Theorems · Theorem · commutative algebra
UniqueFactorizationMonoid.dvd_iff_normalizedFactors_le_normalizedFactors
∀ {α : Type u_1} [inst : CommMonoidWithZero α] [inst_1 : NormalizationMonoid α] [inst_2 : UniqueFactorizationMonoid α]
{x y : α},
x ≠ 0 →
y ≠ 0 → (x ∣ y ↔ UniqueFactorizationMonoid.normalizedFactors x ≤ UniqueFactorizationMonoid.normalizedFactors y)- Cited by
- 11 results in Mathlib
- Foundations
- Depth 55 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.
- Multisetstatement · cited by 2,627
- CommMonoidWithZerostatement and proof · cited by 913
- UniqueFactorizationMonoidstatement and proof · cited by 279
- NormalizationMonoidstatement and proof · cited by 165
- UniqueFactorizationMonoid.normalizedFactorsstatement and proof · cited by 151
- right_ne_zero_of_mulproof · cited by 38
- UniqueFactorizationMonoid.prod_normalizedFactorsproof · cited by 23
- Associated.dvd_iff_dvd_rightproof · cited by 11
- UniqueFactorizationMonoid.normalizedFactors_mulproof · cited by 11
- Multiset.prod_dvd_prod_of_leproof · cited by 9
- Associated.dvd_iff_dvd_leftproof · cited by 6
Cited by11
Results whose statement or proof uses this declaration.
- Ideal.IsDedekindDomain.ramificationIdx'_eq_normalizedFactors_countproof · cited by 5
- Ideal.sup_eq_prod_inf_factorsproof · cited by 3
- Ideal.count_le_of_ideal_geproof · cited by 3
- UniqueFactorizationMonoid.radical_dvd_radicalproof · cited by 2
- UniqueFactorizationMonoid.dvd_iff_emultiplicity_leproof · cited by 1
- Ideal.eq_prime_pow_of_succ_lt_of_leproof · cited by 1
- UniqueFactorizationMonoid.exists_dvd_pow_iff_radical_dvdproof · cited by 1
- IsLocalization.OverPrime.mem_normalizedFactors_of_isPrimeproof · cited by 1
- Polynomial.irreducible_of_dvd_cyclotomic_of_natDegreeproof · cited by 1