Theorems · Theorem · linear algebra
Submodule.Quotient.eq
∀ {R : Type u_1} {M : Type u_2} [inst : Ring R] [inst_1 : AddCommGroup M] [inst_2 : Module R M] (p : Submodule R M)
{x y : M}, Submodule.Quotient.mk x = Submodule.Quotient.mk y ↔ x - y ∈ p- Defined in
- Mathlib.LinearAlgebra.Quotient.Defs
- Cited by
- 21 results in Mathlib
- Foundations
- Depth 71 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- RingAddCommGroupModule
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.
- Modulestatement and proof · cited by 20,661
- AddCommGroupstatement and proof · cited by 12,871
- Ringstatement and proof · cited by 7,463
- Submodulestatement and proof · cited by 7,192
- HasQuotient.Quotientstatement · cited by 2,301
- Submodule.Quotient.mkstatement · cited by 184
- QuotientAddGroup.leftRel_applyproof · cited by 20
- Submodule.quotientRel_defproof · cited by 4
- Submodule.Quotient.eq'proof · cited by 2
Cited by21
Results whose statement or proof uses this declaration.
- Ideal.Quotient.eqproof · cited by 22
- SModEq.monoproof · cited by 5
- SModEq.sub_memproof · cited by 5
- Representation.Coinvariants.mk_eq_iffproof · cited by 4
- ModularForm.rank_eq_one_add_rank_cuspFormproof · cited by 3
- SModEq.zeroproof · cited by 3
- Ideal.cotangentIdeal_squareproof · cited by 3
- CategoryTheory.ShortComplex.moduleCat_pOpcycles_eq_iffproof · cited by 3
- lTensor.inverse_of_rightInverse_applyproof · cited by 2
- rTensor.inverse_of_rightInverse_applyproof · cited by 2
- Ring.HasFiniteQuotients.finite_cardQuot_leproof · cited by 2
- Ideal.finrank_quotient_mapproof · cited by 1