Theorem: Polynomial.IsPrimitive.irreducible_iff_irreducible_map_fraction_map
File: Mathlib/RingTheory/Polynomial/GaussLemma.lean

tactic: refine
  ⟨fun hi => ⟨fun h => hi.not_unit (hp.isUnit_iff_isUnit_map.2 h), fun a b hab => ?_⟩,
    hp.irreducible_of_irreducible_map_of_injective (IsFractionRing.injective R K)⟩
state_before: R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
hp : p.IsPrimitive
⊢ Irreducible p ↔ Irreducible (map (algebraMap R K) p)
state_after: R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
hp : p.IsPrimitive
hi : Irreducible p
a b : K[X]
hab : map (algebraMap R K) p = a * b
⊢ IsUnit a ∨ IsUnit b

tactic: obtain ⟨⟨c, c0⟩, hc⟩ := integerNormalization_map_to_map R⁰ a
state_before: R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
hp : p.IsPrimitive
hi : Irreducible p
a b : K[X]
hab : map (algebraMap R K) p = a * b
⊢ IsUnit a ∨ IsUnit b
state_after: case intro.mk
R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
hp : p.IsPrimitive
hi : Irreducible p
a b : K[X]
hab : map (algebraMap R K) p = a * b
c : R
c0 : c ∈ R⁰
hc : map (algebraMap R K) (integerNormalization R⁰ a) = ↑⟨c, c0⟩ • a
⊢ IsUnit a ∨ IsUnit b

tactic: obtain ⟨⟨d, d0⟩, hd⟩ := integerNormalization_map_to_map R⁰ b
state_before: case intro.mk
R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
hp : p.IsPrimitive
hi : Irreducible p
a b : K[X]
hab : map (algebraMap R K) p = a * b
c : R
c0 : c ∈ R⁰
hc : map (algebraMap R K) (integerNormalization R⁰ a) = ↑⟨c, c0⟩ • a
⊢ IsUnit a ∨ IsUnit b
state_after: case intro.mk.intro.mk
R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
hp : p.IsPrimitive
hi : Irreducible p
a b : K[X]
hab : map (algebraMap R K) p = a * b
c : R
c0 : c ∈ R⁰
hc : map (algebraMap R K) (integerNormalization R⁰ a) = ↑⟨c, c0⟩ • a
d : R
d0 : d ∈ R⁰
hd : map (algebraMap R K) (integerNormalization R⁰ b) = ↑⟨d, d0⟩ • b
⊢ IsUnit a ∨ IsUnit b

tactic: rw [Algebra.smul_def, algebraMap_apply, Subtype.coe_mk] at hc hd
state_before: case intro.mk.intro.mk
R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
hp : p.IsPrimitive
hi : Irreducible p
a b : K[X]
hab : map (algebraMap R K) p = a * b
c : R
c0 : c ∈ R⁰
hc : map (algebraMap R K) (integerNormalization R⁰ a) = ↑⟨c, c0⟩ • a
d : R
d0 : d ∈ R⁰
hd : map (algebraMap R K) (integerNormalization R⁰ b) = ↑⟨d, d0⟩ • b
⊢ IsUnit a ∨ IsUnit b
state_after: case intro.mk.intro.mk
R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
hp : p.IsPrimitive
hi : Irreducible p
a b : K[X]
hab : map (algebraMap R K) p = a * b
c : R
c0 : c ∈ R⁰
hc : map (algebraMap R K) (integerNormalization R⁰ a) = C ((algebraMap R K) c) * a
d : R
d0 : d ∈ R⁰
hd : map (algebraMap R K) (integerNormalization R⁰ b) = C ((algebraMap R K) d) * b
⊢ IsUnit a ∨ IsUnit b

tactic: rw [mem_nonZeroDivisors_iff_ne_zero] at c0 d0
state_before: case intro.mk.intro.mk
R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
hp : p.IsPrimitive
hi : Irreducible p
a b : K[X]
hab : map (algebraMap R K) p = a * b
c : R
c0 : c ∈ R⁰
hc : map (algebraMap R K) (integerNormalization R⁰ a) = C ((algebraMap R K) c) * a
d : R
d0 : d ∈ R⁰
hd : map (algebraMap R K) (integerNormalization R⁰ b) = C ((algebraMap R K) d) * b
⊢ IsUnit a ∨ IsUnit b
state_after: case intro.mk.intro.mk
R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
hp : p.IsPrimitive
hi : Irreducible p
a b : K[X]
hab : map (algebraMap R K) p = a * b
c : R
c0 : c ≠ 0
hc : map (algebraMap R K) (integerNormalization R⁰ a) = C ((algebraMap R K) c) * a
d : R
d0 : d ≠ 0
hd : map (algebraMap R K) (integerNormalization R⁰ b) = C ((algebraMap R K) d) * b
⊢ IsUnit a ∨ IsUnit b

