Theorems · Theorem · commutative algebra

Submodule.smul_mem_smul

∀ {R : Type u} [inst : Semiring R] {A : Type v} [inst_1 : Semiring A] [inst_2 : Module R A] {M : Type u_1}
  [inst_3 : AddCommMonoid M] [inst_4 : Module R M] [inst_5 : Module A M] [inst_6 : IsScalarTower R A M]
  {I : Submodule R A} {N : Submodule R M} {r : A} {n : M}, r ∈ I → n ∈ N → r • n ∈ I • N

Defined in

Mathlib.Algebra.Algebra.Operations

Cited by

32 results in Mathlib

Foundations

Depth 69 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringSemiringModuleAddCommMonoidModuleModuleIsScalarTower