Theorem: Rat.lift_binop_eq
File: Mathlib/Data/Rat/Defs.lean

Proof steps:
state_before: q : ℚ
f : ℚ → ℚ → ℚ
f₁ f₂ : ℤ → ℤ → ℤ → ℤ → ℤ
fv :
  ∀ {n₁ : ℤ} {d₁ : ℕ} {h₁ : d₁ ≠ 0} {c₁ : n₁.natAbs.Coprime d₁} {n₂ : ℤ} {d₂ : ℕ} {h₂ : d₂ ≠ 0}
    {c₂ : n₂.natAbs.Coprime d₂},
    f { num := n₁, den := d₁, den_nz := h₁, reduced := c₁ } { num := n₂, den := d₂, den_nz := h₂, reduced := c₂ } =
      f₁ n₁ (↑d₁) n₂ ↑d₂ /. f₂ n₁ (↑d₁) n₂ ↑d₂
f0 : ∀ {n₁ d₁ n₂ d₂ : ℤ}, d₁ ≠ 0 → d₂ ≠ 0 → f₂ n₁ d₁ n₂ d₂ ≠ 0
a b c d : ℤ
b0 : b ≠ 0
d0 : d ≠ 0
H : ∀ {n₁ d₁ n₂ d₂ : ℤ}, a * d₁ = n₁ * b → c * d₂ = n₂ * d → f₁ n₁ d₁ n₂ d₂ * f₂ a b c d = f₁ a b c d * f₂ n₁ d₁ n₂ d₂
⊢ f (a /. b) (c /. d) = f₁ a b c d /. f₂ a b c d
state_after: q : ℚ
f : ℚ → ℚ → ℚ
f₁ f₂ : ℤ → ℤ → ℤ → ℤ → ℤ
fv :
  ∀ {n₁ : ℤ} {d₁ : ℕ} {h₁ : d₁ ≠ 0} {c₁ : n₁.natAbs.Coprime d₁} {n₂ : ℤ} {d₂ : ℕ} {h₂ : d₂ ≠ 0}
    {c₂ : n₂.natAbs.Coprime d₂},
    f { num := n₁, den := d₁, den_nz := h₁, reduced := c₁ } { num := n₂, den := d₂, den_nz := h₂, reduced := c₂ } =
      f₁ n₁ (↑d₁) n₂ ↑d₂ /. f₂ n₁ (↑d₁) n₂ ↑d₂
f0 : ∀ {n₁ d₁ n₂ d₂ : ℤ}, d₁ ≠ 0 → d₂ ≠ 0 → f₂ n₁ d₁ n₂ d₂ ≠ 0
a b c d : ℤ
b0 : b ≠ 0
d0 : d ≠ 0
H : ∀ {n₁ d₁ n₂ d₂ : ℤ}, a * d₁ = n₁ * b → c * d₂ = n₂ * d → f₁ n₁ d₁ n₂ d₂ * f₂ a b c d = f₁ a b c d * f₂ n₁ d₁ n₂ d₂
x : ℚ
ha : a /. b = x
⊢ f x (c /. d) = f₁ a b c d /. f₂ a b c d
tactic: generalize ha : a /. b = x
state_before: q : ℚ
f : ℚ → ℚ → ℚ
f₁ f₂ : ℤ → ℤ → ℤ → ℤ → ℤ
fv :
  ∀ {n₁ : ℤ} {d₁ : ℕ} {h₁ : d₁ ≠ 0} {c₁ : n₁.natAbs.Coprime d₁} {n₂ : ℤ} {d₂ : ℕ} {h₂ : d₂ ≠ 0}
    {c₂ : n₂.natAbs.Coprime d₂},
    f { num := n₁, den := d₁, den_nz := h₁, reduced := c₁ } { num := n₂, den := d₂, den_nz := h₂, reduced := c₂ } =
      f₁ n₁ (↑d₁) n₂ ↑d₂ /. f₂ n₁ (↑d₁) n₂ ↑d₂
f0 : ∀ {n₁ d₁ n₂ d₂ : ℤ}, d₁ ≠ 0 → d₂ ≠ 0 → f₂ n₁ d₁ n₂ d₂ ≠ 0
a b c d : ℤ
b0 : b ≠ 0
d0 : d ≠ 0
H : ∀ {n₁ d₁ n₂ d₂ : ℤ}, a * d₁ = n₁ * b → c * d₂ = n₂ * d → f₁ n₁ d₁ n₂ d₂ * f₂ a b c d = f₁ a b c d * f₂ n₁ d₁ n₂ d₂
x : ℚ
ha : a /. b = x
⊢ f x (c /. d) = f₁ a b c d /. f₂ a b c d
state_after: case mk'
q : ℚ
f : ℚ → ℚ → ℚ
f₁ f₂ : ℤ → ℤ → ℤ → ℤ → ℤ
fv :
  ∀ {n₁ : ℤ} {d₁ : ℕ} {h₁ : d₁ ≠ 0} {c₁ : n₁.natAbs.Coprime d₁} {n₂ : ℤ} {d₂ : ℕ} {h₂ : d₂ ≠ 0}
    {c₂ : n₂.natAbs.Coprime d₂},
    f { num := n₁, den := d₁, den_nz := h₁, reduced := c₁ } { num := n₂, den := d₂, den_nz := h₂, reduced := c₂ } =
      f₁ n₁ (↑d₁) n₂ ↑d₂ /. f₂ n₁ (↑d₁) n₂ ↑d₂
f0 : ∀ {n₁ d₁ n₂ d₂ : ℤ}, d₁ ≠ 0 → d₂ ≠ 0 → f₂ n₁ d₁ n₂ d₂ ≠ 0
a b c d : ℤ
b0 : b ≠ 0
d0 : d ≠ 0
H : ∀ {n₁ d₁ n₂ d₂ : ℤ}, a * d₁ = n₁ * b → c * d₂ = n₂ * d → f₁ n₁ d₁ n₂ d₂ * f₂ a b c d = f₁ a b c d * f₂ n₁ d₁ n₂ d₂
n₁ : ℤ
d₁ : ℕ
h₁ : d₁ ≠ 0
c₁ : n₁.natAbs.Coprime d₁
ha : a /. b = { num := n₁, den := d₁, den_nz := h₁, reduced := c₁ }
⊢ f { num := n₁, den := d₁, den_nz := h₁, reduced := c₁ } (c /. d) = f₁ a b c d /. f₂ a b c d
tactic: cases' x with n₁ d₁ h₁ c₁
state_before: case mk'
q : ℚ
f : ℚ → ℚ → ℚ
f₁ f₂ : ℤ → ℤ → ℤ → ℤ → ℤ
fv :
  ∀ {n₁ : ℤ} {d₁ : ℕ} {h₁ : d₁ ≠ 0} {c₁ : n₁.natAbs.Coprime d₁} {n₂ : ℤ} {d₂ : ℕ} {h₂ : d₂ ≠ 0}
    {c₂ : n₂.natAbs.Coprime d₂},
    f { num := n₁, den := d₁, den_nz := h₁, reduced := c₁ } { num := n₂, den := d₂, den_nz := h₂, reduced := c₂ } =
      f₁ n₁ (↑d₁) n₂ ↑d₂ /. f₂ n₁ (↑d₁) n₂ ↑d₂
f0 : ∀ {n₁ d₁ n₂ d₂ : ℤ}, d₁ ≠ 0 → d₂ ≠ 0 → f₂ n₁ d₁ n₂ d₂ ≠ 0
a b c d : ℤ
b0 : b ≠ 0
d0 : d ≠ 0
H : ∀ {n₁ d₁ n₂ d₂ : ℤ}, a * d₁ = n₁ * b → c * d₂ = n₂ * d → f₁ n₁ d₁ n₂ d₂ * f₂ a b c d = f₁ a b c d * f₂ n₁ d₁ n₂ d₂
n₁ : ℤ
d₁ : ℕ
h₁ : d₁ ≠ 0
c₁ : n₁.natAbs.Coprime d₁
ha : a /. b = { num := n₁, den := d₁, den_nz := h₁, reduced := c₁ }
⊢ f { num := n₁, den := d₁, den_nz := h₁, reduced := c₁ } (c /. d) = f₁ a b c d /. f₂ a b c d
state_after: case mk'
q : ℚ
f : ℚ → ℚ → ℚ
f₁ f₂ : ℤ → ℤ → ℤ → ℤ → ℤ
fv :
  ∀ {n₁ : ℤ} {d₁ : ℕ} {h₁ : d₁ ≠ 0} {c₁ : n₁.natAbs.Coprime d₁} {n₂ : ℤ} {d₂ : ℕ} {h₂ : d₂ ≠ 0}
    {c₂ : n₂.natAbs.Coprime d₂},
    f { num := n₁, den := d₁, den_nz := h₁, reduced := c₁ } { num := n₂, den := d₂, den_nz := h₂, reduced := c₂ } =
