Theorems · Definition · linear algebra

setBasisOfTopLeSpanOfCardEqFinrank

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

A set 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: Semiring OrzechProperty AddCommMonoid Module Fintype

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Cites15

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

Setstatement and proof · cited by 53,352
Modulestatement and proof · cited by 20,661
Semiringstatement and proof · cited by 13,802
AddCommMonoidstatement and proof · cited by 12,281
Top.topstatement and proof · cited by 9,680
Fintypestatement and proof · cited by 7,736
Submodulestatement · cited by 7,192
Set.Elemstatement and proof · cited by 7,166
Finset.cardstatement and proof · cited by 2,327
Module.finrankstatement and proof · cited by 1,770
Submodule.spanstatement and proof · cited by 1,504
Module.Basisstatement · cited by 1,477

Cited by1

Results whose statement or proof uses this declaration.

setBasisOfTopLeSpanOfCardEqFinrank_repr_applystatement and proof · cited by 0