theorem extracted_formal_statement_23.{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) →
      (ConvexOn ℝ univ fun x => ADMM.ρ E₁ E₂ F • f x) → ConvexOn ℝ univ f := 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
⊢ ConvexOn ℝ univ fun x => ADMM.ρ E₁ E₂ F • f x

Tactic: apply ConvexOn.smul (le_of_lt admm.hrho)

State after:
[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
 ⊢ ConvexOn ℝ univ f]

Relevant lemmas:
ConvexOn.smul : ∀ {𝕜 : Type u_1} {E : Type u_2} {β : Type u_5} [inst : OrderedCommSemiring 𝕜] [inst_9 : AddCommMonoid E]
  [inst_10 : OrderedAddCommMonoid β] [inst_11 : SMul 𝕜 E] [inst_12 : Module 𝕜 β] [inst_13 : OrderedSMul 𝕜 β] {s : Set E}
  {f : E → β} {c : 𝕜}, 0 ≤ c → ConvexOn 𝕜 s f → ConvexOn 𝕜 s fun x => c • f x
le_of_lt : ∀ {α : Type u_1} [inst : Preorder α] {a b : α}, a < b → a ≤ b