preComp_epi_of_epi — Mathlib

English

If HasImage f and Epi (image.ι f) and Epi (factorThruImage f) then Epi f.

Русский

Если образ f существует и image.ι f эпиморфизм, и factorThruImage f эпиморфизм, то f эпиморфизм.

LaTeX

$$$ [HasImage f] \\; [Epi (image.ι f)] \\; [Epi (factorThruImage f)] \\Rightarrow Epi f $$$

Lean4

/-- `image.preComp f g` is an epimorphism when `f` is an epimorphism
(we need `C` to have equalizers to prove this).
-/
instance preComp_epi_of_epi [HasImage g] [HasImage (f ≫ g)] [Epi f] : Epi (image.preComp f g) :=
  by
  apply @epi_of_epi_fac _ _ _ _ _ _ _ _ ?_ (image.factorThruImage_preComp _ _)
  exact epi_comp _ _

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