Diffeomorphism from Inverse function theorem I often heard that it is possible to show by using the inverse function theorem that if a function is smooth (i.e. arbitrarily often differentiable), a bijection between open sets, and has a non-singular jacobian, then it is a smooth diffeomorphism. but somehow the inverse function theorem that I know and that wikipedia seems to know, only states that if it is a continuously differentiable bijection with nonzero jacobian, then its inverse function is also continuously differentiable. But how do you get from this, to the statement that I proposed above? I don't see the implication.
 A: Let $f(p)=q$, $J_f(p)\ne 0$. Then $f$ maps a suitable neighborhood $U$ of $p$ diffeomorphically onto a neighborhhod $V$ of $q$. Let
$g:\ V\to U$ be the inverse of $f\restriction U$. The statement of the inverse function then contains the formula
$$dg(y)=\left(df\bigl(g(y)\bigr)\right)^{-1}\qquad(y\in V)\ .$$
This can be interpreted as
$$dg=\iota\circ df\circ g\ ,$$
where $\iota:\ L\mapsto L^{-1}$ denotes inversion in $GL({\mathbb R}^n)$.
Assume that $g$ is $r$ times continuously differentiable for some $r\geq1$. As both $f$ and $\iota$ are infinitely differentiable it follows by the chain rule that $dg$ is $r$ times continuously differentiable as well, and this implies that $g$ is in fact $r+1$ times continuouslay differentiable.
A: Let $LI^m$ and $L^m$ be the set of invertible linear transformations of $\mathbb{R}^m$ in itself and the set of linear transformations of $\mathbb{R}^m$ in itself respectively. Define $\operatorname{Inv}: LI^m\to L^m$ by $\operatorname{Inv}(A)$ is the inverse matrix of $A$. First you can note that if $f$ is a function satisyfing your conditions, then $$(f^{-1})'(y)=[f'(g(y))]^{-1}$$
This implies that $(f^{-1})'=(\operatorname{Inv})\circ f'\circ f^{-1}$. Because $\operatorname{Inv},f\in C^\infty$, you can conclude from the last formula (by using chain rule) that $f^{-1}\in C^\infty$. 
Update: I do not understood OP's comment in the bounty, nevertheless I give here more explanations:
Since $f'\neq 0$ we have that $\operatorname{Inv}(f'(g(y)))$ is well defined for all $y$, so the derivative of $(f^{-1})$ dependes on functions that are differentiable again. By a induction argument OP's can finish the proof. Also the proof that $\operatorname{Inv}$ is $C^\infty$ was given in the comments by @ronno. As I have wrote to @Lipschits in the comments, I know another proof of the fact that $\operatorname{Inv}$ is $C^\infty$, but it is really big and @ronnos's proof is more straightforward.
A: A continuous bijection between open sets $U$ and $V$ implies homeomorphism.
But there is a corollary of the inverse function theorem: if the domain $U$ is open, then a global non-singular jacobian of $f\in C^\infty$ implies that $f(U)$ is open. Moreover, if $f\in C^\infty$ is also bijection, then the inverse $g$ exist in $f(U)$ and it's also smooth in the open set $f(U)$, since it's smooth in some open neighborhood of every point of $f(U)$. Thus, $f$ is a smooth diffeomorphism.
