Lemma module_mult_neg1 {R : ring} {M : module R} (x : M) : ringminus1 * x = @grinv _ x. Proof. apply pathsinv0. apply grinv_path_from_op_path. refine (maponpaths (λ y, y * ((_ * x)%module))%multmonoid (!(module_mult_unel2 x)) @ _). now rewrite <- module_mult_is_rdistr, ringrinvax1, module_mult_0_to_0. Defined.
Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Algebra.RigsAndRings. Require Import UniMath.Algebra.Groups. Require Import UniMath.Algebra.Monoids.