∨ Real.sin (θ₁₂ ⟦V⟧) = 0 ∨ Real.sin (θ₁₃ ⟦V⟧) = 0 ∨ Real.sin (θ₂₃ ⟦V⟧) = 0V:CKMMatrix⊢ Real.cos (θ₁₂ ⟦V⟧) * Real.cos (θ₁₃ ⟦V⟧) = 0 ∨
    S₁₃ ⟦V⟧ = 0 ∨
      S₁₂ ⟦V⟧ * Real.cos (θ₁₃ ⟦V⟧) = 0 ∨
        S₂₃ ⟦V⟧ * Real.cos (θ₁₃ ⟦V⟧) = 0 ∨ Real.cos (θ₂₃ ⟦V⟧) * Real.cos (θ₁₃ ⟦V⟧) = 0 ↔
  Real.cos (θ₁₂ ⟦V⟧) = 0 ∨ Real.cos (θ₁₃ ⟦V⟧) = 0 ∨ Real.cos (θ₂₃ ⟦V⟧) = 0 ∨ S₁₂ ⟦V⟧ = 0 ∨ S₁₃ ⟦V⟧ = 0 ∨ S₂₃ ⟦V⟧ = 0 S₁₂_eq_sin_θ₁₂,V:CKMMatrix⊢ Real.cos (θ₁₂ ⟦V⟧) * Real.cos (θ₁₃ ⟦V⟧) = 0 ∨
    S₁₃ ⟦V⟧ = 0 ∨
      S₁₂ ⟦V⟧ * Real.cos (θ₁₃ ⟦V⟧) = 0 ∨
        S₂₃ ⟦V⟧ * Real.cos (θ₁₃ ⟦V⟧) = 0 ∨ Real.cos (θ₂₃ ⟦V⟧) * Real.cos (θ₁₃ ⟦V⟧) = 0 ↔
  Real.cos (θ₁₂ ⟦V⟧) = 0 ∨
    Real.cos (θ₁₃ ⟦V⟧) = 0 ∨ Real.cos (θ₂₃ ⟦V⟧) = 0 ∨ S₁₂ ⟦V⟧ = 0 ∨ Real.sin (θ₁₃ ⟦V⟧) = 0 ∨ Real.sin (θ₂₃ ⟦V⟧) = 0V:CKMMatrix⊢ Real.cos (θ₁₂ ⟦V⟧) * Real.cos (θ₁₃ ⟦V⟧) = 0 ∨
    S₁₃ ⟦V⟧ = 0 ∨
      S₁₂ ⟦V⟧ * Real.cos (θ₁₃ ⟦V⟧) = 0 ∨
        S₂₃ ⟦V⟧ * Real.cos (θ₁₃ ⟦V⟧) = 0 ∨ Real.cos (θ₂₃ ⟦V⟧) * Real.cos (θ₁₃ ⟦V⟧) = 0 ↔
  Real.cos (θ₁₂ ⟦V⟧) = 0 ∨ Real.cos (θ₁₃ ⟦V⟧) = 0 ∨ Real.cos (θ₂₃ ⟦V⟧) = 0 ∨ S₁₂ ⟦V⟧ = 0 ∨ S₁₃ ⟦V⟧ = 0 ∨ S₂₃ ⟦V⟧ = 0 S₂₃_eq_sin_θ₂₃,V:CKMMatrix⊢ Real.cos (θ₁₂ ⟦V⟧) * Real.cos (θ₁₃ ⟦V⟧) = 0 ∨
    S₁₃ ⟦V⟧ = 0 ∨
      S₁₂ ⟦V⟧ * Real.cos (θ₁₃ ⟦V⟧) = 0 ∨
        S₂₃ ⟦V⟧ * Real.cos (θ₁₃ ⟦V⟧) = 0 ∨ Real.cos (θ₂₃ ⟦V⟧) * Real.cos (θ₁₃ ⟦V⟧) = 0 ↔
  Real.cos (θ₁₂ ⟦V⟧) = 0 ∨
    Real.cos (θ₁₃ ⟦V⟧) = 0 ∨ Real.cos (θ₂₃ ⟦V⟧) = 0 ∨ S₁₂ ⟦V⟧ = 0 ∨ Real.sin (θ₁₃ ⟦V⟧) = 0 ∨ S₂₃ ⟦V⟧ = 0V:CKMMatrix⊢ Real.cos (θ₁₂ ⟦V⟧) * Real.cos (θ₁₃ ⟦V⟧) = 0 ∨
    S₁₃ ⟦V⟧ = 0 ∨
      S₁₂ ⟦V⟧ * Real.cos (θ₁₃ ⟦V⟧) = 0 ∨
        S₂₃ ⟦V⟧ * Real.cos (θ₁₃ ⟦V⟧) = 0 ∨ Real.cos (θ₂₃ ⟦V⟧) * Real.cos (θ₁₃ ⟦V⟧) = 0 ↔
  Real.cos (θ₁₂ ⟦V⟧) = 0 ∨ Real.cos (θ₁₃ ⟦V⟧) = 0 ∨ Real.cos (θ₂₃ ⟦V⟧) = 0 ∨ S₁₂ ⟦V⟧ = 0 ∨ S₁₃ ⟦V⟧ = 0 ∨ S₂₃ ⟦V⟧ = 0 S₁₃_eq_sin_θ₁₃V:CKMMatrix⊢ Real.cos (θ₁₂ ⟦V⟧) * Real.cos (θ₁₃ ⟦V⟧) = 0 ∨
    S₁₃ ⟦V⟧ = 0 ∨
      S₁₂ ⟦V⟧ * Real.cos (θ₁₃ ⟦V⟧) = 0 ∨
        S₂₃ ⟦V⟧ * Real.cos (θ₁₃ ⟦V⟧) = 0 ∨ Real.cos (θ₂₃ ⟦V⟧) * Real.cos (θ₁₃ ⟦V⟧) = 0 ↔
  Real.cos (θ₁₂ ⟦V⟧) = 0 ∨ Real.cos (θ₁₃ ⟦V⟧) = 0 ∨ Real.cos (θ₂₃ ⟦V⟧) = 0 ∨ S₁₂ ⟦V⟧ = 0 ∨ S₁₃ ⟦V⟧ = 0 ∨ S₂₃ ⟦V⟧ = 0V:CKMMatrix⊢ Real.cos (θ₁₂ ⟦V⟧) * Real.cos (θ₁₃ ⟦V⟧) = 0 ∨
    S₁₃ ⟦V⟧ = 0 ∨
      S₁₂ ⟦V⟧ * Real.cos (θ₁₃ ⟦V⟧) = 0 ∨
        S₂₃ ⟦V⟧ * Real.cos (θ₁₃ ⟦V⟧) = 0 ∨ Real.cos (θ₂₃ ⟦V⟧) * Real.cos (θ₁₃ ⟦V⟧) = 0 ↔
  Real.cos (θ₁₂ ⟦V⟧) = 0 ∨ Real.cos (θ₁₃ ⟦V⟧) = 0 ∨ Real.cos (θ₂₃ ⟦V⟧) = 0 ∨ S₁₂ ⟦V⟧ = 0 ∨ S₁₃ ⟦V⟧ = 0 ∨ S₂₃ ⟦V⟧ = 0]V:CKMMatrix⊢ Real.cos (θ₁₂ ⟦V⟧) * Real.cos (θ₁₃ ⟦V⟧) = 0 ∨
    S₁₃ ⟦V⟧ = 0 ∨
      S₁₂ ⟦V⟧ * Real.cos (θ₁₃ ⟦V⟧) = 0 ∨
        S₂₃ ⟦V⟧ * Real.cos (θ₁₃ ⟦V⟧) = 0 ∨ Real.cos (θ₂₃ ⟦V⟧) * Real.cos (θ₁₃ ⟦V⟧) = 0 ↔
  Real.cos (θ₁₂ ⟦V⟧) = 0 ∨ Real.cos (θ₁₃ ⟦V⟧) = 0 ∨ Real.cos (θ₂₃ ⟦V⟧) = 0 ∨ S₁₂ ⟦V⟧ = 0 ∨ S₁₃ ⟦V⟧ = 0 ∨ S₂₃ ⟦V⟧ = 0
  aesopAll goals completed! 🐙lemma mulExpδ₁₃_on_param_δ₁₃ (V : CKMMatrix) (δ₁₃ : ℝ) :
    mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ =
    sin (θ₁₂ ⟦V⟧) * cos (θ₁₃ ⟦V⟧) ^ 2 * sin (θ₂₃ ⟦V⟧) * sin (θ₁₃ ⟦V⟧)
    * cos (θ₁₂ ⟦V⟧) * cos (θ₂₃ ⟦V⟧) * cexp (I * δ₁₃) := byV:CKMMatrixδ₁₃:ℝ⊢ mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ =
  sin ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) ^ 2 * sin ↑(θ₂₃ ⟦V⟧) * sin ↑(θ₁₃ ⟦V⟧) * cos ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₂₃ ⟦V⟧) *
    cexp (I * ↑δ₁₃)
  refine mulExpδ₁₃_eq _ _ _ _ ?_ (Real.cos_arcsin_nonneg _) ?_ (Real.cos_arcsin_nonneg _)refine_1V:CKMMatrixδ₁₃:ℝ⊢ 0 ≤ Real.sin (θ₁₂ ⟦V⟧)refine_2V:CKMMatrixδ₁₃:ℝ⊢ 0 ≤ Real.sin (θ₂₃ ⟦V⟧)
  ·refine_1V:CKMMatrixδ₁₃:ℝ⊢ 0 ≤ Real.sin (θ₁₂ ⟦V⟧) exact le_of_le_of_eq (S₁₂_nonneg _) (S₁₂_eq_sin_θ₁₂ _).symmAll goals completed! 🐙
  ·refine_2V:CKMMatrixδ₁₃:ℝ⊢ 0 ≤ Real.sin (θ₂₃ ⟦V⟧) exact le_of_le_of_eq (S₂₃_nonneg _) (S₂₃_eq_sin_θ₂₃ _).symmAll goals completed! 🐙

lemma mulExpδ₁₃_on_param_eq_zero_iff (V : CKMMatrix) (δ₁₃ : ℝ) :
    mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ = 0 ↔
      VudAbs ⟦V⟧ = 0 ∨ VubAbs ⟦V⟧ = 0 ∨ VusAbs ⟦V⟧ = 0 ∨ VcbAbs ⟦V⟧ = 0 ∨ VtbAbs ⟦V⟧ = 0 := byV:CKMMatrixδ₁₃:ℝ⊢ mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ = 0 ↔
  VudAbs ⟦V⟧ = 0 ∨ VubAbs ⟦V⟧ = 0 ∨ VusAbs ⟦V⟧ = 0 ∨ VcbAbs ⟦V⟧ = 0 ∨ VtbAbs ⟦V⟧ = 0
  rw [VudAbs_eq_C₁₂_mul_C₁₃,V:CKMMatrixδ₁₃:ℝ⊢ mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ = 0 ↔
  C₁₂ ⟦V⟧ * C₁₃ ⟦V⟧ = 0 ∨ VubAbs ⟦V⟧ = 0 ∨ VusAbs ⟦V⟧ = 0 ∨ VcbAbs ⟦V⟧ = 0 ∨ VtbAbs ⟦V⟧ = 0V:CKMMatrixδ₁₃:ℝ⊢ mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ = 0 ↔
  ↑(C₁₂ ⟦V⟧ * C₁₃ ⟦V⟧) = ↑0 ∨
    ↑(S₁₃ ⟦V⟧) = ↑0 ∨ ↑(S₁₂ ⟦V⟧ * C₁₃ ⟦V⟧) = ↑0 ∨ ↑(S₂₃ ⟦V⟧ * C₁₃ ⟦V⟧) = ↑0 ∨ ↑(C₂₃ ⟦V⟧ * C₁₃ ⟦V⟧) = ↑0 VubAbs_eq_S₁₃,V:CKMMatrixδ₁₃:ℝ⊢ mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ = 0 ↔
  C₁₂ ⟦V⟧ * C₁₃ ⟦V⟧ = 0 ∨ S₁₃ ⟦V⟧ = 0 ∨ VusAbs ⟦V⟧ = 0 ∨ VcbAbs ⟦V⟧ = 0 ∨ VtbAbs ⟦V⟧ = 0V:CKMMatrixδ₁₃:ℝ⊢ mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ = 0 ↔
  ↑(C₁₂ ⟦V⟧ * C₁₃ ⟦V⟧) = ↑0 ∨
    ↑(S₁₃ ⟦V⟧) = ↑0 ∨ ↑(S₁₂ ⟦V⟧ * C₁₃ ⟦V⟧) = ↑0 ∨ ↑(S₂₃ ⟦V⟧ * C₁₃ ⟦V⟧) = ↑0 ∨ ↑(C₂₃ ⟦V⟧ * C₁₃ ⟦V⟧) = ↑0 VusAbs_eq_S₁₂_mul_C₁₃,V:CKMMatrixδ₁₃:ℝ⊢ mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ = 0 ↔
  C₁₂ ⟦V⟧ * C₁₃ ⟦V⟧ = 0 ∨ S₁₃ ⟦V⟧ = 0 ∨ S₁₂ ⟦V⟧ * C₁₃ ⟦V⟧ = 0 ∨ VcbAbs ⟦V⟧ = 0 ∨ VtbAbs ⟦V⟧ = 0V:CKMMatrixδ₁₃:ℝ⊢ mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ = 0 ↔
  ↑(C₁₂ ⟦V⟧ * C₁₃ ⟦V⟧) = ↑0 ∨
    ↑(S₁₃ ⟦V⟧) = ↑0 ∨ ↑(S₁₂ ⟦V⟧ * C₁₃ ⟦V⟧) = ↑0 ∨ ↑(S₂₃ ⟦V⟧ * C₁₃ ⟦V⟧) = ↑0 ∨ ↑(C₂₃ ⟦V⟧ * C₁₃ ⟦V⟧) = ↑0 VcbAbs_eq_S₂₃_mul_C₁₃,V:CKMMatrixδ₁₃:ℝ⊢ mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ = 0 ↔
  C₁₂ ⟦V⟧ * C₁₃ ⟦V⟧ = 0 ∨ S₁₃ ⟦V⟧ = 0 ∨ S₁₂ ⟦V⟧ * C₁₃ ⟦V⟧ = 0 ∨ S₂₃ ⟦V⟧ * C₁₃ ⟦V⟧ = 0 ∨ VtbAbs ⟦V⟧ = 0V:CKMMatrixδ₁₃:ℝ⊢ mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ = 0 ↔
  ↑(C₁₂ ⟦V⟧ * C₁₃ ⟦V⟧) = ↑0 ∨
    ↑(S₁₃ ⟦V⟧) = ↑0 ∨ ↑(S₁₂ ⟦V⟧ * C₁₃ ⟦V⟧) = ↑0 ∨ ↑(S₂₃ ⟦V⟧ * C₁₃ ⟦V⟧) = ↑0 ∨ ↑(C₂₃ ⟦V⟧ * C₁₃ ⟦V⟧) = ↑0
    VtbAbs_eq_C₂₃_mul_C₁₃,V:CKMMatrixδ₁₃:ℝ⊢ mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ = 0 ↔
  C₁₂ ⟦V⟧ * C₁₃ ⟦V⟧ = 0 ∨ S₁₃ ⟦V⟧ = 0 ∨ S₁₂ ⟦V⟧ * C₁₃ ⟦V⟧ = 0 ∨ S₂₃ ⟦V⟧ * C₁₃ ⟦V⟧ = 0 ∨ C₂₃ ⟦V⟧ * C₁₃ ⟦V⟧ = 0V:CKMMatrixδ₁₃:ℝ⊢ mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ = 0 ↔
  ↑(C₁₂ ⟦V⟧ * C₁₃ ⟦V⟧) = ↑0 ∨
    ↑(S₁₃ ⟦V⟧) = ↑0 ∨ ↑(S₁₂ ⟦V⟧ * C₁₃ ⟦V⟧) = ↑0 ∨ ↑(S₂₃ ⟦V⟧ * C₁₃ ⟦V⟧) = ↑0 ∨ ↑(C₂₃ ⟦V⟧ * C₁₃ ⟦V⟧) = ↑0 ← ofReal_inj,V:CKMMatrixδ₁₃:ℝ⊢ mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ = 0 ↔
  ↑(C₁₂ ⟦V⟧ * C₁₃ ⟦V⟧) = ↑0 ∨ S₁₃ ⟦V⟧ = 0 ∨ S₁₂ ⟦V⟧ * C₁₃ ⟦V⟧ = 0 ∨ S₂₃ ⟦V⟧ * C₁₃ ⟦V⟧ = 0 ∨ C₂₃ ⟦V⟧ * C₁₃ ⟦V⟧ = 0V:CKMMatrixδ₁₃:ℝ⊢ mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ = 0 ↔
  ↑(C₁₂ ⟦V⟧ * C₁₃ ⟦V⟧) = ↑0 ∨
    ↑(S₁₃ ⟦V⟧) = ↑0 ∨ ↑(S₁₂ ⟦V⟧ * C₁₃ ⟦V⟧) = ↑0 ∨ ↑(S₂₃ ⟦V⟧ * C₁₃ ⟦V⟧) = ↑0 ∨ ↑(C₂₃ ⟦V⟧ * C₁₃ ⟦V⟧) = ↑0
    ← ofReal_inj,V:CKMMatrixδ₁₃:ℝ⊢ mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ = 0 ↔
  ↑(C₁₂ ⟦V⟧ * C₁₃ ⟦V⟧) = ↑0 ∨ ↑(S₁₃ ⟦V⟧) = ↑0 ∨ S₁₂ ⟦V⟧ * C₁₃ ⟦V⟧ = 0 ∨ S₂₃ ⟦V⟧ * C₁₃ ⟦V⟧ = 0 ∨ C₂₃ ⟦V⟧ * C₁₃ ⟦V⟧ = 0V:CKMMatrixδ₁₃:ℝ⊢ mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ = 0 ↔
  ↑(C₁₂ ⟦V⟧ * C₁₃ ⟦V⟧) = ↑0 ∨
    ↑(S₁₃ ⟦V⟧) = ↑0 ∨ ↑(S₁₂ ⟦V⟧ * C₁₃ ⟦V⟧) = ↑0 ∨ ↑(S₂₃ ⟦V⟧ * C₁₃ ⟦V⟧) = ↑0 ∨ ↑(C₂₃ ⟦V⟧ * C₁₃ ⟦V⟧) = ↑0 ← ofReal_inj,V:CKMMatrixδ₁₃:ℝ⊢ mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ = 0 ↔
  ↑(C₁₂ ⟦V⟧ * C₁₃ ⟦V⟧) = ↑0 ∨
    ↑(S₁₃ ⟦V⟧) = ↑0 ∨ ↑(S₁₂ ⟦V⟧ * C₁₃ ⟦V⟧) = ↑0 ∨ S₂₃ ⟦V⟧ * C₁₃ ⟦V⟧ = 0 ∨ C₂₃ ⟦V⟧ * C₁₃ ⟦V⟧ = 0V:CKMMatrixδ₁₃:ℝ⊢ mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ = 0 ↔
  ↑(C₁₂ ⟦V⟧ * C₁₃ ⟦V⟧) = ↑0 ∨
    ↑(S₁₃ ⟦V⟧) = ↑0 ∨ ↑(S₁₂ ⟦V⟧ * C₁₃ ⟦V⟧) = ↑0 ∨ ↑(S₂₃ ⟦V⟧ * C₁₃ ⟦V⟧) = ↑0 ∨ ↑(C₂₃ ⟦V⟧ * C₁₃ ⟦V⟧) = ↑0 ← ofReal_inj,V:CKMMatrixδ₁₃:ℝ⊢ mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ = 0 ↔
  ↑(C₁₂ ⟦V⟧ * C₁₃ ⟦V⟧) = ↑0 ∨
    ↑(S₁₃ ⟦V⟧) = ↑0 ∨ ↑(S₁₂ ⟦V⟧ * C₁₃ ⟦V⟧) = ↑0 ∨ ↑(S₂₃ ⟦V⟧ * C₁₃ ⟦V⟧) = ↑0 ∨ C₂₃ ⟦V⟧ * C₁₃ ⟦V⟧ = 0V:CKMMatrixδ₁₃:ℝ⊢ mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ = 0 ↔
  ↑(C₁₂ ⟦V⟧ * C₁₃ ⟦V⟧) = ↑0 ∨
    ↑(S₁₃ ⟦V⟧) = ↑0 ∨ ↑(S₁₂ ⟦V⟧ * C₁₃ ⟦V⟧) = ↑0 ∨ ↑(S₂₃ ⟦V⟧ * C₁₃ ⟦V⟧) = ↑0 ∨ ↑(C₂₃ ⟦V⟧ * C₁₃ ⟦V⟧) = ↑0 ← ofReal_injV:CKMMatrixδ₁₃:ℝ⊢ mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ = 0 ↔
  ↑(C₁₂ ⟦V⟧ * C₁₃ ⟦V⟧) = ↑0 ∨
    ↑(S₁₃ ⟦V⟧) = ↑0 ∨ ↑(S₁₂ ⟦V⟧ * C₁₃ ⟦V⟧) = ↑0 ∨ ↑(S₂₃ ⟦V⟧ * C₁₃ ⟦V⟧) = ↑0 ∨ ↑(C₂₃ ⟦V⟧ * C₁₃ ⟦V⟧) = ↑0V:CKMMatrixδ₁₃:ℝ⊢ mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ = 0 ↔
  ↑(C₁₂ ⟦V⟧ * C₁₃ ⟦V⟧) = ↑0 ∨
    ↑(S₁₃ ⟦V⟧) = ↑0 ∨ ↑(S₁₂ ⟦V⟧ * C₁₃ ⟦V⟧) = ↑0 ∨ ↑(S₂₃ ⟦V⟧ * C₁₃ ⟦V⟧) = ↑0 ∨ ↑(C₂₃ ⟦V⟧ * C₁₃ ⟦V⟧) = ↑0]V:CKMMatrixδ₁₃:ℝ⊢ mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ = 0 ↔
  ↑(C₁₂ ⟦V⟧ * C₁₃ ⟦V⟧) = ↑0 ∨
    ↑(S₁₃ ⟦V⟧) = ↑0 ∨ ↑(S₁₂ ⟦V⟧ * C₁₃ ⟦V⟧) = ↑0 ∨ ↑(S₂₃ ⟦V⟧ * C₁₃ ⟦V⟧) = ↑0 ∨ ↑(C₂₃ ⟦V⟧ * C₁₃ ⟦V⟧) = ↑0
  simp only [ofReal_mul]V:CKMMatrixδ₁₃:ℝ⊢ mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ = 0 ↔
  ↑(C₁₂ ⟦V⟧) * ↑(C₁₃ ⟦V⟧) = ↑0 ∨
    ↑(S₁₃ ⟦V⟧) = ↑0 ∨ ↑(S₁₂ ⟦V⟧) * ↑(C₁₃ ⟦V⟧) = ↑0 ∨ ↑(S₂₃ ⟦V⟧) * ↑(C₁₃ ⟦V⟧) = ↑0 ∨ ↑(C₂₃ ⟦V⟧) * ↑(C₁₃ ⟦V⟧) = ↑0
  rw [← S₁₃_eq_ℂsin_θ₁₃,V:CKMMatrixδ₁₃:ℝ⊢ mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ = 0 ↔
  ↑(C₁₂ ⟦V⟧) * ↑(C₁₃ ⟦V⟧) = ↑0 ∨
    sin ↑(θ₁₃ ⟦V⟧) = ↑0 ∨ ↑(S₁₂ ⟦V⟧) * ↑(C₁₃ ⟦V⟧) = ↑0 ∨ ↑(S₂₃ ⟦V⟧) * ↑(C₁₃ ⟦V⟧) = ↑0 ∨ ↑(C₂₃ ⟦V⟧) * ↑(C₁₃ ⟦V⟧) = ↑0V:CKMMatrixδ₁₃:ℝ⊢ sin ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) ^ 2 * sin ↑(θ₂₃ ⟦V⟧) * sin ↑(θ₁₃ ⟦V⟧) * cos ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₂₃ ⟦V⟧) *
      cexp (I * ↑δ₁₃) =
    0 ↔
  cos ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) = ↑0 ∨
    sin ↑(θ₁₃ ⟦V⟧) = ↑0 ∨
      sin ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) = ↑0 ∨ sin ↑(θ₂₃ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) = ↑0 ∨ cos ↑(θ₂₃ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) = ↑0 ← S₁₂_eq_ℂsin_θ₁₂,V:CKMMatrixδ₁₃:ℝ⊢ mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ = 0 ↔
  ↑(C₁₂ ⟦V⟧) * ↑(C₁₃ ⟦V⟧) = ↑0 ∨
    sin ↑(θ₁₃ ⟦V⟧) = ↑0 ∨ sin ↑(θ₁₂ ⟦V⟧) * ↑(C₁₃ ⟦V⟧) = ↑0 ∨ ↑(S₂₃ ⟦V⟧) * ↑(C₁₃ ⟦V⟧) = ↑0 ∨ ↑(C₂₃ ⟦V⟧) * ↑(C₁₃ ⟦V⟧) = ↑0V:CKMMatrixδ₁₃:ℝ⊢ sin ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) ^ 2 * sin ↑(θ₂₃ ⟦V⟧) * sin ↑(θ₁₃ ⟦V⟧) * cos ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₂₃ ⟦V⟧) *
      cexp (I * ↑δ₁₃) =
    0 ↔
  cos ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) = ↑0 ∨
    sin ↑(θ₁₃ ⟦V⟧) = ↑0 ∨
      sin ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) = ↑0 ∨ sin ↑(θ₂₃ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) = ↑0 ∨ cos ↑(θ₂₃ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) = ↑0 ← S₂₃_eq_ℂsin_θ₂₃,V:CKMMatrixδ₁₃:ℝ⊢ mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ = 0 ↔
  ↑(C₁₂ ⟦V⟧) * ↑(C₁₃ ⟦V⟧) = ↑0 ∨
    sin ↑(θ₁₃ ⟦V⟧) = ↑0 ∨
      sin ↑(θ₁₂ ⟦V⟧) * ↑(C₁₃ ⟦V⟧) = ↑0 ∨ sin ↑(θ₂₃ ⟦V⟧) * ↑(C₁₃ ⟦V⟧) = ↑0 ∨ ↑(C₂₃ ⟦V⟧) * ↑(C₁₃ ⟦V⟧) = ↑0V:CKMMatrixδ₁₃:ℝ⊢ sin ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) ^ 2 * sin ↑(θ₂₃ ⟦V⟧) * sin ↑(θ₁₃ ⟦V⟧) * cos ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₂₃ ⟦V⟧) *
      cexp (I * ↑δ₁₃) =
    0 ↔
  cos ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) = ↑0 ∨
    sin ↑(θ₁₃ ⟦V⟧) = ↑0 ∨
      sin ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) = ↑0 ∨ sin ↑(θ₂₃ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) = ↑0 ∨ cos ↑(θ₂₃ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) = ↑0
    ← C₁₃_eq_ℂcos_θ₁₃,V:CKMMatrixδ₁₃:ℝ⊢ mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ = 0 ↔
  ↑(C₁₂ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) = ↑0 ∨
    sin ↑(θ₁₃ ⟦V⟧) = ↑0 ∨
      sin ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) = ↑0 ∨ sin ↑(θ₂₃ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) = ↑0 ∨ ↑(C₂₃ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) = ↑0V:CKMMatrixδ₁₃:ℝ⊢ sin ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) ^ 2 * sin ↑(θ₂₃ ⟦V⟧) * sin ↑(θ₁₃ ⟦V⟧) * cos ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₂₃ ⟦V⟧) *
      cexp (I * ↑δ₁₃) =
    0 ↔
  cos ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) = ↑0 ∨
    sin ↑(θ₁₃ ⟦V⟧) = ↑0 ∨
      sin ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) = ↑0 ∨ sin ↑(θ₂₃ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) = ↑0 ∨ cos ↑(θ₂₃ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) = ↑0 ← C₂₃_eq_ℂcos_θ₂₃,V:CKMMatrixδ₁₃:ℝ⊢ mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ = 0 ↔
  ↑(C₁₂ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) = ↑0 ∨
    sin ↑(θ₁₃ ⟦V⟧) = ↑0 ∨
      sin ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) = ↑0 ∨ sin ↑(θ₂₃ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) = ↑0 ∨ cos ↑(θ₂₃ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) = ↑0V:CKMMatrixδ₁₃:ℝ⊢ sin ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) ^ 2 * sin ↑(θ₂₃ ⟦V⟧) * sin ↑(θ₁₃ ⟦V⟧) * cos ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₂₃ ⟦V⟧) *
      cexp (I * ↑δ₁₃) =
    0 ↔
  cos ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) = ↑0 ∨
    sin ↑(θ₁₃ ⟦V⟧) = ↑0 ∨
      sin ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) = ↑0 ∨ sin ↑(θ₂₃ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) = ↑0 ∨ cos ↑(θ₂₃ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) = ↑0 ← C₁₂_eq_ℂcos_θ₁₂,V:CKMMatrixδ₁₃:ℝ⊢ mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ = 0 ↔
  cos ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) = ↑0 ∨
    sin ↑(θ₁₃ ⟦V⟧) = ↑0 ∨
      sin ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) = ↑0 ∨ sin ↑(θ₂₃ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) = ↑0 ∨ cos ↑(θ₂₃ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) = ↑0V:CKMMatrixδ₁₃:ℝ⊢ sin ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) ^ 2 * sin ↑(θ₂₃ ⟦V⟧) * sin ↑(θ₁₃ ⟦V⟧) * cos ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₂₃ ⟦V⟧) *
      cexp (I * ↑δ₁₃) =
    0 ↔
  cos ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) = ↑0 ∨
    sin ↑(θ₁₃ ⟦V⟧) = ↑0 ∨
      sin ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) = ↑0 ∨ sin ↑(θ₂₃ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) = ↑0 ∨ cos ↑(θ₂₃ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) = ↑0 mulExpδ₁₃_on_param_δ₁₃V:CKMMatrixδ₁₃:ℝ⊢ sin ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) ^ 2 * sin ↑(θ₂₃ ⟦V⟧) * sin ↑(θ₁₃ ⟦V⟧) * cos ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₂₃ ⟦V⟧) *
      cexp (I * ↑δ₁₃) =
    0 ↔
  cos ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) = ↑0 ∨
    sin ↑(θ₁₃ ⟦V⟧) = ↑0 ∨
      sin ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) = ↑0 ∨ sin ↑(θ₂₃ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) = ↑0 ∨ cos ↑(θ₂₃ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) = ↑0V:CKMMatrixδ₁₃:ℝ⊢ sin ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) ^ 2 * sin ↑(θ₂₃ ⟦V⟧) * sin ↑(θ₁₃ ⟦V⟧) * cos ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₂₃ ⟦V⟧) *
      cexp (I * ↑δ₁₃) =
    0 ↔
  cos ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) = ↑0 ∨
    sin ↑(θ₁₃ ⟦V⟧) = ↑0 ∨
      sin ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) = ↑0 ∨ sin ↑(θ₂₃ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) = ↑0 ∨ cos ↑(θ₂₃ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) = ↑0]V:CKMMatrixδ₁₃:ℝ⊢ sin ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) ^ 2 * sin ↑(θ₂₃ ⟦V⟧) * sin ↑(θ₁₃ ⟦V⟧) * cos ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₂₃ ⟦V⟧) *
      cexp (I * ↑δ₁₃) =
    0 ↔
  cos ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) = ↑0 ∨
    sin ↑(θ₁₃ ⟦V⟧) = ↑0 ∨
      sin ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) = ↑0 ∨ sin ↑(θ₂₃ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) = ↑0 ∨ cos ↑(θ₂₃ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) = ↑0
  simp only [mul_eq_zero, ne_eq, OfNat.ofNat_ne_zero, not_false_eq_true, pow_eq_zero_iff,
    ofReal_zero, exp_ne_zero, or_false]V:CKMMatrixδ₁₃:ℝ⊢ ((((sin ↑(θ₁₂ ⟦V⟧) = 0 ∨ cos ↑(θ₁₃ ⟦V⟧) = 0) ∨ sin ↑(θ₂₃ ⟦V⟧) = 0) ∨ sin ↑(θ₁₃ ⟦V⟧) = 0) ∨ cos ↑(θ₁₂ ⟦V⟧) = 0) ∨
    cos ↑(θ₂₃ ⟦V⟧) = 0 ↔
  (cos ↑(θ₁₂ ⟦V⟧) = 0 ∨ cos ↑(θ₁₃ ⟦V⟧) = 0) ∨
    sin ↑(θ₁₃ ⟦V⟧) = 0 ∨
      (sin ↑(θ₁₂ ⟦V⟧) = 0 ∨ cos ↑(θ₁₃ ⟦V⟧) = 0) ∨
        (sin ↑(θ₂₃ ⟦V⟧) = 0 ∨ cos ↑(θ₁₃ ⟦V⟧) = 0) ∨ cos ↑(θ₂₃ ⟦V⟧) = 0 ∨ cos ↑(θ₁₃ ⟦V⟧) = 0
  aesopAll goals completed! 🐙

lemma mulExpδ₁₃_on_param_abs (V : CKMMatrix) (δ₁₃ : ℝ) :
    norm (mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧) =
    sin (θ₁₂ ⟦V⟧) * cos (θ₁₃ ⟦V⟧) ^ 2 * sin (θ₂₃ ⟦V⟧) * sin (θ₁₃ ⟦V⟧)
    * cos (θ₁₂ ⟦V⟧) * cos (θ₂₃ ⟦V⟧) := byV:CKMMatrixδ₁₃:ℝ⊢ ↑‖mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧‖ =
  sin ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) ^ 2 * sin ↑(θ₂₃ ⟦V⟧) * sin ↑(θ₁₃ ⟦V⟧) * cos ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₂₃ ⟦V⟧)
  rw [mulExpδ₁₃_on_param_δ₁₃V:CKMMatrixδ₁₃:ℝ⊢ ↑‖sin ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) ^ 2 * sin ↑(θ₂₃ ⟦V⟧) * sin ↑(θ₁₃ ⟦V⟧) * cos ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₂₃ ⟦V⟧) *
        cexp (I * ↑δ₁₃)‖ =