tactic: have hcd0 : c * d ≠ 0 := mul_ne_zero c0 d0
state_before: case intro.mk.intro.mk
R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
hp : p.IsPrimitive
hi : Irreducible p
a b : K[X]
hab : map (algebraMap R K) p = a * b
c : R
c0 : c ≠ 0
hc : map (algebraMap R K) (integerNormalization R⁰ a) = C ((algebraMap R K) c) * a
d : R
d0 : d ≠ 0
hd : map (algebraMap R K) (integerNormalization R⁰ b) = C ((algebraMap R K) d) * b
⊢ IsUnit a ∨ IsUnit b
state_after: case intro.mk.intro.mk
R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
hp : p.IsPrimitive
hi : Irreducible p
a b : K[X]
hab : map (algebraMap R K) p = a * b
c : R
c0 : c ≠ 0
hc : map (algebraMap R K) (integerNormalization R⁰ a) = C ((algebraMap R K) c) * a
d : R
d0 : d ≠ 0
hd : map (algebraMap R K) (integerNormalization R⁰ b) = C ((algebraMap R K) d) * b
hcd0 : c * d ≠ 0
⊢ IsUnit a ∨ IsUnit b

tactic: rw [Ne, ← C_eq_zero] at hcd0
state_before: case intro.mk.intro.mk
R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
hp : p.IsPrimitive
hi : Irreducible p
a b : K[X]
hab : map (algebraMap R K) p = a * b
c : R
c0 : c ≠ 0
hc : map (algebraMap R K) (integerNormalization R⁰ a) = C ((algebraMap R K) c) * a
d : R
d0 : d ≠ 0
hd : map (algebraMap R K) (integerNormalization R⁰ b) = C ((algebraMap R K) d) * b
hcd0 : c * d ≠ 0
⊢ IsUnit a ∨ IsUnit b
state_after: case intro.mk.intro.mk
R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
hp : p.IsPrimitive
hi : Irreducible p
a b : K[X]
hab : map (algebraMap R K) p = a * b
c : R
c0 : c ≠ 0
hc : map (algebraMap R K) (integerNormalization R⁰ a) = C ((algebraMap R K) c) * a
d : R
d0 : d ≠ 0
hd : map (algebraMap R K) (integerNormalization R⁰ b) = C ((algebraMap R K) d) * b
hcd0 : ¬C (c * d) = 0
⊢ IsUnit a ∨ IsUnit b

tactic: have h1 : C c * C d * p = integerNormalization R⁰ a * integerNormalization R⁰ b := by
  apply map_injective (algebraMap R K) (IsFractionRing.injective _ _) _
  rw [Polynomial.map_mul, Polynomial.map_mul, Polynomial.map_mul, hc, hd, map_C, map_C, hab]
  ring
state_before: case intro.mk.intro.mk
R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
hp : p.IsPrimitive
hi : Irreducible p
a b : K[X]
hab : map (algebraMap R K) p = a * b
c : R
c0 : c ≠ 0
hc : map (algebraMap R K) (integerNormalization R⁰ a) = C ((algebraMap R K) c) * a
d : R
d0 : d ≠ 0
hd : map (algebraMap R K) (integerNormalization R⁰ b) = C ((algebraMap R K) d) * b
hcd0 : ¬C (c * d) = 0
⊢ IsUnit a ∨ IsUnit b
state_after: case intro.mk.intro.mk
R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
hp : p.IsPrimitive
hi : Irreducible p
a b : K[X]
hab : map (algebraMap R K) p = a * b
c : R
c0 : c ≠ 0
hc : map (algebraMap R K) (integerNormalization R⁰ a) = C ((algebraMap R K) c) * a
d : R
d0 : d ≠ 0
hd : map (algebraMap R K) (integerNormalization R⁰ b) = C ((algebraMap R K) d) * b
hcd0 : ¬C (c * d) = 0
h1 : C c * C d * p = integerNormalization R⁰ a * integerNormalization R⁰ b
⊢ IsUnit a ∨ IsUnit b

tactic: obtain ⟨u, hu⟩ :
  Associated (c * d)
    (content (integerNormalization R⁰ a) * content (integerNormalization R⁰ b)) := by
  rw [← dvd_dvd_iff_associated, ← normalize_eq_normalize_iff, normalize.map_mul,
    normalize.map_mul, normalize_content, normalize_content, ←
    mul_one (normalize c * normalize d), ← hp.content_eq_one, ← content_C, ← content_C, ←
    content_mul, ← content_mul, ← content_mul, h1]
state_before: case intro.mk.intro.mk
R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
hp : p.IsPrimitive
hi : Irreducible p
a b : K[X]
hab : map (algebraMap R K) p = a * b
c : R
c0 : c ≠ 0
hc : map (algebraMap R K) (integerNormalization R⁰ a) = C ((algebraMap R K) c) * a
d : R
d0 : d ≠ 0
hd : map (algebraMap R K) (integerNormalization R⁰ b) = C ((algebraMap R K) d) * b
hcd0 : ¬C (c * d) = 0
h1 : C c * C d * p = integerNormalization R⁰ a * integerNormalization R⁰ b
⊢ IsUnit a ∨ IsUnit b
state_after: case intro.mk.intro.mk.intro
R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
hp : p.IsPrimitive
hi : Irreducible p
a b : K[X]
hab : map (algebraMap R K) p = a * b
c : R
c0 : c ≠ 0
hc : map (algebraMap R K) (integerNormalization R⁰ a) = C ((algebraMap R K) c) * a
d : R
d0 : d ≠ 0
hd : map (algebraMap R K) (integerNormalization R⁰ b) = C ((algebraMap R K) d) * b
hcd0 : ¬C (c * d) = 0
h1 : C c * C d * p = integerNormalization R⁰ a * integerNormalization R⁰ b
u : Rˣ
hu : c * d * ↑u = (integerNormalization R⁰ a).content * (integerNormalization R⁰ b).content
⊢ IsUnit a ∨ IsUnit b

tactic: rw [← RingHom.map_mul, eq_comm, (integerNormalization R⁰ a).eq_C_content_mul_primPart,
  (integerNormalization R⁰ b).eq_C_content_mul_primPart, mul_assoc, mul_comm _ (C _ * _), ←
  mul_assoc, ← mul_assoc, ← RingHom.map_mul, ← hu, RingHom.map_mul, mul_assoc, mul_assoc, ←
  mul_assoc (C (u : R))] at h1
state_before: case intro.mk.intro.mk.intro
R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
hp : p.IsPrimitive
hi : Irreducible p
a b : K[X]
hab : map (algebraMap R K) p = a * b
c : R
c0 : c ≠ 0
hc : map (algebraMap R K) (integerNormalization R⁰ a) = C ((algebraMap R K) c) * a
d : R
d0 : d ≠ 0
hd : map (algebraMap R K) (integerNormalization R⁰ b) = C ((algebraMap R K) d) * b
hcd0 : ¬C (c * d) = 0
h1 : C c * C d * p = integerNormalization R⁰ a * integerNormalization R⁰ b
u : Rˣ
hu : c * d * ↑u = (integerNormalization R⁰ a).content * (integerNormalization R⁰ b).content
⊢ IsUnit a ∨ IsUnit b
state_after: case intro.mk.intro.mk.intro
R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
hp : p.IsPrimitive
hi : Irreducible p
a b : K[X]
hab : map (algebraMap R K) p = a * b
c : R
c0 : c ≠ 0
hc : map (algebraMap R K) (integerNormalization R⁰ a) = C ((algebraMap R K) c) * a
d : R
d0 : d ≠ 0
hd : map (algebraMap R K) (integerNormalization R⁰ b) = C ((algebraMap R K) d) * b
hcd0 : ¬C (c * d) = 0
u : Rˣ
h1 : C (c * d) * (C ↑u * (integerNormalization R⁰ b).primPart * (integerNormalization R⁰ a).primPart) = C (c * d) * p
hu : c * d * ↑u = (integerNormalization R⁰ a).content * (integerNormalization R⁰ b).content
⊢ IsUnit a ∨ IsUnit b

