Theorems · Theorem · commutative algebra

Submodule.spanRank_span_range_of_linearIndependent

∀ {R : Type u_1} {M : Type u} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] [RankCondition R]
  {ι : Type u} {v : ι → M},
  Function.Injective v → LinearIndependent R v → (Submodule.span R (Set.range v)).spanRank = Cardinal.mk ι

Defined in: Mathlib.Algebra.Module.SpanRank
Cited by: 1 results in Mathlib
Foundations: Depth 104 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: Semiring AddCommMonoid Module RankCondition

Mathlib docs Source Loogle LeanSearch

Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

Cites38

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

DFunLike.coeproof · cited by 62,936
Setproof · 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.topproof · cited by 9,680
Set.Elemproof · cited by 7,166
Set.ofPredproof · cited by 6,101
Set.preimageproof · cited by 4,946
Set.rangestatement and proof · cited by 4,705
Cardinalstatement and proof · cited by 2,598
le_antisymmproof · cited by 2,068

Cited by1

Results whose statement or proof uses this declaration.

Submodule.spanRank_span_of_linearIndepOnproof · cited by 1