  sin ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) ^ 2 * sin ↑(θ₂₃ ⟦V⟧) * sin ↑(θ₁₃ ⟦V⟧) * cos ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₂₃ ⟦V⟧)V:CKMMatrixδ₁₃:ℝ⊢ ↑‖sin ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) ^ 2 * sin ↑(θ₂₃ ⟦V⟧) * sin ↑(θ₁₃ ⟦V⟧) * cos ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₂₃ ⟦V⟧) *
        cexp (I * ↑δ₁₃)‖ =
  sin ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) ^ 2 * sin ↑(θ₂₃ ⟦V⟧) * sin ↑(θ₁₃ ⟦V⟧) * cos ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₂₃ ⟦V⟧)]V:CKMMatrixδ₁₃:ℝ⊢ ↑‖sin ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) ^ 2 * sin ↑(θ₂₃ ⟦V⟧) * sin ↑(θ₁₃ ⟦V⟧) * cos ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₂₃ ⟦V⟧) *
        cexp (I * ↑δ₁₃)‖ =
  sin ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) ^ 2 * sin ↑(θ₂₃ ⟦V⟧) * sin ↑(θ₁₃ ⟦V⟧) * cos ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₂₃ ⟦V⟧)
  simp only [Complex.norm_mul, norm_pow, norm_exp, mul_re, I_re, ofReal_re, zero_mul, I_im,
    ofReal_im, mul_zero, sub_self, Real.exp_zero, mul_one, ofReal_mul, ofReal_pow]V:CKMMatrixδ₁₃:ℝ⊢ ↑‖sin ↑(θ₁₂ ⟦V⟧)‖ * ↑‖cos ↑(θ₁₃ ⟦V⟧)‖ ^ 2 * ↑‖sin ↑(θ₂₃ ⟦V⟧)‖ * ↑‖sin ↑(θ₁₃ ⟦V⟧)‖ * ↑‖cos ↑(θ₁₂ ⟦V⟧)‖ *
    ↑‖cos ↑(θ₂₃ ⟦V⟧)‖ =
  sin ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) ^ 2 * sin ↑(θ₂₃ ⟦V⟧) * sin ↑(θ₁₃ ⟦V⟧) * cos ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₂₃ ⟦V⟧)
  rw [complexAbs_sin_θ₁₃,V:CKMMatrixδ₁₃:ℝ⊢ ↑‖sin ↑(θ₁₂ ⟦V⟧)‖ * ↑‖cos ↑(θ₁₃ ⟦V⟧)‖ ^ 2 * ↑‖sin ↑(θ₂₃ ⟦V⟧)‖ * sin ↑(θ₁₃ ⟦V⟧) * ↑‖cos ↑(θ₁₂ ⟦V⟧)‖ * ↑‖cos ↑(θ₂₃ ⟦V⟧)‖ =
  sin ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) ^ 2 * sin ↑(θ₂₃ ⟦V⟧) * sin ↑(θ₁₃ ⟦V⟧) * cos ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₂₃ ⟦V⟧)All goals completed! 🐙 complexAbs_cos_θ₁₃,V:CKMMatrixδ₁₃:ℝ⊢ ↑‖sin ↑(θ₁₂ ⟦V⟧)‖ * cos ↑(θ₁₃ ⟦V⟧) ^ 2 * ↑‖sin ↑(θ₂₃ ⟦V⟧)‖ * sin ↑(θ₁₃ ⟦V⟧) * ↑‖cos ↑(θ₁₂ ⟦V⟧)‖ * ↑‖cos ↑(θ₂₃ ⟦V⟧)‖ =
  sin ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) ^ 2 * sin ↑(θ₂₃ ⟦V⟧) * sin ↑(θ₁₃ ⟦V⟧) * cos ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₂₃ ⟦V⟧)All goals completed! 🐙 complexAbs_sin_θ₁₂,V:CKMMatrixδ₁₃:ℝ⊢ sin ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) ^ 2 * ↑‖sin ↑(θ₂₃ ⟦V⟧)‖ * sin ↑(θ₁₃ ⟦V⟧) * ↑‖cos ↑(θ₁₂ ⟦V⟧)‖ * ↑‖cos ↑(θ₂₃ ⟦V⟧)‖ =
  sin ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) ^ 2 * sin ↑(θ₂₃ ⟦V⟧) * sin ↑(θ₁₃ ⟦V⟧) * cos ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₂₃ ⟦V⟧)All goals completed! 🐙 complexAbs_cos_θ₁₂,V:CKMMatrixδ₁₃:ℝ⊢ sin ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) ^ 2 * ↑‖sin ↑(θ₂₃ ⟦V⟧)‖ * sin ↑(θ₁₃ ⟦V⟧) * cos ↑(θ₁₂ ⟦V⟧) * ↑‖cos ↑(θ₂₃ ⟦V⟧)‖ =
  sin ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) ^ 2 * sin ↑(θ₂₃ ⟦V⟧) * sin ↑(θ₁₃ ⟦V⟧) * cos ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₂₃ ⟦V⟧)All goals completed! 🐙
    complexAbs_sin_θ₂₃,V:CKMMatrixδ₁₃:ℝ⊢ sin ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) ^ 2 * sin ↑(θ₂₃ ⟦V⟧) * sin ↑(θ₁₃ ⟦V⟧) * cos ↑(θ₁₂ ⟦V⟧) * ↑‖cos ↑(θ₂₃ ⟦V⟧)‖ =
  sin ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) ^ 2 * sin ↑(θ₂₃ ⟦V⟧) * sin ↑(θ₁₃ ⟦V⟧) * cos ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₂₃ ⟦V⟧)All goals completed! 🐙 complexAbs_cos_θ₂₃V:CKMMatrixδ₁₃:ℝ⊢ sin ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) ^ 2 * sin ↑(θ₂₃ ⟦V⟧) * sin ↑(θ₁₃ ⟦V⟧) * cos ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₂₃ ⟦V⟧) =
  sin ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) ^ 2 * sin ↑(θ₂₃ ⟦V⟧) * sin ↑(θ₁₃ ⟦V⟧) * cos ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₂₃ ⟦V⟧)All goals completed! 🐙]All goals completed! 🐙

lemma mulExpδ₁₃_on_param_ne_zero_arg (V : CKMMatrix) (δ₁₃ : ℝ)
    (h1 : mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ ≠ 0) :
    cexp (arg (mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧) * I) =
    cexp (δ₁₃ * I) := byV:CKMMatrixδ₁₃:ℝh1:mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ ≠ 0⊢ cexp (↑(mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧).arg * I) = cexp (↑δ₁₃ * I)
  have h2 : mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ = norm
      (mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧) * exp (δ₁₃ * I) := by
    rw [mulExpδ₁₃_on_param_abs V δ₁₃,V:CKMMatrixδ₁₃:ℝh1:mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ ≠ 0⊢ mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ =
  sin ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) ^ 2 * sin ↑(θ₂₃ ⟦V⟧) * sin ↑(θ₁₃ ⟦V⟧) * cos ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₂₃ ⟦V⟧) *
    cexp (↑δ₁₃ * I)V:CKMMatrixδ₁₃:ℝh1:mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ ≠ 0⊢ sin ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) ^ 2 * sin ↑(θ₂₃ ⟦V⟧) * sin ↑(θ₁₃ ⟦V⟧) * cos ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₂₃ ⟦V⟧) *
    cexp (I * ↑δ₁₃) =
  sin ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) ^ 2 * sin ↑(θ₂₃ ⟦V⟧) * sin ↑(θ₁₃ ⟦V⟧) * cos ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₂₃ ⟦V⟧) *
    cexp (↑δ₁₃ * I)V:CKMMatrixδ₁₃:ℝh1:mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ ≠ 0h2:mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ =
  ↑‖mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧‖ * cexp (↑δ₁₃ * I)⊢ cexp (↑(mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧).arg * I) = cexp (↑δ₁₃ * I) mulExpδ₁₃_on_param_δ₁₃ V δ₁₃V:CKMMatrixδ₁₃:ℝh1:mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ ≠ 0⊢ sin ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) ^ 2 * sin ↑(θ₂₃ ⟦V⟧) * sin ↑(θ₁₃ ⟦V⟧) * cos ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₂₃ ⟦V⟧) *
    cexp (I * ↑δ₁₃) =
  sin ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) ^ 2 * sin ↑(θ₂₃ ⟦V⟧) * sin ↑(θ₁₃ ⟦V⟧) * cos ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₂₃ ⟦V⟧) *
    cexp (↑δ₁₃ * I)V:CKMMatrixδ₁₃:ℝh1:mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ ≠ 0⊢ sin ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) ^ 2 * sin ↑(θ₂₃ ⟦V⟧) * sin ↑(θ₁₃ ⟦V⟧) * cos ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₂₃ ⟦V⟧) *
    cexp (I * ↑δ₁₃) =
  sin ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) ^ 2 * sin ↑(θ₂₃ ⟦V⟧) * sin ↑(θ₁₃ ⟦V⟧) * cos ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₂₃ ⟦V⟧) *
    cexp (↑δ₁₃ * I)V:CKMMatrixδ₁₃:ℝh1:mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ ≠ 0h2:mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ =
  ↑‖mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧‖ * cexp (↑δ₁₃ * I)⊢ cexp (↑(mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧).arg * I) = cexp (↑δ₁₃ * I)]V:CKMMatrixδ₁₃:ℝh1:mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ ≠ 0⊢ sin ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) ^ 2 * sin ↑(θ₂₃ ⟦V⟧) * sin ↑(θ₁₃ ⟦V⟧) * cos ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₂₃ ⟦V⟧) *
    cexp (I * ↑δ₁₃) =
  sin ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) ^ 2 * sin ↑(θ₂₃ ⟦V⟧) * sin ↑(θ₁₃ ⟦V⟧) * cos ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₂₃ ⟦V⟧) *
    cexp (↑δ₁₃ * I)V:CKMMatrixδ₁₃:ℝh1:mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ ≠ 0h2:mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ =
  ↑‖mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧‖ * cexp (↑δ₁₃ * I)⊢ cexp (↑(mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧).arg * I) = cexp (↑δ₁₃ * I)
    ring_nfV:CKMMatrixδ₁₃:ℝh1:mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ ≠ 0h2:mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ =
  ↑‖mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧‖ * cexp (↑δ₁₃ * I)⊢ cexp (↑(mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧).arg * I) = cexp (↑δ₁₃ * I)V:CKMMatrixδ₁₃:ℝh1:mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ ≠ 0h2:mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ =
  ↑‖mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧‖ * cexp (↑δ₁₃ * I)⊢ cexp (↑(mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧).arg * I) = cexp (↑δ₁₃ * I)
  nth_rewrite 1 [← norm_mul_exp_arg_mul_I (mulExpδ₁₃
    ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧)] at h2V:CKMMatrixδ₁₃:ℝh1:mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ ≠ 0h2:↑‖mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧‖ *
    cexp (↑(mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧).arg * I) =
  ↑‖mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧‖ * cexp (↑δ₁₃ * I)⊢ cexp (↑(mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧).arg * I) = cexp (↑δ₁₃ * I)
  have habs_ne_zero :
      (norm (mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧) : ℂ) ≠ 0 := by
    simpa only [ne_eq, ofReal_eq_zero, norm_eq_zero] using h1V:CKMMatrixδ₁₃:ℝh1:mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ ≠ 0h2:↑‖mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧‖ *
    cexp (↑(mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧).arg * I) =
  ↑‖mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧‖ * cexp (↑δ₁₃ * I)habs_ne_zero:↑‖mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧‖ ≠ 0⊢ cexp (↑(mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧).arg * I) = cexp (↑δ₁₃ * I)V:CKMMatrixδ₁₃:ℝh1:mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ ≠ 0h2:↑‖mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧‖ *
    cexp (↑(mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧).arg * I) =
  ↑‖mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧‖ * cexp (↑δ₁₃ * I)habs_ne_zero:↑‖mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧‖ ≠ 0⊢ cexp (↑(mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧).arg * I) = cexp (↑δ₁₃ * I)
  rw [← mul_right_inj' habs_ne_zero,V:CKMMatrixδ₁₃:ℝh1:mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ ≠ 0h2:↑‖mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧‖ *
    cexp (↑(mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧).arg * I) =
  ↑‖mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧‖ * cexp (↑δ₁₃ * I)habs_ne_zero:↑‖mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧‖ ≠ 0⊢ ↑‖mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧‖ *
    cexp (↑(mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧).arg * I) =
  ↑‖mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧‖ * cexp (↑δ₁₃ * I)All goals completed! 🐙 ← h2V:CKMMatrixδ₁₃:ℝh1:mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ ≠ 0h2:↑‖mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧‖ *
    cexp (↑(mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧).arg * I) =
  ↑‖mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧‖ * cexp (↑δ₁₃ * I)habs_ne_zero:↑‖mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧‖ ≠ 0⊢ ↑‖mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧‖ *
    cexp (↑(mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧).arg * I) =
  ↑‖mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧‖ *
    cexp (↑(mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧).arg * I)All goals completed! 🐙]All goals completed! 🐙lemma on_param_cos_θ₁₃_eq_zero {V : CKMMatrix} (δ₁₃ : ℝ) (h : Real.cos (θ₁₃ ⟦V⟧) = 0) :
    standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃ ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 := byV:CKMMatrixδ₁₃:ℝh:Real.cos (θ₁₃ ⟦V⟧) = 0⊢ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃ ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0
  have hub := VubAbs_of_cos_θ₁₃_zero hV:CKMMatrixδ₁₃:ℝh:Real.cos (θ₁₃ ⟦V⟧) = 0hub:VubAbs ⟦V⟧ = 1⊢ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃ ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0
  use 0, 0, 0, δ₁₃, 0, -δ₁₃hV:CKMMatrixδ₁₃:ℝh:Real.cos (θ₁₃ ⟦V⟧) = 0hub:VubAbs ⟦V⟧ = 1⊢ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃ =
  phaseShift 0 0 0 * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift δ₁₃ 0 (-δ₁₃)
  ext i jhV:CKMMatrixδ₁₃:ℝh:Real.cos (θ₁₃ ⟦V⟧) = 0hub:VubAbs ⟦V⟧ = 1i:Fin 3j:Fin 3⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) i j =
  ↑(phaseShift 0 0 0 * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift δ₁₃ 0 (-δ₁₃)) i j
  fin_cases ih.«0»V:CKMMatrixδ₁₃:ℝh:Real.cos (θ₁₃ ⟦V⟧) = 0hub:VubAbs ⟦V⟧ = 1j:Fin 3⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨0, ⋯⟩) j =
  ↑(phaseShift 0 0 0 * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift δ₁₃ 0 (-δ₁₃)) ((fun i => i) ⟨0, ⋯⟩) jh.«1»V:CKMMatrixδ₁₃:ℝh:Real.cos (θ₁₃ ⟦V⟧) = 0hub:VubAbs ⟦V⟧ = 1j:Fin 3⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨1, ⋯⟩) j =
  ↑(phaseShift 0 0 0 * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift δ₁₃ 0 (-δ₁₃)) ((fun i => i) ⟨1, ⋯⟩) jh.«2»V:CKMMatrixδ₁₃:ℝh:Real.cos (θ₁₃ ⟦V⟧) = 0hub:VubAbs ⟦V⟧ = 1j:Fin 3⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨2, ⋯⟩) j =
  ↑(phaseShift 0 0 0 * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift δ₁₃ 0 (-δ₁₃)) ((fun i => i) ⟨2, ⋯⟩) j <;>h.«0»V:CKMMatrixδ₁₃:ℝh:Real.cos (θ₁₃ ⟦V⟧) = 0hub:VubAbs ⟦V⟧ = 1j:Fin 3⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨0, ⋯⟩) j =
  ↑(phaseShift 0 0 0 * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift δ₁₃ 0 (-δ₁₃)) ((fun i => i) ⟨0, ⋯⟩) jh.«1»V:CKMMatrixδ₁₃:ℝh:Real.cos (θ₁₃ ⟦V⟧) = 0hub:VubAbs ⟦V⟧ = 1j:Fin 3⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨1, ⋯⟩) j =
  ↑(phaseShift 0 0 0 * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift δ₁₃ 0 (-δ₁₃)) ((fun i => i) ⟨1, ⋯⟩) jh.«2»V:CKMMatrixδ₁₃:ℝh:Real.cos (θ₁₃ ⟦V⟧) = 0hub:VubAbs ⟦V⟧ = 1j:Fin 3⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨2, ⋯⟩) j =
  ↑(phaseShift 0 0 0 * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift δ₁₃ 0 (-δ₁₃)) ((fun i => i) ⟨2, ⋯⟩) j fin_cases jh.«2».«0»V:CKMMatrixδ₁₃:ℝh:Real.cos (θ₁₃ ⟦V⟧) = 0hub:VubAbs ⟦V⟧ = 1⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨2, ⋯⟩) ((fun i => i) ⟨0, ⋯⟩) =
  ↑(phaseShift 0 0 0 * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift δ₁₃ 0 (-δ₁₃)) ((fun i => i) ⟨2, ⋯⟩)
    ((fun i => i) ⟨0, ⋯⟩)h.«2».«1»V:CKMMatrixδ₁₃:ℝh:Real.cos (θ₁₃ ⟦V⟧) = 0hub:VubAbs ⟦V⟧ = 1⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨2, ⋯⟩) ((fun i => i) ⟨1, ⋯⟩) =
  ↑(phaseShift 0 0 0 * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift δ₁₃ 0 (-δ₁₃)) ((fun i => i) ⟨2, ⋯⟩)
    ((fun i => i) ⟨1, ⋯⟩)h.«2».«2»V:CKMMatrixδ₁₃:ℝh:Real.cos (θ₁₃ ⟦V⟧) = 0hub:VubAbs ⟦V⟧ = 1⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨2, ⋯⟩) ((fun i => i) ⟨2, ⋯⟩) =
  ↑(phaseShift 0 0 0 * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift δ₁₃ 0 (-δ₁₃)) ((fun i => i) ⟨2, ⋯⟩)
    ((fun i => i) ⟨2, ⋯⟩) <;>h.«0».«0»V:CKMMatrixδ₁₃:ℝh:Real.cos (θ₁₃ ⟦V⟧) = 0hub:VubAbs ⟦V⟧ = 1⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨0, ⋯⟩) ((fun i => i) ⟨0, ⋯⟩) =
  ↑(phaseShift 0 0 0 * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift δ₁₃ 0 (-δ₁₃)) ((fun i => i) ⟨0, ⋯⟩)
    ((fun i => i) ⟨0, ⋯⟩)h.«0».«1»V:CKMMatrixδ₁₃:ℝh:Real.cos (θ₁₃ ⟦V⟧) = 0hub:VubAbs ⟦V⟧ = 1⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨0, ⋯⟩) ((fun i => i) ⟨1, ⋯⟩) =
  ↑(phaseShift 0 0 0 * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift δ₁₃ 0 (-δ₁₃)) ((fun i => i) ⟨0, ⋯⟩)
    ((fun i => i) ⟨1, ⋯⟩)h.«0».«2»V:CKMMatrixδ₁₃:ℝh:Real.cos (θ₁₃ ⟦V⟧) = 0hub:VubAbs ⟦V⟧ = 1⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨0, ⋯⟩) ((fun i => i) ⟨2, ⋯⟩) =
  ↑(phaseShift 0 0 0 * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift δ₁₃ 0 (-δ₁₃)) ((fun i => i) ⟨0, ⋯⟩)
    ((fun i => i) ⟨2, ⋯⟩)h.«1».«0»V:CKMMatrixδ₁₃:ℝh:Real.cos (θ₁₃ ⟦V⟧) = 0hub:VubAbs ⟦V⟧ = 1⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨1, ⋯⟩) ((fun i => i) ⟨0, ⋯⟩) =
  ↑(phaseShift 0 0 0 * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift δ₁₃ 0 (-δ₁₃)) ((fun i => i) ⟨1, ⋯⟩)
    ((fun i => i) ⟨0, ⋯⟩)h.«1».«1»V:CKMMatrixδ₁₃:ℝh:Real.cos (θ₁₃ ⟦V⟧) = 0hub:VubAbs ⟦V⟧ = 1⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨1, ⋯⟩) ((fun i => i) ⟨1, ⋯⟩) =
  ↑(phaseShift 0 0 0 * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift δ₁₃ 0 (-δ₁₃)) ((fun i => i) ⟨1, ⋯⟩)
    ((fun i => i) ⟨1, ⋯⟩)h.«1».«2»V:CKMMatrixδ₁₃:ℝh:Real.cos (θ₁₃ ⟦V⟧) = 0hub:VubAbs ⟦V⟧ = 1⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨1, ⋯⟩) ((fun i => i) ⟨2, ⋯⟩) =
  ↑(phaseShift 0 0 0 * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift δ₁₃ 0 (-δ₁₃)) ((fun i => i) ⟨1, ⋯⟩)
    ((fun i => i) ⟨2, ⋯⟩)h.«2».«0»V:CKMMatrixδ₁₃:ℝh:Real.cos (θ₁₃ ⟦V⟧) = 0hub:VubAbs ⟦V⟧ = 1⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨2, ⋯⟩) ((fun i => i) ⟨0, ⋯⟩) =
  ↑(phaseShift 0 0 0 * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift δ₁₃ 0 (-δ₁₃)) ((fun i => i) ⟨2, ⋯⟩)
    ((fun i => i) ⟨0, ⋯⟩)h.«2».«1»V:CKMMatrixδ₁₃:ℝh:Real.cos (θ₁₃ ⟦V⟧) = 0hub:VubAbs ⟦V⟧ = 1⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨2, ⋯⟩) ((fun i => i) ⟨1, ⋯⟩) =
  ↑(phaseShift 0 0 0 * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift δ₁₃ 0 (-δ₁₃)) ((fun i => i) ⟨2, ⋯⟩)
    ((fun i => i) ⟨1, ⋯⟩)h.«2».«2»V:CKMMatrixδ₁₃:ℝh:Real.cos (θ₁₃ ⟦V⟧) = 0hub:VubAbs ⟦V⟧ = 1⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨2, ⋯⟩) ((fun i => i) ⟨2, ⋯⟩) =
