Section Submodule.
Variable U : 'M[F]_n.
Definition val_submod m : 'M_(m, \rank U) -> 'M_(m, n) := mulmxr (row_base U).
Definition in_submod m : 'M_(m, n) -> 'M_(m, \rank U) :=
mulmxr (invmx (row_ebase U) *m pid_mx (\rank U)).
Canonical val_submod_linear m := [linear of @val_submod m].
Canonical in_submod_linear m := [linear of @in_submod m].
Definition val_factmod m : _ -> 'M_(m, n) :=
mulmxr (row_base (cokermx U) *m row_ebase U).
Definition in_factmod m : 'M_(m, n) -> _ := mulmxr (col_base (cokermx U)).
Canonical val_factmod_linear m := [linear of @val_factmod m].
Canonical in_factmod_linear m := [linear of @in_factmod m].
Definition submod_mx of mxmodule U :=
fun x => in_submod (val_submod 1%:M *m rG x).
Definition factmod_mx of mxmodule U :=
fun x => in_factmod (val_factmod 1%:M *m rG x).
Hypothesis Umod : mxmodule U.
Canonical submod_repr := MxRepresentation submod_mx_repr.
Canonical factmod_repr := MxRepresentation factmod_mx_repr.
End Submodule.
(* Properties of enveloping algebra as a subspace of 'rV_(n ^ 2). *)
(* Module homomorphisms; any square matrix f defines a module homomorphism   *)
(* over some domain, namely, dom_hom_mx f.                                   *)
Definition dom_hom_mx f : 'M_n :=
kermx (lin1_mx (mxvec \o mulmx (cent_mx_fun E_G f) \o lin_mul_row)).

Lemma hom_mxP m f (W : 'M_(m, n)) :
reflect (forall x, x \in G -> W *m rG x *m f = W *m f *m rG x)
(W <= dom_hom_mx f)%MS.