Theorems · Theorem · commutative algebra
Ideal.Quotient.mk_eq_mk_iff_sub_mem
∀ {R : Type u} [inst : Ring R] {I : Ideal R} [inst_1 : I.IsTwoSided] (x y : R),
(Ideal.Quotient.mk I) x = (Ideal.Quotient.mk I) y ↔ x - y ∈ I- Defined in
- Mathlib.RingTheory.Ideal.Quotient.Defs
- Cited by
- 8 results in Mathlib
- Foundations
- Depth 84 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.
Cites10
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
- RingHomstatement · cited by 10,189
- Ringstatement and proof · cited by 7,463
- Idealstatement and proof · cited by 4,748
- HasQuotient.Quotientstatement and proof · cited by 2,301
- Ideal.Quotient.mkstatement and proof · cited by 610
- map_subproof · cited by 565
- sub_eq_zeroproof · cited by 407
- Ideal.IsTwoSidedstatement and proof · cited by 179
- Ideal.Quotient.eq_zero_iff_memproof · cited by 74
Cited by8
Results whose statement or proof uses this declaration.
- Ideal.mk_mem_cotangentIdealproof · cited by 2
- AdicCompletion.residueField_map_bijective_of_fgproof · cited by 1
- Valued.integer.totallyBounded_iff_finite_residueFieldproof · cited by 1
- Algebra.FormallySmooth.pi_iffproof · cited by 1
- PowerSeries.IsWeierstrassDivisorAt.mk_mod'_eq_selfproof · cited by 0
- Ideal.Quotient.mk_eq_one_iff_sub_memproof · cited by 0
- MvPowerSeries.mk_truncTotal_toAdicCompletionInvproof · cited by 0