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

tactic: obtain ⟨r, hr⟩ := h
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 q : R[X]
hp : p.Monic
hq : q.Monic
h : Polynomial.map (algebraMap R K) q ∣ Polynomial.map (algebraMap R K) p
⊢ q ∣ p
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 q : R[X]
hp : p.Monic
hq : q.Monic
r : K[X]
hr : Polynomial.map (algebraMap R K) p = Polynomial.map (algebraMap R K) q * r
⊢ q ∣ p

tactic: obtain ⟨d', hr'⟩ := IsIntegrallyClosed.eq_map_mul_C_of_dvd K hp (dvd_of_mul_left_eq _ hr.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 q : R[X]
hp : p.Monic
hq : q.Monic
r : K[X]
hr : Polynomial.map (algebraMap R K) p = Polynomial.map (algebraMap R K) q * r
⊢ q ∣ p
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 q : R[X]
hp : p.Monic
hq : q.Monic
r : K[X]
hr : Polynomial.map (algebraMap R K) p = Polynomial.map (algebraMap R K) q * r
d' : R[X]
hr' : Polynomial.map (algebraMap R K) d' * C r.leadingCoeff = r
⊢ q ∣ p

tactic: rw [Monic.leadingCoeff, C_1, mul_one] at hr'
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 q : R[X]
hp : p.Monic
hq : q.Monic
r : K[X]
hr : Polynomial.map (algebraMap R K) p = Polynomial.map (algebraMap R K) q * r
d' : R[X]
hr' : Polynomial.map (algebraMap R K) d' * C r.leadingCoeff = r
⊢ q ∣ p
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 q : R[X]
hp : p.Monic
hq : q.Monic
r : K[X]
hr : Polynomial.map (algebraMap R K) p = Polynomial.map (algebraMap R K) q * r
d' : R[X]
hr' : Polynomial.map (algebraMap R K) d' = r
⊢ q ∣ p

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 q : R[X]
hp : p.Monic
hq : q.Monic
r : K[X]
hr : Polynomial.map (algebraMap R K) p = Polynomial.map (algebraMap R K) q * r
d' : R[X]
hr' : Polynomial.map (algebraMap R K) d' * C r.leadingCoeff = r
⊢ r.Monic

tactic: rw [← hr', ← Polynomial.map_mul] at hr
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 q : R[X]
hp : p.Monic
hq : q.Monic
r : K[X]
hr : Polynomial.map (algebraMap R K) p = Polynomial.map (algebraMap R K) q * r
d' : R[X]
hr' : Polynomial.map (algebraMap R K) d' = r
⊢ q ∣ p
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 q : R[X]
hp : p.Monic
hq : q.Monic
r : K[X]
d' : R[X]
hr : Polynomial.map (algebraMap R K) p = Polynomial.map (algebraMap R K) (q * d')
hr' : Polynomial.map (algebraMap R K) d' = r
⊢ q ∣ p

tactic: exact dvd_of_mul_right_eq _ (Polynomial.map_injective _ (IsFractionRing.injective R K) hr.symm)
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 q : R[X]
hp : p.Monic
hq : q.Monic
r : K[X]
d' : R[X]
hr : Polynomial.map (algebraMap R K) p = Polynomial.map (algebraMap R K) (q * d')
hr' : Polynomial.map (algebraMap R K) d' = r
⊢ q ∣ p
state_after: no goals

tactic: exact Monic.of_mul_monic_left (hq.map (algebraMap R K)) (by simpa [← hr] using hp.map _)
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 q : R[X]
hp : p.Monic
hq : q.Monic
r : K[X]
hr : Polynomial.map (algebraMap R K) p = Polynomial.map (algebraMap R K) q * r
d' : R[X]
hr' : Polynomial.map (algebraMap R K) d' * C r.leadingCoeff = r
⊢ r.Monic
state_after: no goals

tactic: simpa [← hr] using hp.map _
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 q : R[X]
hp : p.Monic
hq : q.Monic
r : K[X]
hr : Polynomial.map (algebraMap R K) p = Polynomial.map (algebraMap R K) q * r
d' : R[X]
hr' : Polynomial.map (algebraMap R K) d' * C r.leadingCoeff = r
⊢ (Polynomial.map (algebraMap R K) q * r).Monic
state_after: no goals