Theorems · Theorem · commutative algebra
Submodule.mem_toAddSubmonoid
∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] (p : Submodule R M)
(x : M), x ∈ p.toAddSubmonoid ↔ x ∈ p- Defined in
- Mathlib.Algebra.Module.Submodule.Defs
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 13 from the axioms · uses no axioms
- Assumes
- SemiringAddCommMonoidModule
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites6
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
- AddSubmonoidstatement · cited by 1,178
- Submodule.toAddSubmonoidstatement · cited by 162
Cited by4
Results whose statement or proof uses this declaration.
- Submodule.iSup_inductionproof · cited by 4
- IsLocalization.isMaximal_iff_isMaximal_disjointproof · cited by 3
- RootPairing.Base.pos_or_neg_of_sum_smul_root_memproof · cited by 2
- Submodule.sup_toAddSubmonoidproof · cited by 1