Theorem: count_span_normalizedFactors_eq
File: Mathlib/RingTheory/DedekindDomain/Ideal.lean

tactic: have := multiplicity_eq_multiplicity_span (R := R) (a := X) (b := r)
state_before: R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁹ : CommRing R
inst✝⁸ : CommRing A
inst✝⁷ : Field K
inst✝⁶ : IsDomain R
inst✝⁵ : IsPrincipalIdealRing R
inst✝⁴ : DecidableEq R
inst✝³ : DecidableEq (Ideal R)
inst✝² : NormalizationMonoid R
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
r X : R
hr : r ≠ 0
hX : Prime X
⊢ Multiset.count (span {X}) (normalizedFactors (span {r})) = Multiset.count (normalize X) (normalizedFactors r)
state_after: R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁹ : CommRing R
inst✝⁸ : CommRing A
inst✝⁷ : Field K
inst✝⁶ : IsDomain R
inst✝⁵ : IsPrincipalIdealRing R
inst✝⁴ : DecidableEq R
inst✝³ : DecidableEq (Ideal R)
inst✝² : NormalizationMonoid R
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
r X : R
hr : r ≠ 0
hX : Prime X
this : multiplicity (span {X}) (span {r}) = multiplicity X r
⊢ Multiset.count (span {X}) (normalizedFactors (span {r})) = Multiset.count (normalize X) (normalizedFactors r)

tactic: rw [multiplicity_eq_count_normalizedFactors (Prime.irreducible hX) hr,
  multiplicity_eq_count_normalizedFactors (Prime.irreducible ?_), normalize_apply,
  normUnit_eq_one, Units.val_one, one_eq_top, mul_top, Nat.cast_inj] at this
state_before: R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁹ : CommRing R
inst✝⁸ : CommRing A
inst✝⁷ : Field K
inst✝⁶ : IsDomain R
inst✝⁵ : IsPrincipalIdealRing R
inst✝⁴ : DecidableEq R
inst✝³ : DecidableEq (Ideal R)
inst✝² : NormalizationMonoid R
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
r X : R
hr : r ≠ 0
hX : Prime X
this : multiplicity (span {X}) (span {r}) = multiplicity X r
⊢ Multiset.count (span {X}) (normalizedFactors (span {r})) = Multiset.count (normalize X) (normalizedFactors r)
state_after: R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁹ : CommRing R
inst✝⁸ : CommRing A
inst✝⁷ : Field K
inst✝⁶ : IsDomain R
inst✝⁵ : IsPrincipalIdealRing R
inst✝⁴ : DecidableEq R
inst✝³ : DecidableEq (Ideal R)
inst✝² : NormalizationMonoid R
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
r X : R
hr : r ≠ 0
hX : Prime X
this : Multiset.count (span {X}) (normalizedFactors (span {r})) = Multiset.count (normalize X) (normalizedFactors r)
⊢ Multiset.count (span {X}) (normalizedFactors (span {r})) = Multiset.count (normalize X) (normalizedFactors r)

R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁹ : CommRing R
inst✝⁸ : CommRing A
inst✝⁷ : Field K
inst✝⁶ : IsDomain R
inst✝⁵ : IsPrincipalIdealRing R
inst✝⁴ : DecidableEq R
inst✝³ : DecidableEq (Ideal R)
inst✝² : NormalizationMonoid R
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
r X : R
hr : r ≠ 0
hX : Prime X
this : multiplicity (span {X}) (span {r}) = ↑(Multiset.count (normalize X) (normalizedFactors r))
⊢ span {r} ≠ 0

R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁹ : CommRing R
inst✝⁸ : CommRing A
inst✝⁷ : Field K
inst✝⁶ : IsDomain R
inst✝⁵ : IsPrincipalIdealRing R
inst✝⁴ : DecidableEq R
inst✝³ : DecidableEq (Ideal R)
inst✝² : NormalizationMonoid R
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
r X : R
hr : r ≠ 0
hX : Prime X
this : multiplicity (span {X}) (span {r}) = ↑(Multiset.count (normalize X) (normalizedFactors r))
⊢ Prime (span {X})

tactic: simp only [normalize_apply, this]
state_before: R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁹ : CommRing R
inst✝⁸ : CommRing A
inst✝⁷ : Field K
inst✝⁶ : IsDomain R
inst✝⁵ : IsPrincipalIdealRing R
inst✝⁴ : DecidableEq R
inst✝³ : DecidableEq (Ideal R)
inst✝² : NormalizationMonoid R
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
r X : R
hr : r ≠ 0
hX : Prime X
this : Multiset.count (span {X}) (normalizedFactors (span {r})) = Multiset.count (normalize X) (normalizedFactors r)
⊢ Multiset.count (span {X}) (normalizedFactors (span {r})) = Multiset.count (normalize X) (normalizedFactors r)

