Theorems · Theorem · group theory
Submonoid.LocalizationMap.eq_iff_exists
∀ {M : Type u_1} [inst : CommMonoid M] {S : Submonoid M} {N : Type u_2} [inst_1 : CommMonoid N]
(f : S.LocalizationMap N) {x y : M}, f x = f y ↔ ∃ c, ↑c * x = ↑c * y- Cited by
- 9 results in Mathlib
- Foundations
- Depth 21 from the axioms · uses propext, Quot.sound
- Assumes
- CommMonoidCommMonoid
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites9
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
- Submonoidstatement and proof · cited by 3,086
- CommMonoidstatement and proof · cited by 2,264
- map_mulproof · cited by 1,137
- Submonoid.LocalizationMapstatement and proof · cited by 147
- Submonoid.LocalizationMap.map_unitsproof · cited by 46
- IsUnit.mul_right_injproof · cited by 13
- Submonoid.IsLocalizationMap.exists_of_eqproof · cited by 6
- Submonoid.LocalizationMap.isLocalizationMapproof · cited by 5
Cited by9
Results whose statement or proof uses this declaration.
- IsLocalization.eq_iff_existsproof · cited by 30
- Submonoid.LocalizationMap.eq_of_eqproof · cited by 5
- Submonoid.LocalizationMap.eqproof · cited by 3
- Submonoid.LocalizationMap.eq_iff_eqproof · cited by 1
- Submonoid.LocalizationMap.map_dvd_mapproof · cited by 1
- Submonoid.LocalizationMap.map_injective_of_surjOn_or_injectiveproof · cited by 1
- Submonoid.LocalizationMap.map_isUnit_iffproof · cited by 1
- Submonoid.LocalizationMap.exists_of_sec_mk'proof · cited by 0
- Submonoid.LocalizationMap.mk'_eq_of_sameproof · cited by 0