Theorems · Theorem · linear algebra

linearIndependent_of_top_le_span_of_card_le_finrank

∀ {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 → LinearIndependent R b

Defined in: Mathlib.LinearAlgebra.Dimension.OrzechProperty
Cited by: 1 results in Mathlib
Foundations: Depth 98 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: Semiring OrzechProperty AddCommMonoid Module Fintype

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Cites24

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

DFunLike.coeproof · cited by 62,936
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 and proof · cited by 7,192
Set.rangestatement and proof · cited by 4,705
Module.finrankstatement and proof · cited by 1,770
Submodule.spanstatement and proof · cited by 1,504
Fintype.cardstatement and proof · cited by 1,386
Module.Finiteproof · cited by 1,032

Cited by1

Results whose statement or proof uses this declaration.

linearIndependent_of_top_le_span_of_card_eq_finrankproof · cited by 4