Valuation Ring of P-adic Norm on Rationals/Corollary 1

Theorem
Let $\norm {\,\cdot\,}_p$ be the $p$-adic norm on the rationals $\Q$ for some prime $p$.

Let $\mathcal O$ be the induced valuation ring on $\struct {\Q,\norm {\,\cdot\,}_p}$.

The set of integers $\Z$ Is a subring of $\mathcal O$.

Proof
By Valuation Ring of P-adic Norm on Rationals the induced valuation ring $\mathcal O$ is the set:
 * $\mathcal O = \Z_{(p)} = \set{ \dfrac a b \in \Q : p \nmid b }$

Since $p \nmid 1$ then for all $a \in \Z$, $a = \dfrac a 1 \in \mathcal O$.

Hence $\Z \subseteq \mathcal O$.

By Valuation Ring of Non-Archimedean Division Ring is Subring then $\mathcal O$ is a subring of $\Q$.

By Integers form Subdomain of Rationals then $\Z$ is a subring of $\Q$.

By Intersection of Subrings is Largest Subring Contained in all Subrings then $\Z \cap \mathcal O = \Z$ is a subring of $\mathcal O$.