R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁹ : CommRing R
inst✝⁸ : CommRing A
inst✝⁷ : Field K
inst✝⁶ : IsDomain R
inst✝⁵ : IsPrincipalIdealRing R
inst✝⁴ : DecidableEq R
inst✝³ : DecidableEq (Ideal R)
inst✝² : NormalizationMonoid R
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
r X : R
hr : r ≠ 0
hX : Prime X
this : multiplicity (span {X}) (span {r}) = ↑(Multiset.count (normalize X) (normalizedFactors r))
⊢ span {r} ≠ 0

R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁹ : CommRing R
inst✝⁸ : CommRing A
inst✝⁷ : Field K
inst✝⁶ : IsDomain R
inst✝⁵ : IsPrincipalIdealRing R
inst✝⁴ : DecidableEq R
inst✝³ : DecidableEq (Ideal R)
inst✝² : NormalizationMonoid R
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
r X : R
hr : r ≠ 0
hX : Prime X
this : multiplicity (span {X}) (span {r}) = ↑(Multiset.count (normalize X) (normalizedFactors r))
⊢ Prime (span {X})
state_after: R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁹ : CommRing R
inst✝⁸ : CommRing A
inst✝⁷ : Field K
inst✝⁶ : IsDomain R
inst✝⁵ : IsPrincipalIdealRing R
inst✝⁴ : DecidableEq R
inst✝³ : DecidableEq (Ideal R)
inst✝² : NormalizationMonoid R
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
r X : R
hr : r ≠ 0
hX : Prime X
this : multiplicity (span {X}) (span {r}) = ↑(Multiset.count (normalize X) (normalizedFactors r))
⊢ span {r} ≠ 0

R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁹ : CommRing R
inst✝⁸ : CommRing A
inst✝⁷ : Field K
inst✝⁶ : IsDomain R
inst✝⁵ : IsPrincipalIdealRing R
inst✝⁴ : DecidableEq R
inst✝³ : DecidableEq (Ideal R)
inst✝² : NormalizationMonoid R
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
r X : R
hr : r ≠ 0
hX : Prime X
this : multiplicity (span {X}) (span {r}) = ↑(Multiset.count (normalize X) (normalizedFactors r))
⊢ Prime (span {X})

tactic: simp only [Submodule.zero_eq_bot, ne_eq, span_singleton_eq_bot, hr, not_false_eq_true]
state_before: R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁹ : CommRing R
inst✝⁸ : CommRing A
inst✝⁷ : Field K
inst✝⁶ : IsDomain R
inst✝⁵ : IsPrincipalIdealRing R
inst✝⁴ : DecidableEq R
inst✝³ : DecidableEq (Ideal R)
inst✝² : NormalizationMonoid R
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
r X : R
hr : r ≠ 0
hX : Prime X
this : multiplicity (span {X}) (span {r}) = ↑(Multiset.count (normalize X) (normalizedFactors r))
⊢ span {r} ≠ 0
state_after: no goals

tactic: simpa only [prime_span_singleton_iff]
state_before: R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁹ : CommRing R
inst✝⁸ : CommRing A
inst✝⁷ : Field K
inst✝⁶ : IsDomain R
inst✝⁵ : IsPrincipalIdealRing R
inst✝⁴ : DecidableEq R
inst✝³ : DecidableEq (Ideal R)
inst✝² : NormalizationMonoid R
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
r X : R
hr : r ≠ 0
hX : Prime X
this : multiplicity (span {X}) (span {r}) = ↑(Multiset.count (normalize X) (normalizedFactors r))
⊢ Prime (span {X})
state_after: no goals