Theorems · Theorem · linear algebra
finrank_span_eq_card
∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] [StrongRankCondition R]
[Nontrivial R] {ι : Type u_2} [inst_5 : Fintype ι] {b : ι → M},
LinearIndependent R b → Module.finrank R ↥(Submodule.span R (Set.range b)) = Fintype.card ι- Cited by
- 7 results in Mathlib
- Foundations
- Depth 114 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites25
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Modulestatement and proof · cited by 20,661
- Semiringstatement and proof · cited by 13,802
- AddCommMonoidstatement and proof · cited by 12,281
- Fintypestatement and proof · cited by 7,736
- Submodulestatement · cited by 7,192
- Set.Elemproof · cited by 7,166
- Set.rangestatement and proof · cited by 4,705
- Cardinalproof · cited by 2,598
- Nontrivialstatement and proof · cited by 2,416
- Module.finrankstatement · cited by 1,770
- Submodule.spanstatement and proof · cited by 1,504
- Fintype.cardstatement and proof · cited by 1,386
Cited by7
Results whose statement or proof uses this declaration.
- linearIndependent_iff_card_eq_finrank_spanproof · cited by 5
- LinearIndependent.span_eq_top_of_card_eq_finrank'proof · cited by 2
- EuclideanGeometry.eq_of_dist_eq_of_dist_eq_of_mem_of_finrank_eq_twoproof · cited by 2
- Projectivization.isCollinear_pairproof · cited by 1
- card_linearIndependentproof · cited by 1
- Projectivization.line_unique'proof · cited by 1
- GeneralSchauderBasis.finrank_range_projproof · cited by 1