Show that $\bar mf=f\bar m$ for any isometry $f$ of $S^2$

Question

Let $\bar m$ be the isometry sending a point $P$ to its opposite point $-P$, show that $\bar m$ commutes with any isometry of $S^2$

I know I can use the fact that isometries preserve distances, but do I need to individually apply this to each type of isometry?

Answer 1

What you want to prove is that, for each $P\in S^2$, $-f(P)=f(-P)$. Bu this is what was proved here.