      f₁ n₁ (↑d₁) n₂ ↑d₂ /. f₂ n₁ (↑d₁) n₂ ↑d₂
f0 : ∀ {n₁ d₁ n₂ d₂ : ℤ}, d₁ ≠ 0 → d₂ ≠ 0 → f₂ n₁ d₁ n₂ d₂ ≠ 0
a b c d : ℤ
b0 : b ≠ 0
d0 : d ≠ 0
H : ∀ {n₁ d₁ n₂ d₂ : ℤ}, a * d₁ = n₁ * b → c * d₂ = n₂ * d → f₁ n₁ d₁ n₂ d₂ * f₂ a b c d = f₁ a b c d * f₂ n₁ d₁ n₂ d₂
n₁ : ℤ
d₁ : ℕ
h₁ : d₁ ≠ 0
c₁ : n₁.natAbs.Coprime d₁
ha : a /. b = n₁ /. ↑d₁
⊢ f { num := n₁, den := d₁, den_nz := h₁, reduced := c₁ } (c /. d) = f₁ a b c d /. f₂ a b c d
tactic: rw [mk'_eq_divInt] at ha
state_before: case mk'
q : ℚ
f : ℚ → ℚ → ℚ
f₁ f₂ : ℤ → ℤ → ℤ → ℤ → ℤ
fv :
  ∀ {n₁ : ℤ} {d₁ : ℕ} {h₁ : d₁ ≠ 0} {c₁ : n₁.natAbs.Coprime d₁} {n₂ : ℤ} {d₂ : ℕ} {h₂ : d₂ ≠ 0}
    {c₂ : n₂.natAbs.Coprime d₂},
    f { num := n₁, den := d₁, den_nz := h₁, reduced := c₁ } { num := n₂, den := d₂, den_nz := h₂, reduced := c₂ } =
      f₁ n₁ (↑d₁) n₂ ↑d₂ /. f₂ n₁ (↑d₁) n₂ ↑d₂
f0 : ∀ {n₁ d₁ n₂ d₂ : ℤ}, d₁ ≠ 0 → d₂ ≠ 0 → f₂ n₁ d₁ n₂ d₂ ≠ 0
a b c d : ℤ
b0 : b ≠ 0
d0 : d ≠ 0
H : ∀ {n₁ d₁ n₂ d₂ : ℤ}, a * d₁ = n₁ * b → c * d₂ = n₂ * d → f₁ n₁ d₁ n₂ d₂ * f₂ a b c d = f₁ a b c d * f₂ n₁ d₁ n₂ d₂
n₁ : ℤ
d₁ : ℕ
h₁ : d₁ ≠ 0
c₁ : n₁.natAbs.Coprime d₁
ha : a /. b = n₁ /. ↑d₁
⊢ f { num := n₁, den := d₁, den_nz := h₁, reduced := c₁ } (c /. d) = f₁ a b c d /. f₂ a b c d
state_after: case mk'
q : ℚ
f : ℚ → ℚ → ℚ
f₁ f₂ : ℤ → ℤ → ℤ → ℤ → ℤ
fv :
  ∀ {n₁ : ℤ} {d₁ : ℕ} {h₁ : d₁ ≠ 0} {c₁ : n₁.natAbs.Coprime d₁} {n₂ : ℤ} {d₂ : ℕ} {h₂ : d₂ ≠ 0}
    {c₂ : n₂.natAbs.Coprime d₂},
    f { num := n₁, den := d₁, den_nz := h₁, reduced := c₁ } { num := n₂, den := d₂, den_nz := h₂, reduced := c₂ } =
      f₁ n₁ (↑d₁) n₂ ↑d₂ /. f₂ n₁ (↑d₁) n₂ ↑d₂
f0 : ∀ {n₁ d₁ n₂ d₂ : ℤ}, d₁ ≠ 0 → d₂ ≠ 0 → f₂ n₁ d₁ n₂ d₂ ≠ 0
a b c d : ℤ
b0 : b ≠ 0
d0 : d ≠ 0
H : ∀ {n₁ d₁ n₂ d₂ : ℤ}, a * d₁ = n₁ * b → c * d₂ = n₂ * d → f₁ n₁ d₁ n₂ d₂ * f₂ a b c d = f₁ a b c d * f₂ n₁ d₁ n₂ d₂
n₁ : ℤ
d₁ : ℕ
h₁ : d₁ ≠ 0
c₁ : n₁.natAbs.Coprime d₁
ha : a /. b = n₁ /. ↑d₁
x : ℚ
hc : c /. d = x
