proposition: Lemma MappingCylinderisoHomot_eq3 {C1 C2 : Complex A} (f : (ComplexPreCat_Additive A)⟦C1, C2⟧) (i : hz) : let DS1 := to_BinDirectSums A (C1 (i + 1)) (C2 i) in let DS2 := to_BinDirectSums A (C1 i) DS1 in to_binop DS2 DS2 (to_binop DS2 DS2 (to_Pr1 DS2 · to_In1 DS2) (to_Pr2 DS2 · to_Pr1 DS1 · to_In1 DS1 · to_In2 DS2)) (to_inv (to_Pr1 DS2 · MMor f i · to_In2 DS1 · to_In2 DS2)) = to_binop DS2 DS2 (MappingCylinderIsoHomot_mor1 f i) (MappingCylinderIsoHomot_mor2 f i).
proof: Proof. intros DS1 DS2. unfold MappingCylinderIsoHomot_mor1. unfold MappingCylinderIsoHomot_mor2. cbn. fold DS1 DS2. cbn. rewrite assoc. rewrite assoc. rewrite assoc. rewrite assoc. rewrite assoc. rewrite assoc. rewrite assoc. rewrite assoc. rewrite to_assoc. rewrite to_assoc. apply maponpaths. rewrite to_commax'. rewrite (to_commax' _ _ ((to_Pr1 DS2) · (to_inv (MMor f i)) · (to_In2 DS1) · (to_In2 DS2))). rewrite to_assoc. rewrite <- PreAdditive_invrcomp. rewrite <- PreAdditive_invlcomp. rewrite <- PreAdditive_invlcomp. apply maponpaths. rewrite <- to_assoc. rewrite <- PreAdditive_invrcomp. rewrite <- PreAdditive_invlcomp. rewrite <- PreAdditive_invlcomp. rewrite (@to_rinvax' A (Additive.to_Zero A)). rewrite to_lunax''. 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