Theorems · Theorem · linear algebra

Submodule.span_le_restrictScalars

∀ (R : Type u_1) {M : Type u_4} (S : Type u_7) [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]
  (s : Set M) [inst_3 : Semiring S] [inst_4 : SMul R S] [inst_5 : Module S M] [inst_6 : IsScalarTower R S M],
  Submodule.span R s ≤ Submodule.restrictScalars R (Submodule.span S s)

If R is "smaller" ring than S then the span by R is smaller than the span by S.

Defined in

Mathlib.LinearAlgebra.Span.Basic

Cited by

10 results in Mathlib

Foundations

Depth 25 from the axioms · uses propext, Quot.sound

Assumes

SemiringAddCommMonoidModuleSemiringSMulModuleIsScalarTower