proposition: Lemma MappingCylinderIso_eq1' {C1 C2 : Complex A} (f : (ComplexPreCat_Additive A)⟦C1, C2⟧) : (MappingCylinderMor1 f) · (MappingCylinderMor2 f) = identity _.
proof: Proof. use MorphismEq. intros i. cbn. unfold MappingCylinderMor1_mor. unfold MappingCylinderMor2_mor. cbn. set (DS1 := to_BinDirectSums A (C1 (i + 1)) (C2 i)). set (DS2 := to_BinDirectSums A (C1 i) DS1). rewrite to_premor_linear'. rewrite assoc. rewrite <- (assoc _ _ (to_Pr1 DS2)). rewrite (to_Unel2' DS2). rewrite ZeroArrow_comp_right. rewrite ZeroArrow_comp_left. rewrite to_lunax''. rewrite assoc. rewrite <- (assoc _ _ (to_Pr2 DS2)). rewrite (to_IdIn2 DS2). rewrite id_right. rewrite (to_IdIn2 DS1). apply idpath. Qed.
imports: 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.Core.TransportMorphisms 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.Adjunctions.Core UniMath.CategoryTheory.Equivalences.Core UniMath.CategoryTheory.CategoriesWithBinOps UniMath.CategoryTheory.PrecategoriesWithAbgrops UniMath.CategoryTheory.PreAdditive UniMath.CategoryTheory.Additive UniMath.CategoryTheory.Abelian UniMath.CategoryTheory.AbelianToAdditive UniMath.CategoryTheory.AdditiveFunctors UniMath.HomologicalAlgebra.Complexes UniMath.HomologicalAlgebra.KA UniMath.HomologicalAlgebra.TranslationFunctors UniMath.HomologicalAlgebra.MappingCone