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A smooth principal fiber bundle is a smooth fiber bundle $\pi: E \to M$ together with a Lie group $G$ and a fiber preserving right action $E \times G \to E$ which restricts to each fiber freely and transitively.

How prove that the map $g_{\alpha \beta}: U_{\alpha}\cap U_{\beta}\to G$ is smooth, using the implicit function theorem?

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Let $F$ denote the fiber of the bundle, and fix some $f\in F$. Let $$\varphi_\alpha:E|_{U_\alpha}\to U_\alpha\times F,\:\varphi_\beta:E|_{U_\beta}\to U_\beta\times F$$denote the relevant local trivializations. By definition, we have for every $p\in U_\alpha\cap U_\beta$ $$\varphi_\alpha\circ\varphi_\beta^{-1}(p,f)=(p, g_{\alpha\beta}(p)(f)),$$and now, as said in the question, smoothness follows from the implicit function theorem.

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