Leray-Hirsch theorem for cohomology

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Statement

Let $p:E\to B$ be a fiber bundle with fiber space $F$, and $R$ be a commutative unital ring, such that the following hold:

• $H^{n}(F;R)$ is a finitely generated free $R$-module for every $n$
• There exist classes $c_{j}\in H^{k_{j}}(E;R)$ whose restrictions $i^{*}(c_{j})$ form a basis for $H^{*}(F;R)$ in each fiber $F$, via the inclusion of the fiber in $E$ (the choice of these classes needs to be made independent of the fiber)

Then the map:

$\Phi :H^{*}(B;R)\otimes _{R}H^{*}(F;R)\to H^{*}(E;R)$

given by:

$\sum _{jk}b_{j}\otimes i^{*}(c_{k})\mapsto \sum _{jk}p^{*}(b_{j})\smile c_{j}$

is an isomorphism.