proposition: Lemma Ohm1Eprime G : 'Ohm_1(G) = <<[set x in G | prime #[x]]>>.
proof: Proof. rewrite -['Ohm_1(G)](genD1 (group1 _)); congr <<_>>. apply/setP=> x; rewrite !inE andbCA -order_dvdn -order_gt1; congr (_ && _). apply/andP/idP=> [[p_gt1] | p_pr]; last by rewrite prime_gt1 ?pdiv_id. set p := pdiv _ => ox_p; have p_pr: prime p by rewrite pdiv_prime. by have [_ dv_p] := primeP p_pr; case/pred2P: (dv_p _ ox_p) p_gt1 => ->. 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)