mono_of_zero_of_map_zero — Mathlib

English

An epimorphism is surjective on pseudoelements.

Русский

Эпиморфизм на псевдоэлементам отображает на все элементы псевдоэлементов.

LaTeX

$$$f\\text{ epi} \\Rightarrow \\mathrm{pseudoApply}(f)$ is surjective$$

Lean4

/-- A morphism that only maps the zero pseudoelement to zero is a monomorphism. -/
theorem mono_of_zero_of_map_zero {P Q : C} (f : P ⟶ Q) : (∀ a, f a = 0 → a = 0) → Mono f := fun h =>
  (mono_iff_cancel_zero _).2 fun _ g hg =>
    (pseudoZero_iff (g : Over P)).1 <| h _ <| show f g = 0 from (pseudoZero_iff (g ≫ f : Over Q)).2 hg

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