Proof of The Norm Distance from the Origin Is the Norm

Let $y\in V$. By The Norm Metric of a Real Normed Space (i) and The Zero Vector, Negatives and Differences in a Vector Space (iii),

$$d_{V}(0_{V},y)=\lVert 0_{V}-y\rVert=\lVert 0_{V}+(-y)\rVert .$$

Here $0_{V}+(-y)=-y$ by The Zero Vector, Negatives and Differences in a Vector Space (i) and commutativity, Vector Space over a Field (ii). By Elementary Identities in a Vector Space (iv), $-y=(-1)y$, and absolute homogeneity of the norm, Norms and Real Normed Spaces (iii), gives

$$\lVert -y\rVert=\lVert(-1)y\rVert=|-1|\,\lVert y\rVert .$$

As $|-1|=1$, we conclude

$$d_{V}(0_{V},y)=1\cdot\lVert y\rVert=\lVert y\rVert .$$