Theorems · Definition · linear algebra

finsetBasisOfTopLeSpanOfCardEqFinrank

{R : Type u_1} →
  {M : Type u_2} →
    [inst : Semiring R] →
      [OrzechProperty R] →
        [inst_2 : AddCommMonoid M] →
          [inst_3 : Module R M] →
            {s : Finset M} → ⊤ ≤ Submodule.span R ↑s → s.card = Module.finrank R M → Module.Basis (↥s) R M

A finset 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

1 results in Mathlib

Foundations

Depth 101 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringOrzechPropertyAddCommMonoidModule