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

tactic: refine
  ⟨fun hp =>
    irreducible_iff.mpr
      ⟨hp.not_unit.imp h.isPrimitive.isUnit_iff_isUnit_map.mpr, fun a b H =>
        or_iff_not_imp_left.mpr fun hₐ => ?_⟩,
    fun hp =>
    h.isPrimitive.irreducible_of_irreducible_map_of_injective (IsFractionRing.injective R K) hp⟩
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✝ : IsIntegrallyClosed R
p : R[X]
h : p.Monic
⊢ Irreducible p ↔ Irreducible (Polynomial.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✝ : IsIntegrallyClosed R
p : R[X]
h : p.Monic
hp : Irreducible p
a b : K[X]
H : Polynomial.map (algebraMap R K) p = a * b
hₐ : ¬IsUnit a
⊢ IsUnit b

tactic: obtain ⟨a', ha⟩ := eq_map_mul_C_of_dvd K h (dvd_of_mul_right_eq b H.symm)
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✝ : IsIntegrallyClosed R
p : R[X]
h : p.Monic
hp : Irreducible p
a b : K[X]
H : Polynomial.map (algebraMap R K) p = a * b
hₐ : ¬IsUnit a
⊢ IsUnit b
state_after: case 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✝ : IsIntegrallyClosed R
p : R[X]
h : p.Monic
hp : Irreducible p
a b : K[X]
H : Polynomial.map (algebraMap R K) p = a * b
hₐ : ¬IsUnit a
a' : R[X]
ha : Polynomial.map (algebraMap R K) a' * C a.leadingCoeff = a
⊢ IsUnit b

tactic: obtain ⟨b', hb⟩ := eq_map_mul_C_of_dvd K h (dvd_of_mul_left_eq a H.symm)
state_before: case 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✝ : IsIntegrallyClosed R
p : R[X]
h : p.Monic
hp : Irreducible p
a b : K[X]
H : Polynomial.map (algebraMap R K) p = a * b
hₐ : ¬IsUnit a
a' : R[X]
ha : Polynomial.map (algebraMap R K) a' * C a.leadingCoeff = a
⊢ IsUnit b
state_after: case intro.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✝ : IsIntegrallyClosed R
p : R[X]
h : p.Monic
hp : Irreducible p
a b : K[X]
H : Polynomial.map (algebraMap R K) p = a * b
hₐ : ¬IsUnit a
a' : R[X]
ha : Polynomial.map (algebraMap R K) a' * C a.leadingCoeff = a
b' : R[X]
hb : Polynomial.map (algebraMap R K) b' * C b.leadingCoeff = b
⊢ IsUnit b

tactic: have : a.leadingCoeff * b.leadingCoeff = 1 := by
  rw [← leadingCoeff_mul, ← H, Monic.leadingCoeff (h.map <| algebraMap R K)]
state_before: case intro.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✝ : IsIntegrallyClosed R
p : R[X]
h : p.Monic
hp : Irreducible p
a b : K[X]
H : Polynomial.map (algebraMap R K) p = a * b
hₐ : ¬IsUnit a
a' : R[X]
ha : Polynomial.map (algebraMap R K) a' * C a.leadingCoeff = a
b' : R[X]
hb : Polynomial.map (algebraMap R K) b' * C b.leadingCoeff = b
⊢ IsUnit b
state_after: case intro.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✝ : IsIntegrallyClosed R
p : R[X]
h : p.Monic
hp : Irreducible p
a b : K[X]
H : Polynomial.map (algebraMap R K) p = a * b
hₐ : ¬IsUnit a
a' : R[X]
ha : Polynomial.map (algebraMap R K) a' * C a.leadingCoeff = a
b' : R[X]
hb : Polynomial.map (algebraMap R K) b' * C b.leadingCoeff = b
this : a.leadingCoeff * b.leadingCoeff = 1
⊢ IsUnit b

