Theorems · Theorem · commutative algebra
Ideal.Quotient.eq
∀ {R : Type u} [inst : Ring R] {I : Ideal R} {x y : R} [inst_1 : I.IsTwoSided],
(Ideal.Quotient.mk I) x = (Ideal.Quotient.mk I) y ↔ x - y ∈ I- Defined in
- Mathlib.RingTheory.Ideal.Quotient.Defs
- Cited by
- 22 results in Mathlib
- Foundations
- Depth 83 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- RingIdeal.IsTwoSided
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites8
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement · cited by 62,936
- RingHomstatement · cited by 10,189
- Ringstatement and proof · cited by 7,463
- Idealstatement and proof · cited by 4,748
- HasQuotient.Quotientstatement · cited by 2,301
- Ideal.Quotient.mkstatement · cited by 610
- Ideal.IsTwoSidedstatement and proof · cited by 179
- Submodule.Quotient.eqproof · cited by 21
Cited by22
Results whose statement or proof uses this declaration.
- AdjoinRoot.mk_eq_mkproof · cited by 6
- ModP.preVal_mkproof · cited by 5
- Ideal.algebraMap_quotient_injectiveproof · cited by 5
- AlgHom.IsArithFrobAt.mk_applyproof · cited by 4
- Ideal.Quotient.maximal_of_isFieldproof · cited by 4
- IsNilpotent.isUnit_quotient_mk_iffproof · cited by 3
- Algebra.FormallyUnramified.lift_uniqueproof · cited by 3
- dvd_geom_sum₂_iff_of_dvd_subproof · cited by 2
- Ring.DirectLimit.of_fproof · cited by 2
- IsLocalRing.adjoin_residue_eq_top_iff_adjoin_eq_topproof · cited by 2
- CharP.quotient'proof · cited by 1
- exists_integral_inj_algHom_of_quotientproof · cited by 1