Theorems · Theorem · commutative algebra
Submodule.span_smul_span
∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] (S : Set R) (T : Set M)
[(Ideal.span S).IsTwoSided], Ideal.span S • Submodule.span R T = Submodule.span R (S • T)- Defined in
- Mathlib.RingTheory.Ideal.Operations
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 70 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites26
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
- Submodulestatement · cited by 7,192
- Idealstatement · cited by 4,748
- le_antisymmproof · cited by 2,068
- Submodule.spanstatement and proof · cited by 1,504
- Ideal.spanstatement and proof · cited by 948
- zero_smulproof · cited by 716
- smul_zeroproof · cited by 665
- SemigroupAction.mul_smulproof · cited by 291
Cited by4
Results whose statement or proof uses this declaration.
- Submodule.ideal_span_singleton_smulproof · cited by 7
- Ideal.span_mul_spanproof · cited by 5
- Submodule.FG.smulproof · cited by 2
- Submodule.mem_smul_spanproof · cited by 1