proposition: Lemma sigma_disp_cat_monoidal_braiding_laws : disp_braiding_laws (sigma_disp_cat_monoidal EM E_prop) sigma_disp_cat_monoidal_braiding_data sigma_disp_cat_monoidal_braiding_data.
proof: Proof. repeat split ; try (intro ; intros) ; use total2_paths_f ; try (apply E_prop). - etrans. { apply pr1_transportf. } apply (disp_braiding_to_naturality_left DM DS). - etrans. { apply pr1_transportf. } apply (disp_braiding_to_naturality_right DM DS). - etrans. { apply (disp_braiding_to_inverses DM DS). } apply pathsinv0, pr1_transportf. - etrans. { apply (disp_braiding_to_inverses DM DS). } etrans. 2: { apply pathsinv0, pr1_transportf. } unfold transportb. apply maponpaths_2. apply homset_property. - etrans. 2: { apply (disp_braiding_to_hexagon1 _ DS). } apply pr1_transportf. - etrans. 2: { apply (disp_braiding_to_hexagon2 _ DS). } apply pr1_transportf. Qed.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.DisplayedCats.Core UniMath.CategoryTheory.DisplayedCats.Total UniMath.CategoryTheory.DisplayedCats.Constructions UniMath.CategoryTheory.DisplayedCats.Projection UniMath.CategoryTheory.DisplayedCats.Examples.Sigma UniMath.CategoryTheory.PrecategoryBinProduct UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors UniMath.CategoryTheory.Monoidal.Categories UniMath.CategoryTheory.Monoidal.Functors UniMath.CategoryTheory.Monoidal.Displayed.WhiskeredDisplayedBifunctors UniMath.CategoryTheory.Monoidal.Displayed.Monoidal UniMath.CategoryTheory.Monoidal.Displayed.TotalMonoidal UniMath.CategoryTheory.Monoidal.Displayed.Symmetric UniMath.CategoryTheory.Monoidal.Examples.Fullsub UniMath.CategoryTheory.Monoidal.Structure.Symmetric