Translation Mapping is Isometry

Theorem

Let $\Gamma = \R^n$ denote the real Euclidean space of $n$ dimensions.

Let $\tau_\mathbf x$ be a translation on $\Gamma$:

$\forall \mathbf y \in \R^n: \map {\tau_\mathbf x} {\mathbf y} = \mathbf y - \mathbf x$

where $\mathbf x$ is a vector in $\R^n$.

Then $\tau_\mathbf x$ is an isometry.

Proof

From Translation Mapping is Bijection, $\tau_\mathbf x$ is a bijection.

From Euclidean Metric on Real Number Space is Translation Invariant, $\tau_\mathbf x$ is distance-preserving on $\Gamma$.

The result follows by definition of isometry.

$\blacksquare$