(* Matrix multiplication using bigops. *)
Definition mulmx {m n p} (A : 'M_(m, n)) (B : 'M_(n, p)) : 'M[R]_(m, p) :=
\matrix[mulmx_key]_(i, k) \sum_j (A i j * B j k).
Local Notation "A *m B" := (mulmx A B) : ring_scope.

Lemma scalemxAl m n p a (A : 'M_(m, n)) (B : 'M_(n, p)) :
a *: (A *m B) = (a *: A) *m B.