theorem extracted_formal_statement_3 {n m : ℕ+} {A : Matrix (Fin ↑m) (Fin ↑n) ℝ} :
  let normA := ‖(toEuclideanLin ≪≫ₗ LinearMap.toContinuousLinearMap) A‖;
  (∀ (x : EuclideanSpace ℝ (Fin ↑n)), ‖A *ᵥ x‖ ≤ normA * ‖x‖) → normA ≤ 0 → 0 ≤ normA → normA = 0 → A = 0 := sorry

State before:
n m : ℕ+
A : Matrix (Fin ↑m) (Fin ↑n) ℝ
x : EuclideanSpace ℝ (Fin ↑n)
ε : ℝ
εpos : ε > 0
normA : ℝ := ‖(toEuclideanLin ≪≫ₗ LinearMap.toContinuousLinearMap) A‖
norm_mul : ∀ (x : EuclideanSpace ℝ (Fin ↑n)), ‖A *ᵥ x‖ ≤ normA * ‖x‖
hA : normA ≤ 0
hA' : 0 ≤ normA
hA'' : normA = 0
⊢ A = 0

Tactic: rw [norm_eq_zero] at hA''
    simp at hA''
    exact hA''

State after:

Relevant lemmas:
norm_eq_zero : ∀ {E : Type u_5} [inst : NormedAddGroup E] {a : E}, ‖a‖ = 0 ↔ a = 0