finite_of_isNoetherian — Mathlib

English

In a Noetherian module M over a ring R with R nontrivial, any linearly independent family v: ι → M is finite.

Русский

В модулярном пространстве М над кольцом R, где R ненулевое, любая линейно независимая семья вьюшек конечна.

LaTeX

$$If hv : LinearIndependent R v, then Finite ι.$$

Lean4

/-- A linearly-independent family of vectors in a module over a non-trivial ring must be finite if
the module is Noetherian. -/
theorem finite_of_isNoetherian [Nontrivial R] {ι} {v : ι → M} (hv : LinearIndependent R v) : Finite ι :=
  by
  refine WellFoundedGT.finite_of_iSupIndep hv.iSupIndep_span_singleton fun i contra => ?_
  apply hv.ne_zero i
  have : v i ∈ R ∙ v i := Submodule.mem_span_singleton_self (v i)
  rwa [contra, Submodule.mem_bot] at this

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