Suppose $P_1$ and $P_2$ are $n \times n$ matrices satisfying $P_1^2 = P_1$, $P_2^2 = P_2$, and $P_1 P_2 = P_2 P_1$. Prove that $\operatorname{range}(P_1P_2) =\operatorname{range}P_1\cap\operatorname{range}P_2\,$.
Proving that $\operatorname{range}(P_1 P_2)\subseteq\operatorname{range}P_1\cap \operatorname{range}P_2$ is fairly straightforward.
I'm not seeing the trick to show the reverse direction, however. I let $y\in\operatorname{range}P_1\cap\operatorname{range}P_2$, and I've managed to prove that $P_1 y = P_2 y$, but I can't seem to figure out how to get to the conclusion
$$\operatorname{range}(P_1P_2)\supseteq\operatorname{range}P_1\cap \operatorname{range}P_2\,.$$
Let $y\in\operatorname{Range}P_1\cap\operatorname{Range}P_2$.
Then pre-images $\,x_1, x_2\,$ exist with $y=P_1x_1=P_2x_2$, and $\,y=P_1y=P_2y\,$ holds by idempotence. Thus $$y=P_2P_1x_1=P_1P_2x_2\,.$$