Valuation Ring of P-adic Norm is Subring of P-adic Integers/Corollary 1

Theorem
Let $p$ be a prime number.

Let $\Z_p$ be the $p$-adic integers.

The set of integers $\Z$ is a subring of $\Z_p$.

Proof
Let $\Z_{(p)}$ be the valuation ring induced by $\norm {\,\cdot\,}_p$ on $\Q$.

By Integers form Subring of Valuation Ring of P-adic Norm on Rationals then:
 * $\Z$ Is a subring of $\Z_{(p)}$

By Valuation Ring of P-adic Norm is Subring of P-adic Integers then:
 * $\Z_{(p)}$ Is a subring of $\Z_p$

By Subset is Subring of Subring Iff it is Subring then $\Z$ is a subring of $\Z_p$.