tactic: rw [← ha, ← hb, mul_comm _ (C b.leadingCoeff), mul_assoc, ← mul_assoc (C a.leadingCoeff), ←
  C_mul, this, C_1, one_mul, ← Polynomial.map_mul] at H
state_before: case intro.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✝ : IsIntegrallyClosed R
p : R[X]
h : p.Monic
hp : Irreducible p
a b : K[X]
H : Polynomial.map (algebraMap R K) p = a * b
hₐ : ¬IsUnit a
a' : R[X]
ha : Polynomial.map (algebraMap R K) a' * C a.leadingCoeff = a
b' : R[X]
hb : Polynomial.map (algebraMap R K) b' * C b.leadingCoeff = b
this : a.leadingCoeff * b.leadingCoeff = 1
⊢ IsUnit b
state_after: case intro.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✝ : IsIntegrallyClosed R
p : R[X]
h : p.Monic
hp : Irreducible p
a b : K[X]
hₐ : ¬IsUnit a
a' : R[X]
ha : Polynomial.map (algebraMap R K) a' * C a.leadingCoeff = a
b' : R[X]
H : Polynomial.map (algebraMap R K) p = Polynomial.map (algebraMap R K) (a' * b')
hb : Polynomial.map (algebraMap R K) b' * C b.leadingCoeff = b
this : a.leadingCoeff * b.leadingCoeff = 1
⊢ IsUnit b

tactic: rw [← hb, ← Polynomial.coe_mapRingHom]
state_before: case intro.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✝ : IsIntegrallyClosed R
p : R[X]
h : p.Monic
hp : Irreducible p
a b : K[X]
hₐ : ¬IsUnit a
a' : R[X]
ha : Polynomial.map (algebraMap R K) a' * C a.leadingCoeff = a
b' : R[X]
H : Polynomial.map (algebraMap R K) p = Polynomial.map (algebraMap R K) (a' * b')
hb : Polynomial.map (algebraMap R K) b' * C b.leadingCoeff = b
this : a.leadingCoeff * b.leadingCoeff = 1
⊢ IsUnit b
state_after: case intro.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✝ : IsIntegrallyClosed R
p : R[X]
h : p.Monic
hp : Irreducible p
a b : K[X]
hₐ : ¬IsUnit a
a' : R[X]
ha : Polynomial.map (algebraMap R K) a' * C a.leadingCoeff = a
b' : R[X]
H : Polynomial.map (algebraMap R K) p = Polynomial.map (algebraMap R K) (a' * b')
hb : Polynomial.map (algebraMap R K) b' * C b.leadingCoeff = b
this : a.leadingCoeff * b.leadingCoeff = 1
⊢ IsUnit ((mapRingHom (algebraMap R K)) b' * C b.leadingCoeff)

tactic: refine
  IsUnit.mul (IsUnit.map _ (Or.resolve_left (hp.isUnit_or_isUnit ?_) (show ¬IsUnit a' from ?_)))
    (isUnit_iff_exists_inv'.mpr
      (Exists.intro (C a.leadingCoeff) <| by rw [← C_mul, this, C_1]))
state_before: case intro.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✝ : IsIntegrallyClosed R
p : R[X]
h : p.Monic
hp : Irreducible p
a b : K[X]
hₐ : ¬IsUnit a
a' : R[X]
ha : Polynomial.map (algebraMap R K) a' * C a.leadingCoeff = a
b' : R[X]
H : Polynomial.map (algebraMap R K) p = Polynomial.map (algebraMap R K) (a' * b')
hb : Polynomial.map (algebraMap R K) b' * C b.leadingCoeff = b
this : a.leadingCoeff * b.leadingCoeff = 1
⊢ IsUnit ((mapRingHom (algebraMap R K)) b' * C b.leadingCoeff)
state_after: case intro.intro.refine_1
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✝ : IsIntegrallyClosed R
p : R[X]
h : p.Monic
hp : Irreducible p
a b : K[X]
hₐ : ¬IsUnit a
a' : R[X]
ha : Polynomial.map (algebraMap R K) a' * C a.leadingCoeff = a
b' : R[X]
H : Polynomial.map (algebraMap R K) p = Polynomial.map (algebraMap R K) (a' * b')
hb : Polynomial.map (algebraMap R K) b' * C b.leadingCoeff = b
this : a.leadingCoeff * b.leadingCoeff = 1
⊢ p = a' * b'

