proposition: Lemma TriangulatedFiveLemma {D1 D2 : @DTri PT} (M : TriMor D1 D2) (H1 : is_z_isomorphism (MPMor1 M)) (H2 : is_z_isomorphism (MPMor2 M)) : is_z_isomorphism (MPMor3 M).
proof: Proof. set (Mor1 := TriangulatedMorphism_from_object M (Ob3 D2)). set (Mor2 := TriangulatedMorphism_to_object M (Ob3 D1)). assert (e1 : is_z_isomorphism (@FMor3 abgr_Abelian _ _ Mor1)). { use FiveLemma. - exact (@abgr_Additive_is_iso_postmor PT _ _ _ _ H1). - exact (@abgr_Additive_is_iso_postmor PT _ _ _ _ H2). - exact (abgr_Additive_is_iso_postmor (Ob3 D2) _ _ (functor_on_is_z_isomorphism (AddEquiv1 (@Trans PT)) H1)). - exact (abgr_Additive_is_iso_postmor (Ob3 D2) _ _ (functor_on_is_z_isomorphism (AddEquiv1 (@Trans PT)) H2)). } assert (e2 : is_z_isomorphism (@FMor3 abgr_Abelian _ _ Mor2)). { use FiveLemma. - exact (abgr_Additive_is_iso_premor _ _ (Ob3 D1) (functor_on_is_z_isomorphism (AddEquiv1 (@Trans PT)) H2)). - exact (abgr_Additive_is_iso_premor _ _ (Ob3 D1) (functor_on_is_z_isomorphism (AddEquiv1 (@Trans PT)) H1)). - exact (@abgr_Additive_is_iso_premor PT _ _ _ _ H2). - exact (@abgr_Additive_is_iso_premor PT _ _ _ _ H1). } exact (@abgr_Additive_premor_postmor_is_iso PT _ _ (MPMor3 M) e2 e1). 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