Theorems · Theorem · commutative algebra
UniqueFactorizationMonoid.mem_normalizedFactors_iff
∀ {α : Type u_1} [inst : CommMonoidWithZero α] [inst_1 : NormalizationMonoid α] [inst_2 : UniqueFactorizationMonoid α]
[Subsingleton αˣ] {p x : α}, x ≠ 0 → (p ∈ UniqueFactorizationMonoid.normalizedFactors x ↔ Prime p ∧ p ∣ x)- Cited by
- 1 results in Mathlib
- Foundations
- Depth 28 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.
- Unitsstatement and proof · cited by 2,804
- Multisetstatement · cited by 2,627
- CommMonoidWithZerostatement and proof · cited by 913
- Associatedproof · cited by 296
- UniqueFactorizationMonoidstatement and proof · cited by 279
- Primestatement and proof · cited by 277
- NormalizationMonoidstatement and proof · cited by 165
- UniqueFactorizationMonoid.normalizedFactorsstatement and proof · cited by 151
- Prime.irreducibleproof · cited by 54
- associated_iff_eqproof · cited by 28
- UniqueFactorizationMonoid.prime_of_normalized_factorproof · cited by 23
- UniqueFactorizationMonoid.dvd_of_mem_normalizedFactorsproof · cited by 20
Cited by1
Results whose statement or proof uses this declaration.
- Ideal.mem_normalizedFactors_iffproof · cited by 3