Theorems · Theorem · commutative algebra
UniqueFactorizationMonoid.factors_unique
- #80 of the 100 theorems: The Fundamental Theorem of Arithmetic
∀ {α : Type u_1} [inst : CommMonoidWithZero α] [UniqueFactorizationMonoid α] {f g : Multiset α},
(∀ x ∈ f, Irreducible x) → (∀ x ∈ g, Irreducible x) → Associated f.prod g.prod → Multiset.Rel Associated f g- Cited by
- 14 results in Mathlib
- Foundations
- Depth 26 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites9
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.prodstatement and proof · cited by 528
- Irreduciblestatement and proof · cited by 496
- Associatedstatement and proof · cited by 296
- UniqueFactorizationMonoidstatement and proof · cited by 279
- Multiset.Relstatement · cited by 47
- UniqueFactorizationMonoid.irreducible_iff_primeproof · cited by 8
- prime_factors_uniqueproof · cited by 3
Cited by14
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.exists_mem_factors_of_dvdproof · cited by 4
- UniqueFactorizationMonoid.factors_oneproof · cited by 2
- Associates.unique'proof · cited by 2
- UniqueFactorizationMonoid.factors_rel_of_associatedproof · cited by 2
- UniqueFactorizationMonoid.factors_mulproof · cited by 1
- IsDiscreteValuationRing.unit_mul_pow_congr_powproof · cited by 1
- Nat.divisors_filter_squarefreeproof · cited by 1
- Associated.card_factors_eqproof · cited by 1