I was reading this introductory text on diffeology by Patrick Iglesias-Zemmour and it is claimed in page $18$ that $T_\alpha$ is diffeomorphic $\mathbb R/(\mathbb Z + \alpha \mathbb Z)$, where $\alpha$ is an irrational number. The space $T_\alpha$ is the irrational torus defined as $\mathbb T^2/\Delta_\alpha$, where $\Delta_\alpha$ is the dense subgroup $\{[x,\alpha x], x\in \mathbb R\}$, and $\mathbb T^2 = \mathbb R^2/\mathbb Z^2$, the usual torus. So far I was able to prove the map $$\Lambda:T_\alpha \to \mathbb R/(\mathbb Z + \alpha \mathbb Z), $$ given by $\Lambda[x,y] = [y-\alpha x]$, is smooth (in the sense of diffeology) and bijective. Nevertheless, I don't have the inverse map explicitly and I have no idea how to verify this inverse is smooth. So, here is my question:
How do I prove $\Lambda^{-1}$ is smooth?