case step =>
  intro (n' : ℕ) (ih : (p^n')⁻¹ ≃ (p⁻¹)^n')
  show (p^(step n'))⁻¹ ≃ (p⁻¹)^(step n')
  calc
    _ ≃ (p^(step n'))⁻¹ := eqv_refl
    _ ≃ (p^n' * p)⁻¹    := recip_subst pow_step
    _ ≃ (p^n')⁻¹ * p⁻¹  := recip_compat_mul
    _ ≃ (p⁻¹)^n' * p⁻¹  := mul_substL ih
    _ ≃ (p⁻¹)^(step n') := eqv_symm pow_step
ℕ ℤ : Type
inst✝¹³ : Natural ℕ
inst✝¹² : Integer ℤ
ℚ : Type
inst✝¹¹ : Core ℚ
inst✝¹⁰ : Addition ℚ
inst✝⁹ : Multiplication ℚ
inst✝⁸ : Negation ℚ
inst✝⁷ : Subtraction ℚ
inst✝⁶ : Reciprocation ℚ
inst✝⁵ : Division ℚ
inst✝⁴ : Sign ℚ
inst✝³ : Order ℚ
inst✝² : Metric ℚ
inst✝¹ : Natural.Exponentiation ℕ fun x x_1 => x * x_1
p : ℚ
n : ℕ
inst✝ : AP (p ≄ 0)
⊢ ∀ (m : ℕ), (p ^ m)⁻¹ ≃ p⁻¹ ^ m → (p ^ step m)⁻¹ ≃ p⁻¹ ^ step m
no goals