proposition: Lemma ahom_inP {f : 'Hom(aT, rT)} {U : {vspace aT}} : reflect ({in U &, {morph f : x y / x * y >-> x * y}} * (f 1 = 1)) (ahom_in U f).
proof: Proof. apply: (iffP andP) => [[/allrelP fM /eqP f1] | [fM f1]]; last first. rewrite f1; split=> //; apply/allrelP => x y Ax Ay. by rewrite fM // vbasis_mem. split=> // x y /coord_vbasis -> /coord_vbasis ->. rewrite !mulr_suml ![f _]linear_sum mulr_suml; apply: eq_bigr => i _ /=. rewrite !mulr_sumr linear_sum; apply: eq_bigr => j _ /=. rewrite !linearZ -!scalerAr -!scalerAl 2!linearZ /=; congr (_ *: (_ *: _)). by apply/eqP/fM; apply: memt_nth. Qed.
imports: HB(structures) mathcomp(ssreflect ssrfun ssrbool eqtype ssrnat seq path) mathcomp(choice fintype div tuple finfun bigop ssralg) mathcomp(finalg zmodp matrix vector poly)