kernelSubobject_comp_mono — Mathlib

English

If h is a monomorphism, then the kernel subobject does not change under postcomposition: kernelSubobject (f ≫ h) = kernelSubobject f.

Русский

Если h моноодино, то ядро не меняется при постпостпроекции: kernelSubobject(f ≫ h) = kernelSubobject f.

LaTeX

$$$\operatorname{kernelSubobject}(f \circ h) = \operatorname{kernelSubobject}(f)$, when $h$ is mono$$

Lean4

/-- Postcomposing by a monomorphism does not change the kernel subobject. -/
@[simp]
theorem kernelSubobject_comp_mono (f : X ⟶ Y) [HasKernel f] {Z : C} (h : Y ⟶ Z) [Mono h] :
    kernelSubobject (f ≫ h) = kernelSubobject f :=
  le_antisymm (le_kernelSubobject _ _ ((cancel_mono h).mp (by simp))) (kernelSubobject_comp_le f h)

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