-- Mathlib/RingTheory/IntegralClosure/IntegrallyClosed.lean
theorem pow_dvd_pow_iff [IsDomain R] [IsIntegrallyClosed R]
    {n : ℕ} (hn : n ≠ 0) {a b : R} : a ^ n ∣ b ^ n ↔ a ∣ b

Proof state:
case neg
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : IsIntegrallyClosed R
n : ℕ
hn : n ≠ 0
a b : R
x✝ : a ^ n ∣ b ^ n
x : R
hx : b ^ n = a ^ n * x
ha : ¬a = 0
⊢ a ∣ b

Tactic:
let K := FractionRing R

Result:
case neg
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : IsIntegrallyClosed R
n : ℕ
hn : n ≠ 0
a b : R
x✝ : a ^ n ∣ b ^ n
x : R
hx : b ^ n = a ^ n * x
ha : ¬a = 0
K : Type u_1 := FractionRing R
⊢ a ∣ b

Proof state:
case neg
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : IsIntegrallyClosed R
n : ℕ
hn : n ≠ 0
a b : R
x✝ : a ^ n ∣ b ^ n
x : R
hx : b ^ n = a ^ n * x
K : Type u_1 := FractionRing R
ha : (algebraMap R K) a ≠ 0
y : K := (algebraMap R K) b / (algebraMap R K) a
⊢ a ∣ b

Tactic:
have hy : IsIntegral R y := by
  refine ⟨X ^ n - C x, monic_X_pow_sub_C _ hn, ?_⟩
  simp only [y, map_pow, eval₂_sub, eval₂_X_pow, div_pow, eval₂_pow', eval₂_C]
  replace hx := congr_arg (algebraMap R K) hx
  rw [map_pow] at hx
  field_simp [hx, ha]

Result:
case neg
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : IsIntegrallyClosed R
n : ℕ
hn : n ≠ 0
a b : R
x✝ : a ^ n ∣ b ^ n
x : R
hx : b ^ n = a ^ n * x
K : Type u_1 := FractionRing R
ha : (algebraMap R K) a ≠ 0
y : K := (algebraMap R K) b / (algebraMap R K) a
hy : IsIntegral R y
⊢ a ∣ b

Proof state:
case neg.intro
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : IsIntegrallyClosed R
n : ℕ
hn : n ≠ 0
a b : R
x✝ : a ^ n ∣ b ^ n
x : R
hx : b ^ n = a ^ n * x
K : Type u_1 := FractionRing R
ha : (algebraMap R K) a ≠ 0
y : K := (algebraMap R K) b / (algebraMap R K) a
hy : IsIntegral R y
k : R
hk : (algebraMap R K) k = y
⊢ (algebraMap R K) b = (algebraMap R K) (a * k)

Tactic:
rw [map_mul, hk, mul_div_cancel₀ _ ha]

Result:
no goals

Proof state:
case neg
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : IsIntegrallyClosed R
n : ℕ
hn : n ≠ 0
a b : R
x✝ : a ^ n ∣ b ^ n
x : R
hx : b ^ n = a ^ n * x
K : Type u_1 := FractionRing R
ha : (algebraMap R K) a ≠ 0
⊢ a ∣ b

Tactic:
let y := (algebraMap R K b) / (algebraMap R K a)

Result:
case neg
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : IsIntegrallyClosed R
n : ℕ
hn : n ≠ 0
a b : R
x✝ : a ^ n ∣ b ^ n
x : R
hx : b ^ n = a ^ n * x
K : Type u_1 := FractionRing R
ha : (algebraMap R K) a ≠ 0
y : K := (algebraMap R K) b / (algebraMap R K) a
⊢ a ∣ b

Proof state:
case neg
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : IsIntegrallyClosed R
n : ℕ
hn : n ≠ 0
a b : R
x✝ : a ^ n ∣ b ^ n
x : R
hx : b ^ n = a ^ n * x
ha : ¬a = 0
K : Type u_1 := FractionRing R
⊢ a ∣ b

Tactic:
replace ha : algebraMap R K a ≠ 0 := fun h ↦
  ha <| (injective_iff_map_eq_zero _).1 (IsFractionRing.injective R K) _ h

Result:
case neg
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : IsIntegrallyClosed R
n : ℕ
hn : n ≠ 0
a b : R
x✝ : a ^ n ∣ b ^ n
x : R
hx : b ^ n = a ^ n * x
K : Type u_1 := FractionRing R
ha : (algebraMap R K) a ≠ 0
⊢ a ∣ b

Proof state:
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : IsIntegrallyClosed R
n : ℕ
hn : n ≠ 0
a b : R
x✝ : a ^ n ∣ b ^ n
x : R
hx : b ^ n = a ^ n * x
K : Type u_1 := FractionRing R
ha : (algebraMap R K) a ≠ 0
y : K := (algebraMap R K) b / (algebraMap R K) a
⊢ IsIntegral R y

Tactic:
refine ⟨X ^ n - C x, monic_X_pow_sub_C _ hn, ?_⟩

Result:
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : IsIntegrallyClosed R
n : ℕ
hn : n ≠ 0
a b : R
x✝ : a ^ n ∣ b ^ n
x : R
hx : b ^ n = a ^ n * x
K : Type u_1 := FractionRing R
ha : (algebraMap R K) a ≠ 0
y : K := (algebraMap R K) b / (algebraMap R K) a
⊢ eval₂ (algebraMap R K) y (X ^ n - C x) = 0

Proof state:
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : IsIntegrallyClosed R
n : ℕ
hn : n ≠ 0
a b : R
x✝ : a ^ n ∣ b ^ n
x : R
hx : b ^ n = a ^ n * x
K : Type u_1 := FractionRing R
ha : (algebraMap R K) a ≠ 0
y : K := (algebraMap R K) b / (algebraMap R K) a
⊢ eval₂ (algebraMap R K) y (X ^ n - C x) = 0

Tactic:
simp only [y, map_pow, eval₂_sub, eval₂_X_pow, div_pow, eval₂_pow', eval₂_C]

Result:
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : IsIntegrallyClosed R
n : ℕ
hn : n ≠ 0
a b : R
x✝ : a ^ n ∣ b ^ n
x : R
hx : b ^ n = a ^ n * x
K : Type u_1 := FractionRing R
ha : (algebraMap R K) a ≠ 0
y : K := (algebraMap R K) b / (algebraMap R K) a
⊢ (algebraMap R K) b ^ n / (algebraMap R K) a ^ n - (algebraMap R K) x = 0

Proof state:
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : IsIntegrallyClosed R
n : ℕ
hn : n ≠ 0
a b : R
x✝ : a ^ n ∣ b ^ n
x : R
hx : b ^ n = a ^ n * x
K : Type u_1 := FractionRing R
ha : (algebraMap R K) a ≠ 0
y : K := (algebraMap R K) b / (algebraMap R K) a
⊢ (algebraMap R K) b ^ n / (algebraMap R K) a ^ n - (algebraMap R K) x = 0

Tactic:
replace hx := congr_arg (algebraMap R K) hx

Result:
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : IsIntegrallyClosed R
n : ℕ
hn : n ≠ 0
a b : R
x✝ : a ^ n ∣ b ^ n
x : R
K : Type u_1 := FractionRing R
ha : (algebraMap R K) a ≠ 0
y : K := (algebraMap R K) b / (algebraMap R K) a
hx : (algebraMap R K) (b ^ n) = (algebraMap R K) (a ^ n * x)
⊢ (algebraMap R K) b ^ n / (algebraMap R K) a ^ n - (algebraMap R K) x = 0

Proof state:
case neg
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : IsIntegrallyClosed R
n : ℕ
hn : n ≠ 0
a b : R
x✝ : a ^ n ∣ b ^ n
x : R
hx : b ^ n = a ^ n * x
K : Type u_1 := FractionRing R
ha : (algebraMap R K) a ≠ 0
y : K := (algebraMap R K) b / (algebraMap R K) a
hy : IsIntegral R y
⊢ a ∣ b

Tactic:
obtain ⟨k, hk⟩ := algebraMap_eq_of_integral hy

Result:
case neg.intro
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : IsIntegrallyClosed R
n : ℕ
hn : n ≠ 0
a b : R
x✝ : a ^ n ∣ b ^ n
x : R
hx : b ^ n = a ^ n * x
K : Type u_1 := FractionRing R
ha : (algebraMap R K) a ≠ 0
y : K := (algebraMap R K) b / (algebraMap R K) a
hy : IsIntegral R y
k : R
hk : (algebraMap R K) k = y
⊢ a ∣ b

Proof state:
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : IsIntegrallyClosed R
n : ℕ
hn : n ≠ 0
a b : R
x✝ : a ^ n ∣ b ^ n
x : R
K : Type u_1 := FractionRing R
ha : (algebraMap R K) a ≠ 0
y : K := (algebraMap R K) b / (algebraMap R K) a
hx : (algebraMap R K) b ^ n = (algebraMap R K) (a ^ n * x)
⊢ (algebraMap R K) b ^ n / (algebraMap R K) a ^ n - (algebraMap R K) x = 0

Tactic:
field_simp [hx, ha]

Result:
no goals

Proof state:
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : IsIntegrallyClosed R
n : ℕ
hn : n ≠ 0
a b : R
x✝ : a ^ n ∣ b ^ n
x : R
K : Type u_1 := FractionRing R
ha : (algebraMap R K) a ≠ 0
y : K := (algebraMap R K) b / (algebraMap R K) a
hx : (algebraMap R K) (b ^ n) = (algebraMap R K) (a ^ n * x)
⊢ (algebraMap R K) b ^ n / (algebraMap R K) a ^ n - (algebraMap R K) x = 0

Tactic:
rw [map_pow] at hx

Result:
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : IsIntegrallyClosed R
n : ℕ
hn : n ≠ 0
a b : R
x✝ : a ^ n ∣ b ^ n
x : R
K : Type u_1 := FractionRing R
ha : (algebraMap R K) a ≠ 0
y : K := (algebraMap R K) b / (algebraMap R K) a
hx : (algebraMap R K) b ^ n = (algebraMap R K) (a ^ n * x)
⊢ (algebraMap R K) b ^ n / (algebraMap R K) a ^ n - (algebraMap R K) x = 0

Proof state:
case pos
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : IsIntegrallyClosed R
n : ℕ
hn : n ≠ 0
a b : R
x✝ : a ^ n ∣ b ^ n
x : R
hx : b ^ n = a ^ n * x
ha : a = 0
⊢ a ∣ b

Tactic:
simpa [ha, hn] using hx

Result:
no goals

Proof state:
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : IsIntegrallyClosed R
n : ℕ
hn : n ≠ 0
a b : R
⊢ a ^ n ∣ b ^ n ↔ a ∣ b

Tactic:
refine ⟨fun ⟨x, hx⟩ ↦ ?_, fun h ↦ pow_dvd_pow_of_dvd h n⟩

Result:
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : IsIntegrallyClosed R
n : ℕ
hn : n ≠ 0
a b : R
x✝ : a ^ n ∣ b ^ n
x : R
hx : b ^ n = a ^ n * x
⊢ a ∣ b