Theorems · Theorem · linear algebra
Module.Basis.finite_index_of_rank_lt_aleph0
∀ {R : Type u} {M : Type v} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] [StrongRankCondition R]
{ι : Type u_1} {s : Set ι} (b : Module.Basis (↑s) R M), Module.rank R M < Cardinal.aleph0 → s.FiniteIf a module has a finite dimension, all bases are indexed by a finite set.
- Defined in
- Mathlib.LinearAlgebra.Dimension.Finite
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 114 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites13
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement and proof · cited by 53,352
- Modulestatement and proof · cited by 20,661
- Semiringstatement and proof · cited by 13,802
- AddCommMonoidstatement and proof · cited by 12,281
- Set.Elemstatement and proof · cited by 7,166
- Cardinalstatement · cited by 2,598
- Set.Finitestatement · cited by 1,814
- Module.Basisstatement and proof · cited by 1,477
- Cardinal.aleph0statement and proof · cited by 521
- Module.rankstatement and proof · cited by 496
- StrongRankConditionstatement and proof · cited by 286
- Set.finite_defproof · cited by 4
Cited by1
Results whose statement or proof uses this declaration.
- IsNoetherian.finite_basis_indexproof · cited by 0