Prove that if matrix $A$ is an $m\times n$ and $B$ is $n\times p$, then $\operatorname{rank} AB$ is less than or equal to $\operatorname{rank} B$
How to prove and interpret $\operatorname{rank}(AB) \leq \operatorname{min}(\operatorname{rank}(A), \operatorname{rank}(B))$?