Local Notation sr modV := (submod_repr modV).

Lemma mxtrace_dsum_mod (I : finType) (P : pred I) U W
(modU : forall i, mG (U i)) (modW : mG W) :
let S := (\sum_(i | P i) U i)%MS in (S :=: W)%MS -> mxdirect S ->
{in G, forall x, \sum_(i | P i) \tr (sr (modU i) x) = \tr (sr modW x)}.