tactic: have h0 : a ≠ 0 ∧ b ≠ 0 := by
  classical
  rw [Ne, Ne, ← not_or, ← mul_eq_zero, ← hab]
  intro con
  apply hp.ne_zero (map_injective (algebraMap R K) (IsFractionRing.injective _ _) _)
  simp [con]
state_before: case intro.mk.intro.mk.intro
R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
hp : p.IsPrimitive
hi : Irreducible p
a b : K[X]
hab : map (algebraMap R K) p = a * b
c : R
c0 : c ≠ 0
hc : map (algebraMap R K) (integerNormalization R⁰ a) = C ((algebraMap R K) c) * a
d : R
d0 : d ≠ 0
hd : map (algebraMap R K) (integerNormalization R⁰ b) = C ((algebraMap R K) d) * b
hcd0 : ¬C (c * d) = 0
u : Rˣ
h1 : C (c * d) * (C ↑u * (integerNormalization R⁰ b).primPart * (integerNormalization R⁰ a).primPart) = C (c * d) * p
hu : c * d * ↑u = (integerNormalization R⁰ a).content * (integerNormalization R⁰ b).content
⊢ IsUnit a ∨ IsUnit b
state_after: case intro.mk.intro.mk.intro
R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
hp : p.IsPrimitive
hi : Irreducible p
a b : K[X]
hab : map (algebraMap R K) p = a * b
c : R
c0 : c ≠ 0
hc : map (algebraMap R K) (integerNormalization R⁰ a) = C ((algebraMap R K) c) * a
d : R
d0 : d ≠ 0
hd : map (algebraMap R K) (integerNormalization R⁰ b) = C ((algebraMap R K) d) * b
hcd0 : ¬C (c * d) = 0
u : Rˣ
h1 : C (c * d) * (C ↑u * (integerNormalization R⁰ b).primPart * (integerNormalization R⁰ a).primPart) = C (c * d) * p
hu : c * d * ↑u = (integerNormalization R⁰ a).content * (integerNormalization R⁰ b).content
h0 : a ≠ 0 ∧ b ≠ 0
⊢ IsUnit a ∨ IsUnit b

tactic: rcases hi.isUnit_or_isUnit (mul_left_cancel₀ hcd0 h1).symm with (h | h)
state_before: case intro.mk.intro.mk.intro
R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
hp : p.IsPrimitive
hi : Irreducible p
a b : K[X]
hab : map (algebraMap R K) p = a * b
c : R
c0 : c ≠ 0
hc : map (algebraMap R K) (integerNormalization R⁰ a) = C ((algebraMap R K) c) * a
d : R
d0 : d ≠ 0
hd : map (algebraMap R K) (integerNormalization R⁰ b) = C ((algebraMap R K) d) * b
hcd0 : ¬C (c * d) = 0
u : Rˣ
h1 : C (c * d) * (C ↑u * (integerNormalization R⁰ b).primPart * (integerNormalization R⁰ a).primPart) = C (c * d) * p
hu : c * d * ↑u = (integerNormalization R⁰ a).content * (integerNormalization R⁰ b).content
h0 : a ≠ 0 ∧ b ≠ 0
⊢ IsUnit a ∨ IsUnit b
state_after: case intro.mk.intro.mk.intro.inl
R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
hp : p.IsPrimitive
hi : Irreducible p
a b : K[X]
hab : map (algebraMap R K) p = a * b
c : R
c0 : c ≠ 0
hc : map (algebraMap R K) (integerNormalization R⁰ a) = C ((algebraMap R K) c) * a
d : R
d0 : d ≠ 0
hd : map (algebraMap R K) (integerNormalization R⁰ b) = C ((algebraMap R K) d) * b
hcd0 : ¬C (c * d) = 0
u : Rˣ
h1 : C (c * d) * (C ↑u * (integerNormalization R⁰ b).primPart * (integerNormalization R⁰ a).primPart) = C (c * d) * p
hu : c * d * ↑u = (integerNormalization R⁰ a).content * (integerNormalization R⁰ b).content
h0 : a ≠ 0 ∧ b ≠ 0
h : IsUnit (C ↑u * (integerNormalization R⁰ b).primPart)
⊢ IsUnit a ∨ IsUnit b

case intro.mk.intro.mk.intro.inr
R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
hp : p.IsPrimitive
hi : Irreducible p
a b : K[X]
hab : map (algebraMap R K) p = a * b
c : R
c0 : c ≠ 0
hc : map (algebraMap R K) (integerNormalization R⁰ a) = C ((algebraMap R K) c) * a
d : R
d0 : d ≠ 0
hd : map (algebraMap R K) (integerNormalization R⁰ b) = C ((algebraMap R K) d) * b
hcd0 : ¬C (c * d) = 0
u : Rˣ
h1 : C (c * d) * (C ↑u * (integerNormalization R⁰ b).primPart * (integerNormalization R⁰ a).primPart) = C (c * d) * p
hu : c * d * ↑u = (integerNormalization R⁰ a).content * (integerNormalization R⁰ b).content
h0 : a ≠ 0 ∧ b ≠ 0
h : IsUnit (integerNormalization R⁰ a).primPart
⊢ IsUnit a ∨ IsUnit b

