Theorems · Theorem · commutative algebra

Submodule.smul_le

∀ {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 P : Submodule R M}, I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P

Defined in

Mathlib.Algebra.Algebra.Operations

Cited by

20 results in Mathlib

Foundations

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

Assumes

SemiringSemiringModuleAddCommMonoidModuleModuleIsScalarTower