Theorems · Theorem · commutative algebra
UniqueFactorizationMonoid.dvd_iff_emultiplicity_le
∀ {R : Type u_2} [inst : CommMonoidWithZero R] [UniqueFactorizationMonoid R] {a b : R},
a ≠ 0 → (a ∣ b ↔ ∀ (p : R), Prime p → emultiplicity p a ≤ emultiplicity p b)- Cited by
- 1 results in Mathlib
- Foundations
- Depth 57 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.
- ENatstatement · cited by 4,985
- CommMonoidWithZerostatement and proof · cited by 913
- Multiset.countproof · cited by 302
- UniqueFactorizationMonoidstatement and proof · cited by 279
- Primestatement and proof · cited by 277
- emultiplicitystatement and proof · cited by 156
- UniqueFactorizationMonoid.normalizedFactorsproof · cited by 151
- normalizeproof · cited by 137
- Prime.irreducibleproof · cited by 54
- Multiset.count_eq_zero_of_notMemproof · cited by 27
- UniqueFactorizationMonoid.prime_of_normalized_factorproof · cited by 23
- Multiset.le_iff_countproof · cited by 14
Cited by1
Results whose statement or proof uses this declaration.
- UniqueFactorizationMonoid.pow_dvd_pow_iff_dvdproof · cited by 0