I would do it like this:
A is a mxn matrix of rank m, now it is obvius that , why?
So A represents a linear map with rank .
B is a nxp matrix of rank n, it is equally obvius that .
So B represents a linear map with of rank n.
Now AB represents linear , or
So g has a domain of p dimensions, and the codomain has n dimensions.
f has domain of n dimensions, and the codomain has m dimensions.
So represents a map , where V is p dimensional and W is m dimensional. So AB has rank m.
This is what I would do, but there are probably nicer ways. Assuming ofcourse that this is correct.