Leray-Hirsch analogue of algebraic geometry

Question

I want to use a Leray-Hirsch analogue of algebraic geometry to construct the chern classes. I am not sure how to prove the statement. Suppose that $E$ is a locally free sheaf of rank $r$ on $X$. I want to show that if $P(E)$ is the associated projection, then $A(P(E))$ is a free $A(X)$ module generated by $1, z, ..., z^{r-1}$ where $z$ is the first Chern class. How does one prove this statement? I want to use this statement to construct the chern classes. A good reference would be great. Or an answer or partial answer. Thanks for the help!