theorem extracted_formal_statement_12.{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+) ⦃xx : E₁⦄ (a : xx ∈ univ) ⦃yy : E₁⦄
  (a_1 : yy ∈ univ) ⦃aa bb : ℝ⦄ (a_2 : 0 ≤ aa) (a_3 : 0 ≤ bb) (abh : aa + bb = 1)
  (this :
    (OptProblem.A₁ E₂) (aa • xx + bb • yy) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂ =
      aa • ((OptProblem.A₁ E₂) xx + ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂)) +
        bb • ((OptProblem.A₁ E₂) yy + ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂))) :
  aa *
        ⟪ADMM.y E₁ E₂ n.natPred,
          (OptProblem.A₁ E₂) xx + ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂)⟫_ℝ +
      ⟪ADMM.y E₁ E₂ n.natPred,
        bb • ((OptProblem.A₁ E₂) yy + ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂))⟫_ℝ =
    aa *
        ⟪ADMM.y E₁ E₂ n.natPred,
          (OptProblem.A₁ E₂) xx + ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂)⟫_ℝ +
      bb *
        ⟪ADMM.y E₁ E₂ n.natPred,
          (OptProblem.A₁ E₂) yy + ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂)⟫_ℝ := 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+
xx : E₁
a✝³ : xx ∈ univ
yy : E₁
a✝² : yy ∈ univ
aa bb : ℝ
a✝¹ : 0 ≤ aa
a✝ : 0 ≤ bb
abh : aa + bb = 1
this :
  (OptProblem.A₁ E₂) (aa • xx + bb • yy) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂ =
    aa • ((OptProblem.A₁ E₂) xx + ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂)) +
      bb • ((OptProblem.A₁ E₂) yy + ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂))
⊢ aa *
        ⟪ADMM.y E₁ E₂ n.natPred,
          (OptProblem.A₁ E₂) xx + ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂)⟫_ℝ +
      ⟪ADMM.y E₁ E₂ n.natPred,
        bb • ((OptProblem.A₁ E₂) yy + ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂))⟫_ℝ =
    aa *
        ⟪ADMM.y E₁ E₂ n.natPred,
          (OptProblem.A₁ E₂) xx + ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂)⟫_ℝ +
      bb *
        ⟪ADMM.y E₁ E₂ n.natPred,
          (OptProblem.A₁ E₂) yy + ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂)⟫_ℝ

Tactic: rw [inner_smul_right]

State after:

Relevant lemmas:
inner_smul_right : ∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_9 : SeminormedAddCommGroup E]
  [inst_10 : InnerProductSpace 𝕜 E] (x y : E) (r : 𝕜), ⟪x, r • y⟫_𝕜 = r * ⟪x, y⟫_𝕜