Theorems · Theorem · commutative algebra

Module.Finite.finite_basis

∀ {R : Type u_1} {M : Type u_2} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] [Nontrivial R]
  {ι : Type u_3} [Module.Finite R M] (b : Module.Basis ι R M), Finite ι

If a free module is finite, then any arbitrary basis is finite.

Defined in: Mathlib.RingTheory.Finiteness.Cardinality
Cited by: 8 results in Mathlib
Foundations: Depth 84 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: Semiring AddCommMonoid Module Nontrivial Module.Finite

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Cites13

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
Finsetproof · cited by 13,712
AddCommMonoidstatement and proof · cited by 12,281
Top.topproof · cited by 9,680
SetLike.coeproof · cited by 8,199
Finitestatement and proof · cited by 3,029
Nontrivialstatement and proof · cited by 2,416
Submodule.spanproof · cited by 1,504
Module.Basisstatement and proof · cited by 1,477
Module.Finitestatement and proof · cited by 1,032
Finset.finite_toSetproof · cited by 210

Cited by8

Results whose statement or proof uses this declaration.

LinearMap.BilinForm.Nondegenerate.ofSeparatingLeftproof · cited by 2
ZLattice.exists_forall_abs_repr_le_normproof · cited by 2
Algebra.IsStandardSmooth.of_basis_kaehlerDifferentialproof · cited by 2
Algebra.IsSmoothAt.exists_notMem_isStandardSmoothproof · cited by 2
Module.not_finite_of_infinite_basisproof · cited by 1
Module.exists_basis_of_basis_baseChangeproof · cited by 1
LinearMap.exists_basis_basis_of_span_eq_top_of_mem_algebraMapproof · cited by 1
Module.Basis.nonempty_unique_index_of_finrank_eq_oneproof · cited by 0