Theorems · Theorem · linear algebra

maximal_linearIndependent_eq_infinite_basis

∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] [StrongRankCondition R]
  {ι : Type w} (b : Module.Basis ι R M) [Infinite ι] {κ : Type w} (v : κ → M) (i : LinearIndependent R v),
  i.Maximal → Cardinal.mk κ = Cardinal.mk ι

Over any ring R satisfying the strong rank condition, if b is an infinite basis for a module M, then every maximal linearly independent set has the same cardinality as b. This proof (along with some of the lemmas above) comes from [Les familles libres maximales d'un module ont-elles le meme cardinal?][lazarus1973]

Defined in

Mathlib.LinearAlgebra.Dimension.StrongRankCondition

Cited by

0 results in Mathlib

Foundations

Depth 110 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringAddCommMonoidModuleStrongRankConditionInfinite