tactic: apply map_injective (algebraMap R K) (IsFractionRing.injective _ _) _
state_before: R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
hp : p.IsPrimitive
hi : Irreducible p
a b : K[X]
hab : map (algebraMap R K) p = a * b
c : R
c0 : c ≠ 0
hc : map (algebraMap R K) (integerNormalization R⁰ a) = C ((algebraMap R K) c) * a
d : R
d0 : d ≠ 0
hd : map (algebraMap R K) (integerNormalization R⁰ b) = C ((algebraMap R K) d) * b
hcd0 : ¬C (c * d) = 0
⊢ C c * C d * p = integerNormalization R⁰ a * integerNormalization R⁰ b
state_after: R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
hp : p.IsPrimitive
hi : Irreducible p
a b : K[X]
hab : map (algebraMap R K) p = a * b
c : R
c0 : c ≠ 0
hc : map (algebraMap R K) (integerNormalization R⁰ a) = C ((algebraMap R K) c) * a
d : R
d0 : d ≠ 0
hd : map (algebraMap R K) (integerNormalization R⁰ b) = C ((algebraMap R K) d) * b
hcd0 : ¬C (c * d) = 0
⊢ map (algebraMap R K) (C c * C d * p) = map (algebraMap R K) (integerNormalization R⁰ a * integerNormalization R⁰ b)

tactic: rw [Polynomial.map_mul, Polynomial.map_mul, Polynomial.map_mul, hc, hd, map_C, map_C, hab]
state_before: R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
hp : p.IsPrimitive
hi : Irreducible p
a b : K[X]
hab : map (algebraMap R K) p = a * b
c : R
c0 : c ≠ 0
hc : map (algebraMap R K) (integerNormalization R⁰ a) = C ((algebraMap R K) c) * a
d : R
d0 : d ≠ 0
hd : map (algebraMap R K) (integerNormalization R⁰ b) = C ((algebraMap R K) d) * b
hcd0 : ¬C (c * d) = 0
⊢ map (algebraMap R K) (C c * C d * p) = map (algebraMap R K) (integerNormalization R⁰ a * integerNormalization R⁰ b)
state_after: R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
hp : p.IsPrimitive
hi : Irreducible p
a b : K[X]
hab : map (algebraMap R K) p = a * b
c : R
c0 : c ≠ 0
hc : map (algebraMap R K) (integerNormalization R⁰ a) = C ((algebraMap R K) c) * a
d : R
d0 : d ≠ 0
hd : map (algebraMap R K) (integerNormalization R⁰ b) = C ((algebraMap R K) d) * b
hcd0 : ¬C (c * d) = 0
⊢ C ((algebraMap R K) c) * C ((algebraMap R K) d) * (a * b) = C ((algebraMap R K) c) * a * (C ((algebraMap R K) d) * b)

tactic: ring
state_before: R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
hp : p.IsPrimitive
hi : Irreducible p
a b : K[X]
hab : map (algebraMap R K) p = a * b
c : R
c0 : c ≠ 0
hc : map (algebraMap R K) (integerNormalization R⁰ a) = C ((algebraMap R K) c) * a
d : R
d0 : d ≠ 0
hd : map (algebraMap R K) (integerNormalization R⁰ b) = C ((algebraMap R K) d) * b
hcd0 : ¬C (c * d) = 0
⊢ C ((algebraMap R K) c) * C ((algebraMap R K) d) * (a * b) = C ((algebraMap R K) c) * a * (C ((algebraMap R K) d) * b)
state_after: no goals

tactic: rw [← dvd_dvd_iff_associated, ← normalize_eq_normalize_iff, normalize.map_mul,
  normalize.map_mul, normalize_content, normalize_content, ←
  mul_one (normalize c * normalize d), ← hp.content_eq_one, ← content_C, ← content_C, ←
  content_mul, ← content_mul, ← content_mul, h1]
state_before: R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
hp : p.IsPrimitive
hi : Irreducible p
a b : K[X]
hab : map (algebraMap R K) p = a * b
c : R
c0 : c ≠ 0
hc : map (algebraMap R K) (integerNormalization R⁰ a) = C ((algebraMap R K) c) * a
d : R
d0 : d ≠ 0
hd : map (algebraMap R K) (integerNormalization R⁰ b) = C ((algebraMap R K) d) * b
hcd0 : ¬C (c * d) = 0
h1 : C c * C d * p = integerNormalization R⁰ a * integerNormalization R⁰ b
⊢ Associated (c * d) ((integerNormalization R⁰ a).content * (integerNormalization R⁰ b).content)
state_after: no goals

