Residue Field of P-adic Norm on Rationals/Lemma 3

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

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

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

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

Then:
 * $\phi$ is onto $\Z_{(p)}/p\Z_{(p)}$

Proof
Let $\eqclass {a/b} {} \in \Z_{(p)}/p\Z_{(p)}$, where $a/b$ are in canonical form.

Then $p \nmid b$

Since $\Z/p\Z$ is a field then:
 * $\exists b' \in \Z: bb' \equiv 1 \pmod p$

Then $\forall a \in \Z$:
 * $abb' - a \equiv a - a = 0 \pmod p$

So:
 * $p \divides abb' - a$

Hence:
 * $ab' - \dfrac a b = \dfrac {abb' - a} b \in p\Z_{(p)}$

By then:
 * $\map \phi {ab'} = \eqclass {ab'} {} = \eqclass {a/b} {}$

That is:
 * $\phi$ is onto $\Z_{(p)}/p\Z_{(p)}$