Theorems · Theorem · linear algebra
linearIndependent_le_basis
∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] [StrongRankCondition R]
{ι : Type w} (b : Module.Basis ι R M) {κ : Type w} (v : κ → M), LinearIndependent R v → Cardinal.mk κ ≤ Cardinal.mk ιOver any ring R satisfying the strong rank condition,
if b is a basis for a module M,
and s is a linearly independent set,
then the cardinality of s is bounded by the cardinality of b.
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 109 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites22
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
- Fintypeproof · cited by 7,736
- 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
- Cardinal.mkstatement and proof · cited by 942
- LinearIndependentstatement and proof · cited by 560
- Infiniteproof · cited by 352
Cited by4
Results whose statement or proof uses this declaration.
- Module.Basis.mk_eq_rank''proof · cited by 23
- card_le_of_injective''proof · cited by 1
- maximal_linearIndependent_eq_infinite_basisproof · cited by 0
- Basis.card_le_card_of_linearIndependent_auxproof · cited by 0