\begin{align*} |\psi> & = |0>\otimes|1>\\ \hat{I} & = |0><0|+|1><1| \text{ identity}\\ \hat{P} & = \hat{I}\otimes |1><1|=\left(|0><0|+|1><1|\right)\otimes |1><1|\\ \hat{P}|\psi> & = \left[\left(|0><0|+|1><1|\right)\otimes |1><1|\right]\left[|0>\otimes|1>\right] \text{tensor product simp}\\ & = \left[\left(|0><0|+|1><1|\right)|0>\right]\otimes\left[|1><1|1>\right] \text{qapply} \\ & = \left[|0><0|0>+|1><1|0>\right]\otimes |1> \\ & = |0>\otimes|1> = |\psi> \end{align*}