Equivalence of atlases of vector bundles is an equivalence relation

Question

I'm trying to show that equivalence between atlases of vector bundles is indeed an equivalence relation.

Definitions

Let $ V\subset \mathbb R^n $ be an open set and let $ F $ be a (real, finite dimensional) vector space.

For the purpose of this post define a local vector bundle as the product $ V\times F $.

If $ V_1\times F_1 $ and $ V_2\times F_2 $ are two local vector bundles, call a function $ f\colon V_1\times F_1\to V_2\times F_2 $ a morphism of local vector bundles if there exist two smooth (say, $ C^\infty $) functions $ \alpha_1\colon V_1\to V_2 $ and $ \alpha_2\colon V_1\to \hom(F_1, F_2) $ such that $ f(x,\eta) = (\alpha_1(x),\alpha_2(x)\eta) $.

If $ S $ is a set, call a pair $ (U,\phi) $ where $ U\subset S $ is a subset and $ \phi $ is a bijection from $ U $ to a local vector bundle a local vector bundle chart on $ S $.

Define now an atlas of local vector bundle charts on $ S $ as a collection $ \mathscr B $ of (guess what...) local vector bundle charts on $ S $ such that:

1. $ S = \bigcup_{(U,\phi)\in \mathscr B}U $;
2. for every pair $ (U_1,\phi_1) $ and $ (U_1,\phi_2) $ of charts of $ \mathscr B $ such that $ U_{12} = U_1\cap U_2\neq \emptyset $, the images $ \phi_1(U_{12}) $ and $ \phi_2(U_{12}) $ are local vector bundles and the composite of the mappings $$ \require{AMScd}\begin{CD} \phi_1(U_{12}) @>{\phi_1^{-1}{\restriction_{\phi_1(U_{12})}^{U_{12}}}}>> U_{12} @>{\phi_2{\restriction_{U_{12}}^{\phi_2(U_{12})}}}>> \phi_2(U_{12}) \end{CD} $$ and $$ \require{AMScd}\begin{CD} \phi_2(U_{12}) @>{\phi_2^{-1}{\restriction_{\phi_2(U_{12})}^{U_{12}}}}>> U_{12} @>{\phi_1{\restriction_{U_{12}}^{\phi_1(U_{12})}}}>> \phi_1(U_{12}) \end{CD} $$ are morphisms of local vector bundles.

The main definition is the following. Two atlases $ \mathscr B_1 $ and $ \mathscr B_2 $ are said to be equivalent (graphically, $ \mathscr B_1\sim \mathscr B_2 $) if their union $ \mathscr B_1\cup \mathscr B_2 $ is again an atlas.

I'm trying to show that, if $ \mathscr B_1\sim \mathscr B_2 $ and $ \mathscr B_2\sim \mathscr B_3 $, then $ \mathscr B_1\sim \mathscr B_3 $, where the $ \mathscr B_j $s are all atlases on a fixed set $ S $.

What I tried so far

Let $ (U_1,\phi_1\colon U_1\to V_1\times F_1) $ and $ (U_3,\phi_3\colon U_3\to V_3\times F_3) $ be charts respectively of $ \mathscr B_1 $ and $ \mathscr B_3 $ such that $ U_{13}\neq \emptyset $.

I tried to imitate the proof of the fact that equivalence of atlases smooth manifold is an equivalence relation, but things started to get messy. How to deal with all the $ \alpha_1 $ and $ \alpha_2 $s?

Answer 1

Hint: consider $(U_1,\varphi_1)\in\mathscr{B}_1,(U_2,\varphi_2)\in\mathscr{B}_2,(U_3,\varphi_3)\in\mathscr{B}_3$ charts on $X$ such that $U_1\cap U_2\cap U_3\neq\emptyset$. Of course $\mathscr{B}_1,\mathscr{B}_2,\mathscr{B}_3$ are atlases such that $\mathscr{B}_1\sim\mathscr{B}_2$ and $\mathscr{B}_2\sim\mathscr{B}_3$.