Theorems · Theorem · commutative algebra

Submodule.mem_set_smul_def

∀ {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] (s : Set S) (N : Submodule R M) (x : M),
  x ∈ s • N ↔ x ∈ sInf {p | ∀ ⦃r : S⦄ {n : M}, r ∈ s → n ∈ N → r • n ∈ p}

Defined in

Mathlib.Algebra.Module.Submodule.Pointwise

Cited by

5 results in Mathlib

Foundations

Depth 18 from the axioms · uses propext, Quot.sound

Assumes

SemiringAddCommMonoidModuleMonoidDistribMulAction