Residue Field of P-adic Norm on Rationals/Lemma 2

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

Let $\Z_{\ideal p}$ be the induced valuation ring on $\struct {\Q, \norm {\,\cdot\,}_p}$.

Let $p \Z_{\ideal p}$ be the induced valuation ideal on $\struct {\Q, \norm {\,\cdot\,}_p}$.

Let $\phi : \Z \to \Z_{\ideal p} / p \Z_{\ideal p}$ be the mapping defined by:
 * $\forall a \in \Z: \map \phi a = a + p \Z_{\ideal p}$

Then:
 * $p \Z = \map \ker \phi$

Proof
Let $\map \ker \phi$ denote the kernel of $\phi$.

Then:

Hence:
 * $p \Z = \map \ker \phi$