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

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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

From Inclusion Mapping on Subring is Monomorphism, $i$ is a ring monomorphism.

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:

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


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

$\blacksquare$