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_mxblock_mxrow m
(B_ : forall i j, 'M[R]_(q_ i, p_ j)) (C_ : forall i, 'M[R]_(p_ i, m)) :
\mxblock_(i, j) B_ i j *m \mxcol_j C_ j = \mxcol_i (\sum_j (B_ i j *m C_ j)).