Theorems · Definition · linear algebra

basisOfTopLeSpanOfCardEqFinrank

{R : Type u_1} →
  {M : Type u_2} →
    [inst : Semiring R] →
      [OrzechProperty R] →
        [inst_2 : AddCommMonoid M] →
          [inst_3 : Module R M] →
            {ι : Type u_3} →
              [inst_4 : Fintype ι] →
                (b : ι → M) →
                  ⊤ ≤ Submodule.span R (Set.range b) → Fintype.card ι = Module.finrank R M → Module.Basis ι R M

A family of finrank R M vectors forms a basis if they span the whole space, provided R satisfies the Orzech property.

Defined in

Mathlib.LinearAlgebra.Dimension.OrzechProperty

Cited by

2 results in Mathlib

Foundations

Depth 100 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringOrzechPropertyAddCommMonoidModuleFintype