Section BlockMatrixRing.
Context {R : ringType} {p q : nat} {p_ : 'I_p -> nat} {q_ : 'I_q -> nat}.
Notation sp := (\sum_i p_ i)%N.
Notation sq := (\sum_i q_ i)%N.

Lemma mul_mxrow_mxcol m n
(R_ : forall j, 'M[R]_(m, p_ j)) (C_ : forall i, 'M[R]_(p_ i, n)) :
\mxrow_j R_ j *m \mxcol_i C_ i = \sum_i (R_ i *m C_ i).