Lemma isbinop2funtoringdiff (X : rig) : @isbinopfun (rigmultmonoid X) (ringmultmonoid (rigtoring X)) (toringdiff X). Proof. intros. unfold isbinopfun. intros x x'. change (setquotpr (eqrelrigtoring X) (x * x' ,, 0) = setquotpr _ (rigtoringop2int X (x ,, 0) (x' ,, 0))). apply (maponpaths (setquotpr _)). unfold rigtoringop2int. simpl. apply pathsdirprod. - rewrite (rigmultx0 X _). rewrite (rigrunax1 X _). apply idpath. - rewrite (rigmult0x X _). rewrite (rigmultx0 X _). rewrite (rigrunax1 X _). apply idpath. Defined.
Require Import UniMath.Algebra.Groups. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders.