Theorems · Theorem · linear algebra

Module.Basis.le_span

∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {ι : Type w}
  [RankCondition R] {J : Set M} (v : Module.Basis ι R M),
  Submodule.span R J = ⊤ → Cardinal.mk ↑(Set.range ⇑v) ≤ Cardinal.mk ↑J

If R satisfies the rank condition, then the cardinality of any basis is bounded by the cardinality of any spanning set.

Defined in

Mathlib.LinearAlgebra.Dimension.StrongRankCondition

Cited by

1 results in Mathlib

Foundations

Depth 103 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringAddCommMonoidModuleRankCondition