Theorems · Theorem · commutative algebra

IsLocalization.mem_span_map

∀ {R : Type u_1} [inst : CommSemiring R] (M : Submonoid R) {S : Type u_2} [inst_1 : CommSemiring S]
  [inst_2 : Algebra R S] [inst_3 : IsLocalization M S] {x : S} {a : Set R},
  x ∈ Ideal.span (⇑(algebraMap R S) '' a) ↔ ∃ y ∈ Ideal.span a, ∃ z, x = IsLocalization.mk' S y z

Defined in: Mathlib.RingTheory.Localization.Submodule
Cited by: 1 results in Mathlib
Foundations: Depth 31 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CommSemiring CommSemiring Algebra IsLocalization

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

DFunLike.coestatement and proof · cited by 62,936
Setstatement and proof · cited by 53,352
Algebrastatement and proof · cited by 11,388
CommSemiringstatement and proof · cited by 10,911
RingHomstatement · cited by 10,189
SetLike.coeproof · cited by 8,199
Submoduleproof · cited by 7,192
Set.imagestatement · cited by 5,609
Idealstatement · cited by 4,748
Algebra.algebraMapstatement and proof · cited by 4,706
mul_oneproof · cited by 3,885
Submonoidstatement and proof · cited by 3,086

Cited by1

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IsLocalization.Away.span_range_mulNumerator_eq_topproof · cited by 1