Theorems · Theorem · commutative algebra

Submodule.mem_singleton_set_smul

∀ {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {S : Type u_4}
  [inst_3 : Monoid S] [inst_4 : DistribMulAction S M] (N : Submodule R M) [SMulCommClass R S M] (r : S) (x : M),
  x ∈ {r} • N ↔ ∃ m ∈ N, x = r • m

Defined in

Mathlib.Algebra.Module.Submodule.Pointwise

Cited by

0 results in Mathlib

Foundations

Depth 66 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringAddCommMonoidModuleMonoidDistribMulActionSMulCommClass