Theorems · Theorem · commutative algebra
Submodule.mem_of_span_top_of_smul_mem
∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] (M' : Submodule R M)
(s : Set R), Ideal.span s = ⊤ → ∀ (x : M), (∀ (r : ↑s), ↑r • x ∈ M') → x ∈ M'- Defined in
- Mathlib.RingTheory.Ideal.Operations
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 29 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- SemiringAddCommMonoidModule
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites19
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- Setstatement and proof · cited by 53,352
- Modulestatement and proof · cited by 20,661
- Semiringstatement and proof · cited by 13,802
- AddCommMonoidstatement and proof · cited by 12,281
- Top.topstatement and proof · cited by 9,680
- Submodulestatement and proof · cited by 7,192
- Set.Elemstatement and proof · cited by 7,166
- Idealstatement and proof · cited by 4,748
- Ideal.spanstatement and proof · cited by 948
- LinearMap.rangeproof · cited by 893
- Submodule.mapproof · cited by 614
Cited by2
Results whose statement or proof uses this declaration.
- Submodule.mem_of_span_eq_top_of_smul_pow_memproof · cited by 6
- Subalgebra.mem_of_finsetSum_eq_one_of_pow_smul_memproof · cited by 2