Translation of Open Ball in Invariant Pseudometric on Vector Space

Theorem
Let $K$ be a field.

Let $X$ be a vector space over $K$.

Let $d$ be an invariant metric on $X$.

For $x \in X$ and $\epsilon > 0$, let $\map {B_\epsilon} x$ be the open ball centered at $x$ with radius $\epsilon$.

Let $y \in X$.

Then:
 * $\map {B_\epsilon} x + y = \map {B_\epsilon} {x + y}$

Proof
Let $z \in X$.

Then we have $z \in \map {B_\epsilon} x + y$ $z = u + y$ for $u \in \map {B_\epsilon} x$.

That is, $z - y \in \map {B_\epsilon} x$.

This is equivalent to:
 * $\map d {z - y, x} < \epsilon$

Since $d$ is invariant, this is equivalent to:
 * $\map d {z, x + y} < \epsilon$

So, we have $z \in \map {B_\epsilon} x + y$ $z \in \map {B_\epsilon} {x + y}$.

Hence $\map {B_\epsilon} x + y = \map {B_\epsilon} {x + y}$.