Lemma islinv_weq_bck {X Y : UU} (H : X ≃ Y) (opp : binop Y) (x0 : Y) (inv : Y → Y) : islinv opp x0 inv → islinv (binop_weq_bck H opp) (invmap H x0) (λ y : X, invmap H (inv (H y))). Proof. intros is y. unfold binop_weq_bck. apply maponpaths. refine (_ @ is _). apply (maponpaths (λ x, opp x _)). apply homotweqinvweq. Defined.