Composition of Isometries is Isometry

Theorem

Let:

$\struct{X_1, d_1}$
$\struct{X_2, d_2}$
$\struct{X_3, d_3}$

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.

$\blacksquare$