proposition: Lemma ComplexIsoIndexIso {C1 C2 : Complex A} (f : ComplexPreCat⟦C1, C2⟧) (H : ∏ (i : hz), is_iso (MMor f i)) : is_iso f.
proof: Proof. use is_iso_qinv. - use make_Morphism. + intros i. exact (iso_inv_from_is_iso _ (H i)). + intros i. cbn. use (post_comp_with_iso_is_inj _ _ _ (H (i + 1))). use (pre_comp_with_iso_is_inj _ _ _ _ (H i)). assert (e0 : MMor f i · inv_from_iso (MMor f i,, H i) = identity _). { apply (iso_inv_after_iso (make_iso _ (H i))). } rewrite assoc. rewrite assoc. rewrite e0. rewrite id_left. rewrite <- (MComm f i). apply cancel_precomposition. assert (e1 : inv_from_iso (MMor f (i + 1),, H (i + 1)) · MMor f (i + 1) = identity _). { apply (iso_after_iso_inv (make_iso _ (H (i + 1)))). } rewrite <- assoc. rewrite e1. rewrite id_right. apply idpath. - split. + use MorphismEq. intros i. cbn. apply (iso_inv_after_iso (make_iso _ (H i))). + use MorphismEq. intros i. cbn. apply (iso_after_iso_inv (make_iso _ (H i))). Qed.
imports: UniMath.Tactics.EnsureStructuredProofs UniMath.Foundations.UnivalenceAxiom UniMath.Foundations.PartD UniMath.Foundations.Propositions UniMath.Foundations.Sets UniMath.Foundations.NaturalNumbers UniMath.MoreFoundations.PartA UniMath.Algebra.BinaryOperations UniMath.Algebra.Monoids UniMath.Algebra.Groups 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.CategoriesWithBinOps UniMath.CategoryTheory.PrecategoriesWithAbgrops UniMath.CategoryTheory.PreAdditive UniMath.CategoryTheory.Additive UniMath.CategoryTheory.Abelian UniMath.CategoryTheory.AbelianToAdditive