Theorems · Theorem · commutative algebra
Submodule.span_singleton_mul
∀ {R : Type u} [inst : CommSemiring R] {A : Type v} [inst_1 : Semiring A] [inst_2 : Algebra R A] {x : A}
{p : Submodule R A}, (R ∙ x) * p = x • p- Defined in
- Mathlib.Algebra.Algebra.Operations
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 73 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommSemiringSemiringAlgebra
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites9
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- Semiringstatement and proof · cited by 13,802
- Algebrastatement and proof · cited by 11,388
- CommSemiringstatement and proof · cited by 10,911
- Submodulestatement and proof · cited by 7,192
- Submodule.spanstatement · cited by 1,504
- Submodule.extproof · cited by 204
- Submodule.pointwiseDistribMulActionstatement · cited by 105
- Submodule.mem_span_singleton_mulproof · cited by 1
Cited by2
Results whose statement or proof uses this declaration.
- FractionalIdeal.den_mul_self_eq_num'proof · cited by 5
- conductor_mul_differentIdealproof · cited by 1