  ↑(phaseShift 0 0 0 * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift δ₁₃ 0 (-δ₁₃)) ((fun i => i) ⟨2, ⋯⟩)
    ((fun i => i) ⟨2, ⋯⟩)
    simp [standParam, standParamAsMatrix, phaseShift, phaseShiftMatrix, mul_apply,
      Fin.sum_univ_three, ofReal_cos, ofReal_sin, S₁₃_eq_ℂsin_θ₁₃, C₁₂_eq_ℂcos_θ₁₂,
      S₁₂_eq_ℂsin_θ₁₂, S₁₃_of_Vub_one hub, C₁₂_of_Vub_one hub, S₁₂_of_Vub_one hub, h, exp_neg]All goals completed! 🐙lemma on_param_cos_θ₁₂_eq_zero {V : CKMMatrix} (δ₁₃ : ℝ) (h : Real.cos (θ₁₂ ⟦V⟧) = 0) :
    standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃ ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 := byV:CKMMatrixδ₁₃:ℝh:Real.cos (θ₁₂ ⟦V⟧) = 0⊢ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃ ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0
  use 0, δ₁₃, δ₁₃, -δ₁₃, 0, - δ₁₃hV:CKMMatrixδ₁₃:ℝh:Real.cos (θ₁₂ ⟦V⟧) = 0⊢ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃ =
  phaseShift 0 δ₁₃ δ₁₃ * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift (-δ₁₃) 0 (-δ₁₃)
  ext i jhV:CKMMatrixδ₁₃:ℝh:Real.cos (θ₁₂ ⟦V⟧) = 0i:Fin 3j:Fin 3⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) i j =
  ↑(phaseShift 0 δ₁₃ δ₁₃ * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift (-δ₁₃) 0 (-δ₁₃)) i j
  fin_cases ih.«0»V:CKMMatrixδ₁₃:ℝh:Real.cos (θ₁₂ ⟦V⟧) = 0j:Fin 3⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨0, ⋯⟩) j =
  ↑(phaseShift 0 δ₁₃ δ₁₃ * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift (-δ₁₃) 0 (-δ₁₃))
    ((fun i => i) ⟨0, ⋯⟩) jh.«1»V:CKMMatrixδ₁₃:ℝh:Real.cos (θ₁₂ ⟦V⟧) = 0j:Fin 3⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨1, ⋯⟩) j =
  ↑(phaseShift 0 δ₁₃ δ₁₃ * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift (-δ₁₃) 0 (-δ₁₃))
    ((fun i => i) ⟨1, ⋯⟩) jh.«2»V:CKMMatrixδ₁₃:ℝh:Real.cos (θ₁₂ ⟦V⟧) = 0j:Fin 3⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨2, ⋯⟩) j =
  ↑(phaseShift 0 δ₁₃ δ₁₃ * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift (-δ₁₃) 0 (-δ₁₃))
    ((fun i => i) ⟨2, ⋯⟩) j <;>h.«0»V:CKMMatrixδ₁₃:ℝh:Real.cos (θ₁₂ ⟦V⟧) = 0j:Fin 3⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨0, ⋯⟩) j =
  ↑(phaseShift 0 δ₁₃ δ₁₃ * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift (-δ₁₃) 0 (-δ₁₃))
    ((fun i => i) ⟨0, ⋯⟩) jh.«1»V:CKMMatrixδ₁₃:ℝh:Real.cos (θ₁₂ ⟦V⟧) = 0j:Fin 3⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨1, ⋯⟩) j =
  ↑(phaseShift 0 δ₁₃ δ₁₃ * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift (-δ₁₃) 0 (-δ₁₃))
    ((fun i => i) ⟨1, ⋯⟩) jh.«2»V:CKMMatrixδ₁₃:ℝh:Real.cos (θ₁₂ ⟦V⟧) = 0j:Fin 3⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨2, ⋯⟩) j =
  ↑(phaseShift 0 δ₁₃ δ₁₃ * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift (-δ₁₃) 0 (-δ₁₃))
    ((fun i => i) ⟨2, ⋯⟩) j fin_cases jh.«2».«0»V:CKMMatrixδ₁₃:ℝh:Real.cos (θ₁₂ ⟦V⟧) = 0⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨2, ⋯⟩) ((fun i => i) ⟨0, ⋯⟩) =
  ↑(phaseShift 0 δ₁₃ δ₁₃ * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift (-δ₁₃) 0 (-δ₁₃))
    ((fun i => i) ⟨2, ⋯⟩) ((fun i => i) ⟨0, ⋯⟩)h.«2».«1»V:CKMMatrixδ₁₃:ℝh:Real.cos (θ₁₂ ⟦V⟧) = 0⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨2, ⋯⟩) ((fun i => i) ⟨1, ⋯⟩) =
  ↑(phaseShift 0 δ₁₃ δ₁₃ * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift (-δ₁₃) 0 (-δ₁₃))
    ((fun i => i) ⟨2, ⋯⟩) ((fun i => i) ⟨1, ⋯⟩)h.«2».«2»V:CKMMatrixδ₁₃:ℝh:Real.cos (θ₁₂ ⟦V⟧) = 0⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨2, ⋯⟩) ((fun i => i) ⟨2, ⋯⟩) =
  ↑(phaseShift 0 δ₁₃ δ₁₃ * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift (-δ₁₃) 0 (-δ₁₃))
    ((fun i => i) ⟨2, ⋯⟩) ((fun i => i) ⟨2, ⋯⟩) <;>h.«0».«0»V:CKMMatrixδ₁₃:ℝh:Real.cos (θ₁₂ ⟦V⟧) = 0⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨0, ⋯⟩) ((fun i => i) ⟨0, ⋯⟩) =
  ↑(phaseShift 0 δ₁₃ δ₁₃ * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift (-δ₁₃) 0 (-δ₁₃))
    ((fun i => i) ⟨0, ⋯⟩) ((fun i => i) ⟨0, ⋯⟩)h.«0».«1»V:CKMMatrixδ₁₃:ℝh:Real.cos (θ₁₂ ⟦V⟧) = 0⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨0, ⋯⟩) ((fun i => i) ⟨1, ⋯⟩) =
  ↑(phaseShift 0 δ₁₃ δ₁₃ * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift (-δ₁₃) 0 (-δ₁₃))
    ((fun i => i) ⟨0, ⋯⟩) ((fun i => i) ⟨1, ⋯⟩)h.«0».«2»V:CKMMatrixδ₁₃:ℝh:Real.cos (θ₁₂ ⟦V⟧) = 0⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨0, ⋯⟩) ((fun i => i) ⟨2, ⋯⟩) =
  ↑(phaseShift 0 δ₁₃ δ₁₃ * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift (-δ₁₃) 0 (-δ₁₃))
    ((fun i => i) ⟨0, ⋯⟩) ((fun i => i) ⟨2, ⋯⟩)h.«1».«0»V:CKMMatrixδ₁₃:ℝh:Real.cos (θ₁₂ ⟦V⟧) = 0⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨1, ⋯⟩) ((fun i => i) ⟨0, ⋯⟩) =
  ↑(phaseShift 0 δ₁₃ δ₁₃ * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift (-δ₁₃) 0 (-δ₁₃))
    ((fun i => i) ⟨1, ⋯⟩) ((fun i => i) ⟨0, ⋯⟩)h.«1».«1»V:CKMMatrixδ₁₃:ℝh:Real.cos (θ₁₂ ⟦V⟧) = 0⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨1, ⋯⟩) ((fun i => i) ⟨1, ⋯⟩) =
  ↑(phaseShift 0 δ₁₃ δ₁₃ * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift (-δ₁₃) 0 (-δ₁₃))
    ((fun i => i) ⟨1, ⋯⟩) ((fun i => i) ⟨1, ⋯⟩)h.«1».«2»V:CKMMatrixδ₁₃:ℝh:Real.cos (θ₁₂ ⟦V⟧) = 0⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨1, ⋯⟩) ((fun i => i) ⟨2, ⋯⟩) =
  ↑(phaseShift 0 δ₁₃ δ₁₃ * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift (-δ₁₃) 0 (-δ₁₃))
    ((fun i => i) ⟨1, ⋯⟩) ((fun i => i) ⟨2, ⋯⟩)h.«2».«0»V:CKMMatrixδ₁₃:ℝh:Real.cos (θ₁₂ ⟦V⟧) = 0⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨2, ⋯⟩) ((fun i => i) ⟨0, ⋯⟩) =
  ↑(phaseShift 0 δ₁₃ δ₁₃ * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift (-δ₁₃) 0 (-δ₁₃))
    ((fun i => i) ⟨2, ⋯⟩) ((fun i => i) ⟨0, ⋯⟩)h.«2».«1»V:CKMMatrixδ₁₃:ℝh:Real.cos (θ₁₂ ⟦V⟧) = 0⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨2, ⋯⟩) ((fun i => i) ⟨1, ⋯⟩) =
  ↑(phaseShift 0 δ₁₃ δ₁₃ * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift (-δ₁₃) 0 (-δ₁₃))
    ((fun i => i) ⟨2, ⋯⟩) ((fun i => i) ⟨1, ⋯⟩)h.«2».«2»V:CKMMatrixδ₁₃:ℝh:Real.cos (θ₁₂ ⟦V⟧) = 0⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨2, ⋯⟩) ((fun i => i) ⟨2, ⋯⟩) =
  ↑(phaseShift 0 δ₁₃ δ₁₃ * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift (-δ₁₃) 0 (-δ₁₃))
    ((fun i => i) ⟨2, ⋯⟩) ((fun i => i) ⟨2, ⋯⟩)
    simp [standParam, standParamAsMatrix, phaseShift, phaseShiftMatrix, mul_apply,
      Fin.sum_univ_three, h, exp_neg]h.«2».«2»V:CKMMatrixδ₁₃:ℝh:Real.cos (θ₁₂ ⟦V⟧) = 0⊢ cos ↑(θ₂₃ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) = cexp (I * ↑δ₁₃) * (cos ↑(θ₂₃ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧)) * (cexp (I * ↑δ₁₃))⁻¹ <;>h.«1».«0»V:CKMMatrixδ₁₃:ℝh:Real.cos (θ₁₂ ⟦V⟧) = 0⊢ sin ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₂₃ ⟦V⟧) = cexp (I * ↑δ₁₃) * (sin ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₂₃ ⟦V⟧)) * (cexp (I * ↑δ₁₃))⁻¹h.«1».«1»V:CKMMatrixδ₁₃:ℝh:Real.cos (θ₁₂ ⟦V⟧) = 0⊢ sin ↑(θ₁₂ ⟦V⟧) * sin ↑(θ₁₃ ⟦V⟧) * sin ↑(θ₂₃ ⟦V⟧) * cexp (I * ↑δ₁₃) =
  cexp (I * ↑δ₁₃) * (sin ↑(θ₁₂ ⟦V⟧) * sin ↑(θ₁₃ ⟦V⟧) * sin ↑(θ₂₃ ⟦V⟧))h.«1».«2»V:CKMMatrixδ₁₃:ℝh:Real.cos (θ₁₂ ⟦V⟧) = 0⊢ sin ↑(θ₂₃ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) = cexp (I * ↑δ₁₃) * (sin ↑(θ₂₃ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧)) * (cexp (I * ↑δ₁₃))⁻¹h.«2».«0»V:CKMMatrixδ₁₃:ℝh:Real.cos (θ₁₂ ⟦V⟧) = 0⊢ sin ↑(θ₁₂ ⟦V⟧) * sin ↑(θ₂₃ ⟦V⟧) = cexp (I * ↑δ₁₃) * (sin ↑(θ₁₂ ⟦V⟧) * sin ↑(θ₂₃ ⟦V⟧)) * (cexp (I * ↑δ₁₃))⁻¹h.«2».«1»V:CKMMatrixδ₁₃:ℝh:Real.cos (θ₁₂ ⟦V⟧) = 0⊢ sin ↑(θ₁₂ ⟦V⟧) * sin ↑(θ₁₃ ⟦V⟧) * cos ↑(θ₂₃ ⟦V⟧) * cexp (I * ↑δ₁₃) =
  cexp (I * ↑δ₁₃) * (sin ↑(θ₁₂ ⟦V⟧) * sin ↑(θ₁₃ ⟦V⟧) * cos ↑(θ₂₃ ⟦V⟧))h.«2».«2»V:CKMMatrixδ₁₃:ℝh:Real.cos (θ₁₂ ⟦V⟧) = 0⊢ cos ↑(θ₂₃ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) = cexp (I * ↑δ₁₃) * (cos ↑(θ₂₃ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧)) * (cexp (I * ↑δ₁₃))⁻¹
    field_simpAll goals completed! 🐙lemma on_param_cos_θ₂₃_eq_zero {V : CKMMatrix} (δ₁₃ : ℝ) (h : Real.cos (θ₂₃ ⟦V⟧) = 0) :
    standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃ ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 := byV:CKMMatrixδ₁₃:ℝh:Real.cos (θ₂₃ ⟦V⟧) = 0⊢ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃ ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0
  use 0, δ₁₃, 0, 0, 0, - δ₁₃hV:CKMMatrixδ₁₃:ℝh:Real.cos (θ₂₃ ⟦V⟧) = 0⊢ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃ =
  phaseShift 0 δ₁₃ 0 * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 0 (-δ₁₃)
  ext i jhV:CKMMatrixδ₁₃:ℝh:Real.cos (θ₂₃ ⟦V⟧) = 0i:Fin 3j:Fin 3⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) i j =
  ↑(phaseShift 0 δ₁₃ 0 * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 0 (-δ₁₃)) i j
  fin_cases ih.«0»V:CKMMatrixδ₁₃:ℝh:Real.cos (θ₂₃ ⟦V⟧) = 0j:Fin 3⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨0, ⋯⟩) j =
  ↑(phaseShift 0 δ₁₃ 0 * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 0 (-δ₁₃)) ((fun i => i) ⟨0, ⋯⟩) jh.«1»V:CKMMatrixδ₁₃:ℝh:Real.cos (θ₂₃ ⟦V⟧) = 0j:Fin 3⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨1, ⋯⟩) j =
  ↑(phaseShift 0 δ₁₃ 0 * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 0 (-δ₁₃)) ((fun i => i) ⟨1, ⋯⟩) jh.«2»V:CKMMatrixδ₁₃:ℝh:Real.cos (θ₂₃ ⟦V⟧) = 0j:Fin 3⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨2, ⋯⟩) j =
  ↑(phaseShift 0 δ₁₃ 0 * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 0 (-δ₁₃)) ((fun i => i) ⟨2, ⋯⟩) j <;>h.«0»V:CKMMatrixδ₁₃:ℝh:Real.cos (θ₂₃ ⟦V⟧) = 0j:Fin 3⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨0, ⋯⟩) j =
  ↑(phaseShift 0 δ₁₃ 0 * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 0 (-δ₁₃)) ((fun i => i) ⟨0, ⋯⟩) jh.«1»V:CKMMatrixδ₁₃:ℝh:Real.cos (θ₂₃ ⟦V⟧) = 0j:Fin 3⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨1, ⋯⟩) j =
  ↑(phaseShift 0 δ₁₃ 0 * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 0 (-δ₁₃)) ((fun i => i) ⟨1, ⋯⟩) jh.«2»V:CKMMatrixδ₁₃:ℝh:Real.cos (θ₂₃ ⟦V⟧) = 0j:Fin 3⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨2, ⋯⟩) j =
  ↑(phaseShift 0 δ₁₃ 0 * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 0 (-δ₁₃)) ((fun i => i) ⟨2, ⋯⟩) j fin_cases jh.«2».«0»V:CKMMatrixδ₁₃:ℝh:Real.cos (θ₂₃ ⟦V⟧) = 0⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨2, ⋯⟩) ((fun i => i) ⟨0, ⋯⟩) =
  ↑(phaseShift 0 δ₁₃ 0 * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 0 (-δ₁₃)) ((fun i => i) ⟨2, ⋯⟩)
    ((fun i => i) ⟨0, ⋯⟩)h.«2».«1»V:CKMMatrixδ₁₃:ℝh:Real.cos (θ₂₃ ⟦V⟧) = 0⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨2, ⋯⟩) ((fun i => i) ⟨1, ⋯⟩) =
  ↑(phaseShift 0 δ₁₃ 0 * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 0 (-δ₁₃)) ((fun i => i) ⟨2, ⋯⟩)
    ((fun i => i) ⟨1, ⋯⟩)h.«2».«2»V:CKMMatrixδ₁₃:ℝh:Real.cos (θ₂₃ ⟦V⟧) = 0⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨2, ⋯⟩) ((fun i => i) ⟨2, ⋯⟩) =
  ↑(phaseShift 0 δ₁₃ 0 * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 0 (-δ₁₃)) ((fun i => i) ⟨2, ⋯⟩)
    ((fun i => i) ⟨2, ⋯⟩) <;>h.«0».«0»V:CKMMatrixδ₁₃:ℝh:Real.cos (θ₂₃ ⟦V⟧) = 0⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨0, ⋯⟩) ((fun i => i) ⟨0, ⋯⟩) =
  ↑(phaseShift 0 δ₁₃ 0 * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 0 (-δ₁₃)) ((fun i => i) ⟨0, ⋯⟩)
    ((fun i => i) ⟨0, ⋯⟩)h.«0».«1»V:CKMMatrixδ₁₃:ℝh:Real.cos (θ₂₃ ⟦V⟧) = 0⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨0, ⋯⟩) ((fun i => i) ⟨1, ⋯⟩) =
  ↑(phaseShift 0 δ₁₃ 0 * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 0 (-δ₁₃)) ((fun i => i) ⟨0, ⋯⟩)
    ((fun i => i) ⟨1, ⋯⟩)h.«0».«2»V:CKMMatrixδ₁₃:ℝh:Real.cos (θ₂₃ ⟦V⟧) = 0⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨0, ⋯⟩) ((fun i => i) ⟨2, ⋯⟩) =
  ↑(phaseShift 0 δ₁₃ 0 * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 0 (-δ₁₃)) ((fun i => i) ⟨0, ⋯⟩)
    ((fun i => i) ⟨2, ⋯⟩)h.«1».«0»V:CKMMatrixδ₁₃:ℝh:Real.cos (θ₂₃ ⟦V⟧) = 0⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨1, ⋯⟩) ((fun i => i) ⟨0, ⋯⟩) =
  ↑(phaseShift 0 δ₁₃ 0 * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 0 (-δ₁₃)) ((fun i => i) ⟨1, ⋯⟩)
    ((fun i => i) ⟨0, ⋯⟩)h.«1».«1»V:CKMMatrixδ₁₃:ℝh:Real.cos (θ₂₃ ⟦V⟧) = 0⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨1, ⋯⟩) ((fun i => i) ⟨1, ⋯⟩) =
  ↑(phaseShift 0 δ₁₃ 0 * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 0 (-δ₁₃)) ((fun i => i) ⟨1, ⋯⟩)
    ((fun i => i) ⟨1, ⋯⟩)h.«1».«2»V:CKMMatrixδ₁₃:ℝh:Real.cos (θ₂₃ ⟦V⟧) = 0⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨1, ⋯⟩) ((fun i => i) ⟨2, ⋯⟩) =
  ↑(phaseShift 0 δ₁₃ 0 * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 0 (-δ₁₃)) ((fun i => i) ⟨1, ⋯⟩)
    ((fun i => i) ⟨2, ⋯⟩)h.«2».«0»V:CKMMatrixδ₁₃:ℝh:Real.cos (θ₂₃ ⟦V⟧) = 0⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨2, ⋯⟩) ((fun i => i) ⟨0, ⋯⟩) =
  ↑(phaseShift 0 δ₁₃ 0 * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 0 (-δ₁₃)) ((fun i => i) ⟨2, ⋯⟩)
    ((fun i => i) ⟨0, ⋯⟩)h.«2».«1»V:CKMMatrixδ₁₃:ℝh:Real.cos (θ₂₃ ⟦V⟧) = 0⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨2, ⋯⟩) ((fun i => i) ⟨1, ⋯⟩) =
  ↑(phaseShift 0 δ₁₃ 0 * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 0 (-δ₁₃)) ((fun i => i) ⟨2, ⋯⟩)
    ((fun i => i) ⟨1, ⋯⟩)h.«2».«2»V:CKMMatrixδ₁₃:ℝh:Real.cos (θ₂₃ ⟦V⟧) = 0⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨2, ⋯⟩) ((fun i => i) ⟨2, ⋯⟩) =
  ↑(phaseShift 0 δ₁₃ 0 * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 0 (-δ₁₃)) ((fun i => i) ⟨2, ⋯⟩)
    ((fun i => i) ⟨2, ⋯⟩)
    simp [standParam, standParamAsMatrix, phaseShift, phaseShiftMatrix, mul_apply,
      Fin.sum_univ_three, h, exp_neg]All goals completed! 🐙 <;>h.«1».«0»V:CKMMatrixδ₁₃:ℝh:Real.cos (θ₂₃ ⟦V⟧) = 0⊢ cos ↑(θ₁₂ ⟦V⟧) * sin ↑(θ₁₃ ⟦V⟧) * sin ↑(θ₂₃ ⟦V⟧) * cexp (I * ↑δ₁₃) =
  cexp (I * ↑δ₁₃) * (cos ↑(θ₁₂ ⟦V⟧) * sin ↑(θ₁₃ ⟦V⟧) * sin ↑(θ₂₃ ⟦V⟧))h.«1».«1»V:CKMMatrixδ₁₃:ℝh:Real.cos (θ₂₃ ⟦V⟧) = 0⊢ sin ↑(θ₁₂ ⟦V⟧) * sin ↑(θ₁₃ ⟦V⟧) * sin ↑(θ₂₃ ⟦V⟧) * cexp (I * ↑δ₁₃) =
  cexp (I * ↑δ₁₃) * (sin ↑(θ₁₂ ⟦V⟧) * sin ↑(θ₁₃ ⟦V⟧) * sin ↑(θ₂₃ ⟦V⟧))h.«1».«2»V:CKMMatrixδ₁₃:ℝh:Real.cos (θ₂₃ ⟦V⟧) = 0⊢ sin ↑(θ₂₃ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) = cexp (I * ↑δ₁₃) * (sin ↑(θ₂₃ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧)) * (cexp (I * ↑δ₁₃))⁻¹
    field_simpAll goals completed! 🐙lemma on_param_sin_θ₁₃_eq_zero {V : CKMMatrix} (δ₁₃ : ℝ) (h : Real.sin (θ₁₃ ⟦V⟧) = 0) :
    standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃ ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 := byV:CKMMatrixδ₁₃:ℝh:Real.sin (θ₁₃ ⟦V⟧) = 0⊢ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃ ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0
  use 0, 0, 0, 0, 0, 0hV:CKMMatrixδ₁₃:ℝh:Real.sin (θ₁₃ ⟦V⟧) = 0⊢ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃ =
  phaseShift 0 0 0 * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 0 0
  ext i jhV:CKMMatrixδ₁₃:ℝh:Real.sin (θ₁₃ ⟦V⟧) = 0i:Fin 3j:Fin 3⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) i j =
  ↑(phaseShift 0 0 0 * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 0 0) i j
  fin_cases ih.«0»V:CKMMatrixδ₁₃:ℝh:Real.sin (θ₁₃ ⟦V⟧) = 0j:Fin 3⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨0, ⋯⟩) j =
  ↑(phaseShift 0 0 0 * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 0 0) ((fun i => i) ⟨0, ⋯⟩) jh.«1»V:CKMMatrixδ₁₃:ℝh:Real.sin (θ₁₃ ⟦V⟧) = 0j:Fin 3⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨1, ⋯⟩) j =
  ↑(phaseShift 0 0 0 * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 0 0) ((fun i => i) ⟨1, ⋯⟩) jh.«2»V:CKMMatrixδ₁₃:ℝh:Real.sin (θ₁₃ ⟦V⟧) = 0j:Fin 3⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨2, ⋯⟩) j =
  ↑(phaseShift 0 0 0 * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 0 0) ((fun i => i) ⟨2, ⋯⟩) j <;>h.«0»V:CKMMatrixδ₁₃:ℝh:Real.sin (θ₁₃ ⟦V⟧) = 0j:Fin 3⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨0, ⋯⟩) j =
  ↑(phaseShift 0 0 0 * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 0 0) ((fun i => i) ⟨0, ⋯⟩) jh.«1»V:CKMMatrixδ₁₃:ℝh:Real.sin (θ₁₃ ⟦V⟧) = 0j:Fin 3⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨1, ⋯⟩) j =
  ↑(phaseShift 0 0 0 * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 0 0) ((fun i => i) ⟨1, ⋯⟩) jh.«2»V:CKMMatrixδ₁₃:ℝh:Real.sin (θ₁₃ ⟦V⟧) = 0j:Fin 3⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨2, ⋯⟩) j =
  ↑(phaseShift 0 0 0 * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 0 0) ((fun i => i) ⟨2, ⋯⟩) j fin_cases jh.«2».«0»V:CKMMatrixδ₁₃:ℝh:Real.sin (θ₁₃ ⟦V⟧) = 0⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨2, ⋯⟩) ((fun i => i) ⟨0, ⋯⟩) =
  ↑(phaseShift 0 0 0 * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 0 0) ((fun i => i) ⟨2, ⋯⟩)
    ((fun i => i) ⟨0, ⋯⟩)h.«2».«1»V:CKMMatrixδ₁₃:ℝh:Real.sin (θ₁₃ ⟦V⟧) = 0⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨2, ⋯⟩) ((fun i => i) ⟨1, ⋯⟩) =
  ↑(phaseShift 0 0 0 * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 0 0) ((fun i => i) ⟨2, ⋯⟩)
    ((fun i => i) ⟨1, ⋯⟩)h.«2».«2»V:CKMMatrixδ₁₃:ℝh:Real.sin (θ₁₃ ⟦V⟧) = 0⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨2, ⋯⟩) ((fun i => i) ⟨2, ⋯⟩) =
