theorem extracted_formal_statement_74.{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} (fullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂))
  (fullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)) [inst_9 : Setting E₁ E₂ F admm (admm_kkt₁ fullrank₁ fullrank₂)]
  (n : ℕ) (this : ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F * Φ n = Φ n * (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F))
  (h_original_goal : ‖ey n‖ ^ 2 ≤ ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F * Φ n) :
  ‖ey n‖ ^ 2 / (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F) ≤ Φ 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
fullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂)
fullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)
inst✝ : Setting E₁ E₂ F admm (admm_kkt₁ fullrank₁ fullrank₂)
n : ℕ
this : ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F * Φ n = Φ n * (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F)
⊢ ‖ey n‖ ^ 2 ≤ ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F * Φ n

Tactic: rw [this]
    have : (τ * ρ) > 0 := by apply mul_pos; linarith [admm.htau.1]; linarith [admm.hrho]
    apply (div_le_iff₀ this).mp

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
 fullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂)
 fullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)
 inst✝ : Setting E₁ E₂ F admm (admm_kkt₁ fullrank₁ fullrank₂)
 n : ℕ
 this✝ : ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F * Φ n = Φ n * (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F)
 this : ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F > 0
 ⊢ ‖ey n‖ ^ 2 / (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F) ≤ Φ n]

Relevant lemmas:
div_le_iff₀ : ∀ {G₀ : Type u_2} [inst : GroupWithZero G₀] [inst_10 : PartialOrder G₀] [inst_11 : ZeroLEOneClass G₀]
  [inst_12 : PosMulReflectLT G₀] {a b c : G₀} [inst_13 : MulPosMono G₀], 0 < c → (b / c ≤ a ↔ b ≤ a * c)
mul_pos : ∀ {α : Type u_3} {a b : α} [inst : MulZeroClass α] [inst_10 : Preorder α] [inst_11 : PosMulStrictMono α],
  0 < a → 0 < b → 0 < a * b