Quotient Metric on Vector Space is Invariant Pseudometric

Theorem
Let $K$ be a field.

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

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

Let $N$ be a vector subspace of $X$.

Let $X/N$ be the quotient vector space of $X$ modulo $N$.

Let $\pi : X \to X/N$ be the quotient mapping.

Let $d_N$ be the quotient metric on $X/N$ induced by $d$.

Then $d_N$ is an invariant pseudometric.

Proof of
Let $x, y \in X$.

Then, we have:
 * $\ds \map {d_N} {\map \pi x, \map \pi x} = \inf_{z \mathop \in N} \map d {x - x, z} = \inf_{z \mathop \in N} \map d { {\mathbf 0}_X, z}$

Since $N$ is a vector subspace, we have ${\mathbf 0}_X \in N$.

From, we have $\map d { {\mathbf 0}_X, {\mathbf 0}_X} = 0$, and so:
 * $\ds \inf_{z \mathop \in N} \map d { {\mathbf 0}_X, z} \le 0$

Since we also have:
 * $\ds \inf_{z \mathop \in N} \map d { {\mathbf 0}_X, z} \ge 0$

we obtain:
 * $\ds \inf_{z \mathop \in N} \map d { {\mathbf 0}_X, z} = 0$

Hence we have proved for $d_N$.

Proof of
Let $x, y, z \in X$.

Applying to $d$, we have:
 * $\map d {x, z + n} \le \map d {x, y + n'} + \map d {y + n', z + n}$

for each $n, n' \in N$.

That is, using the translation invariance of $d$:
 * $\map d {x - z, n} \le \map d {x - y, n'} + \map d {y - z, n - n'}$

for each $n, n' \in N$.

Taking the infimum over $n \in N$, we have:

Taking the infimum over $n' \in N$, using Infimum preserves Inequalities and Infimum Plus Constant, we have:
 * $\map {d_N} {\map \pi x, \map \pi z} \le \map {d_N} {\map \pi x, \map \pi y} + \map {d_N} {\map \pi y, \map \pi z}$

Hence we have proved for $d_N$.

Proof of
Let $x, y \in X$.

Then we have:

Hence we have proved for $d_N$.

Proof of translation invariance
Let $\map \pi x, \map \pi y, \map \pi z \in X/N$ for $x, y, z \in X$.

We have:

So $d_N$ is an invariant pseudometric.