Three Points in Ultrametric Space have Two Equal Distances/Corollary 5

Theorem
Let $\norm {\, \cdot \,}$ be a non-trivial non-Archimedean norm on the rational numbers $\Q$.

Let $p$ be the prime number defined by $p = \min \set {n \in N : \norm {n} < 1}$.

Let $q$ be a prime number

Then:
 * $\norm q = 1$

Proof
that $\norm {q} < 1$.

By Prime not Divisor implies Coprime then:
 * $p \perp q$

By Bézout's Identity then:
 * $\exists n, m \in \Z : m p + n q = 1$

By Characterisation of Non-Archimedean Division Ring Norms then:
 * $\forall n \in \N: \norm {n} \le 1$

Let $z \in \Z$, then $\size z \in \N$.

By Norm of Negative then:
 * $\norm {z} = \norm {\size{z}} \le 1$

Hence:
 * $\forall z \in \Z: \norm {z} \le 1$

It follows that:

Similarly $\norm {n q} \lt 1$.

Hence:

This is a contradiction.

Hence $\norm {q} = 1$.