theorem extracted_formal_statement_16.{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 := ADMM.y E₁ E₂ n.natPred;
  let a := (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) - OptProblem.b E₁ E₂;
  ((fun x =>
        ⟪ADMM.y E₁ E₂ n.natPred, (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) + (OptProblem.A₂ E₁) x - OptProblem.b E₁ E₂⟫_ℝ) =
      fun x =>
      ⟪ADMM.y E₁ E₂ n.natPred, (OptProblem.A₂ E₁) x + ((OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) - OptProblem.b E₁ E₂)⟫_ℝ) →
    ((fun x => ⟪c, (OptProblem.A₂ E₁) x + a⟫_ℝ) = fun x => ⟪c, (OptProblem.A₂ E₁) x⟫_ℝ + ⟪c, a⟫_ℝ) →
      ∀ (u v : E₂),
        (fun x => ⟪c, (OptProblem.A₂ E₁) x⟫_ℝ) (u + v) =
            (fun x => ⟪c, (OptProblem.A₂ E₁) x⟫_ℝ) u + (fun x => ⟪c, (OptProblem.A₂ E₁) x⟫_ℝ) v →
          ⟪c, (OptProblem.A₂ E₁) u + (OptProblem.A₂ E₁) v⟫_ℝ =
            ⟪c, (OptProblem.A₂ E₁) u⟫_ℝ + ⟪c, (OptProblem.A₂ E₁) v⟫_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+
c : F := ADMM.y E₁ E₂ n.natPred
a : F := (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) - OptProblem.b E₁ E₂
this :
  (fun x =>
      ⟪ADMM.y E₁ E₂ n.natPred, (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) + (OptProblem.A₂ E₁) x - OptProblem.b E₁ E₂⟫_ℝ) =
    fun x =>
    ⟪ADMM.y E₁ E₂ n.natPred, (OptProblem.A₂ E₁) x + ((OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) - OptProblem.b E₁ E₂)⟫_ℝ
h : (fun x => ⟪c, (OptProblem.A₂ E₁) x + a⟫_ℝ) = fun x => ⟪c, (OptProblem.A₂ E₁) x⟫_ℝ + ⟪c, a⟫_ℝ
p : ℝ := ⟪c, a⟫_ℝ
u v : E₂
⊢ (fun x => ⟪c, (OptProblem.A₂ E₁) x⟫_ℝ) (u + v) =
    (fun x => ⟪c, (OptProblem.A₂ E₁) x⟫_ℝ) u + (fun x => ⟪c, (OptProblem.A₂ E₁) x⟫_ℝ) v

Tactic: simp only [map_add]

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 := ADMM.y E₁ E₂ n.natPred
 a : F := (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) - OptProblem.b E₁ E₂
 this :
   (fun x =>
       ⟪ADMM.y E₁ E₂ n.natPred, (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) + (OptProblem.A₂ E₁) x - OptProblem.b E₁ E₂⟫_ℝ) =
     fun x =>
     ⟪ADMM.y E₁ E₂ n.natPred, (OptProblem.A₂ E₁) x + ((OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) - OptProblem.b E₁ E₂)⟫_ℝ
 h : (fun x => ⟪c, (OptProblem.A₂ E₁) x + a⟫_ℝ) = fun x => ⟪c, (OptProblem.A₂ E₁) x⟫_ℝ + ⟪c, a⟫_ℝ
 p : ℝ := ⟪c, a⟫_ℝ
 u v : E₂
 ⊢ ⟪c, (OptProblem.A₂ E₁) u + (OptProblem.A₂ E₁) v⟫_ℝ = ⟪c, (OptProblem.A₂ E₁) u⟫_ℝ + ⟪c, (OptProblem.A₂ E₁) v⟫_ℝ]

Relevant lemmas:
map_add : ∀ {M : Type u_4} {N : Type u_5} {F : Type u_9} [inst : Add M] [inst_9 : Add N] [inst_10 : FunLike F M N]
  [inst_11 : AddHomClass F M N] (f : F) (x y : M), f (x + y) = f x + f y