Section Bigdproduct.
Variables (gT : finGroupType) (I : finType) (P : pred I).
Variables (A : I -> {group gT}) (G : {group gT}).
Hypothesis defG : \big[dprod/1%g]_(i | P i) A i = G.
Let sAG i : P i -> A i \subset G.
Definition cfBigdprodi i := cfDprodl (cfBigdprodi_subproof i) \o 'Res[_, A i].
Canonical cfBigdprodi_additive i := [additive of @cfBigdprodi i].
Canonical cfBigdprodi_linear i := [linear of @cfBigdprodi i].
Canonical cfBigdprodi_rmorphism i := [rmorphism of @cfBigdprodi i].
Canonical cfBigdprodi_lrmorphism i := [lrmorphism of @cfBigdprodi i].

Lemma cfdot_bigdprod phi psi :
'[cfBigdprod phi, cfBigdprod psi] = \prod_(i | P i) '[phi i, psi i].