Theorems · Theorem · linear algebra
rank_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.rank R M = ↑(Fintype.card ι)If a vector space has a finite basis, then its dimension (seen as a cardinal) is equal to the cardinality of the basis.
- Cited by
- 8 results in Mathlib
- Foundations
- Depth 112 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites18
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
- Fintypestatement and proof · cited by 7,736
- Set.Elemproof · cited by 7,166
- Set.rangeproof · cited by 4,705
- Cardinalstatement and proof · cited by 2,598
- Nontrivialproof · cited by 2,416
- Module.Basisstatement and proof · cited by 1,477
- Fintype.cardstatement and proof · cited by 1,386
- Module.rankstatement · cited by 496
Cited by8
Results whose statement or proof uses this declaration.
- Module.finrank_eq_card_basisproof · cited by 40
- rank_eq_oneproof · cited by 3
- Module.Basis.card_le_card_of_linearIndependentproof · cited by 3
- QuaternionAlgebra.rank_eq_fourproof · cited by 2
- Algebra.SubmersivePresentation.rank_kaehlerDifferentialproof · cited by 2
- QuadraticAlgebra.rank_eq_twoproof · cited by 1
- Submodule.IsLattice.rank'proof · cited by 1
- LinearRecurrence.solSpace_rankproof · cited by 0