Metric Induced by Norm on Normed Division Ring is Metric

Theorem
Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring.

Let $d$ be the metric induced by $\norm{\,\cdot\,}$.

Then $d$ is a metric.

Proof of and
Let $x, y \in R$.

Then $\map d {x, y} = \norm {x - y} \ge 0$, and furthermore:

Proof of
Let $x, y, z \in R$.

Then:

Proof of
Let $x, y \in R$.

Then:

As $d$ satisfies the metric space axioms, it is a metric.