proposition: Lemma AbelianPreCat_homset_zero_left {X Y : A} (f : X --> Y) : Abelian_op _ _ (AbelianPreCat_homset_zero X Y) f = f.
proof: Proof. rewrite <- (AbelianPreCat_homset_inv_left' f). set (tmp := AbelianPreCat_homset_zero_right' f). apply (maponpaths (λ g : _, Abelian_op X Y (Abelian_minus_op f f) g)) in tmp. apply pathsinv0 in tmp. use (pathscomp0 tmp). unfold Abelian_op. rewrite AbelianPreCat_op_eq3. rewrite AbelianPreCat_homset_zero_right'. rewrite AbelianPreCat_homset_zero_right'. rewrite AbelianPreCat_homset_inv_left'. rewrite AbelianPreCat_homset_zero_right'. apply idpath. Qed.
imports: UniMath.Foundations.PartD UniMath.Foundations.Propositions UniMath.Foundations.Sets UniMath.Algebra.Monoids UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.PrecategoryBinProduct UniMath.CategoryTheory.Limits.Zero UniMath.CategoryTheory.Limits.BinProducts UniMath.CategoryTheory.Limits.BinCoproducts UniMath.CategoryTheory.Limits.Equalizers UniMath.CategoryTheory.Limits.Coequalizers UniMath.CategoryTheory.Limits.Kernels UniMath.CategoryTheory.Limits.Cokernels UniMath.CategoryTheory.Limits.Pushouts UniMath.CategoryTheory.Limits.Pullbacks UniMath.CategoryTheory.Limits.BinDirectSums UniMath.CategoryTheory.Monics UniMath.CategoryTheory.Epis UniMath.CategoryTheory.CategoriesWithBinOps UniMath.CategoryTheory.PrecategoriesWithAbgrops UniMath.CategoryTheory.PreAdditive UniMath.CategoryTheory.Additive UniMath.CategoryTheory.Abelian