antitone_chain_condition_of_isArtinianObject

English

Assuming X is Artinian, every decreasing sequence of subobjects of X stabilizes after finitely many steps.

Русский

Полагаем, что X артинян, тогда любая нисходящая последовательность подобъектов X стабилизируется за finitely many steps.

LaTeX

$$$[IsArtinianObject\\; X] \\Rightarrow \\exists n\\; \\forall m\\ge n, f(n)=f(m)$ for every chain $f: \\mathbb{N} \\to (\\mathrm{Subobject}(X))$$$

Lean4

theorem antitone_chain_condition_of_isArtinianObject [IsArtinianObject X] (f : ℕ →o (Subobject X)ᵒᵈ) :
    ∃ (n : ℕ), ∀ (m : ℕ), n ≤ m → f n = f m :=
  (isArtinianObject_iff_antitone_chain_condition X).1 inferInstance f

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