Let rVabelemJmx v x : x \in G -> rV_E (v *m r x) = (rV_E v) ^ x.
Fact abelem_mx_repr : mx_repr G r.
Canonical abelem_repr := MxRepresentation abelem_mx_repr.
Let rG := abelem_repr.

(* Specifications: *)
Lemma rVabelemJ v x : x \in G -> rV_E (v *m rG x) = (rV_E v) ^ x.
Lemma abelem_rowgJ m (A : 'M_(m, n)) x :
x \in G -> rV_E @* rowg (A *m rG x) = (rV_E @* rowg A) :^ x.
Lemma rV_abelem_sJ (L : {group gT}) x :
x \in G -> L \subset E -> ErV @* (L :^ x) = rowg (rowg_mx (ErV @* L) *m rG x).
Lemma rstab_abelem m (A : 'M_(m, n)) : rstab rG A = 'C_G(rV_E @* rowg A).
Lemma rstabs_abelem m (A : 'M_(m, n)) : rstabs rG A = 'N_G(rV_E @* rowg A).
Lemma rstabs_abelemG (L : {group gT}) :
L \subset E -> rstabs rG (rowg_mx (ErV @* L)) = 'N_G(L).
Lemma mxsimple_abelemGP (L : {group gT}) :
L \subset E -> reflect (mxsimple rG (rowg_mx (ErV @* L))) (minnormal L G).
Lemma abelem_mx_irrP : reflect (mx_irreducible rG) (minnormal E G).
Lemma rfix_abelem (H : {set gT}) :
H \subset G -> (rfix_mx rG H :=: rowg_mx (ErV @* 'C_E(H)%g))%MS.
Lemma rker_abelem : rker rG = 'C_G(E).
Lemma rsim_abelem_subg : mx_rsim rHG rH.
Lemma mxmodule_abelem_subg m (U : 'M_(m, n)) : mxmodule rHG U = mxmodule rH U.
Lemma mxsimple_abelem_subg U : mxsimple rHG U <-> mxsimple rH U.
End SubGroup.