calc
  _ = p^(a + b)                         := rfl
  _ ≃ p^(((n + k : ℕ):ℤ) - (m + j : ℕ)) := pow_substR ‹a+b ≃ (n+k:ℕ)-(m+j:ℕ)›
  _ ≃ p^(n + k) / p^(m + j)             := pow_diff
  _ ≃ (p^n * p^k) / p^(m + j)           := div_substL Natural.pow_compatL_add
  _ ≃ (p^n * p^k) / (p^m * p^j)         := div_substR Natural.pow_compatL_add
  _ ≃ (p^n / p^m) * (p^k / p^j)         := Rel.symm div_mul_swap
  _ ≃ p^a * (p^k / p^j)                 := mul_substL (pow_lift ‹a ≃ n - m›)
  _ ≃ p^a * p^b                         := mul_substR (pow_lift ‹b ≃ k - j›)
ℕ ℤ : Type
inst✝¹¹ : Natural ℕ
inst✝¹⁰ : Integer ℤ
ℚ : Type
inst✝⁹ : Core ℚ
inst✝⁸ : Addition ℚ
inst✝⁷ : Multiplication ℚ
inst✝⁶ : Negation ℚ
inst✝⁵ : Reciprocation ℚ
inst✝⁴ : Division ℚ
inst✝³ : Sign ℚ
inst✝² : Natural.Exponentiation ℕ fun x x_1 => x * x_1
inst✝¹ : Exponentiation ℚ
p : ℚ
inst✝ : AP (p ≄ 0)
a b : ℤ
n m : ℕ
h✝¹ : a ≃ Coe.coe n - Coe.coe m
k j : ℕ
h✝ : b ≃ Coe.coe k - Coe.coe j
this : a + b ≃ Coe.coe (n + k) - Coe.coe (m + j)
pow_lift : ∀ {x y : ℕ} {z : ℤ}, z ≃ Coe.coe x - Coe.coe y → p ^ x / p ^ y ≃ p ^ z
⊢ p ^ (a + b) ≃ p ^ a * p ^ b
no goals