theorem extracted_formal_statement_176.{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]
  (fullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂)) (M : ℝ)
  (h₁ : {x | ∃ «y», (⇑(OptProblem.A₁ E₂) ∘ e₁) «y» = x} ⊆ ball 0 M) (C : ℝ) (C_pos : C > 0)
  (h₂ : ∀ (n : ℕ), ‖e₁ n‖ ≤ C * ‖(OptProblem.A₁ E₂) (e₁ n)‖) ⦃x : E₁⦄ (n : ℕ) (eq : ADMM.x₁ E₂ F n = x) :
  (OptProblem.A₁ E₂) (e₁ n) ∈ {x | ∃ n, (OptProblem.A₁ E₂) (e₁ n) = x} := 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
fullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂)
M : ℝ
h₁ : {x | ∃ «y», (⇑(OptProblem.A₁ E₂) ∘ e₁) «y» = x} ⊆ ball 0 M
C : ℝ
C_pos : C > 0
h₂ : ∀ (n : ℕ), ‖e₁ n‖ ≤ C * ‖(OptProblem.A₁ E₂) (e₁ n)‖
x : E₁
n : ℕ
eq : ADMM.x₁ E₂ F n = x
⊢ (OptProblem.A₁ E₂) (e₁ n) ∈ {x | ∃ n, (OptProblem.A₁ E₂) (e₁ n) = x}

Tactic: simp

State after:

Relevant lemmas: