injective_of_mono — Mathlib

English

In AddCommGrpCat, any mono morphism f: G ⟶ H is injective as a function.

Русский

В AddCommGrpCat любая моно-моромафия f: G ⟶ H инъективна как функция.

LaTeX

$$$$ \forall G,H,\; (f: G \to H)\ [\mathrm{Mono} f] \Rightarrow \mathrm{Injective}(f). $$$$

Lean4

theorem injective_of_mono {G H : AddCommGrpCat.{0}} (f : G ⟶ H) [Mono f] : Function.Injective f := fun g₁ g₂ h =>
  by
  have t0 : asHom g₁ ≫ f = asHom g₂ ≫ f := by cat_disch
  have t1 : asHom g₁ = asHom g₂ := (cancel_mono _).1 t0
  apply asHom_injective t1

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