Given $X$, and a locally free sheaf $\mathcal{F}$ of rank $n$ on it. We have the induced map $f:\mathbb{P}(\mathcal{F})\rightarrow X$. Let $\phi$ be the divisor associated to the line bundle $\mathcal{O}_{\mathbb{P}(\mathcal{F})}(1) $.

Is it true $f_{\ast}(\phi^{n-1})=[X]$ in the Chow ring $A^\ast(X)$? Also, is trace map $A^{top}(.)\rightarrow\mathbb{Q}$ compatible with pushforward $f_{\ast}$?

Thank you.