  ↑(phaseShift 0 0 0 * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 0 0) ((fun i => i) ⟨2, ⋯⟩)
    ((fun i => i) ⟨2, ⋯⟩) <;>h.«0».«0»V:CKMMatrixδ₁₃:ℝh:Real.sin (θ₁₃ ⟦V⟧) = 0⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨0, ⋯⟩) ((fun i => i) ⟨0, ⋯⟩) =
  ↑(phaseShift 0 0 0 * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 0 0) ((fun i => i) ⟨0, ⋯⟩)
    ((fun i => i) ⟨0, ⋯⟩)h.«0».«1»V:CKMMatrixδ₁₃:ℝh:Real.sin (θ₁₃ ⟦V⟧) = 0⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨0, ⋯⟩) ((fun i => i) ⟨1, ⋯⟩) =
  ↑(phaseShift 0 0 0 * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 0 0) ((fun i => i) ⟨0, ⋯⟩)
    ((fun i => i) ⟨1, ⋯⟩)h.«0».«2»V:CKMMatrixδ₁₃:ℝh:Real.sin (θ₁₃ ⟦V⟧) = 0⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨0, ⋯⟩) ((fun i => i) ⟨2, ⋯⟩) =
  ↑(phaseShift 0 0 0 * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 0 0) ((fun i => i) ⟨0, ⋯⟩)
    ((fun i => i) ⟨2, ⋯⟩)h.«1».«0»V:CKMMatrixδ₁₃:ℝh:Real.sin (θ₁₃ ⟦V⟧) = 0⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨1, ⋯⟩) ((fun i => i) ⟨0, ⋯⟩) =
  ↑(phaseShift 0 0 0 * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 0 0) ((fun i => i) ⟨1, ⋯⟩)
    ((fun i => i) ⟨0, ⋯⟩)h.«1».«1»V:CKMMatrixδ₁₃:ℝh:Real.sin (θ₁₃ ⟦V⟧) = 0⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨1, ⋯⟩) ((fun i => i) ⟨1, ⋯⟩) =
  ↑(phaseShift 0 0 0 * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 0 0) ((fun i => i) ⟨1, ⋯⟩)
    ((fun i => i) ⟨1, ⋯⟩)h.«1».«2»V:CKMMatrixδ₁₃:ℝh:Real.sin (θ₁₃ ⟦V⟧) = 0⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨1, ⋯⟩) ((fun i => i) ⟨2, ⋯⟩) =
  ↑(phaseShift 0 0 0 * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 0 0) ((fun i => i) ⟨1, ⋯⟩)
    ((fun i => i) ⟨2, ⋯⟩)h.«2».«0»V:CKMMatrixδ₁₃:ℝh:Real.sin (θ₁₃ ⟦V⟧) = 0⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨2, ⋯⟩) ((fun i => i) ⟨0, ⋯⟩) =
  ↑(phaseShift 0 0 0 * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 0 0) ((fun i => i) ⟨2, ⋯⟩)
    ((fun i => i) ⟨0, ⋯⟩)h.«2».«1»V:CKMMatrixδ₁₃:ℝh:Real.sin (θ₁₃ ⟦V⟧) = 0⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨2, ⋯⟩) ((fun i => i) ⟨1, ⋯⟩) =
  ↑(phaseShift 0 0 0 * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 0 0) ((fun i => i) ⟨2, ⋯⟩)
    ((fun i => i) ⟨1, ⋯⟩)h.«2».«2»V:CKMMatrixδ₁₃:ℝh:Real.sin (θ₁₃ ⟦V⟧) = 0⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨2, ⋯⟩) ((fun i => i) ⟨2, ⋯⟩) =
  ↑(phaseShift 0 0 0 * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 0 0) ((fun i => i) ⟨2, ⋯⟩)
    ((fun i => i) ⟨2, ⋯⟩)
    simp [standParam, standParamAsMatrix, phaseShift, phaseShiftMatrix, mul_apply,
      Fin.sum_univ_three, h, exp_neg]All goals completed! 🐙lemma on_param_sin_θ₁₂_eq_zero {V : CKMMatrix} (δ₁₃ : ℝ) (h : Real.sin (θ₁₂ ⟦V⟧) = 0) :
    standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃ ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 := byV:CKMMatrixδ₁₃:ℝh:Real.sin (θ₁₂ ⟦V⟧) = 0⊢ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃ ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0
  use 0, δ₁₃, δ₁₃, 0, -δ₁₃, - δ₁₃hV:CKMMatrixδ₁₃:ℝh:Real.sin (θ₁₂ ⟦V⟧) = 0⊢ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃ =
  phaseShift 0 δ₁₃ δ₁₃ * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 (-δ₁₃) (-δ₁₃)
  ext i jhV:CKMMatrixδ₁₃:ℝh:Real.sin (θ₁₂ ⟦V⟧) = 0i:Fin 3j:Fin 3⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) i j =
  ↑(phaseShift 0 δ₁₃ δ₁₃ * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 (-δ₁₃) (-δ₁₃)) i j
  fin_cases ih.«0»V:CKMMatrixδ₁₃:ℝh:Real.sin (θ₁₂ ⟦V⟧) = 0j:Fin 3⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨0, ⋯⟩) j =
  ↑(phaseShift 0 δ₁₃ δ₁₃ * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 (-δ₁₃) (-δ₁₃))
    ((fun i => i) ⟨0, ⋯⟩) jh.«1»V:CKMMatrixδ₁₃:ℝh:Real.sin (θ₁₂ ⟦V⟧) = 0j:Fin 3⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨1, ⋯⟩) j =
  ↑(phaseShift 0 δ₁₃ δ₁₃ * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 (-δ₁₃) (-δ₁₃))
    ((fun i => i) ⟨1, ⋯⟩) jh.«2»V:CKMMatrixδ₁₃:ℝh:Real.sin (θ₁₂ ⟦V⟧) = 0j:Fin 3⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨2, ⋯⟩) j =
  ↑(phaseShift 0 δ₁₃ δ₁₃ * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 (-δ₁₃) (-δ₁₃))
    ((fun i => i) ⟨2, ⋯⟩) j <;>h.«0»V:CKMMatrixδ₁₃:ℝh:Real.sin (θ₁₂ ⟦V⟧) = 0j:Fin 3⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨0, ⋯⟩) j =
  ↑(phaseShift 0 δ₁₃ δ₁₃ * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 (-δ₁₃) (-δ₁₃))
    ((fun i => i) ⟨0, ⋯⟩) jh.«1»V:CKMMatrixδ₁₃:ℝh:Real.sin (θ₁₂ ⟦V⟧) = 0j:Fin 3⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨1, ⋯⟩) j =
  ↑(phaseShift 0 δ₁₃ δ₁₃ * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 (-δ₁₃) (-δ₁₃))
    ((fun i => i) ⟨1, ⋯⟩) jh.«2»V:CKMMatrixδ₁₃:ℝh:Real.sin (θ₁₂ ⟦V⟧) = 0j:Fin 3⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨2, ⋯⟩) j =
  ↑(phaseShift 0 δ₁₃ δ₁₃ * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 (-δ₁₃) (-δ₁₃))
    ((fun i => i) ⟨2, ⋯⟩) j fin_cases jh.«2».«0»V:CKMMatrixδ₁₃:ℝh:Real.sin (θ₁₂ ⟦V⟧) = 0⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨2, ⋯⟩) ((fun i => i) ⟨0, ⋯⟩) =
  ↑(phaseShift 0 δ₁₃ δ₁₃ * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 (-δ₁₃) (-δ₁₃))
    ((fun i => i) ⟨2, ⋯⟩) ((fun i => i) ⟨0, ⋯⟩)h.«2».«1»V:CKMMatrixδ₁₃:ℝh:Real.sin (θ₁₂ ⟦V⟧) = 0⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨2, ⋯⟩) ((fun i => i) ⟨1, ⋯⟩) =
  ↑(phaseShift 0 δ₁₃ δ₁₃ * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 (-δ₁₃) (-δ₁₃))
    ((fun i => i) ⟨2, ⋯⟩) ((fun i => i) ⟨1, ⋯⟩)h.«2».«2»V:CKMMatrixδ₁₃:ℝh:Real.sin (θ₁₂ ⟦V⟧) = 0⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨2, ⋯⟩) ((fun i => i) ⟨2, ⋯⟩) =
  ↑(phaseShift 0 δ₁₃ δ₁₃ * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 (-δ₁₃) (-δ₁₃))
    ((fun i => i) ⟨2, ⋯⟩) ((fun i => i) ⟨2, ⋯⟩) <;>h.«0».«0»V:CKMMatrixδ₁₃:ℝh:Real.sin (θ₁₂ ⟦V⟧) = 0⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨0, ⋯⟩) ((fun i => i) ⟨0, ⋯⟩) =
  ↑(phaseShift 0 δ₁₃ δ₁₃ * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 (-δ₁₃) (-δ₁₃))
    ((fun i => i) ⟨0, ⋯⟩) ((fun i => i) ⟨0, ⋯⟩)h.«0».«1»V:CKMMatrixδ₁₃:ℝh:Real.sin (θ₁₂ ⟦V⟧) = 0⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨0, ⋯⟩) ((fun i => i) ⟨1, ⋯⟩) =
  ↑(phaseShift 0 δ₁₃ δ₁₃ * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 (-δ₁₃) (-δ₁₃))
    ((fun i => i) ⟨0, ⋯⟩) ((fun i => i) ⟨1, ⋯⟩)h.«0».«2»V:CKMMatrixδ₁₃:ℝh:Real.sin (θ₁₂ ⟦V⟧) = 0⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨0, ⋯⟩) ((fun i => i) ⟨2, ⋯⟩) =
  ↑(phaseShift 0 δ₁₃ δ₁₃ * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 (-δ₁₃) (-δ₁₃))
    ((fun i => i) ⟨0, ⋯⟩) ((fun i => i) ⟨2, ⋯⟩)h.«1».«0»V:CKMMatrixδ₁₃:ℝh:Real.sin (θ₁₂ ⟦V⟧) = 0⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨1, ⋯⟩) ((fun i => i) ⟨0, ⋯⟩) =
  ↑(phaseShift 0 δ₁₃ δ₁₃ * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 (-δ₁₃) (-δ₁₃))
    ((fun i => i) ⟨1, ⋯⟩) ((fun i => i) ⟨0, ⋯⟩)h.«1».«1»V:CKMMatrixδ₁₃:ℝh:Real.sin (θ₁₂ ⟦V⟧) = 0⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨1, ⋯⟩) ((fun i => i) ⟨1, ⋯⟩) =
  ↑(phaseShift 0 δ₁₃ δ₁₃ * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 (-δ₁₃) (-δ₁₃))
    ((fun i => i) ⟨1, ⋯⟩) ((fun i => i) ⟨1, ⋯⟩)h.«1».«2»V:CKMMatrixδ₁₃:ℝh:Real.sin (θ₁₂ ⟦V⟧) = 0⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨1, ⋯⟩) ((fun i => i) ⟨2, ⋯⟩) =
  ↑(phaseShift 0 δ₁₃ δ₁₃ * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 (-δ₁₃) (-δ₁₃))
    ((fun i => i) ⟨1, ⋯⟩) ((fun i => i) ⟨2, ⋯⟩)h.«2».«0»V:CKMMatrixδ₁₃:ℝh:Real.sin (θ₁₂ ⟦V⟧) = 0⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨2, ⋯⟩) ((fun i => i) ⟨0, ⋯⟩) =
  ↑(phaseShift 0 δ₁₃ δ₁₃ * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 (-δ₁₃) (-δ₁₃))
    ((fun i => i) ⟨2, ⋯⟩) ((fun i => i) ⟨0, ⋯⟩)h.«2».«1»V:CKMMatrixδ₁₃:ℝh:Real.sin (θ₁₂ ⟦V⟧) = 0⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨2, ⋯⟩) ((fun i => i) ⟨1, ⋯⟩) =
  ↑(phaseShift 0 δ₁₃ δ₁₃ * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 (-δ₁₃) (-δ₁₃))
    ((fun i => i) ⟨2, ⋯⟩) ((fun i => i) ⟨1, ⋯⟩)h.«2».«2»V:CKMMatrixδ₁₃:ℝh:Real.sin (θ₁₂ ⟦V⟧) = 0⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨2, ⋯⟩) ((fun i => i) ⟨2, ⋯⟩) =
  ↑(phaseShift 0 δ₁₃ δ₁₃ * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 (-δ₁₃) (-δ₁₃))
    ((fun i => i) ⟨2, ⋯⟩) ((fun i => i) ⟨2, ⋯⟩)
    simp [standParam, standParamAsMatrix, phaseShift, phaseShiftMatrix, mul_apply,
      Fin.sum_univ_three, h, exp_neg]h.«2».«2»V:CKMMatrixδ₁₃:ℝh:Real.sin (θ₁₂ ⟦V⟧) = 0⊢ cos ↑(θ₂₃ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) = cexp (I * ↑δ₁₃) * (cos ↑(θ₂₃ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧)) * (cexp (I * ↑δ₁₃))⁻¹ <;>h.«1».«0»V:CKMMatrixδ₁₃:ℝh:Real.sin (θ₁₂ ⟦V⟧) = 0⊢ cos ↑(θ₁₂ ⟦V⟧) * sin ↑(θ₁₃ ⟦V⟧) * sin ↑(θ₂₃ ⟦V⟧) * cexp (I * ↑δ₁₃) =
  cexp (I * ↑δ₁₃) * (cos ↑(θ₁₂ ⟦V⟧) * sin ↑(θ₁₃ ⟦V⟧) * sin ↑(θ₂₃ ⟦V⟧))h.«1».«1»V:CKMMatrixδ₁₃:ℝh:Real.sin (θ₁₂ ⟦V⟧) = 0⊢ cos ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₂₃ ⟦V⟧) = cexp (I * ↑δ₁₃) * (cos ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₂₃ ⟦V⟧)) * (cexp (I * ↑δ₁₃))⁻¹h.«1».«2»V:CKMMatrixδ₁₃:ℝh:Real.sin (θ₁₂ ⟦V⟧) = 0⊢ sin ↑(θ₂₃ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) = cexp (I * ↑δ₁₃) * (sin ↑(θ₂₃ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧)) * (cexp (I * ↑δ₁₃))⁻¹h.«2».«0»V:CKMMatrixδ₁₃:ℝh:Real.sin (θ₁₂ ⟦V⟧) = 0⊢ cos ↑(θ₁₂ ⟦V⟧) * sin ↑(θ₁₃ ⟦V⟧) * cos ↑(θ₂₃ ⟦V⟧) * cexp (I * ↑δ₁₃) =
  cexp (I * ↑δ₁₃) * (cos ↑(θ₁₂ ⟦V⟧) * sin ↑(θ₁₃ ⟦V⟧) * cos ↑(θ₂₃ ⟦V⟧))h.«2».«1»V:CKMMatrixδ₁₃:ℝh:Real.sin (θ₁₂ ⟦V⟧) = 0⊢ cos ↑(θ₁₂ ⟦V⟧) * sin ↑(θ₂₃ ⟦V⟧) = cexp (I * ↑δ₁₃) * (cos ↑(θ₁₂ ⟦V⟧) * sin ↑(θ₂₃ ⟦V⟧)) * (cexp (I * ↑δ₁₃))⁻¹h.«2».«2»V:CKMMatrixδ₁₃:ℝh:Real.sin (θ₁₂ ⟦V⟧) = 0⊢ cos ↑(θ₂₃ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) = cexp (I * ↑δ₁₃) * (cos ↑(θ₂₃ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧)) * (cexp (I * ↑δ₁₃))⁻¹
    field_simpAll goals completed! 🐙lemma on_param_sin_θ₂₃_eq_zero {V : CKMMatrix} (δ₁₃ : ℝ) (h : Real.sin (θ₂₃ ⟦V⟧) = 0) :
    standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃ ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 := byV:CKMMatrixδ₁₃:ℝh:Real.sin (θ₂₃ ⟦V⟧) = 0⊢ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃ ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0
  use 0, 0, δ₁₃, 0, 0, - δ₁₃hV:CKMMatrixδ₁₃:ℝh:Real.sin (θ₂₃ ⟦V⟧) = 0⊢ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃ =
  phaseShift 0 0 δ₁₃ * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 0 (-δ₁₃)
  ext i jhV:CKMMatrixδ₁₃:ℝh:Real.sin (θ₂₃ ⟦V⟧) = 0i:Fin 3j:Fin 3⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) i j =
  ↑(phaseShift 0 0 δ₁₃ * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 0 (-δ₁₃)) i j
  fin_cases ih.«0»V:CKMMatrixδ₁₃:ℝh:Real.sin (θ₂₃ ⟦V⟧) = 0j:Fin 3⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨0, ⋯⟩) j =
  ↑(phaseShift 0 0 δ₁₃ * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 0 (-δ₁₃)) ((fun i => i) ⟨0, ⋯⟩) jh.«1»V:CKMMatrixδ₁₃:ℝh:Real.sin (θ₂₃ ⟦V⟧) = 0j:Fin 3⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨1, ⋯⟩) j =
  ↑(phaseShift 0 0 δ₁₃ * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 0 (-δ₁₃)) ((fun i => i) ⟨1, ⋯⟩) jh.«2»V:CKMMatrixδ₁₃:ℝh:Real.sin (θ₂₃ ⟦V⟧) = 0j:Fin 3⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨2, ⋯⟩) j =
  ↑(phaseShift 0 0 δ₁₃ * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 0 (-δ₁₃)) ((fun i => i) ⟨2, ⋯⟩) j <;>h.«0»V:CKMMatrixδ₁₃:ℝh:Real.sin (θ₂₃ ⟦V⟧) = 0j:Fin 3⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨0, ⋯⟩) j =
  ↑(phaseShift 0 0 δ₁₃ * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 0 (-δ₁₃)) ((fun i => i) ⟨0, ⋯⟩) jh.«1»V:CKMMatrixδ₁₃:ℝh:Real.sin (θ₂₃ ⟦V⟧) = 0j:Fin 3⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨1, ⋯⟩) j =
  ↑(phaseShift 0 0 δ₁₃ * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 0 (-δ₁₃)) ((fun i => i) ⟨1, ⋯⟩) jh.«2»V:CKMMatrixδ₁₃:ℝh:Real.sin (θ₂₃ ⟦V⟧) = 0j:Fin 3⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨2, ⋯⟩) j =
  ↑(phaseShift 0 0 δ₁₃ * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 0 (-δ₁₃)) ((fun i => i) ⟨2, ⋯⟩) j fin_cases jh.«2».«0»V:CKMMatrixδ₁₃:ℝh:Real.sin (θ₂₃ ⟦V⟧) = 0⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨2, ⋯⟩) ((fun i => i) ⟨0, ⋯⟩) =
  ↑(phaseShift 0 0 δ₁₃ * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 0 (-δ₁₃)) ((fun i => i) ⟨2, ⋯⟩)
    ((fun i => i) ⟨0, ⋯⟩)h.«2».«1»V:CKMMatrixδ₁₃:ℝh:Real.sin (θ₂₃ ⟦V⟧) = 0⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨2, ⋯⟩) ((fun i => i) ⟨1, ⋯⟩) =
  ↑(phaseShift 0 0 δ₁₃ * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 0 (-δ₁₃)) ((fun i => i) ⟨2, ⋯⟩)
    ((fun i => i) ⟨1, ⋯⟩)h.«2».«2»V:CKMMatrixδ₁₃:ℝh:Real.sin (θ₂₃ ⟦V⟧) = 0⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨2, ⋯⟩) ((fun i => i) ⟨2, ⋯⟩) =
  ↑(phaseShift 0 0 δ₁₃ * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 0 (-δ₁₃)) ((fun i => i) ⟨2, ⋯⟩)
    ((fun i => i) ⟨2, ⋯⟩) <;>h.«0».«0»V:CKMMatrixδ₁₃:ℝh:Real.sin (θ₂₃ ⟦V⟧) = 0⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨0, ⋯⟩) ((fun i => i) ⟨0, ⋯⟩) =
  ↑(phaseShift 0 0 δ₁₃ * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 0 (-δ₁₃)) ((fun i => i) ⟨0, ⋯⟩)
    ((fun i => i) ⟨0, ⋯⟩)h.«0».«1»V:CKMMatrixδ₁₃:ℝh:Real.sin (θ₂₃ ⟦V⟧) = 0⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨0, ⋯⟩) ((fun i => i) ⟨1, ⋯⟩) =
  ↑(phaseShift 0 0 δ₁₃ * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 0 (-δ₁₃)) ((fun i => i) ⟨0, ⋯⟩)
    ((fun i => i) ⟨1, ⋯⟩)h.«0».«2»V:CKMMatrixδ₁₃:ℝh:Real.sin (θ₂₃ ⟦V⟧) = 0⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨0, ⋯⟩) ((fun i => i) ⟨2, ⋯⟩) =
  ↑(phaseShift 0 0 δ₁₃ * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 0 (-δ₁₃)) ((fun i => i) ⟨0, ⋯⟩)
    ((fun i => i) ⟨2, ⋯⟩)h.«1».«0»V:CKMMatrixδ₁₃:ℝh:Real.sin (θ₂₃ ⟦V⟧) = 0⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨1, ⋯⟩) ((fun i => i) ⟨0, ⋯⟩) =
  ↑(phaseShift 0 0 δ₁₃ * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 0 (-δ₁₃)) ((fun i => i) ⟨1, ⋯⟩)
    ((fun i => i) ⟨0, ⋯⟩)h.«1».«1»V:CKMMatrixδ₁₃:ℝh:Real.sin (θ₂₃ ⟦V⟧) = 0⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨1, ⋯⟩) ((fun i => i) ⟨1, ⋯⟩) =
  ↑(phaseShift 0 0 δ₁₃ * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 0 (-δ₁₃)) ((fun i => i) ⟨1, ⋯⟩)
    ((fun i => i) ⟨1, ⋯⟩)h.«1».«2»V:CKMMatrixδ₁₃:ℝh:Real.sin (θ₂₃ ⟦V⟧) = 0⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨1, ⋯⟩) ((fun i => i) ⟨2, ⋯⟩) =
  ↑(phaseShift 0 0 δ₁₃ * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 0 (-δ₁₃)) ((fun i => i) ⟨1, ⋯⟩)
    ((fun i => i) ⟨2, ⋯⟩)h.«2».«0»V:CKMMatrixδ₁₃:ℝh:Real.sin (θ₂₃ ⟦V⟧) = 0⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨2, ⋯⟩) ((fun i => i) ⟨0, ⋯⟩) =
  ↑(phaseShift 0 0 δ₁₃ * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 0 (-δ₁₃)) ((fun i => i) ⟨2, ⋯⟩)
    ((fun i => i) ⟨0, ⋯⟩)h.«2».«1»V:CKMMatrixδ₁₃:ℝh:Real.sin (θ₂₃ ⟦V⟧) = 0⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨2, ⋯⟩) ((fun i => i) ⟨1, ⋯⟩) =
  ↑(phaseShift 0 0 δ₁₃ * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 0 (-δ₁₃)) ((fun i => i) ⟨2, ⋯⟩)
    ((fun i => i) ⟨1, ⋯⟩)h.«2».«2»V:CKMMatrixδ₁₃:ℝh:Real.sin (θ₂₃ ⟦V⟧) = 0⊢ ↑(standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃) ((fun i => i) ⟨2, ⋯⟩) ((fun i => i) ⟨2, ⋯⟩) =
  ↑(phaseShift 0 0 δ₁₃ * standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 * phaseShift 0 0 (-δ₁₃)) ((fun i => i) ⟨2, ⋯⟩)
    ((fun i => i) ⟨2, ⋯⟩)
    simp [standParam, standParamAsMatrix, phaseShift, phaseShiftMatrix, mul_apply,
      Fin.sum_univ_three, h, exp_neg]h.«2».«2»V:CKMMatrixδ₁₃:ℝh:Real.sin (θ₂₃ ⟦V⟧) = 0⊢ cos ↑(θ₂₃ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) = cexp (I * ↑δ₁₃) * (cos ↑(θ₂₃ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧)) * (cexp (I * ↑δ₁₃))⁻¹ <;>h.«2».«0»V:CKMMatrixδ₁₃:ℝh:Real.sin (θ₂₃ ⟦V⟧) = 0⊢ cos ↑(θ₁₂ ⟦V⟧) * sin ↑(θ₁₃ ⟦V⟧) * cos ↑(θ₂₃ ⟦V⟧) * cexp (I * ↑δ₁₃) =
  cexp (I * ↑δ₁₃) * (cos ↑(θ₁₂ ⟦V⟧) * sin ↑(θ₁₃ ⟦V⟧) * cos ↑(θ₂₃ ⟦V⟧))h.«2».«1»V:CKMMatrixδ₁₃:ℝh:Real.sin (θ₂₃ ⟦V⟧) = 0⊢ sin ↑(θ₁₂ ⟦V⟧) * sin ↑(θ₁₃ ⟦V⟧) * cos ↑(θ₂₃ ⟦V⟧) * cexp (I * ↑δ₁₃) =
  cexp (I * ↑δ₁₃) * (sin ↑(θ₁₂ ⟦V⟧) * sin ↑(θ₁₃ ⟦V⟧) * cos ↑(θ₂₃ ⟦V⟧))h.«2».«2»V:CKMMatrixδ₁₃:ℝh:Real.sin (θ₂₃ ⟦V⟧) = 0⊢ cos ↑(θ₂₃ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧) = cexp (I * ↑δ₁₃) * (cos ↑(θ₂₃ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧)) * (cexp (I * ↑δ₁₃))⁻¹
    field_simpAll goals completed! 🐙

lemma eq_standParam_of_fstRowThdColRealCond {V : CKMMatrix} (hb : [V]ud ≠ 0 ∨ [V]us ≠ 0)
    (hV : FstRowThdColRealCond V) : V = standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) (- arg [V]ub) := byV:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCond⊢ V = standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) (-(↑V 0 2).arg)
