Theorems · Theorem · commutative algebra

UniqueFactorizationMonoid.primeFactors_mul_eq_union

∀ {M : Type u_1} [inst : CommMonoidWithZero M] [inst_1 : NormalizationMonoid M] [inst_2 : UniqueFactorizationMonoid M]
  {a b : M} [inst_3 : DecidableEq M],
  a ≠ 0 →
    b ≠ 0 →
      UniqueFactorizationMonoid.primeFactors (a * b) =
        UniqueFactorizationMonoid.primeFactors a ∪ UniqueFactorizationMonoid.primeFactors b

Defined in: Mathlib.RingTheory.Radical.Basic
Cited by: 2 results in Mathlib
Foundations: Depth 77 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CommMonoidWithZero NormalizationMonoid UniqueFactorizationMonoid DecidableEq

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Cites9

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

Finsetstatement · cited by 13,712
CommMonoidWithZerostatement and proof · cited by 913
Finset.extproof · cited by 565
Irreducibleproof · cited by 496
UniqueFactorizationMonoidstatement and proof · cited by 279
NormalizationMonoidstatement and proof · cited by 165
normalizeproof · cited by 137
UniqueFactorizationMonoid.primeFactorsstatement · cited by 32
UniqueFactorizationMonoid.normalizedFactors_mulproof · cited by 11

Cited by2

Results whose statement or proof uses this declaration.

UniqueFactorizationMonoid.primeFactors_mul_eq_disjUnionproof · cited by 1
UniqueFactorizationMonoid.radical_mul_dvdproof · cited by 1