theorem extracted_formal_statement_33.{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} (n : ℕ+) :
  let c := -((OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) - OptProblem.b E₁ E₂);
  ((fun x =>
        ADMM.ρ E₁ E₂ F / 2 * ‖(OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) + (OptProblem.A₂ E₁) x - OptProblem.b E₁ E₂‖ ^ 2) =
      fun x => ADMM.ρ E₁ E₂ F * ‖(OptProblem.A₂ E₁) x - c‖ ^ 2 / 2) →
    let f := fun x => ‖(OptProblem.A₂ E₁) x - c‖ ^ 2 / 2;
    ((fun x => ADMM.ρ E₁ E₂ F • ‖(OptProblem.A₂ E₁) x - c‖ ^ 2 / 2) = fun x => ADMM.ρ E₁ E₂ F • f x) →
      let u := fun x => ‖x - c‖ ^ 2 / 2;
      let g := OptProblem.A₂ E₁;
      u ∘ ⇑g = f → ⇑g ⁻¹' univ = univ → Convex ℝ univ := 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+
c : F := -((OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) - OptProblem.b E₁ E₂)
h :
  (fun x =>
      ADMM.ρ E₁ E₂ F / 2 * ‖(OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) + (OptProblem.A₂ E₁) x - OptProblem.b E₁ E₂‖ ^ 2) =
    fun x => ADMM.ρ E₁ E₂ F * ‖(OptProblem.A₂ E₁) x - c‖ ^ 2 / 2
f : E₂ → ℝ := fun x => ‖(OptProblem.A₂ E₁) x - c‖ ^ 2 / 2
h1 : (fun x => ADMM.ρ E₁ E₂ F • ‖(OptProblem.A₂ E₁) x - c‖ ^ 2 / 2) = fun x => ADMM.ρ E₁ E₂ F • f x
u : F → ℝ := fun x => ‖x - c‖ ^ 2 / 2
g : E₂ →L[ℝ] F := OptProblem.A₂ E₁
h2 : u ∘ ⇑g = f
h3 : ⇑g ⁻¹' univ = univ
⊢ Convex ℝ univ

Tactic: apply convex_univ

State after:

Relevant lemmas:
convex_univ : ∀ {𝕜 : Type u_1} {E : Type u_2} [inst : OrderedSemiring 𝕜] [inst_9 : AddCommMonoid E]
  [inst_10 : SMul 𝕜 E], Convex 𝕜 univ