theorem extracted_formal_statement_297.{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_kkt]
  (this :
    (fun n => ADMM.ρ E₁ E₂ F • (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (↑(φ₁ n) - 1) - ADMM.x₂ E₁ F ↑(φ₁ n))) = fun n =>
      -ADMM.ρ E₁ E₂ F • (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑(φ₁ n) - ADMM.x₂ E₁ F (↑(φ₁ n) - 1)))
  (h_original_goal :
    Tendsto (fun n => ADMM.ρ E₁ E₂ F • (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (↑(φ₁ n) - 1) - ADMM.x₂ E₁ F ↑(φ₁ n))) atTop
      (𝓝 0)) :
  Tendsto (fun n => -ADMM.ρ E₁ E₂ F • (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑(φ₁ n) - ADMM.x₂ E₁ F (↑(φ₁ n) - 1))) atTop
    (𝓝 (-ADMM.ρ E₁ E₂ F • 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
fullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂)
fullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)
inst✝ : Setting E₁ E₂ F admm admm_kkt
this :
  (fun n => ADMM.ρ E₁ E₂ F • (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (↑(φ₁ n) - 1) - ADMM.x₂ E₁ F ↑(φ₁ n))) = fun n =>
    -ADMM.ρ E₁ E₂ F • (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑(φ₁ n) - ADMM.x₂ E₁ F (↑(φ₁ n) - 1))
⊢ Tendsto (fun n => ADMM.ρ E₁ E₂ F • (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (↑(φ₁ n) - 1) - ADMM.x₂ E₁ F ↑(φ₁ n))) atTop (𝓝 0)

Tactic: rw [this, ← smul_zero (-ρ)]

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_kkt
 this :
   (fun n => ADMM.ρ E₁ E₂ F • (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (↑(φ₁ n) - 1) - ADMM.x₂ E₁ F ↑(φ₁ n))) = fun n =>
     -ADMM.ρ E₁ E₂ F • (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑(φ₁ n) - ADMM.x₂ E₁ F (↑(φ₁ n) - 1))
 ⊢ Tendsto (fun n => -ADMM.ρ E₁ E₂ F • (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑(φ₁ n) - ADMM.x₂ E₁ F (↑(φ₁ n) - 1))) atTop
     (𝓝 (-ADMM.ρ E₁ E₂ F • 0))]

Relevant lemmas:
smul_zero : ∀ {M : Type u_1} {A : Type u_7} [inst : Zero A] [inst_10 : SMulZeroClass M A] (a : M), a • 0 = 0