theorem extracted_formal_statement_422.{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} (fullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂))
  (fullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)) [inst_9 : Setting E₁ E₂ F admm (admm_kkt₁ fullrank₁ fullrank₂)]
  (h :
    ∀ (n : ℕ),
      ‖(OptProblem.A₁ E₂) (e₁ n)‖ ≤
        ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖ + ‖(OptProblem.A₂ E₁) (e₂ n)‖)
  (n : ℕ) : ‖‖(OptProblem.A₁ E₂) (e₁ n)‖‖ = ‖(OptProblem.A₁ E₂) (e₁ n)‖ := 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
fullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂)
fullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)
inst✝ : Setting E₁ E₂ F admm (admm_kkt₁ fullrank₁ fullrank₂)
h :
  ∀ (n : ℕ),
    ‖(OptProblem.A₁ E₂) (e₁ n)‖ ≤ ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖ + ‖(OptProblem.A₂ E₁) (e₂ n)‖
n : ℕ
⊢ ‖‖(OptProblem.A₁ E₂) (e₁ n)‖‖ = ‖(OptProblem.A₁ E₂) (e₁ n)‖

Tactic: simp only [norm_norm]

State after:

Relevant lemmas:
norm_norm : ∀ {E : Type u_5} [inst : SeminormedAddCommGroup E] (x : E), ‖‖x‖‖ = ‖x‖