tactic: classical
rw [Ne, Ne, ← not_or, ← mul_eq_zero, ← hab]
intro con
apply hp.ne_zero (map_injective (algebraMap R K) (IsFractionRing.injective _ _) _)
simp [con]
state_before: R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
hp : p.IsPrimitive
hi : Irreducible p
a b : K[X]
hab : map (algebraMap R K) p = a * b
c : R
c0 : c ≠ 0
hc : map (algebraMap R K) (integerNormalization R⁰ a) = C ((algebraMap R K) c) * a
d : R
d0 : d ≠ 0
hd : map (algebraMap R K) (integerNormalization R⁰ b) = C ((algebraMap R K) d) * b
hcd0 : ¬C (c * d) = 0
u : Rˣ
h1 : C (c * d) * (C ↑u * (integerNormalization R⁰ b).primPart * (integerNormalization R⁰ a).primPart) = C (c * d) * p
hu : c * d * ↑u = (integerNormalization R⁰ a).content * (integerNormalization R⁰ b).content
⊢ a ≠ 0 ∧ b ≠ 0
state_after: no goals

tactic: rw [Ne, Ne, ← not_or, ← mul_eq_zero, ← hab]
state_before: R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
hp : p.IsPrimitive
hi : Irreducible p
a b : K[X]
hab : map (algebraMap R K) p = a * b
c : R
c0 : c ≠ 0
hc : map (algebraMap R K) (integerNormalization R⁰ a) = C ((algebraMap R K) c) * a
d : R
d0 : d ≠ 0
hd : map (algebraMap R K) (integerNormalization R⁰ b) = C ((algebraMap R K) d) * b
hcd0 : ¬C (c * d) = 0
u : Rˣ
h1 : C (c * d) * (C ↑u * (integerNormalization R⁰ b).primPart * (integerNormalization R⁰ a).primPart) = C (c * d) * p
hu : c * d * ↑u = (integerNormalization R⁰ a).content * (integerNormalization R⁰ b).content
⊢ a ≠ 0 ∧ b ≠ 0
state_after: R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
hp : p.IsPrimitive
hi : Irreducible p
a b : K[X]
hab : map (algebraMap R K) p = a * b
c : R
c0 : c ≠ 0
hc : map (algebraMap R K) (integerNormalization R⁰ a) = C ((algebraMap R K) c) * a
d : R
d0 : d ≠ 0
hd : map (algebraMap R K) (integerNormalization R⁰ b) = C ((algebraMap R K) d) * b
hcd0 : ¬C (c * d) = 0
u : Rˣ
h1 : C (c * d) * (C ↑u * (integerNormalization R⁰ b).primPart * (integerNormalization R⁰ a).primPart) = C (c * d) * p
hu : c * d * ↑u = (integerNormalization R⁰ a).content * (integerNormalization R⁰ b).content
⊢ ¬map (algebraMap R K) p = 0

tactic: intro con
state_before: R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
hp : p.IsPrimitive
hi : Irreducible p
a b : K[X]
hab : map (algebraMap R K) p = a * b
c : R
c0 : c ≠ 0
hc : map (algebraMap R K) (integerNormalization R⁰ a) = C ((algebraMap R K) c) * a
d : R
d0 : d ≠ 0
hd : map (algebraMap R K) (integerNormalization R⁰ b) = C ((algebraMap R K) d) * b
hcd0 : ¬C (c * d) = 0
u : Rˣ
h1 : C (c * d) * (C ↑u * (integerNormalization R⁰ b).primPart * (integerNormalization R⁰ a).primPart) = C (c * d) * p
hu : c * d * ↑u = (integerNormalization R⁰ a).content * (integerNormalization R⁰ b).content
⊢ ¬map (algebraMap R K) p = 0
state_after: R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
hp : p.IsPrimitive
hi : Irreducible p
a b : K[X]
hab : map (algebraMap R K) p = a * b
c : R
c0 : c ≠ 0
hc : map (algebraMap R K) (integerNormalization R⁰ a) = C ((algebraMap R K) c) * a
d : R
d0 : d ≠ 0
hd : map (algebraMap R K) (integerNormalization R⁰ b) = C ((algebraMap R K) d) * b
hcd0 : ¬C (c * d) = 0
u : Rˣ
h1 : C (c * d) * (C ↑u * (integerNormalization R⁰ b).primPart * (integerNormalization R⁰ a).primPart) = C (c * d) * p
hu : c * d * ↑u = (integerNormalization R⁰ a).content * (integerNormalization R⁰ b).content
con : map (algebraMap R K) p = 0
⊢ False

