Theorems · Theorem · commutative algebra
UniqueFactorizationMonoid.exists_mem_normalizedFactors_of_dvd
∀ {α : Type u_1} [inst : CommMonoidWithZero α] [inst_1 : NormalizationMonoid α] [inst_2 : UniqueFactorizationMonoid α]
{a p : α}, a ≠ 0 → Irreducible p → p ∣ a → ∃ q ∈ UniqueFactorizationMonoid.normalizedFactors a, Associated p q- Cited by
- 8 results in Mathlib
- Foundations
- Depth 27 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites18
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Multisetstatement · cited by 2,627
- MulZeroClass.mul_zeroproof · cited by 2,091
- CommMonoidWithZerostatement and proof · cited by 913
- Irreduciblestatement and proof · cited by 496
- Multiset.consproof · cited by 313
- Associatedstatement and proof · cited by 296
- UniqueFactorizationMonoidstatement and proof · cited by 279
- NormalizationMonoidstatement and proof · cited by 165
- UniqueFactorizationMonoid.normalizedFactorsstatement and proof · cited by 151
- Associated.symmproof · cited by 87
- Multiset.prod_consproof · cited by 68
- Multiset.Relproof · cited by 47
Cited by8
Results whose statement or proof uses this declaration.
- Ideal.IsDedekindDomain.ramificationIdx'_ne_zeroproof · cited by 3
- UniqueFactorizationMonoid.exists_mem_normalizedFactorsproof · cited by 2
- emultiplicity_factor_dvd_iso_eq_emultiplicity_of_mem_normalizedFactorsproof · cited by 1
- UniqueFactorizationMonoid.mem_normalizedFactors_iffproof · cited by 1
- UniqueFactorizationMonoid.dvd_radical_iff_of_irreducibleproof · cited by 1
- Ideal.singleton_span_mem_normalizedFactors_of_mem_normalizedFactorsproof · cited by 1
- mem_normalizedFactors_factor_dvd_iso_of_mem_normalizedFactorsproof · cited by 1
- mem_normalizedFactors_factor_orderIso_of_mem_normalizedFactorsproof · cited by 1