Let $\mathbb{A} = \widehat{\mathbb{Z}} \otimes \mathbb{Q} \times \mathbb{R}$ be the adeles over $\mathbb{Q}$. In Deligne's article "Formes modulaires et representations de GL(2)" he states without proof the following result: there is a bijection between $\mathbb{Q}$-lattices in $\mathbb{A}^n$ (i.e. discrete and cocompact $\mathbb{Q}$-vector subspaces) and $\mathbb{Z}$-lattices $L \subset \mathbb{R}^n$ equipped with an isomorphism $L \otimes \widehat{\mathbb{Z}} \to \widehat{\mathbb{Z}}^n$.
Does anyone know of a place where this is actually proved? I don't think it's difficult, but there are a few details I can't figure out. For example: given a $\mathbb{Q}$-lattice $R \subset \mathbb{A}^n$, why does the second projection make $L = R \cap (\widehat{\mathbb{Z}} \times \mathbb{R})^n$ a $\mathbb{Z}$-lattice in $\mathbb{R}^n$, and why does the first projection induce an isomorphism $L \otimes \widehat{\mathbb{Z}} \to \widehat{\mathbb{Z}}^n$?
Edit: A friend explained to me how one proves that the projection $L \to \mathbb{R}^n$ is injective. The kernel $L \cap \widehat{\mathbb{Z}}^n$ is compact and discrete, i.e. finite, but $L \subset R$ is torsion-free.