Canonical capmx_monoid n :=
Monoid.Law (@capmxA n n n n) (@cap1mx n) (@capmx1 n).
Canonical capmx_comoid n := Monoid.ComLaw (@capmxC n n n).

Lemma mxrank_mul_ker m n p (A : 'M_(m, n)) (B : 'M_(n, p)) :
(\rank (A *m B) + \rank (A :&: kermx B))%N = \rank A.