⊢ f { num := n₁, den := d₁, den_nz := h₁, reduced := c₁ } x = f₁ a b c d /. f₂ a b c d
tactic: generalize hc : c /. d = x
state_before: case mk'
q : ℚ
f : ℚ → ℚ → ℚ
f₁ f₂ : ℤ → ℤ → ℤ → ℤ → ℤ
fv :
  ∀ {n₁ : ℤ} {d₁ : ℕ} {h₁ : d₁ ≠ 0} {c₁ : n₁.natAbs.Coprime d₁} {n₂ : ℤ} {d₂ : ℕ} {h₂ : d₂ ≠ 0}
    {c₂ : n₂.natAbs.Coprime d₂},
    f { num := n₁, den := d₁, den_nz := h₁, reduced := c₁ } { num := n₂, den := d₂, den_nz := h₂, reduced := c₂ } =
      f₁ n₁ (↑d₁) n₂ ↑d₂ /. f₂ n₁ (↑d₁) n₂ ↑d₂
f0 : ∀ {n₁ d₁ n₂ d₂ : ℤ}, d₁ ≠ 0 → d₂ ≠ 0 → f₂ n₁ d₁ n₂ d₂ ≠ 0
a b c d : ℤ
b0 : b ≠ 0
d0 : d ≠ 0
H : ∀ {n₁ d₁ n₂ d₂ : ℤ}, a * d₁ = n₁ * b → c * d₂ = n₂ * d → f₁ n₁ d₁ n₂ d₂ * f₂ a b c d = f₁ a b c d * f₂ n₁ d₁ n₂ d₂
n₁ : ℤ
d₁ : ℕ
h₁ : d₁ ≠ 0
c₁ : n₁.natAbs.Coprime d₁
ha : a /. b = n₁ /. ↑d₁
x : ℚ
hc : c /. d = x
⊢ f { num := n₁, den := d₁, den_nz := h₁, reduced := c₁ } x = f₁ a b c d /. f₂ a b c d
state_after: case mk'.mk'
q : ℚ
f : ℚ → ℚ → ℚ
f₁ f₂ : ℤ → ℤ → ℤ → ℤ → ℤ
fv :
  ∀ {n₁ : ℤ} {d₁ : ℕ} {h₁ : d₁ ≠ 0} {c₁ : n₁.natAbs.Coprime d₁} {n₂ : ℤ} {d₂ : ℕ} {h₂ : d₂ ≠ 0}
    {c₂ : n₂.natAbs.Coprime d₂},
    f { num := n₁, den := d₁, den_nz := h₁, reduced := c₁ } { num := n₂, den := d₂, den_nz := h₂, reduced := c₂ } =
      f₁ n₁ (↑d₁) n₂ ↑d₂ /. f₂ n₁ (↑d₁) n₂ ↑d₂
f0 : ∀ {n₁ d₁ n₂ d₂ : ℤ}, d₁ ≠ 0 → d₂ ≠ 0 → f₂ n₁ d₁ n₂ d₂ ≠ 0
a b c d : ℤ
b0 : b ≠ 0
d0 : d ≠ 0
H : ∀ {n₁ d₁ n₂ d₂ : ℤ}, a * d₁ = n₁ * b → c * d₂ = n₂ * d → f₁ n₁ d₁ n₂ d₂ * f₂ a b c d = f₁ a b c d * f₂ n₁ d₁ n₂ d₂
n₁ : ℤ
d₁ : ℕ
h₁ : d₁ ≠ 0
c₁ : n₁.natAbs.Coprime d₁
ha : a /. b = n₁ /. ↑d₁
n₂ : ℤ
d₂ : ℕ
h₂ : d₂ ≠ 0
c₂ : n₂.natAbs.Coprime d₂
hc : c /. d = { num := n₂, den := d₂, den_nz := h₂, reduced := c₂ }
⊢ f { num := n₁, den := d₁, den_nz := h₁, reduced := c₁ } { num := n₂, den := d₂, den_nz := h₂, reduced := c₂ } =
    f₁ a b c d /. f₂ a b c d
tactic: cases' x with n₂ d₂ h₂ c₂
state_before: case mk'.mk'
q : ℚ
f : ℚ → ℚ → ℚ
f₁ f₂ : ℤ → ℤ → ℤ → ℤ → ℤ
fv :
