Theorems · Theorem · linear algebra
Submodule.Quotient.mk_eq_zero
∀ {R : Type u_1} {M : Type u_2} {x : M} [inst : Ring R] [inst_1 : AddCommGroup M] [inst_2 : Module R M]
(p : Submodule R M), Submodule.Quotient.mk x = 0 ↔ x ∈ p- Defined in
- Mathlib.LinearAlgebra.Quotient.Defs
- Cited by
- 35 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.
Cites8
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
- add_zeroproof · cited by 2,707
- HasQuotient.Quotientstatement · cited by 2,301
- Submodule.Quotient.mkstatement and proof · cited by 184
- Submodule.Quotient.eq'proof · cited by 2
Cited by35
Results whose statement or proof uses this declaration.
- Ideal.Quotient.eq_zero_iff_memproof · cited by 74
- Module.isTorsionBySet_quotient_iffproof · cited by 4
- Submodule.exists_dual_map_eq_bot_of_notMemproof · cited by 4
- ModularForm.rank_eq_one_add_rank_cuspFormproof · cited by 3
- Algebra.isEpi_iff_surjective_algebraMap_of_finiteproof · cited by 2
- Module.Basis.sumQuot_repr_inrproof · cited by 2
- Subspace.dualAnnihilator_dualCoannihilator_eqproof · cited by 2
- KaehlerDifferential.kerTotal_mkQ_single_algebraMapproof · cited by 2
- Module.isTorsionBy_quotient_iffproof · cited by 2
- LieSubmodule.Quotient.mk_eq_zeroproof · cited by 2
- Ideal.cotangentToQuotientSquare_injectiveproof · cited by 2
- CharP.quotient_iffproof · cited by 1