Let $ (M,\mathcal{O},A_{\infty})$ be a $C^\infty$-Differentiable dimension-d Manifold.
Recall that this setup means : $M$ is a set, $\mathcal{O}$ a topology, $A_{\infty}$ an atlas of charts "$x:M\to\mathbb{R}^d$", for which any two charts $x,y$ are $C^\infty$ compatible.
I have been told in a $C^\infty$ manifold chart maps are differentiable. Am I right in thinking this is true since $x\circ x^{-1}=\text{id}$, (the identity map $\text{id}:\mathbb{R}^d\to\mathbb{R}^d$), which is clearly infinitely differentiable.
Also I could make this exact same statement about a $C^k$-differentiable manifold right?