tactic: apply hp.ne_zero (map_injective (algebraMap R K) (IsFractionRing.injective _ _) _)
state_before: R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
hp : p.IsPrimitive
hi : Irreducible p
a b : K[X]
hab : map (algebraMap R K) p = a * b
c : R
c0 : c ≠ 0
hc : map (algebraMap R K) (integerNormalization R⁰ a) = C ((algebraMap R K) c) * a
d : R
d0 : d ≠ 0
hd : map (algebraMap R K) (integerNormalization R⁰ b) = C ((algebraMap R K) d) * b
hcd0 : ¬C (c * d) = 0
u : Rˣ
h1 : C (c * d) * (C ↑u * (integerNormalization R⁰ b).primPart * (integerNormalization R⁰ a).primPart) = C (c * d) * p
hu : c * d * ↑u = (integerNormalization R⁰ a).content * (integerNormalization R⁰ b).content
con : map (algebraMap R K) p = 0
⊢ False
state_after: R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
hp : p.IsPrimitive
hi : Irreducible p
a b : K[X]
hab : map (algebraMap R K) p = a * b
c : R
c0 : c ≠ 0
hc : map (algebraMap R K) (integerNormalization R⁰ a) = C ((algebraMap R K) c) * a
d : R
d0 : d ≠ 0
hd : map (algebraMap R K) (integerNormalization R⁰ b) = C ((algebraMap R K) d) * b
hcd0 : ¬C (c * d) = 0
u : Rˣ
h1 : C (c * d) * (C ↑u * (integerNormalization R⁰ b).primPart * (integerNormalization R⁰ a).primPart) = C (c * d) * p
hu : c * d * ↑u = (integerNormalization R⁰ a).content * (integerNormalization R⁰ b).content
con : map (algebraMap R K) p = 0
⊢ map (algebraMap R K) p = map (algebraMap R K) 0

tactic: simp [con]
state_before: R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
hp : p.IsPrimitive
hi : Irreducible p
a b : K[X]
hab : map (algebraMap R K) p = a * b
c : R
c0 : c ≠ 0
hc : map (algebraMap R K) (integerNormalization R⁰ a) = C ((algebraMap R K) c) * a
d : R
d0 : d ≠ 0
hd : map (algebraMap R K) (integerNormalization R⁰ b) = C ((algebraMap R K) d) * b
hcd0 : ¬C (c * d) = 0
u : Rˣ
h1 : C (c * d) * (C ↑u * (integerNormalization R⁰ b).primPart * (integerNormalization R⁰ a).primPart) = C (c * d) * p
hu : c * d * ↑u = (integerNormalization R⁰ a).content * (integerNormalization R⁰ b).content
con : map (algebraMap R K) p = 0
⊢ map (algebraMap R K) p = map (algebraMap R K) 0
state_after: no goals

tactic: right
state_before: case intro.mk.intro.mk.intro.inl
R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
hp : p.IsPrimitive
hi : Irreducible p
a b : K[X]
hab : map (algebraMap R K) p = a * b
c : R
c0 : c ≠ 0
hc : map (algebraMap R K) (integerNormalization R⁰ a) = C ((algebraMap R K) c) * a
d : R
d0 : d ≠ 0
hd : map (algebraMap R K) (integerNormalization R⁰ b) = C ((algebraMap R K) d) * b
hcd0 : ¬C (c * d) = 0
u : Rˣ
h1 : C (c * d) * (C ↑u * (integerNormalization R⁰ b).primPart * (integerNormalization R⁰ a).primPart) = C (c * d) * p
hu : c * d * ↑u = (integerNormalization R⁰ a).content * (integerNormalization R⁰ b).content
h0 : a ≠ 0 ∧ b ≠ 0
h : IsUnit (C ↑u * (integerNormalization R⁰ b).primPart)
⊢ IsUnit a ∨ IsUnit b
state_after: case intro.mk.intro.mk.intro.inl.h
R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
hp : p.IsPrimitive
hi : Irreducible p
a b : K[X]
hab : map (algebraMap R K) p = a * b
c : R
c0 : c ≠ 0
hc : map (algebraMap R K) (integerNormalization R⁰ a) = C ((algebraMap R K) c) * a
d : R
d0 : d ≠ 0
hd : map (algebraMap R K) (integerNormalization R⁰ b) = C ((algebraMap R K) d) * b
hcd0 : ¬C (c * d) = 0
u : Rˣ
h1 : C (c * d) * (C ↑u * (integerNormalization R⁰ b).primPart * (integerNormalization R⁰ a).primPart) = C (c * d) * p
hu : c * d * ↑u = (integerNormalization R⁰ a).content * (integerNormalization R⁰ b).content
h0 : a ≠ 0 ∧ b ≠ 0
h : IsUnit (C ↑u * (integerNormalization R⁰ b).primPart)
⊢ IsUnit b

