Theorems · Theorem · linear algebra
Module.finrank_eq_card_chooseBasisIndex
∀ (R : Type u) (M : Type v) [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] [inst_3 : Module.Free R M] [StrongRankCondition R] [inst_5 : Module.Finite R M], Module.finrank R M = Fintype.card (Module.Free.ChooseBasisIndex R M)
The finrank of a free module M over R is the cardinality of ChooseBasisIndex R M.
- Defined in
- Mathlib.LinearAlgebra.Dimension.Free
- Cited by
- 21 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.
Cites14
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
- Module.finrankstatement · cited by 1,770
- Fintype.cardstatement and proof · cited by 1,386
- Module.Finitestatement and proof · cited by 1,032
- Module.Freestatement and proof · cited by 597
- StrongRankConditionstatement and proof · cited by 286
- Cardinal.toNatproof · cited by 153
- Module.Free.ChooseBasisIndexstatement and proof · cited by 133
- Cardinal.mk_fintypeproof · cited by 81
Cited by21
Results whose statement or proof uses this declaration.
- ZLattice.rankproof · cited by 5
- LinearMap.trace_oneproof · cited by 4
- LinearMap.exists_isNilRegular_of_finrank_le_cardproof · cited by 3
- Algebra.norm_eq_norm_adjoinproof · cited by 3
- LinearMap.charpoly_natDegreeproof · cited by 3
- LinearMap.det_eq_sign_charpoly_coeffproof · cited by 2
- LinearMap.polyCharpoly_coeff_isHomogeneousproof · cited by 2
- LinearMap.polyCharpoly_natDegreeproof · cited by 2
- Module.rank_linearMapproof · cited by 2
- LinearMap.nilRank_le_finrankproof · cited by 2
- IsNilpotent.charpoly_eq_X_pow_finrankproof · cited by 2