theorem extracted_formal_statement_45.{u_3, u_2, u_1} {E₁ : Type u_1} {E₂ : Type u_2} {F : Type u_3}
  [inst : NormedAddCommGroup E₁] [inst_1 : InnerProductSpace ℝ E₁] [inst_2 : FiniteDimensional ℝ E₁]
  [inst_3 : NormedAddCommGroup E₂] [inst_4 : InnerProductSpace ℝ E₂] [inst_5 : FiniteDimensional ℝ E₂]
  [inst_6 : NormedAddCommGroup F] [inst_7 : InnerProductSpace ℝ F] [inst_8 : FiniteDimensional ℝ F]
  {admm : ADMM E₁ E₂ F} {admm_kkt : Existance_of_kkt admm} [inst_9 : Setting E₁ E₂ F admm admm_kkt] (n : ℕ)
  (h_original_goal :
    (ADMM.τ E₁ E₂ F * (ADMM.ρ E₁ E₂ F * (ADMM.ρ E₁ E₂ F)⁻¹ * (ADMM.τ E₁ E₂ F)⁻¹)) • (ey (n + 1) - ey n) =
      ey (n + 1) - ey n) :
  ADMM.ρ E₁ E₂ F ≠ 0 := sorry

State before:
E₁ : Type u_1
E₂ : Type u_2
F : Type u_3
inst✝⁹ : NormedAddCommGroup E₁
inst✝⁸ : InnerProductSpace ℝ E₁
inst✝⁷ : FiniteDimensional ℝ E₁
inst✝⁶ : NormedAddCommGroup E₂
inst✝⁵ : InnerProductSpace ℝ E₂
inst✝⁴ : FiniteDimensional ℝ E₂
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace ℝ F
inst✝¹ : FiniteDimensional ℝ F
admm : ADMM E₁ E₂ F
admm_kkt : Existance_of_kkt admm
inst✝ : Setting E₁ E₂ F admm admm_kkt
n : ℕ
⊢ (ADMM.τ E₁ E₂ F * (ADMM.ρ E₁ E₂ F * (ADMM.ρ E₁ E₂ F)⁻¹ * (ADMM.τ E₁ E₂ F)⁻¹)) • (ey (n + 1) - ey n) =
    ey (n + 1) - ey n

Tactic: rw [mul_inv_cancel₀, one_mul, mul_inv_cancel₀, one_smul]
    rcases admm.htau with ⟨h₁, _⟩
    apply ne_of_gt h₁

State after:
[E₁ : Type u_1
 E₂ : Type u_2
 F : Type u_3
 inst✝⁹ : NormedAddCommGroup E₁
 inst✝⁸ : InnerProductSpace ℝ E₁
 inst✝⁷ : FiniteDimensional ℝ E₁
 inst✝⁶ : NormedAddCommGroup E₂
 inst✝⁵ : InnerProductSpace ℝ E₂
 inst✝⁴ : FiniteDimensional ℝ E₂
 inst✝³ : NormedAddCommGroup F
 inst✝² : InnerProductSpace ℝ F
 inst✝¹ : FiniteDimensional ℝ F
 admm : ADMM E₁ E₂ F
 admm_kkt : Existance_of_kkt admm
 inst✝ : Setting E₁ E₂ F admm admm_kkt
 n : ℕ
 ⊢ ADMM.ρ E₁ E₂ F ≠ 0]

Relevant lemmas:
mul_inv_cancel₀ : ∀ {G₀ : Type u} [inst : GroupWithZero G₀] {a : G₀}, a ≠ 0 → a * a⁻¹ = 1
one_smul : ∀ (M : Type u_1) {α : Type u_5} [inst : Monoid M] [inst_10 : MulAction M α] (b : α), 1 • b = b
ne_of_gt : ∀ {α : Type u_1} [inst : Preorder α] {a b : α}, b < a → a ≠ b
one_mul : ∀ {M : Type u} [inst : MulOneClass M] (a : M), 1 * a = a