proposition: Lemma AbelianPreCat_homset_zero_right' {X Y : A} (f : X --> Y) : Abelian_minus_op f (AbelianPreCat_homset_zero X Y) = f.
proof: Proof. unfold AbelianPreCat_homset_zero. unfold Abelian_minus_op. set (tmp := Abelian_op_eq_IdZero Y). unfold IdZeroMap in tmp. assert (e : BinProductArrow A (to_BinProducts A Y Y) f (ZeroArrow (to_Zero A) X Y) = f · BinProductArrow A (to_BinProducts A Y Y) (identity _) (ZeroArrow (to_Zero A) Y Y)). { apply BinProductArrowsEq. - rewrite BinProductPr1Commutes. rewrite <- assoc. rewrite BinProductPr1Commutes. rewrite id_right. apply idpath. - rewrite BinProductPr2Commutes. rewrite <- assoc. rewrite BinProductPr2Commutes. rewrite ZeroArrow_comp_right. apply idpath. } rewrite e. clear e. rewrite <- assoc. rewrite tmp. apply id_right. 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