  have hb' : VubAbs ⟦V⟧ ≠ 1 := (ud_us_ne_zero_iff_ub_ne_one V).mp hbV:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1⊢ V = standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) (-(↑V 0 2).arg)
  have h1 : ofRealHom (√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) *
    ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2)) = ofRealHom (VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) := by
    rw [Real.mul_self_sqrtV:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1⊢ ofRealHom (VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) = ofRealHom (VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2)V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1⊢ 0 ≤ VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1⊢ 0 ≤ VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:ofRealHom (√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * √(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2)) =
  ofRealHom (VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2)⊢ V = standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) (-(↑V 0 2).arg)]V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1⊢ 0 ≤ VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:ofRealHom (√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * √(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2)) =
  ofRealHom (VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2)⊢ V = standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) (-(↑V 0 2).arg)
    apply add_nonneg (sq_nonneg _) (sq_nonneg _)V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:ofRealHom (√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * √(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2)) =
  ofRealHom (VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2)⊢ V = standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) (-(↑V 0 2).arg)V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:ofRealHom (√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * √(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2)) =
  ofRealHom (VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2)⊢ V = standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) (-(↑V 0 2).arg)
  simp only [Fin.isValue, _root_.map_mul, ofRealHom_eq_coe, map_add, map_pow] at h1V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2⊢ V = standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) (-(↑V 0 2).arg)
  have hx := Vabs_sq_add_ne_zero hbV:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ V = standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) (-(↑V 0 2).arg)
  refine eq_rows V ?_ ?_ hV.2.2.2.2refine_1V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ [V]u = [standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) (-(↑V 0 2).arg)]urefine_2V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ [V]c = [standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) (-(↑V 0 2).arg)]c
  ·refine_1V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ [V]u = [standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) (-(↑V 0 2).arg)]u funext irefine_1V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0i:Fin 3⊢ [V]u i = [standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) (-(↑V 0 2).arg)]u i
    fin_cases irefine_1.«0»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ [V]u ((fun i => i) ⟨0, ⋯⟩) = [standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) (-(↑V 0 2).arg)]u ((fun i => i) ⟨0, ⋯⟩)refine_1.«1»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ [V]u ((fun i => i) ⟨1, ⋯⟩) = [standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) (-(↑V 0 2).arg)]u ((fun i => i) ⟨1, ⋯⟩)refine_1.«2»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ [V]u ((fun i => i) ⟨2, ⋯⟩) = [standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) (-(↑V 0 2).arg)]u ((fun i => i) ⟨2, ⋯⟩)
    ·refine_1.«0»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ [V]u ((fun i => i) ⟨0, ⋯⟩) = [standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) (-(↑V 0 2).arg)]u ((fun i => i) ⟨0, ⋯⟩) simp [uRow, standParam, standParamAsMatrix, hV.1, VudAbs_eq_C₁₂_mul_C₁₃ ⟦V⟧, C₁₂, C₁₃]All goals completed! 🐙
    ·refine_1.«1»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ [V]u ((fun i => i) ⟨1, ⋯⟩) = [standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) (-(↑V 0 2).arg)]u ((fun i => i) ⟨1, ⋯⟩) simp [uRow, standParam, standParamAsMatrix, hV.2.1, VusAbs_eq_S₁₂_mul_C₁₃ ⟦V⟧,
        ← S₁₂_eq_sin_θ₁₂ ⟦V⟧, C₁₃]All goals completed! 🐙
    ·refine_1.«2»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ [V]u ((fun i => i) ⟨2, ⋯⟩) = [standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) (-(↑V 0 2).arg)]u ((fun i => i) ⟨2, ⋯⟩) simp only [uRow, Fin.isValue, Fin.reduceFinMk, cons_val_two, Nat.succ_eq_add_one,
      Nat.reduceAdd, tail_cons, head_cons, standParam, standParamAsMatrix, ofReal_cos, ofReal_sin,
      ofReal_neg, mul_neg, neg_mul, neg_neg, cons_val', cons_val_zero, empty_val', cons_val_fin_one,
      cons_val_one]refine_1.«2»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ ↑V 0 2 = sin ↑(θ₁₃ ⟦V⟧) * cexp (I * ↑(↑V 0 2).arg)
      nth_rewrite 1 [← norm_mul_exp_arg_mul_I (V.1 0 2)]refine_1.«2»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ ↑‖↑V 0 2‖ * cexp (↑(↑V 0 2).arg * I) = sin ↑(θ₁₃ ⟦V⟧) * cexp (I * ↑(↑V 0 2).arg)
      rw [show norm (V.1 0 2) = VubAbs ⟦V⟧ from rfl,refine_1.«2»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ ↑(VubAbs ⟦V⟧) * cexp (↑(↑V 0 2).arg * I) = sin ↑(θ₁₃ ⟦V⟧) * cexp (I * ↑(↑V 0 2).arg)refine_1.«2»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ ↑(Real.sin (θ₁₃ ⟦V⟧)) * cexp (↑(↑V 0 2).arg * I) = sin ↑(θ₁₃ ⟦V⟧) * cexp (I * ↑(↑V 0 2).arg) VubAbs_eq_S₁₃,refine_1.«2»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ ↑(S₁₃ ⟦V⟧) * cexp (↑(↑V 0 2).arg * I) = sin ↑(θ₁₃ ⟦V⟧) * cexp (I * ↑(↑V 0 2).arg)refine_1.«2»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ ↑(Real.sin (θ₁₃ ⟦V⟧)) * cexp (↑(↑V 0 2).arg * I) = sin ↑(θ₁₃ ⟦V⟧) * cexp (I * ↑(↑V 0 2).arg) ← S₁₃_eq_sin_θ₁₃ ⟦V⟧refine_1.«2»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ ↑(Real.sin (θ₁₃ ⟦V⟧)) * cexp (↑(↑V 0 2).arg * I) = sin ↑(θ₁₃ ⟦V⟧) * cexp (I * ↑(↑V 0 2).arg)refine_1.«2»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ ↑(Real.sin (θ₁₃ ⟦V⟧)) * cexp (↑(↑V 0 2).arg * I) = sin ↑(θ₁₃ ⟦V⟧) * cexp (I * ↑(↑V 0 2).arg)]refine_1.«2»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ ↑(Real.sin (θ₁₃ ⟦V⟧)) * cexp (↑(↑V 0 2).arg * I) = sin ↑(θ₁₃ ⟦V⟧) * cexp (I * ↑(↑V 0 2).arg)
      simp only [ofReal_sin, Fin.isValue, mul_eq_mul_left_iff]refine_1.«2»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ cexp (↑(↑V 0 2).arg * I) = cexp (I * ↑(↑V 0 2).arg) ∨ sin ↑(θ₁₃ ⟦V⟧) = 0
      ring_nfrefine_1.«2»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ True ∨ sin ↑(θ₁₃ ⟦V⟧) = 0
      simp only [true_or]All goals completed! 🐙
  ·refine_2V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ [V]c = [standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) (-(↑V 0 2).arg)]c funext irefine_2V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0i:Fin 3⊢ [V]c i = [standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) (-(↑V 0 2).arg)]c i
    fin_cases irefine_2.«0»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ [V]c ((fun i => i) ⟨0, ⋯⟩) = [standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) (-(↑V 0 2).arg)]c ((fun i => i) ⟨0, ⋯⟩)refine_2.«1»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ [V]c ((fun i => i) ⟨1, ⋯⟩) = [standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) (-(↑V 0 2).arg)]c ((fun i => i) ⟨1, ⋯⟩)refine_2.«2»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ [V]c ((fun i => i) ⟨2, ⋯⟩) = [standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) (-(↑V 0 2).arg)]c ((fun i => i) ⟨2, ⋯⟩)
    ·refine_2.«0»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ [V]c ((fun i => i) ⟨0, ⋯⟩) = [standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) (-(↑V 0 2).arg)]c ((fun i => i) ⟨0, ⋯⟩) simp only [cRow, Fin.isValue, Fin.zero_eta, cons_val_zero, standParam, standParamAsMatrix,
      ofReal_cos, ofReal_sin, ofReal_neg, mul_neg, neg_mul, neg_neg, cons_val', empty_val',
      cons_val_fin_one, cons_val_one, head_cons, cons_val_two, Nat.succ_eq_add_one, Nat.reduceAdd,
      tail_cons]refine_2.«0»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ ↑V 1 0 =
  -(sin ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₂₃ ⟦V⟧)) - cos ↑(θ₁₂ ⟦V⟧) * sin ↑(θ₁₃ ⟦V⟧) * sin ↑(θ₂₃ ⟦V⟧) * cexp (-(I * ↑(↑V 0 2).arg))
      rw [cd_of_fstRowThdColRealCond hb hV,refine_2.«0»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ -↑(VtbAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) +
    -↑(VubAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) * ↑(VcbAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) *
      cexp (-↑(↑V 0 2).arg * I) =
  -(sin ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₂₃ ⟦V⟧)) - cos ↑(θ₁₂ ⟦V⟧) * sin ↑(θ₁₃ ⟦V⟧) * sin ↑(θ₂₃ ⟦V⟧) * cexp (-(I * ↑(↑V 0 2).arg))refine_2.«0»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ -↑(VtbAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) +
    -↑(VubAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) * ↑(VcbAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) *
      cexp (-↑(↑V 0 2).arg * I) =
  -(↑(VusAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * ↑(VtbAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2))) -
    ↑(VudAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * ↑(VubAbs ⟦V⟧) *
        ↑(VcbAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) *
      cexp (-(I * ↑(↑V 0 2).arg)) S₁₂_eq_ℂsin_θ₁₂ ⟦V⟧,refine_2.«0»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ -↑(VtbAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) +
    -↑(VubAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) * ↑(VcbAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) *
      cexp (-↑(↑V 0 2).arg * I) =
  -(↑(S₁₂ ⟦V⟧) * cos ↑(θ₂₃ ⟦V⟧)) - cos ↑(θ₁₂ ⟦V⟧) * sin ↑(θ₁₃ ⟦V⟧) * sin ↑(θ₂₃ ⟦V⟧) * cexp (-(I * ↑(↑V 0 2).arg))refine_2.«0»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ -↑(VtbAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) +
    -↑(VubAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) * ↑(VcbAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) *
      cexp (-↑(↑V 0 2).arg * I) =
  -(↑(VusAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * ↑(VtbAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2))) -
    ↑(VudAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * ↑(VubAbs ⟦V⟧) *
        ↑(VcbAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) *
      cexp (-(I * ↑(↑V 0 2).arg)) S₁₂,refine_2.«0»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ -↑(VtbAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) +
    -↑(VubAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) * ↑(VcbAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) *
      cexp (-↑(↑V 0 2).arg * I) =
  -(↑(VusAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * cos ↑(θ₂₃ ⟦V⟧)) -
    cos ↑(θ₁₂ ⟦V⟧) * sin ↑(θ₁₃ ⟦V⟧) * sin ↑(θ₂₃ ⟦V⟧) * cexp (-(I * ↑(↑V 0 2).arg))refine_2.«0»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ -↑(VtbAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) +
    -↑(VubAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) * ↑(VcbAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) *
      cexp (-↑(↑V 0 2).arg * I) =
  -(↑(VusAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * ↑(VtbAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2))) -
    ↑(VudAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * ↑(VubAbs ⟦V⟧) *
        ↑(VcbAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) *
      cexp (-(I * ↑(↑V 0 2).arg)) C₁₂_eq_ℂcos_θ₁₂ ⟦V⟧,refine_2.«0»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ -↑(VtbAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) +
    -↑(VubAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) * ↑(VcbAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) *
      cexp (-↑(↑V 0 2).arg * I) =
  -(↑(VusAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * cos ↑(θ₂₃ ⟦V⟧)) -
    ↑(C₁₂ ⟦V⟧) * sin ↑(θ₁₃ ⟦V⟧) * sin ↑(θ₂₃ ⟦V⟧) * cexp (-(I * ↑(↑V 0 2).arg))refine_2.«0»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ -↑(VtbAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) +
    -↑(VubAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) * ↑(VcbAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) *
      cexp (-↑(↑V 0 2).arg * I) =
  -(↑(VusAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * ↑(VtbAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2))) -
    ↑(VudAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * ↑(VubAbs ⟦V⟧) *
        ↑(VcbAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) *
      cexp (-(I * ↑(↑V 0 2).arg))
        C₁₂_eq_Vud_div_sqrt hb',refine_2.«0»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ -↑(VtbAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) +
    -↑(VubAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) * ↑(VcbAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) *
      cexp (-↑(↑V 0 2).arg * I) =
  -(↑(VusAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * cos ↑(θ₂₃ ⟦V⟧)) -
    ↑(VudAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * sin ↑(θ₁₃ ⟦V⟧) * sin ↑(θ₂₃ ⟦V⟧) * cexp (-(I * ↑(↑V 0 2).arg))refine_2.«0»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ -↑(VtbAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) +
    -↑(VubAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) * ↑(VcbAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) *
      cexp (-↑(↑V 0 2).arg * I) =
  -(↑(VusAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * ↑(VtbAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2))) -
    ↑(VudAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * ↑(VubAbs ⟦V⟧) *
        ↑(VcbAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) *
      cexp (-(I * ↑(↑V 0 2).arg)) S₂₃_eq_ℂsin_θ₂₃ ⟦V⟧,refine_2.«0»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ -↑(VtbAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) +
    -↑(VubAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) * ↑(VcbAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) *
      cexp (-↑(↑V 0 2).arg * I) =
  -(↑(VusAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * cos ↑(θ₂₃ ⟦V⟧)) -
    ↑(VudAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * sin ↑(θ₁₃ ⟦V⟧) * ↑(S₂₃ ⟦V⟧) * cexp (-(I * ↑(↑V 0 2).arg))refine_2.«0»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ -↑(VtbAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) +
    -↑(VubAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) * ↑(VcbAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) *
      cexp (-↑(↑V 0 2).arg * I) =
  -(↑(VusAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * ↑(VtbAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2))) -
    ↑(VudAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * ↑(VubAbs ⟦V⟧) *
        ↑(VcbAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) *
      cexp (-(I * ↑(↑V 0 2).arg)) S₂₃_of_Vub_ne_one hb',refine_2.«0»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ -↑(VtbAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) +
    -↑(VubAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) * ↑(VcbAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) *
      cexp (-↑(↑V 0 2).arg * I) =
  -(↑(VusAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * cos ↑(θ₂₃ ⟦V⟧)) -
    ↑(VudAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * sin ↑(θ₁₃ ⟦V⟧) *
        ↑(VcbAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) *
      cexp (-(I * ↑(↑V 0 2).arg))refine_2.«0»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ -↑(VtbAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) +
    -↑(VubAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) * ↑(VcbAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) *
      cexp (-↑(↑V 0 2).arg * I) =
  -(↑(VusAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * ↑(VtbAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2))) -
    ↑(VudAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * ↑(VubAbs ⟦V⟧) *
        ↑(VcbAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) *
      cexp (-(I * ↑(↑V 0 2).arg)) C₂₃_eq_ℂcos_θ₂₃ ⟦V⟧,refine_2.«0»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ -↑(VtbAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) +
    -↑(VubAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) * ↑(VcbAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) *
      cexp (-↑(↑V 0 2).arg * I) =
  -(↑(VusAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * ↑(C₂₃ ⟦V⟧)) -
    ↑(VudAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * sin ↑(θ₁₃ ⟦V⟧) *
        ↑(VcbAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) *
