theorem extracted_formal_statement_22.{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+) :
  ⟪v ↑n - -(ContinuousLinearMap.adjoint (OptProblem.A₂ E₁)) admm_kkt.y, ADMM.x₂ E₁ F ↑n - admm_kkt.x₂⟫_ℝ ≥ 0 := 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+
⊢ ⟪v ↑n - -(ContinuousLinearMap.adjoint (OptProblem.A₂ E₁)) admm_kkt.y, ADMM.x₂ E₁ F ↑n - admm_kkt.x₂⟫_ℝ ≥ 0

Tactic: apply subgradientAt_mono
    apply v_inthesubgradient
    exact admm_kkt.h.subgrad₂

State after:

Relevant lemmas:
subgradientAt_mono : ∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_10 : InnerProductSpace ℝ E] {x u v : E}
  {f : E → ℝ} {y : E}, u ∈ SubderivAt f x → v ∈ SubderivAt f y → ⟪u - v, x - y⟫_ℝ ≥ 0