TITLE: Analytic implicit function theorem
QUESTION [13 upvotes]: I'm looking for a proof of the analytic implicit function theorem (IFT). The only related proof I could find was the holomorphic inverse function theorem (by Henri Cartan). On Wikipedia, the analytic IFT is mentioned casually in the general article "Implicit function theorem", saying that "Similarly, if f is analytic inside U×V, then the same holds true for the explicit function g inside U. This generalization is called the analytic implicit function theorem." Mmmh, that's fast...
A sketch of the proof may be the following :

use analytic continuation to transform f into a holomorphic function
use the holomorphic inverse function theorem (Cartan) to prove a holomorphic IFT
restriction : g is holomorphic on $\mathbb{C}$, therefore analytic on $\mathbb{R}$.

But it seems weird and I don't think it would work (I have no idea whether a so-called holomorphic IFT exists or not). What would be an efficient proof of the theorem ? Thanks a lot by advance.

REPLY [5 votes]: The implicit function theorem is deduced from the inverse function theorem in most standard texts, such as Spivak's "Calculus on Manifolds", and Guillemin and Pollack's "Differential Topology". Basically you just add coordinate functions until the hypotheses of the inverse function theorem hold.