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:
 * $\forall a \in \Z: \map \phi a = a + p\Z_{(p)}$

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

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

Then $p \nmid b$

Let $\F_p$ be the field of integers modulo $p$.

By the definition of a field then:
 * $\exists b' \in \Z: b b' \equiv 1 \pmod p$

By the definition of congruence modulo $p$ then:
 * $p \divides b b' - 1$

By Divisor Divides Multiple then $\forall a \in \Z$:
 * $p \divides a b b' - a$

By Valuation Ideal of P-adic Norm on Rationals then:
 * $ab' - \dfrac a b = \dfrac {abb' - a} b \in p\Z_{(p)}$

By Element in Left Coset iff Product with Inverse in Subgroup then:
 * $\map \phi {ab'} = ab' + p\Z_{(p)} = a/b + p\Z_{(p)}$

It follows that:
 * $\phi$ is onto $\Z_{(p)}/p\Z_{(p)}$