Theorems · Theorem · commutative algebra
Submodule.zero_mem
∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] {module_M : Module R M} (p : Submodule R M),
0 ∈ p- Defined in
- Mathlib.Algebra.Module.Submodule.Defs
- Cited by
- 58 results in Mathlib
- Foundations
- Depth 14 from the axioms · uses no axioms
- Assumes
- SemiringAddCommMonoid
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites5
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
- Semiringstatement and proof · cited by 13,802
- AddCommMonoidstatement and proof · cited by 12,281
- Submodulestatement and proof · cited by 7,192
- ZeroMemClass.zero_memproof · cited by 162
Cited by59
Results whose statement or proof uses this declaration.
- Submodule.restrictScalarsproof · cited by 180
- Submodule.span_inductionstatement and proof · cited by 77
- Ideal.zero_memproof · cited by 37
- Profinite.NobelingProof.Products.prop_of_isGoodproof · cited by 10
- Submodule.subsingleton_iff_eq_botproof · cited by 10
- TensorProduct.span_tmul_eq_topproof · cited by 8
- KaehlerDifferential.span_range_derivationproof · cited by 7
- Ideal.comap_bot_le_of_injectiveproof · cited by 4
- FractionalIdeal.zero_memproof · cited by 4
- Submodule.set_smul_inductionOnstatement and proof · cited by 4
- Submodule.nonemptyproof · cited by 3
- Submodule.map_zeroproof · cited by 3