case mp =>
  intro (_ : p/q ≃ 0)
  show p ≃ 0
  have : p * q⁻¹ ≃ 0 := eqv_trans (eqv_symm div_mul_recip) ‹p/q ≃ 0›
  have : p ≃ 0 ∨ q⁻¹ ≃ 0 := mul_split_zero.mp this
  match this with
  | Or.inl (_ : p ≃ 0) =>
    exact ‹p ≃ 0›
  | Or.inr (_ : q⁻¹ ≃ 0) =>
    have : q⁻¹ ≄ 0 := recip_preserves_nonzero
    exact absurd ‹q⁻¹ ≃ 0› ‹q⁻¹ ≄ 0›
ℕ ℤ ℚ : Type
inst✝¹⁰ : Natural ℕ
inst✝⁹ : Integer ℤ
inst✝⁸ : Core ℚ
inst✝⁷ : Addition ℚ
inst✝⁶ : Multiplication ℚ
inst✝⁵ : Negation ℚ
inst✝⁴ : Subtraction ℚ
inst✝³ : Reciprocation ℚ
inst✝² : Division ℚ
inst✝¹ : Sign ℚ
p q : ℚ
inst✝ : AP (q ≄ 0)
⊢ p / q ≃ 0 → p ≃ 0
no goals