# Leigh.Samphier/Sandbox/Inclusion Mapping on Normed Division Subring is Distance Preserving Monomorphism

## Theorem

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

Let $\struct {S, \norm {\, \cdot \,}_S}$ be a normed division subring of $\struct {R, \norm {\, \cdot \,}}$.

Then the inclusion mapping $i : S \to R$ is a distance-preserving ring monomorphism.

## Proof

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

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

Thus:

 $\, \ds \forall x, y \in S : \,$ $\ds \map d {\map i x, \map i y}$ $=$ $\ds \map d {x, y}$ Definition of Inclusion Mapping $\ds$ $=$ $\ds \norm {x - y}$ Definition of Metric Induced by Norm on Division Ring $\ds$ $=$ $\ds \norm {x - y}_S$ Definition of Norm on Subring $\ds$ $=$ $\ds \map {d_S} {x, y}$ Definition of Metric Induced by Norm on Division Ring

It follows that $i$ is distance-preserving by definition.

$\blacksquare$