Theorems · Theorem · commutative algebra
UniqueFactorizationMonoid.irreducible_of_normalized_factor
∀ {α : Type u_1} [inst : CommMonoidWithZero α] [inst_1 : NormalizationMonoid α] [inst_2 : UniqueFactorizationMonoid α]
{a : α}, ∀ x ∈ UniqueFactorizationMonoid.normalizedFactors a, Irreducible x- Cited by
- 15 results in Mathlib
- Foundations
- Depth 23 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites8
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Multisetstatement · cited by 2,627
- CommMonoidWithZerostatement and proof · cited by 913
- Irreduciblestatement · cited by 496
- UniqueFactorizationMonoidstatement and proof · cited by 279
- NormalizationMonoidstatement and proof · cited by 165
- UniqueFactorizationMonoid.normalizedFactorsstatement and proof · cited by 151
- Prime.irreducibleproof · cited by 54
- UniqueFactorizationMonoid.prime_of_normalized_factorproof · cited by 23
Cited by15
Results whose statement or proof uses this declaration.
- UniqueFactorizationMonoid.normalizedFactors_mulproof · cited by 11
- UniqueFactorizationMonoid.normalizedFactors_oneproof · cited by 8
- UniqueFactorizationMonoid.exists_mem_normalizedFactors_of_dvdproof · cited by 8
- Nat.factors_eqproof · cited by 5
- UniqueFactorizationMonoid.isRadical_radicalproof · cited by 3
- UniqueFactorizationMonoid.mem_normalizedFactors_iff'proof · cited by 3
- UniqueFactorizationMonoid.squarefree_iff_nodup_normalizedFactorsproof · cited by 3
- map_prime_of_factor_orderIsoproof · cited by 2
- Polynomial.natDegree_of_mem_normalizedFactors_cyclotomicproof · cited by 1
- Nat.divisors_filter_squarefreeproof · cited by 1
- Polynomial.irreducible_of_dvd_cyclotomic_of_natDegreeproof · cited by 1