Let $M$ be a smooth manifold, let $(U_i)_{i \in I}$ be an open cover of $M$, and let $k \in \mathbb{N}$. Let $E \to M$ and $E' \to M$ be smooth real vector bundles of rank $k$ with local trivializations over the same cover $(U_i)_{i \in I}$, and suppose their transition functions are smooth maps
Then $E$ and $E'$ are isomorphic by a smooth vector bundle isomorphism $\Phi:E \to E'$ over $\operatorname{id}_M$ whose local representative over each $U_i$ is a smooth map