Composition of Isometries is Isometry
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Theorem
Let:
- $\struct {X_1, d_1}$
- $\struct {X_2, d_2}$
- $\struct {X_3, d_3}$
be metric spaces.
Let:
- $\phi: \struct {X_1, d_1} \to \struct {X_2, d_2}$
- $\psi: \struct {X_2, d_2} \to \struct {X_3, d_3}$
be isometries.
Then the composite of $\phi$ and $\psi$ is also an isometry.
Proof
An isometry is a distance-preserving mapping which is also a bijection.
From Composition of Distance-Preserving Mappings is Distance-Preserving, $\psi \circ \phi$ is a distance-preserving mapping.
From Composite of Bijections is Bijection, $\psi \circ \phi$ is a bijection.
$\blacksquare$