It is well known that given a compact smooth boundaryless manifold $M$, the set of diffeomorphisms $Diff^{r}(M)$ of $M$ for $r \geq 1$, is open in $C^{r}(M)$, the set of continuous functions (for example, Thm. 1.7, in Differential Topology, M. Hirsch).
My question is: does it follow that $Diff^{r + \alpha}(M) \subset C^{r + \alpha}(M)$ is open w.r.t. $C^{r + \alpha}$ topology for $0 < \alpha \leq 1$? Can someone give me some reference?