This note uses some recent powerful tools related with semidirect products V * W of pseudovarieties of semigroups, particularly when the second factor is the pseudovariety G of all ÿnite groups, to give syntactic proofs of equalities of the form V * G=EV, where EV denotes the pseudovariety consisting of the ÿnite semigroups whose idempotents generate subsemigroups from V. Subpseudovarieties V of DS are considered, including all subpseudovarieties of LI; DA; DS itself, and J, the latter providing a new proof of a result of Henckell and Rhodes.