Here's some context before my question.
Let $\mathbb{V}$ be a topological vector space, which is Hausdorff and such that its topology is generated by some arbitrary family of seminorms $\{\rho_{\alpha}\}_{\alpha \in I}$; this means that $\mathbb{V}$ is locally convex. Now, if $I$ turns out to be countable (or if we can reduce the family $\{\rho_{\alpha}\}_{\alpha \in I}$ to a countable one, while keeping the same topology in $\mathbb{V}$), we can define a metric in $\mathbb{V}$ by $d(u, v) = \sum_{i = 1}^{\infty} \frac{1}{2^i} \frac{\rho_i(u - v)}{1 + \rho_i(u - v)},$ where $\{\rho_i\}_{i \in \mathbb{N}}$ is some enumeration of $\{\rho_{\alpha}\}_{\alpha \in I}$, so that its topology is metrizable. I've been told that the converse is also true, which leads to my question.
QUESTION: Let $\mathbb{V}$ be a topological vector space having a metrizable topology, generated by some metric $d$. How can I prove that $\mathbb{V}$ admits a countable family of seminorms generating its topology? Also, do I need to impose the condition that $d$ is translation invariant (since this happens in the above construction)?
Thanks.