case intro.intro.refine_2
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✝ : IsIntegrallyClosed R
p : R[X]
h : p.Monic
hp : Irreducible p
a b : K[X]
hₐ : ¬IsUnit a
a' : R[X]
ha : Polynomial.map (algebraMap R K) a' * C a.leadingCoeff = a
b' : R[X]
H : Polynomial.map (algebraMap R K) p = Polynomial.map (algebraMap R K) (a' * b')
hb : Polynomial.map (algebraMap R K) b' * C b.leadingCoeff = b
this : a.leadingCoeff * b.leadingCoeff = 1
⊢ ¬IsUnit a'

tactic: rw [← leadingCoeff_mul, ← H, Monic.leadingCoeff (h.map <| algebraMap 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✝ : IsIntegrallyClosed R
p : R[X]
h : p.Monic
hp : Irreducible p
a b : K[X]
H : Polynomial.map (algebraMap R K) p = a * b
hₐ : ¬IsUnit a
a' : R[X]
ha : Polynomial.map (algebraMap R K) a' * C a.leadingCoeff = a
b' : R[X]
hb : Polynomial.map (algebraMap R K) b' * C b.leadingCoeff = b
⊢ a.leadingCoeff * b.leadingCoeff = 1
state_after: no goals

tactic: rw [← C_mul, this, C_1]
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✝ : IsIntegrallyClosed R
p : R[X]
h : p.Monic
hp : Irreducible p
a b : K[X]
hₐ : ¬IsUnit a
a' : R[X]
ha : Polynomial.map (algebraMap R K) a' * C a.leadingCoeff = a
b' : R[X]
H : Polynomial.map (algebraMap R K) p = Polynomial.map (algebraMap R K) (a' * b')
hb : Polynomial.map (algebraMap R K) b' * C b.leadingCoeff = b
this : a.leadingCoeff * b.leadingCoeff = 1
⊢ C a.leadingCoeff * C b.leadingCoeff = 1
state_after: no goals

tactic: exact Polynomial.map_injective _ (IsFractionRing.injective R K) H
state_before: case intro.intro.refine_1
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✝ : IsIntegrallyClosed R
p : R[X]
h : p.Monic
hp : Irreducible p
a b : K[X]
hₐ : ¬IsUnit a
a' : R[X]
ha : Polynomial.map (algebraMap R K) a' * C a.leadingCoeff = a
b' : R[X]
H : Polynomial.map (algebraMap R K) p = Polynomial.map (algebraMap R K) (a' * b')
hb : Polynomial.map (algebraMap R K) b' * C b.leadingCoeff = b
this : a.leadingCoeff * b.leadingCoeff = 1
⊢ p = a' * b'
state_after: no goals

tactic: by_contra h_contra
state_before: case intro.intro.refine_2
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✝ : IsIntegrallyClosed R
p : R[X]
h : p.Monic
hp : Irreducible p
a b : K[X]
hₐ : ¬IsUnit a
a' : R[X]
ha : Polynomial.map (algebraMap R K) a' * C a.leadingCoeff = a
b' : R[X]
H : Polynomial.map (algebraMap R K) p = Polynomial.map (algebraMap R K) (a' * b')
hb : Polynomial.map (algebraMap R K) b' * C b.leadingCoeff = b
this : a.leadingCoeff * b.leadingCoeff = 1
⊢ ¬IsUnit a'
state_after: case intro.intro.refine_2
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✝ : IsIntegrallyClosed R
p : R[X]
h : p.Monic
hp : Irreducible p
a b : K[X]
hₐ : ¬IsUnit a
a' : R[X]
ha : Polynomial.map (algebraMap R K) a' * C a.leadingCoeff = a
b' : R[X]
H : Polynomial.map (algebraMap R K) p = Polynomial.map (algebraMap R K) (a' * b')
hb : Polynomial.map (algebraMap R K) b' * C b.leadingCoeff = b
this : a.leadingCoeff * b.leadingCoeff = 1
h_contra : IsUnit a'
⊢ False

