theorem extracted_formal_statement_17.{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 : ℝ) (v : E₂),
        { toFun := fun x => ⟪c, (OptProblem.A₂ E₁) x⟫_ℝ,
                  map_add' := fun u v =>
                    Eq.mpr
                      (id
                        (congrArg (fun x => ⟪c, x⟫_ℝ = ⟪c, (OptProblem.A₂ E₁) u⟫_ℝ + ⟪c, (OptProblem.A₂ E₁) v⟫_ℝ)
                          (map_add (OptProblem.A₂ E₁) u v)))
                      (Eq.mpr
                        (id
                          (congrArg (fun _a => _a = ⟪c, (OptProblem.A₂ E₁) u⟫_ℝ + ⟪c, (OptProblem.A₂ E₁) v⟫_ℝ)
                            (inner_add_right c ((OptProblem.A₂ E₁) u) ((OptProblem.A₂ E₁) v))))
                        (Eq.refl (⟪c, (OptProblem.A₂ E₁) u⟫_ℝ + ⟪c, (OptProblem.A₂ E₁) v⟫_ℝ))) }.toFun
              (u • v) =
            (RingHom.id ℝ) u •
              { toFun := fun x => ⟪c, (OptProblem.A₂ E₁) x⟫_ℝ,
                    map_add' := fun u v =>
                      Eq.mpr
                        (id
                          (congrArg (fun x => ⟪c, x⟫_ℝ = ⟪c, (OptProblem.A₂ E₁) u⟫_ℝ + ⟪c, (OptProblem.A₂ E₁) v⟫_ℝ)
                            (map_add (OptProblem.A₂ E₁) u v)))
                        (Eq.mpr
                          (id
                            (congrArg (fun _a => _a = ⟪c, (OptProblem.A₂ E₁) u⟫_ℝ + ⟪c, (OptProblem.A₂ E₁) v⟫_ℝ)
                              (inner_add_right c ((OptProblem.A₂ E₁) u) ((OptProblem.A₂ E₁) v))))
                          (Eq.refl (⟪c, (OptProblem.A₂ E₁) u⟫_ℝ + ⟪c, (OptProblem.A₂ E₁) v⟫_ℝ))) }.toFun
                v →
          ⟪c, u • (OptProblem.A₂ E₁) v⟫_ℝ = u * ⟪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 : ℝ
v : E₂
⊢ { toFun := fun x => ⟪c, (OptProblem.A₂ E₁) x⟫_ℝ, map_add' := ⋯ }.toFun (u • v) =
    (RingHom.id ℝ) u • { toFun := fun x => ⟪c, (OptProblem.A₂ E₁) x⟫_ℝ, map_add' := ⋯ }.toFun v

Tactic: simp only [LinearMapClass.map_smul, RingHom.id_apply, smul_eq_mul]

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 : ℝ
 v : E₂
 ⊢ ⟪c, u • (OptProblem.A₂ E₁) v⟫_ℝ = u * ⟪c, (OptProblem.A₂ E₁) v⟫_ℝ]

Relevant lemmas:
LinearMapClass.map_smul : ∀ {R : outParam (Type u_14)} {M : outParam (Type u_15)} {M₂ : outParam (Type u_16)}
  [inst : Semiring R] [inst_9 : AddCommMonoid M] [inst_10 : AddCommMonoid M₂] [inst_11 : Module R M]
  [inst_12 : Module R M₂] {F : Type u_17} [inst_13 : FunLike F M M₂] [inst_14 : LinearMapClass F R M M₂] (f : F) (r : R)
  (x : M), f (r • x) = r • f x
smul_eq_mul : ∀ {α : Type u_9} [inst : Mul α] (a b : α), a • b = a * b
RingHom.id_apply : ∀ {α : Type u_2} {x : NonAssocSemiring α} (x_1 : α), (RingHom.id α) x_1 = x_1