Theorems · Theorem · linear algebra
Module.Free.rank_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], Module.rank R M = Cardinal.mk (Module.Free.ChooseBasisIndex R M)
The rank of a free module M over R is the cardinality of ChooseBasisIndex R M.
- Defined in
- Mathlib.LinearAlgebra.Dimension.Free
- Cited by
- 10 results in Mathlib
- Foundations
- Depth 111 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites11
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
- Cardinalstatement · cited by 2,598
- Cardinal.mkstatement · cited by 942
- Module.Freestatement and proof · cited by 597
- Module.rankstatement · cited by 496
- StrongRankConditionstatement and proof · cited by 286
- Module.Free.ChooseBasisIndexstatement · cited by 133
- Module.Free.chooseBasisproof · cited by 121
- Module.Basis.mk_eq_rank''proof · cited by 23
Cited by10
Results whose statement or proof uses this declaration.
- Module.finrank_eq_card_chooseBasisIndexproof · cited by 21
- Module.rank_lt_aleph0_iffproof · cited by 8
- Module.finrank_pi_fintypeproof · cited by 6
- Module.length_of_freeproof · cited by 3
- rank_prodproof · cited by 2
- Module.finrank_directSumproof · cited by 2
- Subalgebra.LinearDisjoint.adjoin_rank_eq_rank_leftproof · cited by 2
- Subalgebra.adjoin_rank_leproof · cited by 1
- ModuleCat.free_shortExact_rank_addproof · cited by 1
- exists_set_linearIndependent_of_isDomainproof · cited by 0