tactic: apply
  isUnit_or_eq_zero_of_isUnit_integerNormalization_primPart h0.2
    (isUnit_of_mul_isUnit_right h)
state_before: case intro.mk.intro.mk.intro.inl.h
R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
hp : p.IsPrimitive
hi : Irreducible p
a b : K[X]
hab : map (algebraMap R K) p = a * b
c : R
c0 : c ≠ 0
hc : map (algebraMap R K) (integerNormalization R⁰ a) = C ((algebraMap R K) c) * a
d : R
d0 : d ≠ 0
hd : map (algebraMap R K) (integerNormalization R⁰ b) = C ((algebraMap R K) d) * b
hcd0 : ¬C (c * d) = 0
u : Rˣ
h1 : C (c * d) * (C ↑u * (integerNormalization R⁰ b).primPart * (integerNormalization R⁰ a).primPart) = C (c * d) * p
hu : c * d * ↑u = (integerNormalization R⁰ a).content * (integerNormalization R⁰ b).content
h0 : a ≠ 0 ∧ b ≠ 0
h : IsUnit (C ↑u * (integerNormalization R⁰ b).primPart)
⊢ IsUnit b
state_after: no goals

tactic: left
state_before: case intro.mk.intro.mk.intro.inr
R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
hp : p.IsPrimitive
hi : Irreducible p
a b : K[X]
hab : map (algebraMap R K) p = a * b
c : R
c0 : c ≠ 0
hc : map (algebraMap R K) (integerNormalization R⁰ a) = C ((algebraMap R K) c) * a
d : R
d0 : d ≠ 0
hd : map (algebraMap R K) (integerNormalization R⁰ b) = C ((algebraMap R K) d) * b
hcd0 : ¬C (c * d) = 0
u : Rˣ
h1 : C (c * d) * (C ↑u * (integerNormalization R⁰ b).primPart * (integerNormalization R⁰ a).primPart) = C (c * d) * p
hu : c * d * ↑u = (integerNormalization R⁰ a).content * (integerNormalization R⁰ b).content
h0 : a ≠ 0 ∧ b ≠ 0
h : IsUnit (integerNormalization R⁰ a).primPart
⊢ IsUnit a ∨ IsUnit b
state_after: case intro.mk.intro.mk.intro.inr.h
R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
hp : p.IsPrimitive
hi : Irreducible p
a b : K[X]
hab : map (algebraMap R K) p = a * b
c : R
c0 : c ≠ 0
hc : map (algebraMap R K) (integerNormalization R⁰ a) = C ((algebraMap R K) c) * a
d : R
d0 : d ≠ 0
hd : map (algebraMap R K) (integerNormalization R⁰ b) = C ((algebraMap R K) d) * b
hcd0 : ¬C (c * d) = 0
u : Rˣ
h1 : C (c * d) * (C ↑u * (integerNormalization R⁰ b).primPart * (integerNormalization R⁰ a).primPart) = C (c * d) * p
hu : c * d * ↑u = (integerNormalization R⁰ a).content * (integerNormalization R⁰ b).content
h0 : a ≠ 0 ∧ b ≠ 0
h : IsUnit (integerNormalization R⁰ a).primPart
⊢ IsUnit a

tactic: apply isUnit_or_eq_zero_of_isUnit_integerNormalization_primPart h0.1 h
state_before: case intro.mk.intro.mk.intro.inr.h
R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
hp : p.IsPrimitive
hi : Irreducible p
a b : K[X]
hab : map (algebraMap R K) p = a * b
c : R
c0 : c ≠ 0
hc : map (algebraMap R K) (integerNormalization R⁰ a) = C ((algebraMap R K) c) * a
d : R
d0 : d ≠ 0
hd : map (algebraMap R K) (integerNormalization R⁰ b) = C ((algebraMap R K) d) * b
hcd0 : ¬C (c * d) = 0
u : Rˣ
h1 : C (c * d) * (C ↑u * (integerNormalization R⁰ b).primPart * (integerNormalization R⁰ a).primPart) = C (c * d) * p
hu : c * d * ↑u = (integerNormalization R⁰ a).content * (integerNormalization R⁰ b).content
h0 : a ≠ 0 ∧ b ≠ 0
h : IsUnit (integerNormalization R⁰ a).primPart
⊢ IsUnit a
state_after: no goals