      cexp (-(I * ↑(↑V 0 2).arg))refine_2.«0»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ -↑(VtbAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) +
    -↑(VubAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) * ↑(VcbAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) *
      cexp (-↑(↑V 0 2).arg * I) =
  -(↑(VusAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * ↑(VtbAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2))) -
    ↑(VudAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * ↑(VubAbs ⟦V⟧) *
        ↑(VcbAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) *
      cexp (-(I * ↑(↑V 0 2).arg))
        C₂₃_of_Vub_ne_one hb',refine_2.«0»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ -↑(VtbAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) +
    -↑(VubAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) * ↑(VcbAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) *
      cexp (-↑(↑V 0 2).arg * I) =
  -(↑(VusAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * ↑(VtbAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2))) -
    ↑(VudAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * sin ↑(θ₁₃ ⟦V⟧) *
        ↑(VcbAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) *
      cexp (-(I * ↑(↑V 0 2).arg))refine_2.«0»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ -↑(VtbAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) +
    -↑(VubAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) * ↑(VcbAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) *
      cexp (-↑(↑V 0 2).arg * I) =
  -(↑(VusAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * ↑(VtbAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2))) -
    ↑(VudAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * ↑(VubAbs ⟦V⟧) *
        ↑(VcbAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) *
      cexp (-(I * ↑(↑V 0 2).arg)) S₁₃_eq_ℂsin_θ₁₃ ⟦V⟧,refine_2.«0»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ -↑(VtbAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) +
    -↑(VubAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) * ↑(VcbAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) *
      cexp (-↑(↑V 0 2).arg * I) =
  -(↑(VusAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * ↑(VtbAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2))) -
    ↑(VudAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * ↑(S₁₃ ⟦V⟧) *
        ↑(VcbAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) *
      cexp (-(I * ↑(↑V 0 2).arg))refine_2.«0»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ -↑(VtbAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) +
    -↑(VubAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) * ↑(VcbAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) *
      cexp (-↑(↑V 0 2).arg * I) =
  -(↑(VusAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * ↑(VtbAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2))) -
    ↑(VudAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * ↑(VubAbs ⟦V⟧) *
        ↑(VcbAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) *
      cexp (-(I * ↑(↑V 0 2).arg)) S₁₃refine_2.«0»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ -↑(VtbAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) +
    -↑(VubAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) * ↑(VcbAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) *
      cexp (-↑(↑V 0 2).arg * I) =
  -(↑(VusAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * ↑(VtbAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2))) -
    ↑(VudAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * ↑(VubAbs ⟦V⟧) *
        ↑(VcbAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) *
      cexp (-(I * ↑(↑V 0 2).arg))refine_2.«0»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ -↑(VtbAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) +
    -↑(VubAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) * ↑(VcbAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) *
      cexp (-↑(↑V 0 2).arg * I) =
  -(↑(VusAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * ↑(VtbAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2))) -
    ↑(VudAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * ↑(VubAbs ⟦V⟧) *
        ↑(VcbAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) *
      cexp (-(I * ↑(↑V 0 2).arg))]refine_2.«0»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ -↑(VtbAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) +
    -↑(VubAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) * ↑(VcbAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) *
      cexp (-↑(↑V 0 2).arg * I) =
  -(↑(VusAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * ↑(VtbAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2))) -
    ↑(VudAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * ↑(VubAbs ⟦V⟧) *
        ↑(VcbAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) *
      cexp (-(I * ↑(↑V 0 2).arg))
      simp only [VtbAbs, Fin.isValue, VusAbs, neg_mul, VudAbs, VubAbs, VcbAbs, ofReal_div]refine_2.«0»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ -(↑(VAbs 2 2 ⟦V⟧) * ↑(VAbs 0 1 ⟦V⟧)) / (↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2) +
    -(↑(VAbs 0 2 ⟦V⟧) * ↑(VAbs 0 0 ⟦V⟧) * ↑(VAbs 1 2 ⟦V⟧)) / (↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2) *
      cexp (-(↑(↑V 0 2).arg * I)) =
  -(↑(VAbs 0 1 ⟦V⟧) / ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) *
        (↑(VAbs 2 2 ⟦V⟧) / ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2))) -
    ↑(VAbs 0 0 ⟦V⟧) / ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑(VAbs 0 2 ⟦V⟧) *
        (↑(VAbs 1 2 ⟦V⟧) / ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2)) *
      cexp (-(I * ↑(↑V 0 2).arg))
      field_simprefine_2.«0»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ (-(↑(VAbs 2 2 ⟦V⟧) * ↑(VAbs 0 1 ⟦V⟧)) +
      -(↑(VAbs 0 0 ⟦V⟧) * ↑(VAbs 0 2 ⟦V⟧) * ↑(VAbs 1 2 ⟦V⟧) * cexp (-(↑(↑V 0 2).arg * I)))) /
    (↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2) =
  (-(↑(VAbs 2 2 ⟦V⟧) * ↑(VAbs 0 1 ⟦V⟧)) -
      ↑(VAbs 0 0 ⟦V⟧) * ↑(VAbs 0 2 ⟦V⟧) * ↑(VAbs 1 2 ⟦V⟧) * cexp (-(↑(↑V 0 2).arg * I))) /
    ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) ^ 2
      conv_rhs => rw [sq, h1]V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0| (-(↑(VAbs 2 2 ⟦V⟧) * ↑(VAbs 0 1 ⟦V⟧)) -
    ↑(VAbs 0 0 ⟦V⟧) * ↑(VAbs 0 2 ⟦V⟧) * ↑(VAbs 1 2 ⟦V⟧) * cexp (-(↑(↑V 0 2).arg * I))) /
  (↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2)
      simp only [Fin.isValue, sq]refine_2.«0»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ (-(↑(VAbs 2 2 ⟦V⟧) * ↑(VAbs 0 1 ⟦V⟧)) +
      -(↑(VAbs 0 0 ⟦V⟧) * ↑(VAbs 0 2 ⟦V⟧) * ↑(VAbs 1 2 ⟦V⟧) * cexp (-(↑(↑V 0 2).arg * I)))) /
    (↑(VAbs 0 0 ⟦V⟧) * ↑(VAbs 0 0 ⟦V⟧) + ↑(VAbs 0 1 ⟦V⟧) * ↑(VAbs 0 1 ⟦V⟧)) =
  (-(↑(VAbs 2 2 ⟦V⟧) * ↑(VAbs 0 1 ⟦V⟧)) -
      ↑(VAbs 0 0 ⟦V⟧) * ↑(VAbs 0 2 ⟦V⟧) * ↑(VAbs 1 2 ⟦V⟧) * cexp (-(↑(↑V 0 2).arg * I))) /
    (↑(VAbs 0 0 ⟦V⟧) * ↑(VAbs 0 0 ⟦V⟧) + ↑(VAbs 0 1 ⟦V⟧) * ↑(VAbs 0 1 ⟦V⟧))
      field_simprefine_2.«0»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ (-(↑(VAbs 2 2 ⟦V⟧) * ↑(VAbs 0 1 ⟦V⟧)) +
      -(↑(VAbs 0 0 ⟦V⟧) * ↑(VAbs 0 2 ⟦V⟧) * ↑(VAbs 1 2 ⟦V⟧) * cexp (-(↑(↑V 0 2).arg * I)))) /
    (↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2) =
  (-(↑(VAbs 2 2 ⟦V⟧) * ↑(VAbs 0 1 ⟦V⟧)) -
      ↑(VAbs 0 0 ⟦V⟧) * ↑(VAbs 0 2 ⟦V⟧) * ↑(VAbs 1 2 ⟦V⟧) * cexp (-(↑(↑V 0 2).arg * I))) /
    (↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2)
      ring_nfAll goals completed! 🐙
    ·refine_2.«1»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ [V]c ((fun i => i) ⟨1, ⋯⟩) = [standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) (-(↑V 0 2).arg)]c ((fun i => i) ⟨1, ⋯⟩) simp only [cRow, Fin.isValue, Fin.mk_one, cons_val_one, head_cons, standParam,
      standParamAsMatrix, ofReal_cos, ofReal_sin, ofReal_neg, mul_neg, neg_mul, neg_neg, cons_val',
      cons_val_zero, empty_val', cons_val_fin_one, cons_val_two, Nat.succ_eq_add_one, Nat.reduceAdd,
      tail_cons]refine_2.«1»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ ↑V 1 1 =
  cos ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₂₃ ⟦V⟧) - sin ↑(θ₁₂ ⟦V⟧) * sin ↑(θ₁₃ ⟦V⟧) * sin ↑(θ₂₃ ⟦V⟧) * cexp (-(I * ↑(↑V 0 2).arg))
      rw [C₁₂_eq_ℂcos_θ₁₂ ⟦V⟧,refine_2.«1»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ ↑V 1 1 = ↑(C₁₂ ⟦V⟧) * cos ↑(θ₂₃ ⟦V⟧) - sin ↑(θ₁₂ ⟦V⟧) * sin ↑(θ₁₃ ⟦V⟧) * sin ↑(θ₂₃ ⟦V⟧) * cexp (-(I * ↑(↑V 0 2).arg))refine_2.«1»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ ↑(VtbAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) +
    -↑(VubAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) * ↑(VcbAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) *
      cexp (-↑(↑V 0 2).arg * I) =
  ↑(VudAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * ↑(VtbAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) -
    ↑(VusAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * ↑(VubAbs ⟦V⟧) *
        ↑(VcbAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) *
      cexp (-(I * ↑(↑V 0 2).arg)) C₂₃_eq_ℂcos_θ₂₃ ⟦V⟧,refine_2.«1»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ ↑V 1 1 = ↑(C₁₂ ⟦V⟧) * ↑(C₂₃ ⟦V⟧) - sin ↑(θ₁₂ ⟦V⟧) * sin ↑(θ₁₃ ⟦V⟧) * sin ↑(θ₂₃ ⟦V⟧) * cexp (-(I * ↑(↑V 0 2).arg))refine_2.«1»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ ↑(VtbAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) +
    -↑(VubAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) * ↑(VcbAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) *
      cexp (-↑(↑V 0 2).arg * I) =
  ↑(VudAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * ↑(VtbAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) -
    ↑(VusAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * ↑(VubAbs ⟦V⟧) *
        ↑(VcbAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) *
      cexp (-(I * ↑(↑V 0 2).arg)) S₁₂_eq_ℂsin_θ₁₂ ⟦V⟧,refine_2.«1»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ ↑V 1 1 = ↑(C₁₂ ⟦V⟧) * ↑(C₂₃ ⟦V⟧) - ↑(S₁₂ ⟦V⟧) * sin ↑(θ₁₃ ⟦V⟧) * sin ↑(θ₂₃ ⟦V⟧) * cexp (-(I * ↑(↑V 0 2).arg))refine_2.«1»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ ↑(VtbAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) +
    -↑(VubAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) * ↑(VcbAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) *
      cexp (-↑(↑V 0 2).arg * I) =
  ↑(VudAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * ↑(VtbAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) -
    ↑(VusAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * ↑(VubAbs ⟦V⟧) *
        ↑(VcbAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) *
      cexp (-(I * ↑(↑V 0 2).arg))
        S₁₃_eq_ℂsin_θ₁₃ ⟦V⟧,refine_2.«1»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ ↑V 1 1 = ↑(C₁₂ ⟦V⟧) * ↑(C₂₃ ⟦V⟧) - ↑(S₁₂ ⟦V⟧) * ↑(S₁₃ ⟦V⟧) * sin ↑(θ₂₃ ⟦V⟧) * cexp (-(I * ↑(↑V 0 2).arg))refine_2.«1»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ ↑(VtbAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) +
    -↑(VubAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) * ↑(VcbAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) *
      cexp (-↑(↑V 0 2).arg * I) =
  ↑(VudAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * ↑(VtbAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) -
    ↑(VusAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * ↑(VubAbs ⟦V⟧) *
        ↑(VcbAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) *
      cexp (-(I * ↑(↑V 0 2).arg)) S₂₃_eq_ℂsin_θ₂₃ ⟦V⟧,refine_2.«1»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ ↑V 1 1 = ↑(C₁₂ ⟦V⟧) * ↑(C₂₃ ⟦V⟧) - ↑(S₁₂ ⟦V⟧) * ↑(S₁₃ ⟦V⟧) * ↑(S₂₃ ⟦V⟧) * cexp (-(I * ↑(↑V 0 2).arg))refine_2.«1»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ ↑(VtbAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) +
    -↑(VubAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) * ↑(VcbAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) *
      cexp (-↑(↑V 0 2).arg * I) =
  ↑(VudAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * ↑(VtbAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) -
    ↑(VusAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * ↑(VubAbs ⟦V⟧) *
        ↑(VcbAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) *
      cexp (-(I * ↑(↑V 0 2).arg)) C₁₂_eq_Vud_div_sqrt hb',refine_2.«1»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ ↑V 1 1 =
  ↑(VudAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * ↑(C₂₃ ⟦V⟧) -
    ↑(S₁₂ ⟦V⟧) * ↑(S₁₃ ⟦V⟧) * ↑(S₂₃ ⟦V⟧) * cexp (-(I * ↑(↑V 0 2).arg))refine_2.«1»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ ↑(VtbAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) +
    -↑(VubAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) * ↑(VcbAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) *
      cexp (-↑(↑V 0 2).arg * I) =
  ↑(VudAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * ↑(VtbAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) -
    ↑(VusAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * ↑(VubAbs ⟦V⟧) *
        ↑(VcbAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) *
      cexp (-(I * ↑(↑V 0 2).arg)) C₂₃_of_Vub_ne_one hb',refine_2.«1»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ ↑V 1 1 =
  ↑(VudAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * ↑(VtbAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) -
    ↑(S₁₂ ⟦V⟧) * ↑(S₁₃ ⟦V⟧) * ↑(S₂₃ ⟦V⟧) * cexp (-(I * ↑(↑V 0 2).arg))refine_2.«1»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ ↑(VtbAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) +
    -↑(VubAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) * ↑(VcbAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) *
      cexp (-↑(↑V 0 2).arg * I) =
  ↑(VudAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * ↑(VtbAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) -
    ↑(VusAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * ↑(VubAbs ⟦V⟧) *
        ↑(VcbAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) *
      cexp (-(I * ↑(↑V 0 2).arg))
        S₁₂,refine_2.«1»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ ↑V 1 1 =
  ↑(VudAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * ↑(VtbAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) -
    ↑(VusAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * ↑(S₁₃ ⟦V⟧) * ↑(S₂₃ ⟦V⟧) * cexp (-(I * ↑(↑V 0 2).arg))refine_2.«1»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ ↑(VtbAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) +
    -↑(VubAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) * ↑(VcbAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) *
      cexp (-↑(↑V 0 2).arg * I) =
  ↑(VudAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * ↑(VtbAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) -
    ↑(VusAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * ↑(VubAbs ⟦V⟧) *
        ↑(VcbAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) *
      cexp (-(I * ↑(↑V 0 2).arg)) S₁₃,refine_2.«1»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ ↑V 1 1 =
  ↑(VudAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * ↑(VtbAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) -
    ↑(VusAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * ↑(VubAbs ⟦V⟧) * ↑(S₂₃ ⟦V⟧) * cexp (-(I * ↑(↑V 0 2).arg))refine_2.«1»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ ↑(VtbAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) +
    -↑(VubAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) * ↑(VcbAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) *
      cexp (-↑(↑V 0 2).arg * I) =
  ↑(VudAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * ↑(VtbAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) -
    ↑(VusAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * ↑(VubAbs ⟦V⟧) *
        ↑(VcbAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) *
      cexp (-(I * ↑(↑V 0 2).arg)) S₂₃_of_Vub_ne_one hb',refine_2.«1»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ ↑V 1 1 =
  ↑(VudAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * ↑(VtbAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) -
    ↑(VusAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * ↑(VubAbs ⟦V⟧) *
        ↑(VcbAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) *
      cexp (-(I * ↑(↑V 0 2).arg))refine_2.«1»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ ↑(VtbAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) +
    -↑(VubAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) * ↑(VcbAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) *
      cexp (-↑(↑V 0 2).arg * I) =
  ↑(VudAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * ↑(VtbAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) -
