I'm trying to understand the homotopy invariance theorem for smooth fibre bundles. A main step is to show that the interval $[0,1]$ only admits trivial fibre bundles.
I found this proof (see Lemma 3) in the topological setting. In the very last step the author "glues" the previously constructed maps $ \varphi_i $ together to receive a global trivialisation, which is evidently well defined and a homeomorphism. But in the smooth world, I cannot find reason for smoothness here. I know proofs of the theorem for vector bundles by choosing a connection and parallel sections, but I need the statement for general fibre bundles. Anywhere I can find this proof?