  ∀ {n₁ : ℤ} {d₁ : ℕ} {h₁ : d₁ ≠ 0} {c₁ : n₁.natAbs.Coprime d₁} {n₂ : ℤ} {d₂ : ℕ} {h₂ : d₂ ≠ 0}
    {c₂ : n₂.natAbs.Coprime d₂},
    f { num := n₁, den := d₁, den_nz := h₁, reduced := c₁ } { num := n₂, den := d₂, den_nz := h₂, reduced := c₂ } =
      f₁ n₁ (↑d₁) n₂ ↑d₂ /. f₂ n₁ (↑d₁) n₂ ↑d₂
f0 : ∀ {n₁ d₁ n₂ d₂ : ℤ}, d₁ ≠ 0 → d₂ ≠ 0 → f₂ n₁ d₁ n₂ d₂ ≠ 0
a b c d : ℤ
b0 : b ≠ 0
d0 : d ≠ 0
H : ∀ {n₁ d₁ n₂ d₂ : ℤ}, a * d₁ = n₁ * b → c * d₂ = n₂ * d → f₁ n₁ d₁ n₂ d₂ * f₂ a b c d = f₁ a b c d * f₂ n₁ d₁ n₂ d₂
n₁ : ℤ
d₁ : ℕ
h₁ : d₁ ≠ 0
c₁ : n₁.natAbs.Coprime d₁
ha : a /. b = n₁ /. ↑d₁
n₂ : ℤ
d₂ : ℕ
h₂ : d₂ ≠ 0
c₂ : n₂.natAbs.Coprime d₂
hc : c /. d = { num := n₂, den := d₂, den_nz := h₂, reduced := c₂ }
⊢ f { num := n₁, den := d₁, den_nz := h₁, reduced := c₁ } { num := n₂, den := d₂, den_nz := h₂, reduced := c₂ } =
    f₁ a b c d /. f₂ a b c d
state_after: case mk'.mk'
q : ℚ
f : ℚ → ℚ → ℚ
f₁ f₂ : ℤ → ℤ → ℤ → ℤ → ℤ
fv :
  ∀ {n₁ : ℤ} {d₁ : ℕ} {h₁ : d₁ ≠ 0} {c₁ : n₁.natAbs.Coprime d₁} {n₂ : ℤ} {d₂ : ℕ} {h₂ : d₂ ≠ 0}
    {c₂ : n₂.natAbs.Coprime d₂},
    f { num := n₁, den := d₁, den_nz := h₁, reduced := c₁ } { num := n₂, den := d₂, den_nz := h₂, reduced := c₂ } =
      f₁ n₁ (↑d₁) n₂ ↑d₂ /. f₂ n₁ (↑d₁) n₂ ↑d₂
f0 : ∀ {n₁ d₁ n₂ d₂ : ℤ}, d₁ ≠ 0 → d₂ ≠ 0 → f₂ n₁ d₁ n₂ d₂ ≠ 0
a b c d : ℤ
b0 : b ≠ 0
d0 : d ≠ 0
H : ∀ {n₁ d₁ n₂ d₂ : ℤ}, a * d₁ = n₁ * b → c * d₂ = n₂ * d → f₁ n₁ d₁ n₂ d₂ * f₂ a b c d = f₁ a b c d * f₂ n₁ d₁ n₂ d₂
n₁ : ℤ
d₁ : ℕ
h₁ : d₁ ≠ 0
c₁ : n₁.natAbs.Coprime d₁
ha : a /. b = n₁ /. ↑d₁
n₂ : ℤ
d₂ : ℕ
h₂ : d₂ ≠ 0
c₂ : n₂.natAbs.Coprime d₂
hc : c /. d = n₂ /. ↑d₂
⊢ f { num := n₁, den := d₁, den_nz := h₁, reduced := c₁ } { num := n₂, den := d₂, den_nz := h₂, reduced := c₂ } =
    f₁ a b c d /. f₂ a b c d
tactic: rw [mk'_eq_divInt] at hc
state_before: case mk'.mk'
q : ℚ
f : ℚ → ℚ → ℚ
f₁ f₂ : ℤ → ℤ → ℤ → ℤ → ℤ
fv :
  ∀ {n₁ : ℤ} {d₁ : ℕ} {h₁ : d₁ ≠ 0} {c₁ : n₁.natAbs.Coprime d₁} {n₂ : ℤ} {d₂ : ℕ} {h₂ : d₂ ≠ 0}
    {c₂ : n₂.natAbs.Coprime d₂},