tactic: refine hₐ ?_
state_before: case intro.intro.refine_2
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✝ : IsIntegrallyClosed R
p : R[X]
h : p.Monic
hp : Irreducible p
a b : K[X]
hₐ : ¬IsUnit a
a' : R[X]
ha : Polynomial.map (algebraMap R K) a' * C a.leadingCoeff = a
b' : R[X]
H : Polynomial.map (algebraMap R K) p = Polynomial.map (algebraMap R K) (a' * b')
hb : Polynomial.map (algebraMap R K) b' * C b.leadingCoeff = b
this : a.leadingCoeff * b.leadingCoeff = 1
h_contra : IsUnit a'
⊢ False
state_after: case intro.intro.refine_2
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✝ : IsIntegrallyClosed R
p : R[X]
h : p.Monic
hp : Irreducible p
a b : K[X]
hₐ : ¬IsUnit a
a' : R[X]
ha : Polynomial.map (algebraMap R K) a' * C a.leadingCoeff = a
b' : R[X]
H : Polynomial.map (algebraMap R K) p = Polynomial.map (algebraMap R K) (a' * b')
hb : Polynomial.map (algebraMap R K) b' * C b.leadingCoeff = b
this : a.leadingCoeff * b.leadingCoeff = 1
h_contra : IsUnit a'
⊢ IsUnit a

tactic: rw [← ha, ← Polynomial.coe_mapRingHom]
state_before: case intro.intro.refine_2
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✝ : IsIntegrallyClosed R
p : R[X]
h : p.Monic
hp : Irreducible p
a b : K[X]
hₐ : ¬IsUnit a
a' : R[X]
ha : Polynomial.map (algebraMap R K) a' * C a.leadingCoeff = a
b' : R[X]
H : Polynomial.map (algebraMap R K) p = Polynomial.map (algebraMap R K) (a' * b')
hb : Polynomial.map (algebraMap R K) b' * C b.leadingCoeff = b
this : a.leadingCoeff * b.leadingCoeff = 1
h_contra : IsUnit a'
⊢ IsUnit a
state_after: case intro.intro.refine_2
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✝ : IsIntegrallyClosed R
p : R[X]
h : p.Monic
hp : Irreducible p
a b : K[X]
hₐ : ¬IsUnit a
a' : R[X]
ha : Polynomial.map (algebraMap R K) a' * C a.leadingCoeff = a
b' : R[X]
H : Polynomial.map (algebraMap R K) p = Polynomial.map (algebraMap R K) (a' * b')
hb : Polynomial.map (algebraMap R K) b' * C b.leadingCoeff = b
this : a.leadingCoeff * b.leadingCoeff = 1
h_contra : IsUnit a'
⊢ IsUnit ((mapRingHom (algebraMap R K)) a' * C a.leadingCoeff)

tactic: exact
  IsUnit.mul (IsUnit.map _ h_contra)
    (isUnit_iff_exists_inv.mpr
      (Exists.intro (C b.leadingCoeff) <| by rw [← C_mul, this, C_1]))
state_before: case intro.intro.refine_2
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✝ : IsIntegrallyClosed R
p : R[X]
h : p.Monic
hp : Irreducible p
a b : K[X]
hₐ : ¬IsUnit a
a' : R[X]
ha : Polynomial.map (algebraMap R K) a' * C a.leadingCoeff = a
b' : R[X]
H : Polynomial.map (algebraMap R K) p = Polynomial.map (algebraMap R K) (a' * b')
hb : Polynomial.map (algebraMap R K) b' * C b.leadingCoeff = b
this : a.leadingCoeff * b.leadingCoeff = 1
h_contra : IsUnit a'
⊢ IsUnit ((mapRingHom (algebraMap R K)) a' * C a.leadingCoeff)
state_after: no goals

tactic: rw [← C_mul, this, C_1]
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✝ : IsIntegrallyClosed R
p : R[X]
h : p.Monic
hp : Irreducible p
a b : K[X]
hₐ : ¬IsUnit a
a' : R[X]
ha : Polynomial.map (algebraMap R K) a' * C a.leadingCoeff = a
b' : R[X]
H : Polynomial.map (algebraMap R K) p = Polynomial.map (algebraMap R K) (a' * b')
hb : Polynomial.map (algebraMap R K) b' * C b.leadingCoeff = b
this : a.leadingCoeff * b.leadingCoeff = 1
h_contra : IsUnit a'
⊢ C a.leadingCoeff * C b.leadingCoeff = 1
state_after: no goals