Theorems · Theorem · linear algebra

Submodule.mem_set_smul

∀ {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] (sR : Set R)
  (N : Submodule R M) (x : M) [SMulCommClass R R ↥N], x ∈ sR • N ↔ ∃ c, ↑c.support ⊆ sR ∧ x = ↑(c.sum fun r m => r • m)

Defined in

Mathlib.Algebra.Module.Submodule.Finsupp

Cited by

1 results in Mathlib

Foundations

Depth 78 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringAddCommMonoidModuleSMulCommClass