proposition: Lemma total_braiding_iso {B : braiding M} (DB : disp_braiding B) : braiding_iso (total_braiding_data (disp_braiding_to_braiding DB)) (total_braiding_data (disp_braiding_to_braiding_inv DB)).
proof: Proof. intros [x xx] [y yy]. set (i := is_z_iso_total (total_braiding_data (disp_braiding_to_braiding DB) (x,, xx) (y,, yy)) (_,, monoidal_braiding_inverses B x y : is_z_isomorphism (pr1 B x y))). exact (pr2 (i (_ ,, disp_braiding_to_inverses DB _ _ xx yy))). Qed.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.DisplayedCats.Core UniMath.CategoryTheory.DisplayedCats.Functors UniMath.CategoryTheory.DisplayedCats.Isos UniMath.CategoryTheory.DisplayedCats.NaturalTransformations UniMath.CategoryTheory.DisplayedCats.Total UniMath.CategoryTheory.DisplayedCats.Projection 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.TransportLemmas UniMath.CategoryTheory.Monoidal.Structure.Symmetric