proposition: Lemma MappingCylinder_Diff1_Diff2 {C1 C2 : Complex A} (f : Morphism C1 C2) (i : hz) : (MappingCylinderDiff1 f i) · (MappingCylinderDiff2 f (i + 1)) = ZeroArrow (Additive.to_Zero A) _ _.
proof: Proof. unfold MappingCylinderDiff1. unfold MappingCylinderDiff2. cbn. rewrite assoc. rewrite assoc. rewrite assoc. rewrite <- (assoc _ _ (to_Pr2 _)). set (DS := to_BinDirectSums A (C1 (i + 1)) (to_BinDirectSums A (C1 (i + 1 + 1)) (C2 (i + 1)))). rewrite (to_Unel1' DS). rewrite ZeroArrow_comp_right. rewrite ZeroArrow_comp_left. apply ZeroArrow_comp_left. 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