theorem extracted_formal_statement_171.{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] (r1 : ℝ)
  (h_A₂e₂ : range (⇑(OptProblem.A₂ E₁) ∘ e₂) ⊆ ball 0 r1) (r2 : ℝ)
  (h_A₁e₁_A₂e₂ : range (⇑(OptProblem.A₁ E₂) ∘ e₁ + ⇑(OptProblem.A₂ E₁) ∘ e₂) ⊆ ball 0 r2) :
  let r := r1 + r2;
  r = r1 + r2 →
    ∀ (n : ℕ),
      ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖ + ‖(OptProblem.A₂ E₁) (e₂ n)‖ < r1 + r2 →
        ‖(OptProblem.A₁ E₂) (e₁ n)‖ ≤
            ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖ + ‖(OptProblem.A₂ E₁) (e₂ n)‖ →
          ‖(OptProblem.A₁ E₂) (e₁ n)‖ < r →
            (OptProblem.A₁ E₂) (e₁ n) ∈ ball 0 r → (⇑(OptProblem.A₁ E₂) ∘ e₁) n ∈ ball 0 r := sorry

State before:
case h.intro
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
r1 : ℝ
h_A₂e₂ : range (⇑(OptProblem.A₂ E₁) ∘ e₂) ⊆ ball 0 r1
r2 : ℝ
h_A₁e₁_A₂e₂ : range (⇑(OptProblem.A₁ E₂) ∘ e₁ + ⇑(OptProblem.A₂ E₁) ∘ e₂) ⊆ ball 0 r2
r : ℝ := r1 + r2
hr : r = r1 + r2
n : ℕ
h : ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖ + ‖(OptProblem.A₂ E₁) (e₂ n)‖ < r1 + r2
h_ineq :
  ‖(OptProblem.A₁ E₂) (e₁ n)‖ ≤ ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖ + ‖(OptProblem.A₂ E₁) (e₂ n)‖
h_norm : ‖(OptProblem.A₁ E₂) (e₁ n)‖ < r
h_dist : (OptProblem.A₁ E₂) (e₁ n) ∈ ball 0 r
⊢ (⇑(OptProblem.A₁ E₂) ∘ e₁) n ∈ ball 0 r

Tactic: apply h_dist

State after:

Relevant lemmas: