imageToKernel_epi_comp — Mathlib

English

A compatibility under isomorphisms of B with Z yields a transformation of imageToKernel along Iso f i.hom and i.inv g by a standard iso composition.

Русский

Совместимость через изоморфизмы: imageToKernel через изоморфизм B ≅ Z трансформируется аналогично.

LaTeX

$$[HasEqualizers V] [HasImages V] → imageToKernel (f ≫ i.hom) (i.inv ≫ g) w = (imageSubobjectCompIso _ _).hom ≫ imageToKernel f g (by simpa using w) ≫ (kernelSubobjectIsoComp i.inv g).inv$$

Lean4

@[simp]
theorem imageToKernel_epi_comp {Z : V} (h : Z ⟶ A) [Epi h] (w) :
    imageToKernel (h ≫ f) g w =
      Subobject.ofLE _ _ (imageSubobject_comp_le h f) ≫
        imageToKernel f g ((cancel_epi h).mp (by simpa using w : h ≫ f ≫ g = h ≫ 0)) :=
  by
  ext
  simp

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