Theorems · Theorem · commutative algebra
IsLocalization.exist_integer_multiples
∀ {R : Type u_1} [inst : CommSemiring R] (M : Submonoid R) {S : Type u_2} [inst_1 : CommSemiring S]
[inst_2 : Algebra R S] [IsLocalization M S] {ι : Type u_4} (s : Finset ι) (f : ι → S),
∃ b, ∀ i ∈ s, IsLocalization.IsInteger R (↑b • f i)We can clear the denominators of a Finset-indexed family of fractions.
- Defined in
- Mathlib.RingTheory.Localization.Integer
- Cited by
- 4 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.
Cites24
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- Finsetstatement and proof · cited by 13,712
- Algebrastatement and proof · cited by 11,388
- CommSemiringstatement and proof · cited by 10,911
- Algebra.algebraMapproof · cited by 4,706
- Submonoidstatement and proof · cited by 3,086
- Finset.prodproof · cited by 2,356
- mul_commproof · cited by 2,262
- mul_assocproof · cited by 1,667
- map_mulproof · cited by 1,137
- IsLocalizationstatement and proof · cited by 636
- Finset.eraseproof · cited by 455
Cited by5
Results whose statement or proof uses this declaration.
- IsLocalization.commonDenomproof · cited by 4
- IsLocalization.map_integerMultipleproof · cited by 3
- LinearIndependent.of_isLocalizedModuleproof · cited by 2
- IsLocalization.exist_integer_multiples_of_finsetproof · cited by 1
- IsLocalization.exist_integer_multiples_of_finiteproof · cited by 0