Theorems · Theorem · commutative algebra
UniqueFactorizationMonoid.prime_of_normalized_factor
∀ {α : Type u_1} [inst : CommMonoidWithZero α] [inst_1 : NormalizationMonoid α] [inst_2 : UniqueFactorizationMonoid α]
{a : α}, ∀ x ∈ UniqueFactorizationMonoid.normalizedFactors a, Prime x- Cited by
- 23 results in Mathlib
- Foundations
- Depth 22 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites13
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Multisetstatement and proof · cited by 2,627
- CommMonoidWithZerostatement and proof · cited by 913
- Multiset.mapproof · cited by 876
- UniqueFactorizationMonoidstatement and proof · cited by 279
- Primestatement and proof · cited by 277
- Multiset.map_congrproof · cited by 232
- NormalizationMonoidstatement and proof · cited by 165
- UniqueFactorizationMonoid.normalizedFactorsstatement · cited by 151
- normalizeproof · cited by 137
- Multiset.mem_mapproof · cited by 72
- normalize_associatedproof · cited by 7
- UniqueFactorizationMonoid.exists_prime_factorsproof · cited by 6
Cited by23
Results whose statement or proof uses this declaration.
- UniqueFactorizationMonoid.irreducible_of_normalized_factorproof · cited by 15
- UniqueFactorizationMonoid.normalizedFactors_irreducibleproof · cited by 10
- UniqueFactorizationMonoid.zero_notMem_normalizedFactorsproof · cited by 5
- Ideal.eq_prime_pow_mul_coprimeproof · cited by 3
- UniqueFactorizationMonoid.isRadical_radicalproof · cited by 3
- UniqueFactorizationMonoid.normalizedFactors_prod_of_primeproof · cited by 2
- map_prime_of_factor_orderIsoproof · cited by 2
- emultiplicity_prime_le_emultiplicity_image_by_factor_orderIsoproof · cited by 1
- UniqueFactorizationMonoid.normalizedFactors_posproof · cited by 1
- DivisorChain.element_of_chain_eq_pow_second_of_chainproof · cited by 1
- UniqueFactorizationMonoid.normalizedFactors_prod_eq_self_of_subsetproof · cited by 1
- UniqueFactorizationMonoid.induction_on_coprimeproof · cited by 1