I have a line bundle $\{(V,v)\in\mathbb{RP}^n\times \mathbb{R}^{n+1}: v\in V\}$ and I wish to show it is a smooth bundle. I have shown it is a vector bundle as in
Since this is a submanifold of a product manifold and smoothness is local, would it suffice to show that the local trivializations are diffeomorphisms with smooth manifolds? Then the vector bundle would be smooth, and the projection map would be the restriction of a smooth map so by definition a smooth manifold. Or am I making a logic jump.
According to the Wikipedia article, it is already proven that the trivialisation is a homoemorphism. Whether or not it is smooth doesn’t make sense now as it is not a smooth manifold yet. But what is needed is the smoothness of transition maps. This is because, since $\Bbb {RP}^n$ is already a manifold and, shrinking if necessary, $U$ is a coordinate chart, $U \times \Bbb R$ becomes a coordinate chart for the line bundle.