theorem extracted_formal_statement_27.{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.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂);
  ((fun x =>
        ADMM.ρ E₁ E₂ F / 2 *
          ‖(OptProblem.A₁ E₂) x + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂‖ ^ 2) =
      fun x => ADMM.ρ E₁ E₂ F * ‖(OptProblem.A₁ E₂) x - c‖ ^ 2 / 2) →
    let f := fun x => ‖(OptProblem.A₁ E₂) x - c‖ ^ 2 / 2;
    ((fun x => ADMM.ρ E₁ E₂ F • ‖(OptProblem.A₁ E₂) x - c‖ ^ 2 / 2) = fun x => ADMM.ρ E₁ E₂ F • f x) →
      let u := fun x => ‖x - c‖ ^ 2 / 2;
      let g := OptProblem.A₁ E₂;
      u ∘ ⇑g = f → ⇑g ⁻¹' univ = univ → Convex ℝ univ → ConvexOn ℝ (⇑g ⁻¹' univ) (u ∘ ⇑g) := 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.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂)
h :
  (fun x =>
      ADMM.ρ E₁ E₂ F / 2 *
        ‖(OptProblem.A₁ E₂) x + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂‖ ^ 2) =
    fun x => ADMM.ρ E₁ E₂ F * ‖(OptProblem.A₁ E₂) x - c‖ ^ 2 / 2
f : E₁ → ℝ := fun x => ‖(OptProblem.A₁ E₂) x - c‖ ^ 2 / 2
h1 : (fun x => ADMM.ρ E₁ E₂ F • ‖(OptProblem.A₁ E₂) x - c‖ ^ 2 / 2) = fun x => ADMM.ρ E₁ E₂ F • f x
u : F → ℝ := fun x => ‖x - c‖ ^ 2 / 2
g : E₁ →L[ℝ] F := OptProblem.A₁ E₂
h2 : u ∘ ⇑g = f
h3 : ⇑g ⁻¹' univ = univ
h4 : Convex ℝ univ
⊢ ConvexOn ℝ (⇑g ⁻¹' univ) (u ∘ ⇑g)

Tactic: apply ConvexOn.comp_linearMap (convex_of_norm_sq c h4)

State after:

Relevant lemmas:
ConvexOn.comp_linearMap : ∀ {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {β : Type u_5} [inst : OrderedSemiring 𝕜]
  [inst_9 : AddCommMonoid E] [inst_10 : AddCommMonoid F] [inst_11 : OrderedAddCommMonoid β] [inst_12 : Module 𝕜 E]
  [inst_13 : Module 𝕜 F] [inst_14 : SMul 𝕜 β] {f : F → β} {s : Set F},
  ConvexOn 𝕜 s f → ∀ (g : E →ₗ[𝕜] F), ConvexOn 𝕜 (⇑g ⁻¹' s) (f ∘ ⇑g)
convex_of_norm_sq : ∀ {E : Type u_2} [inst : NormedAddCommGroup E] [inst_9 : InnerProductSpace ℝ E] {s : Set E} (x : E),
  Convex ℝ s → ConvexOn ℝ s fun u => ‖u - x‖ ^ 2 / 2