Theorems · Theorem · linear algebra
Module.finrank_eq_card_basis
∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] [StrongRankCondition R]
{ι : Type w} [inst_4 : Fintype ι] (h : Module.Basis ι R M), Module.finrank R M = Fintype.card ιIf a vector space (or module) has a finite basis, then its dimension (or rank) is equal to the cardinality of the basis.
- Cited by
- 40 results in Mathlib
- Foundations
- Depth 113 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites10
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
- Module.finrankstatement · cited by 1,770
- Module.Basisstatement and proof · cited by 1,477
- Fintype.cardstatement · cited by 1,386
- StrongRankConditionstatement and proof · cited by 286
- Module.finrank_eq_of_rank_eqproof · cited by 13
- rank_eq_card_basisproof · cited by 8
Cited by40
Results whose statement or proof uses this declaration.
- Submodule.eq_top_of_finrank_eqproof · cited by 15
- PowerBasis.finrankproof · cited by 14
- Complex.finrank_real_complexproof · cited by 11
- LinearMap.normDet_ne_zero_tfaeproof · cited by 6
- LinearMap.normDet_eq_zero_iff_ker_ne_botproof · cited by 5
- PeriodPair.finrank_latticeproof · cited by 4
- LinearMap.nilRank_le_cardproof · cited by 4
- Module.card_eq_pow_finrankproof · cited by 4
- Ideal.finrank_eq_finrankproof · cited by 3
- LinearMap.det_eq_one_of_finrank_eq_zeroproof · cited by 3
- Algebra.trace_algebraMapproof · cited by 3
- finrank_eq_one_iff_of_nonzeroproof · cited by 3