    ↑(VusAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * ↑(VubAbs ⟦V⟧) *
        ↑(VcbAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) *
      cexp (-(I * ↑(↑V 0 2).arg)) cs_of_fstRowThdColRealCond hb hVrefine_2.«1»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ ↑(VtbAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) +
    -↑(VubAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) * ↑(VcbAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) *
      cexp (-↑(↑V 0 2).arg * I) =
  ↑(VudAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * ↑(VtbAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) -
    ↑(VusAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * ↑(VubAbs ⟦V⟧) *
        ↑(VcbAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) *
      cexp (-(I * ↑(↑V 0 2).arg))refine_2.«1»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ ↑(VtbAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) +
    -↑(VubAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) * ↑(VcbAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) *
      cexp (-↑(↑V 0 2).arg * I) =
  ↑(VudAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * ↑(VtbAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) -
    ↑(VusAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * ↑(VubAbs ⟦V⟧) *
        ↑(VcbAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) *
      cexp (-(I * ↑(↑V 0 2).arg))]refine_2.«1»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ ↑(VtbAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) +
    -↑(VubAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) * ↑(VcbAbs ⟦V⟧) / (↑(VudAbs ⟦V⟧) ^ 2 + ↑(VusAbs ⟦V⟧) ^ 2) *
      cexp (-↑(↑V 0 2).arg * I) =
  ↑(VudAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * ↑(VtbAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) -
    ↑(VusAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) * ↑(VubAbs ⟦V⟧) *
        ↑(VcbAbs ⟦V⟧ / √(VudAbs ⟦V⟧ ^ 2 + VusAbs ⟦V⟧ ^ 2)) *
      cexp (-(I * ↑(↑V 0 2).arg))
      simp only [VtbAbs, Fin.isValue, VudAbs, VusAbs, VubAbs, neg_mul, VcbAbs, ofReal_div]refine_2.«1»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ ↑(VAbs 2 2 ⟦V⟧) * ↑(VAbs 0 0 ⟦V⟧) / (↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2) +
    -(↑(VAbs 0 2 ⟦V⟧) * ↑(VAbs 0 1 ⟦V⟧) * ↑(VAbs 1 2 ⟦V⟧)) / (↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2) *
      cexp (-(↑(↑V 0 2).arg * I)) =
  ↑(VAbs 0 0 ⟦V⟧) / ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) *
      (↑(VAbs 2 2 ⟦V⟧) / ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2)) -
    ↑(VAbs 0 1 ⟦V⟧) / ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑(VAbs 0 2 ⟦V⟧) *
        (↑(VAbs 1 2 ⟦V⟧) / ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2)) *
      cexp (-(I * ↑(↑V 0 2).arg))
      field_simprefine_2.«1»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ (↑(VAbs 2 2 ⟦V⟧) * ↑(VAbs 0 0 ⟦V⟧) +
      -(↑(VAbs 0 1 ⟦V⟧) * ↑(VAbs 0 2 ⟦V⟧) * ↑(VAbs 1 2 ⟦V⟧) * cexp (-(↑(↑V 0 2).arg * I)))) /
    (↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2) =
  (↑(VAbs 2 2 ⟦V⟧) * ↑(VAbs 0 0 ⟦V⟧) -
      ↑(VAbs 0 1 ⟦V⟧) * ↑(VAbs 0 2 ⟦V⟧) * ↑(VAbs 1 2 ⟦V⟧) * cexp (-(↑(↑V 0 2).arg * I))) /
    ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) ^ 2
      conv_rhs => rw [sq, h1]V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0| (↑(VAbs 2 2 ⟦V⟧) * ↑(VAbs 0 0 ⟦V⟧) -
    ↑(VAbs 0 1 ⟦V⟧) * ↑(VAbs 0 2 ⟦V⟧) * ↑(VAbs 1 2 ⟦V⟧) * cexp (-(↑(↑V 0 2).arg * I))) /
  (↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2)
      simp only [Fin.isValue, sq]refine_2.«1»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ (↑(VAbs 2 2 ⟦V⟧) * ↑(VAbs 0 0 ⟦V⟧) +
      -(↑(VAbs 0 1 ⟦V⟧) * ↑(VAbs 0 2 ⟦V⟧) * ↑(VAbs 1 2 ⟦V⟧) * cexp (-(↑(↑V 0 2).arg * I)))) /
    (↑(VAbs 0 0 ⟦V⟧) * ↑(VAbs 0 0 ⟦V⟧) + ↑(VAbs 0 1 ⟦V⟧) * ↑(VAbs 0 1 ⟦V⟧)) =
  (↑(VAbs 2 2 ⟦V⟧) * ↑(VAbs 0 0 ⟦V⟧) -
      ↑(VAbs 0 1 ⟦V⟧) * ↑(VAbs 0 2 ⟦V⟧) * ↑(VAbs 1 2 ⟦V⟧) * cexp (-(↑(↑V 0 2).arg * I))) /
    (↑(VAbs 0 0 ⟦V⟧) * ↑(VAbs 0 0 ⟦V⟧) + ↑(VAbs 0 1 ⟦V⟧) * ↑(VAbs 0 1 ⟦V⟧))
      field_simprefine_2.«1»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ (↑(VAbs 2 2 ⟦V⟧) * ↑(VAbs 0 0 ⟦V⟧) +
      -(↑(VAbs 0 1 ⟦V⟧) * ↑(VAbs 0 2 ⟦V⟧) * ↑(VAbs 1 2 ⟦V⟧) * cexp (-(↑(↑V 0 2).arg * I)))) /
    (↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2) =
  (↑(VAbs 2 2 ⟦V⟧) * ↑(VAbs 0 0 ⟦V⟧) -
      ↑(VAbs 0 1 ⟦V⟧) * ↑(VAbs 0 2 ⟦V⟧) * ↑(VAbs 1 2 ⟦V⟧) * cexp (-(↑(↑V 0 2).arg * I))) /
    (↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2)
      ring_nfAll goals completed! 🐙
    ·refine_2.«2»V:CKMMatrixhb:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0hV:V.FstRowThdColRealCondhb':VubAbs ⟦V⟧ ≠ 1h1:↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) =
  ↑(VAbs 0 0 ⟦V⟧) ^ 2 + ↑(VAbs 0 1 ⟦V⟧) ^ 2hx:↑(VudAbs ⟦V⟧) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧) ≠ 0⊢ [V]c ((fun i => i) ⟨2, ⋯⟩) = [standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) (-(↑V 0 2).arg)]c ((fun i => i) ⟨2, ⋯⟩) simp [cRow, standParam, standParamAsMatrix, hV.2.2.1, VcbAbs_eq_S₂₃_mul_C₁₃ ⟦V⟧,
        S₂₃_eq_ℂsin_θ₂₃ ⟦V⟧, C₁₃]All goals completed! 🐙

lemma eq_standParam_of_ubOnePhaseCond {V : CKMMatrix} (hV : ubOnePhaseCond V) :
    V = standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 := byV:CKMMatrixhV:V.ubOnePhaseCond⊢ V = standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0
  have h1 : VubAbs ⟦V⟧ = 1 := by
    simp only [VubAbs, VAbs, VAbs', Fin.isValue, Quotient.lift_mk, hV.2.2.2.1, norm_one]V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ V = standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ V = standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0
  refine eq_rows V ?_ ?_ hV.2.2.2.2.1refine_1V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ [V]u = [standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0]urefine_2V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ [V]c = [standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0]c
  ·refine_1V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ [V]u = [standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0]u funext irefine_1V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1i:Fin 3⊢ [V]u i = [standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0]u i
    fin_cases irefine_1.«0»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ [V]u ((fun i => i) ⟨0, ⋯⟩) = [standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0]u ((fun i => i) ⟨0, ⋯⟩)refine_1.«1»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ [V]u ((fun i => i) ⟨1, ⋯⟩) = [standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0]u ((fun i => i) ⟨1, ⋯⟩)refine_1.«2»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ [V]u ((fun i => i) ⟨2, ⋯⟩) = [standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0]u ((fun i => i) ⟨2, ⋯⟩)
    ·refine_1.«0»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ [V]u ((fun i => i) ⟨0, ⋯⟩) = [standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0]u ((fun i => i) ⟨0, ⋯⟩) simp only [uRow, Fin.isValue, Fin.zero_eta, cons_val_zero, standParam, standParamAsMatrix,
      ofReal_cos, ofReal_sin, ofReal_zero, mul_zero, exp_zero, mul_one, neg_mul, cons_val',
      empty_val', cons_val_fin_one, cons_val_one, head_cons, cons_val_two, Nat.succ_eq_add_one,
      Nat.reduceAdd, tail_cons]refine_1.«0»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ ↑V 0 0 = cos ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧)
      rw [C₁₃_eq_ℂcos_θ₁₃ ⟦V⟧,refine_1.«0»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ ↑V 0 0 = cos ↑(θ₁₂ ⟦V⟧) * ↑(C₁₃ ⟦V⟧)refine_1.«0»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ 0 = cos ↑(θ₁₂ ⟦V⟧) * ↑0 C₁₃_of_Vub_eq_one h1,refine_1.«0»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ ↑V 0 0 = cos ↑(θ₁₂ ⟦V⟧) * ↑0refine_1.«0»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ 0 = cos ↑(θ₁₂ ⟦V⟧) * ↑0 hV.1refine_1.«0»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ 0 = cos ↑(θ₁₂ ⟦V⟧) * ↑0refine_1.«0»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ 0 = cos ↑(θ₁₂ ⟦V⟧) * ↑0]refine_1.«0»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ 0 = cos ↑(θ₁₂ ⟦V⟧) * ↑0
      exact Eq.symm (mul_eq_zero_of_right (cos ↑(θ₁₂ ⟦V⟧)) rfl)All goals completed! 🐙
    ·refine_1.«1»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ [V]u ((fun i => i) ⟨1, ⋯⟩) = [standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0]u ((fun i => i) ⟨1, ⋯⟩) simp only [uRow, Fin.isValue, Fin.mk_one, cons_val_one, head_cons, standParam,
      standParamAsMatrix, ofReal_cos, ofReal_sin, ofReal_zero, mul_zero, exp_zero, mul_one, neg_mul,
      cons_val', cons_val_zero, empty_val', cons_val_fin_one, cons_val_two, Nat.succ_eq_add_one,
      Nat.reduceAdd, tail_cons]refine_1.«1»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ ↑V 0 1 = sin ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧)
      rw [C₁₃_eq_ℂcos_θ₁₃ ⟦V⟧,refine_1.«1»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ ↑V 0 1 = sin ↑(θ₁₂ ⟦V⟧) * ↑(C₁₃ ⟦V⟧)refine_1.«1»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ 0 = sin ↑(θ₁₂ ⟦V⟧) * ↑0 C₁₃_of_Vub_eq_one h1,refine_1.«1»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ ↑V 0 1 = sin ↑(θ₁₂ ⟦V⟧) * ↑0refine_1.«1»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ 0 = sin ↑(θ₁₂ ⟦V⟧) * ↑0 hV.2.1refine_1.«1»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ 0 = sin ↑(θ₁₂ ⟦V⟧) * ↑0refine_1.«1»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ 0 = sin ↑(θ₁₂ ⟦V⟧) * ↑0]refine_1.«1»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ 0 = sin ↑(θ₁₂ ⟦V⟧) * ↑0
      exact Eq.symm (mul_eq_zero_of_right (sin ↑(θ₁₂ ⟦V⟧)) rfl)All goals completed! 🐙
    ·refine_1.«2»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ [V]u ((fun i => i) ⟨2, ⋯⟩) = [standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0]u ((fun i => i) ⟨2, ⋯⟩) simp only [uRow, Fin.isValue, Fin.reduceFinMk, cons_val_two, Nat.succ_eq_add_one,
      Nat.reduceAdd, tail_cons, head_cons, standParam, standParamAsMatrix, ofReal_cos, ofReal_sin,
      ofReal_zero, mul_zero, exp_zero, mul_one, neg_mul, cons_val', cons_val_zero, empty_val',
      cons_val_fin_one, cons_val_one]refine_1.«2»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ ↑V 0 2 = sin ↑(θ₁₃ ⟦V⟧)
      rw [S₁₃_eq_ℂsin_θ₁₃ ⟦V⟧,refine_1.«2»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ ↑V 0 2 = ↑(S₁₃ ⟦V⟧)refine_1.«2»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ ↑V 0 2 = ↑(VubAbs ⟦V⟧) S₁₃refine_1.«2»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ ↑V 0 2 = ↑(VubAbs ⟦V⟧)refine_1.«2»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ ↑V 0 2 = ↑(VubAbs ⟦V⟧)]refine_1.«2»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ ↑V 0 2 = ↑(VubAbs ⟦V⟧)
      simp only [Fin.isValue, VubAbs, VAbs, VAbs', Quotient.lift_mk]refine_1.«2»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ ↑V 0 2 = ↑‖↑V 0 2‖
      rw [hV.2.2.2.1refine_1.«2»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ 1 = ↑‖1‖refine_1.«2»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ 1 = ↑‖1‖]refine_1.«2»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ 1 = ↑‖1‖
      simpAll goals completed! 🐙
  ·refine_2V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ [V]c = [standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0]c funext irefine_2V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1i:Fin 3⊢ [V]c i = [standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0]c i
    fin_cases irefine_2.«0»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ [V]c ((fun i => i) ⟨0, ⋯⟩) = [standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0]c ((fun i => i) ⟨0, ⋯⟩)refine_2.«1»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ [V]c ((fun i => i) ⟨1, ⋯⟩) = [standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0]c ((fun i => i) ⟨1, ⋯⟩)refine_2.«2»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ [V]c ((fun i => i) ⟨2, ⋯⟩) = [standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0]c ((fun i => i) ⟨2, ⋯⟩)
    ·refine_2.«0»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ [V]c ((fun i => i) ⟨0, ⋯⟩) = [standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0]c ((fun i => i) ⟨0, ⋯⟩) simp only [cRow, Fin.isValue, Fin.zero_eta, cons_val_zero, standParam, standParamAsMatrix,
      ofReal_cos, ofReal_sin, ofReal_zero, mul_zero, exp_zero, mul_one, neg_mul, cons_val',
      empty_val', cons_val_fin_one, cons_val_one, head_cons, cons_val_two, Nat.succ_eq_add_one,
      Nat.reduceAdd, tail_cons]refine_2.«0»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ ↑V 1 0 = -(sin ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₂₃ ⟦V⟧)) - cos ↑(θ₁₂ ⟦V⟧) * sin ↑(θ₁₃ ⟦V⟧) * sin ↑(θ₂₃ ⟦V⟧)
      rw [S₂₃_eq_ℂsin_θ₂₃ ⟦V⟧,refine_2.«0»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ ↑V 1 0 = -(sin ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₂₃ ⟦V⟧)) - cos ↑(θ₁₂ ⟦V⟧) * sin ↑(θ₁₃ ⟦V⟧) * ↑(S₂₃ ⟦V⟧)refine_2.«0»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ -↑(VcdAbs ⟦V⟧) = -(↑0 * cos ↑(θ₂₃ ⟦V⟧)) - ↑1 * ↑1 * ↑(VcdAbs ⟦V⟧) S₂₃_of_Vub_eq_one h1,refine_2.«0»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ ↑V 1 0 = -(sin ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₂₃ ⟦V⟧)) - cos ↑(θ₁₂ ⟦V⟧) * sin ↑(θ₁₃ ⟦V⟧) * ↑(VcdAbs ⟦V⟧)refine_2.«0»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ -↑(VcdAbs ⟦V⟧) = -(↑0 * cos ↑(θ₂₃ ⟦V⟧)) - ↑1 * ↑1 * ↑(VcdAbs ⟦V⟧) S₁₂_eq_ℂsin_θ₁₂ ⟦V⟧,refine_2.«0»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ ↑V 1 0 = -(↑(S₁₂ ⟦V⟧) * cos ↑(θ₂₃ ⟦V⟧)) - cos ↑(θ₁₂ ⟦V⟧) * sin ↑(θ₁₃ ⟦V⟧) * ↑(VcdAbs ⟦V⟧)refine_2.«0»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ -↑(VcdAbs ⟦V⟧) = -(↑0 * cos ↑(θ₂₃ ⟦V⟧)) - ↑1 * ↑1 * ↑(VcdAbs ⟦V⟧) S₁₂_of_Vub_one h1,refine_2.«0»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ ↑V 1 0 = -(↑0 * cos ↑(θ₂₃ ⟦V⟧)) - cos ↑(θ₁₂ ⟦V⟧) * sin ↑(θ₁₃ ⟦V⟧) * ↑(VcdAbs ⟦V⟧)refine_2.«0»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ -↑(VcdAbs ⟦V⟧) = -(↑0 * cos ↑(θ₂₃ ⟦V⟧)) - ↑1 * ↑1 * ↑(VcdAbs ⟦V⟧)
        C₁₂_eq_ℂcos_θ₁₂ ⟦V⟧,refine_2.«0»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ ↑V 1 0 = -(↑0 * cos ↑(θ₂₃ ⟦V⟧)) - ↑(C₁₂ ⟦V⟧) * sin ↑(θ₁₃ ⟦V⟧) * ↑(VcdAbs ⟦V⟧)refine_2.«0»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ -↑(VcdAbs ⟦V⟧) = -(↑0 * cos ↑(θ₂₃ ⟦V⟧)) - ↑1 * ↑1 * ↑(VcdAbs ⟦V⟧) C₁₂_of_Vub_one h1,refine_2.«0»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ ↑V 1 0 = -(↑0 * cos ↑(θ₂₃ ⟦V⟧)) - ↑1 * sin ↑(θ₁₃ ⟦V⟧) * ↑(VcdAbs ⟦V⟧)refine_2.«0»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ -↑(VcdAbs ⟦V⟧) = -(↑0 * cos ↑(θ₂₃ ⟦V⟧)) - ↑1 * ↑1 * ↑(VcdAbs ⟦V⟧) S₁₃_eq_ℂsin_θ₁₃ ⟦V⟧,refine_2.«0»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ ↑V 1 0 = -(↑0 * cos ↑(θ₂₃ ⟦V⟧)) - ↑1 * ↑(S₁₃ ⟦V⟧) * ↑(VcdAbs ⟦V⟧)refine_2.«0»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ -↑(VcdAbs ⟦V⟧) = -(↑0 * cos ↑(θ₂₃ ⟦V⟧)) - ↑1 * ↑1 * ↑(VcdAbs ⟦V⟧) S₁₃_of_Vub_one h1,refine_2.«0»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ ↑V 1 0 = -(↑0 * cos ↑(θ₂₃ ⟦V⟧)) - ↑1 * ↑1 * ↑(VcdAbs ⟦V⟧)refine_2.«0»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ -↑(VcdAbs ⟦V⟧) = -(↑0 * cos ↑(θ₂₃ ⟦V⟧)) - ↑1 * ↑1 * ↑(VcdAbs ⟦V⟧)
        hV.2.2.2.2.2.1refine_2.«0»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ -↑(VcdAbs ⟦V⟧) = -(↑0 * cos ↑(θ₂₃ ⟦V⟧)) - ↑1 * ↑1 * ↑(VcdAbs ⟦V⟧)refine_2.«0»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ -↑(VcdAbs ⟦V⟧) = -(↑0 * cos ↑(θ₂₃ ⟦V⟧)) - ↑1 * ↑1 * ↑(VcdAbs ⟦V⟧)]refine_2.«0»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ -↑(VcdAbs ⟦V⟧) = -(↑0 * cos ↑(θ₂₃ ⟦V⟧)) - ↑1 * ↑1 * ↑(VcdAbs ⟦V⟧)
      simp only [VcdAbs, Fin.isValue, ofReal_zero, zero_mul, neg_zero, ofReal_one, mul_one, one_mul,
        zero_sub]All goals completed! 🐙
    ·refine_2.«1»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ [V]c ((fun i => i) ⟨1, ⋯⟩) = [standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0]c ((fun i => i) ⟨1, ⋯⟩) simp only [cRow, Fin.isValue, Fin.mk_one, cons_val_one, head_cons, standParam,
      standParamAsMatrix, ofReal_cos, ofReal_sin, ofReal_zero, mul_zero, exp_zero, mul_one, neg_mul,
      cons_val', cons_val_zero, empty_val', cons_val_fin_one, cons_val_two, Nat.succ_eq_add_one,
      Nat.reduceAdd, tail_cons]refine_2.«1»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ ↑V 1 1 = cos ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₂₃ ⟦V⟧) - sin ↑(θ₁₂ ⟦V⟧) * sin ↑(θ₁₃ ⟦V⟧) * sin ↑(θ₂₃ ⟦V⟧)
