Theorems · Theorem · commutative algebra
UniqueFactorizationMonoid.factors_eq_normalizedFactors
∀ {M : Type u_2} [inst : CommMonoidWithZero M] [inst_1 : UniqueFactorizationMonoid M] [inst_2 : Subsingleton Mˣ]
(x : M), UniqueFactorizationMonoid.factors x = UniqueFactorizationMonoid.normalizedFactors xAn arbitrary choice of factors of x : M is exactly the (unique) normalized set of factors,
if M has a trivial group of units.
- Cited by
- 7 results in Mathlib
- Foundations
- Depth 28 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites11
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Unitsstatement and proof · cited by 2,804
- Multisetstatement and proof · cited by 2,627
- CommMonoidWithZerostatement and proof · cited by 913
- Multiset.mapproof · cited by 876
- UniqueFactorizationMonoidstatement and proof · cited by 279
- Multiset.map_congrproof · cited by 232
- UniqueFactorizationMonoid.normalizedFactorsstatement · cited by 151
- normalizeproof · cited by 137
- UniqueFactorizationMonoid.factorsstatement and proof · cited by 55
- normalize_eqproof · cited by 26
- Multiset.map_idproof · cited by 20
Cited by7
Results whose statement or proof uses this declaration.
- IsDedekindDomain.coe_primesOverFinsetproof · cited by 4
- IsDedekindDomain.HeightOneSpectrum.maxPowDividing_eq_pow_multiset_countproof · cited by 2
- Ideal.IsDedekindDomain.ramificationIdx_eq_normalizedFactors_countproof · cited by 2
- Ideal.map_algebraMap_eq_finsetProd_powproof · cited by 2
- Ideal.IsDedekindDomain.ramificationIdx'_eq_factors_countproof · cited by 1
- Ideal.factors_span_eqproof · cited by 0
- Nat.moebius_eqproof · cited by 0