proposition: Lemma TriMor_is_iso_to_inv_Comm3 {D1 D2 : Tri} {M : TriMor D1 D2} (Ti : TriMor_is_iso M) : MPMor3 (make_MPMor (make_MPMorMors (is_z_isomorphism_mor (TriMor_is_iso1 Ti)) (is_z_isomorphism_mor (TriMor_is_iso2 Ti)) (is_z_isomorphism_mor (TriMor_is_iso3 Ti))) (TriMor_is_iso_to_inv_Comm Ti)) · Mor3 D1 = (Mor3 D2) · (# (AddEquiv1 T) (MPMor1 (make_MPMor (make_MPMorMors (is_z_isomorphism_mor (TriMor_is_iso1 Ti)) (is_z_isomorphism_mor (TriMor_is_iso2 Ti)) (is_z_isomorphism_mor (TriMor_is_iso3 Ti))) (TriMor_is_iso_to_inv_Comm Ti)))).
proof: Proof. cbn. use (pre_comp_with_z_iso_is_inj (TriMor_is_iso3 Ti)). rewrite assoc. rewrite (is_inverse_in_precat1 (TriMor_is_iso3 Ti)). rewrite id_left. use (post_comp_with_z_iso_is_inj (functor_on_is_z_isomorphism _ (TriMor_is_iso1 Ti))). rewrite <- assoc. rewrite <- assoc. rewrite <- functor_comp. rewrite (is_inverse_in_precat2 (TriMor_is_iso1 Ti)). rewrite functor_id. rewrite id_right. rewrite <- (DComm3 M). apply idpath. Qed.
imports: UniMath.Tactics.EnsureStructuredProofs UniMath.Foundations.UnivalenceAxiom UniMath.Foundations.PartD UniMath.Foundations.Propositions UniMath.Foundations.Sets UniMath.Foundations.NaturalNumbers UniMath.Algebra.BinaryOperations UniMath.Algebra.Monoids UniMath.NumberSystems.Integers UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Isos 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.Core.Functors UniMath.CategoryTheory.Abelian UniMath.CategoryTheory.ShortExactSequences UniMath.CategoryTheory.Categories.Abgr UniMath.CategoryTheory.CategoriesWithBinOps UniMath.CategoryTheory.PrecategoriesWithAbgrops UniMath.CategoryTheory.PreAdditive UniMath.CategoryTheory.Additive UniMath.CategoryTheory.Morphisms UniMath.CategoryTheory.AdditiveFunctors UniMath.CategoryTheory.FiveLemma