Theorems · Theorem · commutative algebra
LocalizedModule.smul_eq_iff_of_mem
∀ {R : Type u} [inst : CommSemiring R] {S : Submonoid R} {M : Type v} [inst_1 : AddCommMonoid M] [inst_2 : Module R M]
(r : R) (hr : r ∈ S) (x y : LocalizedModule S M), r • x = y ↔ x = Localization.mk 1 ⟨r, hr⟩ • y- Cited by
- 1 results in Mathlib
- Foundations
- Depth 35 from the axioms · uses propext, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites18
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Modulestatement and proof · cited by 20,661
- AddCommMonoidstatement and proof · cited by 12,281
- CommSemiringstatement and proof · cited by 10,911
- Submonoidstatement and proof · cited by 3,086
- mul_commproof · cited by 2,262
- one_smulproof · cited by 1,374
- SemigroupAction.mul_smulproof · cited by 291
- Localizationstatement · cited by 270
- LocalizedModulestatement and proof · cited by 154
- SMulCommClass.smul_commproof · cited by 143
- Localization.mkstatement and proof · cited by 110
- LocalizedModule.mkproof · cited by 73
Cited by1
Results whose statement or proof uses this declaration.
- LocalizedModule.eq_zero_of_smul_eq_zeroproof · cited by 0