proposition: Lemma trivg_Mho G : 'Mho^n(G) == 1 -> 'Ohm_n(G) == G.
proof: Proof. rewrite -subG1 gen_subG eqEsubset Ohm_sub /= => Gp1. rewrite -{1}(Sylow_gen G) genS //; apply/bigcupsP=> P. case/SylowP=> p p_pr /and3P[sPG pP _]; apply/subsetP=> x Px. have Gx := subsetP sPG x Px; rewrite inE Gx //=. rewrite (sameP eqP set1P) (subsetP Gp1) ?mem_gen //; apply: imset_f. by rewrite inE Gx; apply: pgroup_p (mem_p_elt pP Px). Qed.
imports: mathcomp(ssreflect ssrbool ssrfun eqtype ssrnat seq path choice) mathcomp(div fintype finfun bigop finset prime binomial) mathcomp(fingroup morphism perm automorphism action) mathcomp(quotient gfunctor gproduct ssralg countalg finalg zmodp) mathcomp(cyclic pgroup gseries nilpotent sylow)