Proof of The Norm of a Negative, and the Norm Distance of a Translate

(i). Let $v\in V$. Since $-v=(-1)v$ (Elementary Identities in a Vector Space (iv)), absolute homogeneity of the norm (Norms and Real Normed Spaces (iii)) gives

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

(ii). Let $a,w\in V$. By the definition of the norm metric (The Norm Metric of a Real Normed Space (i)) and since $(a+w)-a=w$ (Rearranging Sums, Differences and Two-Term Combinations in a Real Vector Space (iv)),

$$d_V(a+w,a)=\lVert (a+w)-a\rVert=\lVert w\rVert .$$