Theorems · Theorem · commutative algebra
Submodule.mem_smul_span
∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {I : Ideal R}
[I.IsTwoSided] {s : Set M} {x : M}, x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (↑I • s)- Defined in
- Mathlib.RingTheory.Ideal.Operations
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 71 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites13
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- 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
- SetLike.coestatement and proof · cited by 8,199
- Submodulestatement and proof · cited by 7,192
- Idealstatement and proof · cited by 4,748
- Submodule.spanstatement and proof · cited by 1,504
- Ideal.spanproof · cited by 948
- Ideal.IsTwoSidedstatement and proof · cited by 179
- Set.smulstatement · cited by 110
- Ideal.span_eqproof · cited by 7
Cited by1
Results whose statement or proof uses this declaration.
- Submodule.mem_ideal_smul_span_iff_exists_sumproof · cited by 3