theorem extracted_formal_statement_187.{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) :
  let A_e_sum := (OptProblem.A₁ E₂) (e₁ (n + 1)) + (OptProblem.A₂ E₁) (e₂ (n + 1));
  (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F - ADMM.ρ E₁ E₂ F + ADMM.ρ E₁ E₂ F + -(ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F)) • A_e_sum =
    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
A_e_sum : F := (OptProblem.A₁ E₂) (e₁ (n + 1)) + (OptProblem.A₂ E₁) (e₂ (n + 1))
A_x_sum : F := -(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n - ADMM.x₂ E₁ F (n + 1))
ey' : F := ey (n + 1)
Ae1 : F := (OptProblem.A₁ E₂) (e₁ (n + 1))
Ae2 : F := (OptProblem.A₂ E₁) (e₂ (n + 1))
⊢ (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F - ADMM.ρ E₁ E₂ F + ADMM.ρ E₁ E₂ F + -(ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F)) • A_e_sum = 0

Tactic: rw [sub_eq_add_neg, add_assoc (τ * ρ) (-ρ) ρ, add_comm (-ρ) ρ, add_neg_cancel, add_zero, add_neg_cancel,
  zero_smul]

State after:

Relevant lemmas:
sub_eq_add_neg : ∀ {G : Type u_1} [inst : SubNegMonoid G] (a b : G), a - b = a + -b
add_assoc : ∀ {G : Type u_1} [inst : AddSemigroup G] (a b c : G), a + b + c = a + (b + c)
zero_smul : ∀ (M₀ : Type u_2) {A : Type u_7} [inst : Zero M₀] [inst_10 : Zero A] [inst_11 : SMulWithZero M₀ A] (m : A),
  0 • m = 0
add_zero : ∀ {M : Type u} [inst : AddZeroClass M] (a : M), a + 0 = a
add_neg_cancel : ∀ {G : Type u_1} [inst : AddGroup G] (a : G), a + -a = 0
add_comm : ∀ {G : Type u_1} [inst : AddCommMagma G] (a b : G), a + b = b + a