isClosed_range_of_stableUnderSpecialization

English

For a ring hom f, if the range of comap f is stable under specialization, then its range is closed.

Русский

Для кольца гомоморфизма f, если образ comap f стабилен по специализации, он является замкнутым.

LaTeX

$$$$ \operatorname{StableUnderSpecialization}( \operatorname{range}(\operatorname{comap} f) ) \Rightarrow \operatorname{IsClosed}( \operatorname{range}(\operatorname{comap} f) ). $$$$

Lean4

@[stacks 00HY]
theorem isClosed_range_of_stableUnderSpecialization (hf : StableUnderSpecialization (Set.range (comap f))) :
    IsClosed (Set.range (comap f)) := by
  rw [← Set.image_univ] at hf ⊢
  exact isClosed_image_of_stableUnderSpecialization _ _ isClosed_univ hf

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