Let $H$ be a definite quaternion algebra defined over $\mathbb{Q}$, unramified at $p$, and ramified at $\infty$. Denote by $D$ the multiplicative group $H^\times$ divided by its center.
In this paper, on page 6, the following double coset space is considered:
$X=D(\mathbb{Q})\backslash D(\mathbb{A})/D(\mathbb{R})\prod_{q\text{ prime}}D(\mathbb{Z}_q)$
Where $\mathbb{A}$ stands for the (rational) adelic ring, and $\mathbb{Z}_q$ stands for the $q$-adic integers.
It is then stated that $X$ can be identified with $D(Z[\frac{1}{p}])\backslash D(\mathbb{Q}_p) / D(\mathbb{Z}_p)$.
This is claimed to be a consequence of "Strong Approximation".
My question:
What is the statement of the appropriate version of strong approximation? How does this identification follow? In particular, for a double coset $D(Z[\frac{1}{p}]) g D(\mathbb{Z}_p)$, what is the corresponding double coset $D(\mathbb{Q})aD(\mathbb{R})\prod_{q\text{ prime}}D(\mathbb{Z}_q)$.