    f { num := n₁, den := d₁, den_nz := h₁, reduced := c₁ } { num := n₂, den := d₂, den_nz := h₂, reduced := c₂ } =
      f₁ n₁ (↑d₁) n₂ ↑d₂ /. f₂ n₁ (↑d₁) n₂ ↑d₂
f0 : ∀ {n₁ d₁ n₂ d₂ : ℤ}, d₁ ≠ 0 → d₂ ≠ 0 → f₂ n₁ d₁ n₂ d₂ ≠ 0
a b c d : ℤ
b0 : b ≠ 0
d0 : d ≠ 0
H : ∀ {n₁ d₁ n₂ d₂ : ℤ}, a * d₁ = n₁ * b → c * d₂ = n₂ * d → f₁ n₁ d₁ n₂ d₂ * f₂ a b c d = f₁ a b c d * f₂ n₁ d₁ n₂ d₂
n₁ : ℤ
d₁ : ℕ
h₁ : d₁ ≠ 0
c₁ : n₁.natAbs.Coprime d₁
ha : a /. b = n₁ /. ↑d₁
n₂ : ℤ
d₂ : ℕ
h₂ : d₂ ≠ 0
c₂ : n₂.natAbs.Coprime d₂
hc : c /. d = n₂ /. ↑d₂
⊢ f { num := n₁, den := d₁, den_nz := h₁, reduced := c₁ } { num := n₂, den := d₂, den_nz := h₂, reduced := c₂ } =
    f₁ a b c d /. f₂ a b c d
state_after: case mk'.mk'
q : ℚ
f : ℚ → ℚ → ℚ
f₁ f₂ : ℤ → ℤ → ℤ → ℤ → ℤ
fv :
  ∀ {n₁ : ℤ} {d₁ : ℕ} {h₁ : d₁ ≠ 0} {c₁ : n₁.natAbs.Coprime d₁} {n₂ : ℤ} {d₂ : ℕ} {h₂ : d₂ ≠ 0}
    {c₂ : n₂.natAbs.Coprime d₂},
    f { num := n₁, den := d₁, den_nz := h₁, reduced := c₁ } { num := n₂, den := d₂, den_nz := h₂, reduced := c₂ } =
      f₁ n₁ (↑d₁) n₂ ↑d₂ /. f₂ n₁ (↑d₁) n₂ ↑d₂
f0 : ∀ {n₁ d₁ n₂ d₂ : ℤ}, d₁ ≠ 0 → d₂ ≠ 0 → f₂ n₁ d₁ n₂ d₂ ≠ 0
a b c d : ℤ
b0 : b ≠ 0
d0 : d ≠ 0
H : ∀ {n₁ d₁ n₂ d₂ : ℤ}, a * d₁ = n₁ * b → c * d₂ = n₂ * d → f₁ n₁ d₁ n₂ d₂ * f₂ a b c d = f₁ a b c d * f₂ n₁ d₁ n₂ d₂
n₁ : ℤ
d₁ : ℕ
h₁ : d₁ ≠ 0
c₁ : n₁.natAbs.Coprime d₁
ha : a /. b = n₁ /. ↑d₁
n₂ : ℤ
d₂ : ℕ
h₂ : d₂ ≠ 0
c₂ : n₂.natAbs.Coprime d₂
hc : c /. d = n₂ /. ↑d₂
⊢ f₁ n₁ (↑d₁) n₂ ↑d₂ /. f₂ n₁ (↑d₁) n₂ ↑d₂ = f₁ a b c d /. f₂ a b c d
tactic: rw [fv]
state_before: case mk'.mk'
q : ℚ
f : ℚ → ℚ → ℚ
f₁ f₂ : ℤ → ℤ → ℤ → ℤ → ℤ
fv :
  ∀ {n₁ : ℤ} {d₁ : ℕ} {h₁ : d₁ ≠ 0} {c₁ : n₁.natAbs.Coprime d₁} {n₂ : ℤ} {d₂ : ℕ} {h₂ : d₂ ≠ 0}
    {c₂ : n₂.natAbs.Coprime d₂},
    f { num := n₁, den := d₁, den_nz := h₁, reduced := c₁ } { num := n₂, den := d₂, den_nz := h₂, reduced := c₂ } =
      f₁ n₁ (↑d₁) n₂ ↑d₂ /. f₂ n₁ (↑d₁) n₂ ↑d₂
f0 : ∀ {n₁ d₁ n₂ d₂ : ℤ}, d₁ ≠ 0 → d₂ ≠ 0 → f₂ n₁ d₁ n₂ d₂ ≠ 0
a b c d : ℤ
b0 : b ≠ 0
