proposition: Lemma AbelianPreCat_homset_inv_left' {X Y : A} (f : X --> Y) : Abelian_minus_op f f = (AbelianPreCat_homset_zero X Y).
proof: Proof. unfold AbelianPreCat_homset_zero. unfold Abelian_minus_op. set (bpY := to_BinProducts A Y Y). assert (e : BinProductArrow A bpY f f = f · BinProductArrow A bpY (identity _ ) (identity _ )). { use BinProductArrowsEq. - rewrite BinProductPr1Commutes. rewrite <- assoc. rewrite BinProductPr1Commutes. rewrite id_right. apply idpath. - rewrite BinProductPr2Commutes. rewrite <- assoc. rewrite BinProductPr2Commutes. rewrite id_right. apply idpath. } rewrite e. rewrite <- assoc. rewrite CokernelCompZero. apply ZeroArrow_comp_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