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_sum_cap m1 m2 n (A : 'M_(m1, n)) (B : 'M_(m2, n)) :
(\rank (A + B) + \rank (A :&: B) = \rank A + \rank B)%N.