d0 : d ≠ 0
H : ∀ {n₁ d₁ n₂ d₂ : ℤ}, a * d₁ = n₁ * b → c * d₂ = n₂ * d → f₁ n₁ d₁ n₂ d₂ * f₂ a b c d = f₁ a b c d * f₂ n₁ d₁ n₂ d₂
n₁ : ℤ
d₁ : ℕ
h₁ : d₁ ≠ 0
c₁ : n₁.natAbs.Coprime d₁
ha : a /. b = n₁ /. ↑d₁
n₂ : ℤ
d₂ : ℕ
h₂ : d₂ ≠ 0
c₂ : n₂.natAbs.Coprime d₂
hc : c /. d = n₂ /. ↑d₂
⊢ f₁ n₁ (↑d₁) n₂ ↑d₂ /. f₂ n₁ (↑d₁) n₂ ↑d₂ = f₁ a b c d /. f₂ a b c d
state_after: case mk'.mk'
q : ℚ
f : ℚ → ℚ → ℚ
f₁ f₂ : ℤ → ℤ → ℤ → ℤ → ℤ
fv :
  ∀ {n₁ : ℤ} {d₁ : ℕ} {h₁ : d₁ ≠ 0} {c₁ : n₁.natAbs.Coprime d₁} {n₂ : ℤ} {d₂ : ℕ} {h₂ : d₂ ≠ 0}
    {c₂ : n₂.natAbs.Coprime d₂},
    f { num := n₁, den := d₁, den_nz := h₁, reduced := c₁ } { num := n₂, den := d₂, den_nz := h₂, reduced := c₂ } =
      f₁ n₁ (↑d₁) n₂ ↑d₂ /. f₂ n₁ (↑d₁) n₂ ↑d₂
f0 : ∀ {n₁ d₁ n₂ d₂ : ℤ}, d₁ ≠ 0 → d₂ ≠ 0 → f₂ n₁ d₁ n₂ d₂ ≠ 0
a b c d : ℤ
b0 : b ≠ 0
d0 : d ≠ 0
H : ∀ {n₁ d₁ n₂ d₂ : ℤ}, a * d₁ = n₁ * b → c * d₂ = n₂ * d → f₁ n₁ d₁ n₂ d₂ * f₂ a b c d = f₁ a b c d * f₂ n₁ d₁ n₂ d₂
n₁ : ℤ
d₁ : ℕ
h₁ : d₁ ≠ 0
c₁ : n₁.natAbs.Coprime d₁
ha : a /. b = n₁ /. ↑d₁
n₂ : ℤ
d₂ : ℕ
h₂ : d₂ ≠ 0
c₂ : n₂.natAbs.Coprime d₂
hc : c /. d = n₂ /. ↑d₂
d₁0 : ↑d₁ ≠ 0
⊢ f₁ n₁ (↑d₁) n₂ ↑d₂ /. f₂ n₁ (↑d₁) n₂ ↑d₂ = f₁ a b c d /. f₂ a b c d
tactic: have d₁0 := Int.ofNat_ne_zero.2 h₁
state_before: case mk'.mk'
q : ℚ
f : ℚ → ℚ → ℚ
f₁ f₂ : ℤ → ℤ → ℤ → ℤ → ℤ
fv :
  ∀ {n₁ : ℤ} {d₁ : ℕ} {h₁ : d₁ ≠ 0} {c₁ : n₁.natAbs.Coprime d₁} {n₂ : ℤ} {d₂ : ℕ} {h₂ : d₂ ≠ 0}
    {c₂ : n₂.natAbs.Coprime d₂},
    f { num := n₁, den := d₁, den_nz := h₁, reduced := c₁ } { num := n₂, den := d₂, den_nz := h₂, reduced := c₂ } =
      f₁ n₁ (↑d₁) n₂ ↑d₂ /. f₂ n₁ (↑d₁) n₂ ↑d₂
f0 : ∀ {n₁ d₁ n₂ d₂ : ℤ}, d₁ ≠ 0 → d₂ ≠ 0 → f₂ n₁ d₁ n₂ d₂ ≠ 0
a b c d : ℤ
b0 : b ≠ 0
d0 : d ≠ 0
H : ∀ {n₁ d₁ n₂ d₂ : ℤ}, a * d₁ = n₁ * b → c * d₂ = n₂ * d → f₁ n₁ d₁ n₂ d₂ * f₂ a b c d = f₁ a b c d * f₂ n₁ d₁ n₂ d₂
n₁ : ℤ
d₁ : ℕ
h₁ : d₁ ≠ 0
c₁ : n₁.natAbs.Coprime d₁
ha : a /. b = n₁ /. ↑d₁
n₂ : ℤ
d₂ : ℕ
h₂ : d₂ ≠ 0
c₂ : n₂.natAbs.Coprime d₂
hc : c /. d = n₂ /. ↑d₂