      rw [S₂₃_eq_ℂsin_θ₂₃ ⟦V⟧,refine_2.«1»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ ↑V 1 1 = cos ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₂₃ ⟦V⟧) - sin ↑(θ₁₂ ⟦V⟧) * sin ↑(θ₁₃ ⟦V⟧) * ↑(S₂₃ ⟦V⟧)refine_2.«1»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ ↑V 1 1 = ↑1 * cos ↑(θ₂₃ ⟦V⟧) - ↑0 * ↑1 * ↑(VcdAbs ⟦V⟧) S₂₃_of_Vub_eq_one h1,refine_2.«1»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ ↑V 1 1 = cos ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₂₃ ⟦V⟧) - sin ↑(θ₁₂ ⟦V⟧) * sin ↑(θ₁₃ ⟦V⟧) * ↑(VcdAbs ⟦V⟧)refine_2.«1»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ ↑V 1 1 = ↑1 * cos ↑(θ₂₃ ⟦V⟧) - ↑0 * ↑1 * ↑(VcdAbs ⟦V⟧) S₁₂_eq_ℂsin_θ₁₂ ⟦V⟧,refine_2.«1»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ ↑V 1 1 = cos ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₂₃ ⟦V⟧) - ↑(S₁₂ ⟦V⟧) * sin ↑(θ₁₃ ⟦V⟧) * ↑(VcdAbs ⟦V⟧)refine_2.«1»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ ↑V 1 1 = ↑1 * cos ↑(θ₂₃ ⟦V⟧) - ↑0 * ↑1 * ↑(VcdAbs ⟦V⟧) S₁₂_of_Vub_one h1,refine_2.«1»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ ↑V 1 1 = cos ↑(θ₁₂ ⟦V⟧) * cos ↑(θ₂₃ ⟦V⟧) - ↑0 * sin ↑(θ₁₃ ⟦V⟧) * ↑(VcdAbs ⟦V⟧)refine_2.«1»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ ↑V 1 1 = ↑1 * cos ↑(θ₂₃ ⟦V⟧) - ↑0 * ↑1 * ↑(VcdAbs ⟦V⟧)
        C₁₂_eq_ℂcos_θ₁₂ ⟦V⟧,refine_2.«1»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ ↑V 1 1 = ↑(C₁₂ ⟦V⟧) * cos ↑(θ₂₃ ⟦V⟧) - ↑0 * sin ↑(θ₁₃ ⟦V⟧) * ↑(VcdAbs ⟦V⟧)refine_2.«1»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ ↑V 1 1 = ↑1 * cos ↑(θ₂₃ ⟦V⟧) - ↑0 * ↑1 * ↑(VcdAbs ⟦V⟧) C₁₂_of_Vub_one h1,refine_2.«1»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ ↑V 1 1 = ↑1 * cos ↑(θ₂₃ ⟦V⟧) - ↑0 * sin ↑(θ₁₃ ⟦V⟧) * ↑(VcdAbs ⟦V⟧)refine_2.«1»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ ↑V 1 1 = ↑1 * cos ↑(θ₂₃ ⟦V⟧) - ↑0 * ↑1 * ↑(VcdAbs ⟦V⟧) S₁₃_eq_ℂsin_θ₁₃ ⟦V⟧,refine_2.«1»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ ↑V 1 1 = ↑1 * cos ↑(θ₂₃ ⟦V⟧) - ↑0 * ↑(S₁₃ ⟦V⟧) * ↑(VcdAbs ⟦V⟧)refine_2.«1»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ ↑V 1 1 = ↑1 * cos ↑(θ₂₃ ⟦V⟧) - ↑0 * ↑1 * ↑(VcdAbs ⟦V⟧) S₁₃_of_Vub_one h1refine_2.«1»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ ↑V 1 1 = ↑1 * cos ↑(θ₂₃ ⟦V⟧) - ↑0 * ↑1 * ↑(VcdAbs ⟦V⟧)refine_2.«1»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ ↑V 1 1 = ↑1 * cos ↑(θ₂₃ ⟦V⟧) - ↑0 * ↑1 * ↑(VcdAbs ⟦V⟧)]refine_2.«1»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ ↑V 1 1 = ↑1 * cos ↑(θ₂₃ ⟦V⟧) - ↑0 * ↑1 * ↑(VcdAbs ⟦V⟧)
      simp only [Fin.isValue, ofReal_one, one_mul, ofReal_zero, mul_one, VcdAbs, zero_mul, sub_zero]refine_2.«1»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ ↑V 1 1 = cos ↑(θ₂₃ ⟦V⟧)
      have h3 : (Real.cos (θ₂₃ ⟦V⟧) : ℂ) = √(1 - S₂₃ ⟦V⟧ ^ 2) := byV:CKMMatrixhV:V.ubOnePhaseCond⊢ V = standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0refine_2.«1»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1h3:↑(Real.cos (θ₂₃ ⟦V⟧)) = ↑√(1 - S₂₃ ⟦V⟧ ^ 2)⊢ ↑V 1 1 = cos ↑(θ₂₃ ⟦V⟧)
        rw [θ₂₃,V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ ↑(Real.cos (Real.arcsin (S₂₃ ⟦V⟧))) = ↑√(1 - S₂₃ ⟦V⟧ ^ 2)refine_2.«1»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1h3:↑(Real.cos (θ₂₃ ⟦V⟧)) = ↑√(1 - S₂₃ ⟦V⟧ ^ 2)⊢ ↑V 1 1 = cos ↑(θ₂₃ ⟦V⟧) Real.cos_arcsinV:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ ↑√(1 - S₂₃ ⟦V⟧ ^ 2) = ↑√(1 - S₂₃ ⟦V⟧ ^ 2)refine_2.«1»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1h3:↑(Real.cos (θ₂₃ ⟦V⟧)) = ↑√(1 - S₂₃ ⟦V⟧ ^ 2)⊢ ↑V 1 1 = cos ↑(θ₂₃ ⟦V⟧)]refine_2.«1»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1h3:↑(Real.cos (θ₂₃ ⟦V⟧)) = ↑√(1 - S₂₃ ⟦V⟧ ^ 2)⊢ ↑V 1 1 = cos ↑(θ₂₃ ⟦V⟧)refine_2.«1»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1h3:↑(Real.cos (θ₂₃ ⟦V⟧)) = ↑√(1 - S₂₃ ⟦V⟧ ^ 2)⊢ ↑V 1 1 = cos ↑(θ₂₃ ⟦V⟧)
      simp only [ofReal_cos] at h3refine_2.«1»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1h3:cos ↑(θ₂₃ ⟦V⟧) = ↑√(1 - S₂₃ ⟦V⟧ ^ 2)⊢ ↑V 1 1 = cos ↑(θ₂₃ ⟦V⟧)
      rw [h3,refine_2.«1»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1h3:cos ↑(θ₂₃ ⟦V⟧) = ↑√(1 - S₂₃ ⟦V⟧ ^ 2)⊢ ↑V 1 1 = ↑√(1 - S₂₃ ⟦V⟧ ^ 2)All goals completed! 🐙 S₂₃_of_Vub_eq_one h1,refine_2.«1»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1h3:cos ↑(θ₂₃ ⟦V⟧) = ↑√(1 - S₂₃ ⟦V⟧ ^ 2)⊢ ↑V 1 1 = ↑√(1 - VcdAbs ⟦V⟧ ^ 2)All goals completed! 🐙 hV.2.2.2.2.2.2refine_2.«1»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1h3:cos ↑(θ₂₃ ⟦V⟧) = ↑√(1 - S₂₃ ⟦V⟧ ^ 2)⊢ ↑√(1 - VcdAbs ⟦V⟧ ^ 2) = ↑√(1 - VcdAbs ⟦V⟧ ^ 2)All goals completed! 🐙]All goals completed! 🐙
    ·refine_2.«2»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ [V]c ((fun i => i) ⟨2, ⋯⟩) = [standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0]c ((fun i => i) ⟨2, ⋯⟩) simp only [cRow, Fin.isValue, Fin.reduceFinMk, cons_val_two, Nat.succ_eq_add_one,
      Nat.reduceAdd, tail_cons, head_cons, standParam, standParamAsMatrix, ofReal_cos, ofReal_sin,
      ofReal_zero, mul_zero, exp_zero, mul_one, neg_mul, cons_val', cons_val_zero, empty_val',
      cons_val_fin_one, cons_val_one]refine_2.«2»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ ↑V 1 2 = sin ↑(θ₂₃ ⟦V⟧) * cos ↑(θ₁₃ ⟦V⟧)
      rw [C₁₃_eq_ℂcos_θ₁₃ ⟦V⟧,refine_2.«2»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ ↑V 1 2 = sin ↑(θ₂₃ ⟦V⟧) * ↑(C₁₃ ⟦V⟧)refine_2.«2»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ 0 = sin ↑(θ₂₃ ⟦V⟧) * ↑0 C₁₃_of_Vub_eq_one h1,refine_2.«2»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ ↑V 1 2 = sin ↑(θ₂₃ ⟦V⟧) * ↑0refine_2.«2»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ 0 = sin ↑(θ₂₃ ⟦V⟧) * ↑0 hV.2.2.1refine_2.«2»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ 0 = sin ↑(θ₂₃ ⟦V⟧) * ↑0refine_2.«2»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ 0 = sin ↑(θ₂₃ ⟦V⟧) * ↑0]refine_2.«2»V:CKMMatrixhV:V.ubOnePhaseCondh1:VubAbs ⟦V⟧ = 1⊢ 0 = sin ↑(θ₂₃ ⟦V⟧) * ↑0
      exact Eq.symm (mul_eq_zero_of_right (sin ↑(θ₂₃ ⟦V⟧)) rfl)All goals completed! 🐙

theorem exists_δ₁₃ (V : CKMMatrix) :
    ∃ (δ₃ : ℝ), V ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₃ := byV:CKMMatrix⊢ ∃ δ₃, V ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₃
  obtain ⟨U, hU⟩ := fstRowThdColRealCond_holds_up_to_equiv VV:CKMMatrixU:CKMMatrixhU:V ≈ U ∧ U.FstRowThdColRealCond⊢ ∃ δ₃, V ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₃
  have hUV : ⟦U⟧ = ⟦V⟧ := (Quotient.eq.mpr (phaseShiftRelation_equiv.symm hU.1))V:CKMMatrixU:CKMMatrixhU:V ≈ U ∧ U.FstRowThdColRealCondhUV:⟦U⟧ = ⟦V⟧⊢ ∃ δ₃, V ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₃
  by_cases ha : [V]ud ≠ 0 ∨ [V]us ≠ 0posV:CKMMatrixU:CKMMatrixhU:V ≈ U ∧ U.FstRowThdColRealCondhUV:⟦U⟧ = ⟦V⟧ha:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0⊢ ∃ δ₃, V ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₃negV:CKMMatrixU:CKMMatrixhU:V ≈ U ∧ U.FstRowThdColRealCondhUV:⟦U⟧ = ⟦V⟧ha:¬(↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0)⊢ ∃ δ₃, V ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₃
  ·posV:CKMMatrixU:CKMMatrixhU:V ≈ U ∧ U.FstRowThdColRealCondhUV:⟦U⟧ = ⟦V⟧ha:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0⊢ ∃ δ₃, V ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₃ have haU : [U]ud ≠ 0 ∨ [U]us ≠ 0 := by
      rw [ud_us_ne_zero_iff_ub_ne_oneV:CKMMatrixU:CKMMatrixhU:V ≈ U ∧ U.FstRowThdColRealCondhUV:⟦U⟧ = ⟦V⟧ha:‖↑V 0 2‖ ≠ 1⊢ ‖↑U 0 2‖ ≠ 1V:CKMMatrixU:CKMMatrixhU:V ≈ U ∧ U.FstRowThdColRealCondhUV:⟦U⟧ = ⟦V⟧ha:‖↑V 0 2‖ ≠ 1⊢ ‖↑U 0 2‖ ≠ 1posV:CKMMatrixU:CKMMatrixhU:V ≈ U ∧ U.FstRowThdColRealCondhUV:⟦U⟧ = ⟦V⟧ha:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0haU:↑U 0 0 ≠ 0 ∨ ↑U 0 1 ≠ 0⊢ ∃ δ₃, V ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₃] at ha ⊢V:CKMMatrixU:CKMMatrixhU:V ≈ U ∧ U.FstRowThdColRealCondhUV:⟦U⟧ = ⟦V⟧ha:‖↑V 0 2‖ ≠ 1⊢ ‖↑U 0 2‖ ≠ 1posV:CKMMatrixU:CKMMatrixhU:V ≈ U ∧ U.FstRowThdColRealCondhUV:⟦U⟧ = ⟦V⟧ha:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0haU:↑U 0 0 ≠ 0 ∨ ↑U 0 1 ≠ 0⊢ ∃ δ₃, V ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₃
      rwa [show norm [U]ub = VubAbs ⟦U⟧ from rfl,V:CKMMatrixU:CKMMatrixhU:V ≈ U ∧ U.FstRowThdColRealCondhUV:⟦U⟧ = ⟦V⟧ha:‖↑V 0 2‖ ≠ 1⊢ VubAbs ⟦U⟧ ≠ 1posV:CKMMatrixU:CKMMatrixhU:V ≈ U ∧ U.FstRowThdColRealCondhUV:⟦U⟧ = ⟦V⟧ha:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0haU:↑U 0 0 ≠ 0 ∨ ↑U 0 1 ≠ 0⊢ ∃ δ₃, V ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₃ hUVV:CKMMatrixU:CKMMatrixhU:V ≈ U ∧ U.FstRowThdColRealCondhUV:⟦U⟧ = ⟦V⟧ha:‖↑V 0 2‖ ≠ 1⊢ VubAbs ⟦V⟧ ≠ 1posV:CKMMatrixU:CKMMatrixhU:V ≈ U ∧ U.FstRowThdColRealCondhUV:⟦U⟧ = ⟦V⟧ha:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0haU:↑U 0 0 ≠ 0 ∨ ↑U 0 1 ≠ 0⊢ ∃ δ₃, V ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₃]V:CKMMatrixU:CKMMatrixhU:V ≈ U ∧ U.FstRowThdColRealCondhUV:⟦U⟧ = ⟦V⟧ha:‖↑V 0 2‖ ≠ 1⊢ VubAbs ⟦V⟧ ≠ 1posV:CKMMatrixU:CKMMatrixhU:V ≈ U ∧ U.FstRowThdColRealCondhUV:⟦U⟧ = ⟦V⟧ha:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0haU:↑U 0 0 ≠ 0 ∨ ↑U 0 1 ≠ 0⊢ ∃ δ₃, V ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₃posV:CKMMatrixU:CKMMatrixhU:V ≈ U ∧ U.FstRowThdColRealCondhUV:⟦U⟧ = ⟦V⟧ha:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0haU:↑U 0 0 ≠ 0 ∨ ↑U 0 1 ≠ 0⊢ ∃ δ₃, V ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₃
    have hU' := eq_standParam_of_fstRowThdColRealCond haU hU.2posV:CKMMatrixU:CKMMatrixhU:V ≈ U ∧ U.FstRowThdColRealCondhUV:⟦U⟧ = ⟦V⟧ha:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0haU:↑U 0 0 ≠ 0 ∨ ↑U 0 1 ≠ 0hU':U = standParam (θ₁₂ ⟦U⟧) (θ₁₃ ⟦U⟧) (θ₂₃ ⟦U⟧) (-(↑U 0 2).arg)⊢ ∃ δ₃, V ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₃
    rw [hU'posV:CKMMatrixU:CKMMatrixhU:V ≈ standParam (θ₁₂ ⟦U⟧) (θ₁₃ ⟦U⟧) (θ₂₃ ⟦U⟧) (-(↑U 0 2).arg) ∧
  (standParam (θ₁₂ ⟦U⟧) (θ₁₃ ⟦U⟧) (θ₂₃ ⟦U⟧) (-(↑U 0 2).arg)).FstRowThdColRealCondhUV:⟦U⟧ = ⟦V⟧ha:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0haU:↑U 0 0 ≠ 0 ∨ ↑U 0 1 ≠ 0hU':U = standParam (θ₁₂ ⟦U⟧) (θ₁₃ ⟦U⟧) (θ₂₃ ⟦U⟧) (-(↑U 0 2).arg)⊢ ∃ δ₃, V ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₃posV:CKMMatrixU:CKMMatrixhU:V ≈ standParam (θ₁₂ ⟦U⟧) (θ₁₃ ⟦U⟧) (θ₂₃ ⟦U⟧) (-(↑U 0 2).arg) ∧
  (standParam (θ₁₂ ⟦U⟧) (θ₁₃ ⟦U⟧) (θ₂₃ ⟦U⟧) (-(↑U 0 2).arg)).FstRowThdColRealCondhUV:⟦U⟧ = ⟦V⟧ha:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0haU:↑U 0 0 ≠ 0 ∨ ↑U 0 1 ≠ 0hU':U = standParam (θ₁₂ ⟦U⟧) (θ₁₃ ⟦U⟧) (θ₂₃ ⟦U⟧) (-(↑U 0 2).arg)⊢ ∃ δ₃, V ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₃] at hUposV:CKMMatrixU:CKMMatrixhU:V ≈ standParam (θ₁₂ ⟦U⟧) (θ₁₃ ⟦U⟧) (θ₂₃ ⟦U⟧) (-(↑U 0 2).arg) ∧
  (standParam (θ₁₂ ⟦U⟧) (θ₁₃ ⟦U⟧) (θ₂₃ ⟦U⟧) (-(↑U 0 2).arg)).FstRowThdColRealCondhUV:⟦U⟧ = ⟦V⟧ha:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0haU:↑U 0 0 ≠ 0 ∨ ↑U 0 1 ≠ 0hU':U = standParam (θ₁₂ ⟦U⟧) (θ₁₃ ⟦U⟧) (θ₂₃ ⟦U⟧) (-(↑U 0 2).arg)⊢ ∃ δ₃, V ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₃
    use (- arg ([U]ub))hV:CKMMatrixU:CKMMatrixhU:V ≈ standParam (θ₁₂ ⟦U⟧) (θ₁₃ ⟦U⟧) (θ₂₃ ⟦U⟧) (-(↑U 0 2).arg) ∧
  (standParam (θ₁₂ ⟦U⟧) (θ₁₃ ⟦U⟧) (θ₂₃ ⟦U⟧) (-(↑U 0 2).arg)).FstRowThdColRealCondhUV:⟦U⟧ = ⟦V⟧ha:↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0haU:↑U 0 0 ≠ 0 ∨ ↑U 0 1 ≠ 0hU':U = standParam (θ₁₂ ⟦U⟧) (θ₁₃ ⟦U⟧) (θ₂₃ ⟦U⟧) (-(↑U 0 2).arg)⊢ V ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) (-(↑U 0 2).arg)
    exact hUV ▸ hU.1All goals completed! 🐙
  ·negV:CKMMatrixU:CKMMatrixhU:V ≈ U ∧ U.FstRowThdColRealCondhUV:⟦U⟧ = ⟦V⟧ha:¬(↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0)⊢ ∃ δ₃, V ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₃ have haU : ¬ ([U]ud ≠ 0 ∨ [U]us ≠ 0) := by
      rw [ud_us_ne_zero_iff_ub_ne_oneV:CKMMatrixU:CKMMatrixhU:V ≈ U ∧ U.FstRowThdColRealCondhUV:⟦U⟧ = ⟦V⟧ha:¬‖↑V 0 2‖ ≠ 1⊢ ¬‖↑U 0 2‖ ≠ 1V:CKMMatrixU:CKMMatrixhU:V ≈ U ∧ U.FstRowThdColRealCondhUV:⟦U⟧ = ⟦V⟧ha:¬‖↑V 0 2‖ ≠ 1⊢ ¬‖↑U 0 2‖ ≠ 1negV:CKMMatrixU:CKMMatrixhU:V ≈ U ∧ U.FstRowThdColRealCondhUV:⟦U⟧ = ⟦V⟧ha:¬(↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0)haU:¬(↑U 0 0 ≠ 0 ∨ ↑U 0 1 ≠ 0)⊢ ∃ δ₃, V ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₃] at ha ⊢V:CKMMatrixU:CKMMatrixhU:V ≈ U ∧ U.FstRowThdColRealCondhUV:⟦U⟧ = ⟦V⟧ha:¬‖↑V 0 2‖ ≠ 1⊢ ¬‖↑U 0 2‖ ≠ 1negV:CKMMatrixU:CKMMatrixhU:V ≈ U ∧ U.FstRowThdColRealCondhUV:⟦U⟧ = ⟦V⟧ha:¬(↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0)haU:¬(↑U 0 0 ≠ 0 ∨ ↑U 0 1 ≠ 0)⊢ ∃ δ₃, V ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₃
      rwa [show norm [U]ub = VubAbs ⟦U⟧ from rfl,V:CKMMatrixU:CKMMatrixhU:V ≈ U ∧ U.FstRowThdColRealCondhUV:⟦U⟧ = ⟦V⟧ha:¬‖↑V 0 2‖ ≠ 1⊢ ¬VubAbs ⟦U⟧ ≠ 1negV:CKMMatrixU:CKMMatrixhU:V ≈ U ∧ U.FstRowThdColRealCondhUV:⟦U⟧ = ⟦V⟧ha:¬(↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0)haU:¬(↑U 0 0 ≠ 0 ∨ ↑U 0 1 ≠ 0)⊢ ∃ δ₃, V ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₃ hUVV:CKMMatrixU:CKMMatrixhU:V ≈ U ∧ U.FstRowThdColRealCondhUV:⟦U⟧ = ⟦V⟧ha:¬‖↑V 0 2‖ ≠ 1⊢ ¬VubAbs ⟦V⟧ ≠ 1negV:CKMMatrixU:CKMMatrixhU:V ≈ U ∧ U.FstRowThdColRealCondhUV:⟦U⟧ = ⟦V⟧ha:¬(↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0)haU:¬(↑U 0 0 ≠ 0 ∨ ↑U 0 1 ≠ 0)⊢ ∃ δ₃, V ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₃]V:CKMMatrixU:CKMMatrixhU:V ≈ U ∧ U.FstRowThdColRealCondhUV:⟦U⟧ = ⟦V⟧ha:¬‖↑V 0 2‖ ≠ 1⊢ ¬VubAbs ⟦V⟧ ≠ 1negV:CKMMatrixU:CKMMatrixhU:V ≈ U ∧ U.FstRowThdColRealCondhUV:⟦U⟧ = ⟦V⟧ha:¬(↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0)haU:¬(↑U 0 0 ≠ 0 ∨ ↑U 0 1 ≠ 0)⊢ ∃ δ₃, V ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₃negV:CKMMatrixU:CKMMatrixhU:V ≈ U ∧ U.FstRowThdColRealCondhUV:⟦U⟧ = ⟦V⟧ha:¬(↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0)haU:¬(↑U 0 0 ≠ 0 ∨ ↑U 0 1 ≠ 0)⊢ ∃ δ₃, V ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₃
    have ⟨U2, hU2⟩ := ubOnePhaseCond_hold_up_to_equiv_of_ub_one haU hU.2negV:CKMMatrixU:CKMMatrixhU:V ≈ U ∧ U.FstRowThdColRealCondhUV:⟦U⟧ = ⟦V⟧ha:¬(↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0)haU:¬(↑U 0 0 ≠ 0 ∨ ↑U 0 1 ≠ 0)U2:CKMMatrixhU2:U ≈ U2 ∧ U2.ubOnePhaseCond⊢ ∃ δ₃, V ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₃
    have hUVa2 : V ≈ U2 := phaseShiftRelation_equiv.trans hU.1 hU2.1negV:CKMMatrixU:CKMMatrixhU:V ≈ U ∧ U.FstRowThdColRealCondhUV:⟦U⟧ = ⟦V⟧ha:¬(↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0)haU:¬(↑U 0 0 ≠ 0 ∨ ↑U 0 1 ≠ 0)U2:CKMMatrixhU2:U ≈ U2 ∧ U2.ubOnePhaseCondhUVa2:V ≈ U2⊢ ∃ δ₃, V ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₃
    have hUV2 : ⟦U2⟧ = ⟦V⟧ := (Quotient.eq.mpr (phaseShiftRelation_equiv.symm hUVa2))negV:CKMMatrixU:CKMMatrixhU:V ≈ U ∧ U.FstRowThdColRealCondhUV:⟦U⟧ = ⟦V⟧ha:¬(↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0)haU:¬(↑U 0 0 ≠ 0 ∨ ↑U 0 1 ≠ 0)U2:CKMMatrixhU2:U ≈ U2 ∧ U2.ubOnePhaseCondhUVa2:V ≈ U2hUV2:⟦U2⟧ = ⟦V⟧⊢ ∃ δ₃, V ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₃
    have hx := eq_standParam_of_ubOnePhaseCond hU2.2negV:CKMMatrixU:CKMMatrixhU:V ≈ U ∧ U.FstRowThdColRealCondhUV:⟦U⟧ = ⟦V⟧ha:¬(↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0)haU:¬(↑U 0 0 ≠ 0 ∨ ↑U 0 1 ≠ 0)U2:CKMMatrixhU2:U ≈ U2 ∧ U2.ubOnePhaseCondhUVa2:V ≈ U2hUV2:⟦U2⟧ = ⟦V⟧hx:U2 = standParam (θ₁₂ ⟦U2⟧) (θ₁₃ ⟦U2⟧) (θ₂₃ ⟦U2⟧) 0⊢ ∃ δ₃, V ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₃
    use 0hV:CKMMatrixU:CKMMatrixhU:V ≈ U ∧ U.FstRowThdColRealCondhUV:⟦U⟧ = ⟦V⟧ha:¬(↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0)haU:¬(↑U 0 0 ≠ 0 ∨ ↑U 0 1 ≠ 0)U2:CKMMatrixhU2:U ≈ U2 ∧ U2.ubOnePhaseCondhUVa2:V ≈ U2hUV2:⟦U2⟧ = ⟦V⟧hx:U2 = standParam (θ₁₂ ⟦U2⟧) (θ₁₃ ⟦U2⟧) (θ₂₃ ⟦U2⟧) 0⊢ V ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0
    rw [← hUV2,hV:CKMMatrixU:CKMMatrixhU:V ≈ U ∧ U.FstRowThdColRealCondhUV:⟦U⟧ = ⟦V⟧ha:¬(↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0)haU:¬(↑U 0 0 ≠ 0 ∨ ↑U 0 1 ≠ 0)U2:CKMMatrixhU2:U ≈ U2 ∧ U2.ubOnePhaseCondhUVa2:V ≈ U2hUV2:⟦U2⟧ = ⟦V⟧hx:U2 = standParam (θ₁₂ ⟦U2⟧) (θ₁₃ ⟦U2⟧) (θ₂₃ ⟦U2⟧) 0⊢ V ≈ standParam (θ₁₂ ⟦U2⟧) (θ₁₃ ⟦U2⟧) (θ₂₃ ⟦U2⟧) 0hV:CKMMatrixU:CKMMatrixhU:V ≈ U ∧ U.FstRowThdColRealCondhUV:⟦U⟧ = ⟦V⟧ha:¬(↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0)haU:¬(↑U 0 0 ≠ 0 ∨ ↑U 0 1 ≠ 0)U2:CKMMatrixhU2:U ≈ U2 ∧ U2.ubOnePhaseCondhUVa2:V ≈ U2hUV2:⟦U2⟧ = ⟦V⟧hx:U2 = standParam (θ₁₂ ⟦U2⟧) (θ₁₃ ⟦U2⟧) (θ₂₃ ⟦U2⟧) 0⊢ V ≈ U2 ← hxhV:CKMMatrixU:CKMMatrixhU:V ≈ U ∧ U.FstRowThdColRealCondhUV:⟦U⟧ = ⟦V⟧ha:¬(↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0)haU:¬(↑U 0 0 ≠ 0 ∨ ↑U 0 1 ≠ 0)U2:CKMMatrixhU2:U ≈ U2 ∧ U2.ubOnePhaseCondhUVa2:V ≈ U2hUV2:⟦U2⟧ = ⟦V⟧hx:U2 = standParam (θ₁₂ ⟦U2⟧) (θ₁₃ ⟦U2⟧) (θ₂₃ ⟦U2⟧) 0⊢ V ≈ U2hV:CKMMatrixU:CKMMatrixhU:V ≈ U ∧ U.FstRowThdColRealCondhUV:⟦U⟧ = ⟦V⟧ha:¬(↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0)haU:¬(↑U 0 0 ≠ 0 ∨ ↑U 0 1 ≠ 0)U2:CKMMatrixhU2:U ≈ U2 ∧ U2.ubOnePhaseCondhUVa2:V ≈ U2hUV2:⟦U2⟧ = ⟦V⟧hx:U2 = standParam (θ₁₂ ⟦U2⟧) (θ₁₃ ⟦U2⟧) (θ₂₃ ⟦U2⟧) 0⊢ V ≈ U2]hV:CKMMatrixU:CKMMatrixhU:V ≈ U ∧ U.FstRowThdColRealCondhUV:⟦U⟧ = ⟦V⟧ha:¬(↑V 0 0 ≠ 0 ∨ ↑V 0 1 ≠ 0)haU:¬(↑U 0 0 ≠ 0 ∨ ↑U 0 1 ≠ 0)U2:CKMMatrixhU2:U ≈ U2 ∧ U2.ubOnePhaseCondhUVa2:V ≈ U2hUV2:⟦U2⟧ = ⟦V⟧hx:U2 = standParam (θ₁₂ ⟦U2⟧) (θ₁₃ ⟦U2⟧) (θ₂₃ ⟦U2⟧) 0⊢ V ≈ U2
    exact hUVa2All goals completed! 🐙theorem exists_for_CKMatrix (V : CKMMatrix) :
    ∃ (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ), V ≈ standParam θ₁₂ θ₁₃ θ₂₃ δ₁₃ := byV:CKMMatrix⊢ ∃ θ₁₂ θ₁₃ θ₂₃ δ₁₃, V ≈ standParam θ₁₂ θ₁₃ θ₂₃ δ₁₃
  use θ₁₂ ⟦V⟧, θ₁₃ ⟦V⟧, θ₂₃ ⟦V⟧, δ₁₃ ⟦V⟧hV:CKMMatrix⊢ V ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) (δ₁₃ ⟦V⟧)
  exact eq_standardParameterization_δ₃ VAll goals completed! 🐙