theorem extracted_formal_statement_229.{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+)
  (this :
    v ↑n =
      -(ContinuousLinearMap.adjoint (OptProblem.A₂ E₁))
          (ADMM.y E₁ E₂ ↑n +
            ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • ((OptProblem.A₁ E₂) (e₁ ↑n) + (OptProblem.A₂ E₁) (e₂ ↑n)))) :
  v ↑n ∈ SubderivAt (OptProblem.f₂ E₁ F) (ADMM.x₂ E₁ F ↑n) := 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+
this :
  v ↑n =
    -(ContinuousLinearMap.adjoint (OptProblem.A₂ E₁))
        (ADMM.y E₁ E₂ ↑n +
          ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • ((OptProblem.A₁ E₂) (e₁ ↑n) + (OptProblem.A₂ E₁) (e₂ ↑n)))
⊢ v ↑n ∈ SubderivAt (OptProblem.f₂ E₁ F) (ADMM.x₂ E₁ F ↑n)

Tactic: apply v_inthesubgradient

State after:

Relevant lemmas: