Define $f:S^n\to S^n$ by $x\to -x$. Prove this is a diffeomorphism.
My attempt: My definition of a smooth map between two manifolds $M,\,N$ is that a map $g:M\to N$ is smooth iff the maps: $$\psi_i\circ g\circ \phi_j^{-1}$$ is smooth (i.e it is smooth in local coordinates) for some maps $\psi_i:V\to\mathbb{R}^n$ and $\phi_j:U\to\mathbb{R}^n$ in the smooth atlases of $N$ and $M$ respectively.
The fact that this map is a hoemomorphism is clear. However, I'm stuck with proving it is smooth. I fixed some maximal smooth atlas $\{(\phi_i,V_i)\}$. For every $x\in\phi_i(V_i\cap V_j)$: $$\phi_j\circ f\circ \phi_i^{-1}(x)=\phi_j(-\phi_i^{-1}(x))$$ I feel like this is obviously smooth but can't seem to find a formal way to prove it. I thought maybe it's worth showing that $\psi_i(x)=\phi_i(-x)=(-\phi_i(x)^{-1})$ is in the maximal atlas, but wasn't able to, or find specific maps that I can work with - but I don't really know any specific maps for $S^n$.
Any help would be appreciated.
Consider the sets: $$U_i^+=\{\vec{x}:x_i>0\},\,U_i^-=\{\vec{x}:x_i<0\}$$ We have maps: $$\phi_i:U_i^+\to\mathbb{R}^n$$ defined by $(x_1,\ldots,x_{n+1})\to (x_1,\ldots,x_{i-1},x_{i+1},\ldots,x_{n+1})$ with inverse $(x_1,\ldots,x_{i-1},x_{i+1},\ldots,x_{n+1})\to (x_1,\ldots,\sqrt{1-\sum_{j\neq i}^{n+1}x_i^2},\ldots,x_{n+1})$ and similarly we have $\psi_i:U_{i}^-\to\mathbb{R}^n$ (with the obvious adaptions). This gives you a smooth atlas (check!). Moreover, for $i=1$ and for every $x$ for which it makes sense to compose (and it is possibile for every $x\in U_i^+$ by the definition of $f$ and $U_i^{\pm}$: $$\psi_1\circ f\circ \phi_1^{-1}(x_1,\ldots,x_{i-1},x_{i+1},\ldots,x_{n+1})=(-x_1,\ldots,-x_{i-1},-x_{i+1},\ldots,-x_{n+1})$$ And this is clearly smooth (as a function from Euclidean space to itself).