Theorems · Theorem · commutative algebra

Submodule.smul_mem_iff

∀ {S : Type u'} {R : Type u} {M : Type v} [inst : DivisionSemiring S] [inst_1 : Semiring R] [inst_2 : AddCommMonoid M]
  [inst_3 : Module R M] [inst_4 : SMul S R] [inst_5 : Module S M] [IsScalarTower S R M] (p : Submodule R M) {s : S}
  {x : M}, s ≠ 0 → (s • x ∈ p ↔ x ∈ p)

Defined in

Mathlib.Algebra.Module.Submodule.Basic

Cited by

19 results in Mathlib

Foundations

Depth 23 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

DivisionSemiringSemiringAddCommMonoidModuleSMulModuleIsScalarTower