I want to show the following:
Let $E \rightarrow B$ be a fiber bundle with fiber $F$. Show that if $B$ and $F$ are manifolds, then so is $E$.
Solving this problem seems easy enough simply by checking manifold conditions on $E$, since the requirements for a manifold behave nicely with the fiber bundle structure. However, I was wondering if there is a slicker way to prove this, aside from this monotonous definition checking? Any help would be appreciated.