Theorems · Theorem · ring theory
OreLocalization.oreDiv_eq_iff
∀ {R : Type u_1} [inst : Monoid R] {S : Submonoid R} [inst_1 : OreLocalization.OreSet S] {X : Type u_2}
[inst_2 : MulAction R X] {r₁ r₂ : X} {s₁ s₂ : ↥S}, r₁ /ₒ s₁ = r₂ /ₒ s₂ ↔ ∃ u v, u • r₂ = v • r₁ ∧ ↑u * ↑s₂ = v * ↑s₁- Cited by
- 9 results in Mathlib
- Foundations
- Depth 22 from the axioms · uses propext, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites7
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Monoidstatement and proof · cited by 3,887
- Submonoidstatement and proof · cited by 3,086
- MulActionstatement and proof · cited by 1,294
- OreLocalization.OreSetstatement and proof · cited by 92
- OreLocalizationstatement · cited by 90
- OreLocalization.oreDivstatement · cited by 72
- Quotient.eq''proof · cited by 44
Cited by9
Results whose statement or proof uses this declaration.
- LocalizedModule.mk_eqproof · cited by 15
- OreLocalization.zero_oreDivproof · cited by 8
- OreLocalization.zero_oreDiv'proof · cited by 3
- OreLocalization.oreDiv_one_surjective_of_finite_leftproof · cited by 2
- OreLocalization.oreDiv_one_surjective_of_finite_rightproof · cited by 2
- OreLocalization.subsingleton_iffproof · cited by 1
- OreLocalization.numeratorHom_injproof · cited by 1
- OreLocalization.div_eq_one'proof · cited by 1
- OreLocalization.cardinalMk_le_lift_cardinalMk_of_commuteproof · cited by 0