(* 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 mulmxA m n p q (A : 'M_(m, n)) (B : 'M_(n, p)) (C : 'M_(p, q)) :
A *m (B *m C) = A *m B *m C.