theorem extracted_formal_statement_204.{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} [inst_9 : Setting E₁ E₂ F admm admm_kkt] (n : ℕ) :
  let pm1 := 1 / (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F);
  pm1 * ⟪ey (n + 1), ey n - ey (n + 1)⟫_ℝ = ⟪ey (n + 1), pm1 • (ey n - ey (n + 1))⟫_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
pm1 : ℝ := 1 / (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F)
pm2 : ℝ := (1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F
⊢ pm1 * ⟪ey (n + 1), ey n - ey (n + 1)⟫_ℝ = ⟪ey (n + 1), pm1 • (ey n - ey (n + 1))⟫_ℝ

Tactic: rw [← real_inner_smul_right (ey (n + 1)) ((ey n) - (ey (n + 1))) pm1]

State after:

Relevant lemmas:
real_inner_smul_right : ∀ {F : Type u_3} [inst : SeminormedAddCommGroup F] [inst_10 : InnerProductSpace ℝ F] (x y : F)
  (r : ℝ), ⟪x, r • y⟫_ℝ = r * ⟪x, y⟫_ℝ