Suppose that $F$ is a field and $A\in M_{m\times k}(F)$, $B\in M_{k\times p}(F)$ and $C\in M_{p\times n}(F)$. Show that: $$\text{rank}(AB)+\text{rank}(BC)\leq \text{rank}(B)+\text{rank}(ABC).$$ I tried to find the row/column space of those matrices and find a basis for them to find the rank. But I wasn't seccessful. Also I tried to find a relation between these spaces and say if they're subspaces of each other, but this also couldn't help me.
Any helps or hints are so much appreciated!
$$rank(B)-rank(AB)=dim (\ker\hat{A} \cap Im \hat{B})$$ $$rank(BC)-rank(ABC)=dim (\ker\hat{A} \cap Im \hat{B}\hat{C})$$