d₁0 : ↑d₁ ≠ 0
⊢ f₁ n₁ (↑d₁) n₂ ↑d₂ /. f₂ n₁ (↑d₁) n₂ ↑d₂ = f₁ a b c d /. f₂ a b c d
state_after: case mk'.mk'
q : ℚ
f : ℚ → ℚ → ℚ
f₁ f₂ : ℤ → ℤ → ℤ → ℤ → ℤ
fv :
  ∀ {n₁ : ℤ} {d₁ : ℕ} {h₁ : d₁ ≠ 0} {c₁ : n₁.natAbs.Coprime d₁} {n₂ : ℤ} {d₂ : ℕ} {h₂ : d₂ ≠ 0}
    {c₂ : n₂.natAbs.Coprime d₂},
    f { num := n₁, den := d₁, den_nz := h₁, reduced := c₁ } { num := n₂, den := d₂, den_nz := h₂, reduced := c₂ } =
      f₁ n₁ (↑d₁) n₂ ↑d₂ /. f₂ n₁ (↑d₁) n₂ ↑d₂
f0 : ∀ {n₁ d₁ n₂ d₂ : ℤ}, d₁ ≠ 0 → d₂ ≠ 0 → f₂ n₁ d₁ n₂ d₂ ≠ 0
a b c d : ℤ
b0 : b ≠ 0
d0 : d ≠ 0
H : ∀ {n₁ d₁ n₂ d₂ : ℤ}, a * d₁ = n₁ * b → c * d₂ = n₂ * d → f₁ n₁ d₁ n₂ d₂ * f₂ a b c d = f₁ a b c d * f₂ n₁ d₁ n₂ d₂
n₁ : ℤ
d₁ : ℕ
h₁ : d₁ ≠ 0
c₁ : n₁.natAbs.Coprime d₁
ha : a /. b = n₁ /. ↑d₁
n₂ : ℤ
d₂ : ℕ
h₂ : d₂ ≠ 0
c₂ : n₂.natAbs.Coprime d₂
hc : c /. d = n₂ /. ↑d₂
d₁0 : ↑d₁ ≠ 0
d₂0 : ↑d₂ ≠ 0
⊢ f₁ n₁ (↑d₁) n₂ ↑d₂ /. f₂ n₁ (↑d₁) n₂ ↑d₂ = f₁ a b c d /. f₂ a b c d
tactic: have d₂0 := Int.ofNat_ne_zero.2 h₂
state_before: case mk'.mk'
q : ℚ
f : ℚ → ℚ → ℚ
f₁ f₂ : ℤ → ℤ → ℤ → ℤ → ℤ
fv :
  ∀ {n₁ : ℤ} {d₁ : ℕ} {h₁ : d₁ ≠ 0} {c₁ : n₁.natAbs.Coprime d₁} {n₂ : ℤ} {d₂ : ℕ} {h₂ : d₂ ≠ 0}
    {c₂ : n₂.natAbs.Coprime d₂},
    f { num := n₁, den := d₁, den_nz := h₁, reduced := c₁ } { num := n₂, den := d₂, den_nz := h₂, reduced := c₂ } =
      f₁ n₁ (↑d₁) n₂ ↑d₂ /. f₂ n₁ (↑d₁) n₂ ↑d₂
f0 : ∀ {n₁ d₁ n₂ d₂ : ℤ}, d₁ ≠ 0 → d₂ ≠ 0 → f₂ n₁ d₁ n₂ d₂ ≠ 0
a b c d : ℤ
b0 : b ≠ 0
d0 : d ≠ 0
H : ∀ {n₁ d₁ n₂ d₂ : ℤ}, a * d₁ = n₁ * b → c * d₂ = n₂ * d → f₁ n₁ d₁ n₂ d₂ * f₂ a b c d = f₁ a b c d * f₂ n₁ d₁ n₂ d₂
n₁ : ℤ
d₁ : ℕ
h₁ : d₁ ≠ 0
c₁ : n₁.natAbs.Coprime d₁
ha : a /. b = n₁ /. ↑d₁
n₂ : ℤ
d₂ : ℕ
h₂ : d₂ ≠ 0
c₂ : n₂.natAbs.Coprime d₂
hc : c /. d = n₂ /. ↑d₂
d₁0 : ↑d₁ ≠ 0
d₂0 : ↑d₂ ≠ 0
⊢ f₁ n₁ (↑d₁) n₂ ↑d₂ /. f₂ n₁ (↑d₁) n₂ ↑d₂ = f₁ a b c d /. f₂ a b c d
state_after: no goals
tactic: exact (divInt_eq_iff (f0 d₁0 d₂0) (f0 b0 d0)).2
    (H ((divInt_eq_iff b0 d₁0).1 ha) ((divInt_eq_iff d0 d₂0).1 hc))