Proving implicit function theorem using Kroneker-Rouchè-Capelli

Question

I'm a physics student facing the implicit function theorem. My professor gave me an unintuitive proof of the implicit function theorem based on Banach fixed-point theorem. I need help formalizing a more intuitive proof that I'm trying to give based on Kronecker-Rouché-Capelli's theorem.

Let the system of equations be written in vector form $$ \mathbf f(\mathbf x, \mathbf y) = \mathbf 0 $$ Let $(\mathbf x_0, \mathbf y_0)$ be a solution of the system, meaning $\mathbf f(\mathbf x_0, \mathbf y_0) = 0$. Let ${\mathbf f}: \mathbb{R}^{n+m} \to \mathbb{R}^m$ be $\mathcal{C}^1$ in an open neighborhood of $(\mathbf x_0, \mathbf y_0)$. Thus we have that $\mathbf f$ is differentiable and by Taylor expansion we have

$$ {\mathbf f}({\mathbf x}, {\mathbf y}) = \underbrace{{\mathbf f}({\mathbf x_0}, {\mathbf y_0})}_{0} + D{\mathbf f}({\mathbf x_0}, {\mathbf y_0})({\mathbf x - \mathbf x_0}, {\mathbf y - \mathbf y_0}) + o(||({\mathbf x - \mathbf x_0}, {\mathbf y - \mathbf y_0})||); \quad \text{for } {(\mathbf x, \mathbf y) \to (\mathbf x_0, \mathbf y_0)} $$

By imposing $\mathbf(x, y)$ to be a solution of the system we get:

$$ \begin{aligned} 0 = &D{\mathbf f}({\mathbf x_0}, {\mathbf y_0})({\mathbf x - \mathbf x_0}, {\mathbf y - \mathbf y_0}) + o(||({\mathbf x - \mathbf x_0}, {\mathbf y - \mathbf y_0})||) \\ &D{\mathbf f}({\mathbf x_0}, {\mathbf y_0})({\mathbf x - \mathbf x_0}, {\mathbf y - \mathbf y_0}) = o(||({\mathbf x - \mathbf x_0}, {\mathbf y - \mathbf y_0})||) \\ &D{\mathbf f}({\mathbf x_0}, {\mathbf y_0})({\mathbf x}, {\mathbf y}) = D{\mathbf f}({\mathbf x_0}, {\mathbf y_0})({\mathbf x_0}, {\mathbf y_0}) + o(||({\mathbf x - \mathbf x_0}, {\mathbf y - \mathbf y_0})||) \\ \end{aligned} $$

Now since the columns of $D\mathbf f$ corresponding to $\mathbf y$ are independent by hypothesis (they give maximum rank $m$) we can use Kronecker-Rouché-Capelli theorem to state that for every fixed value of $\mathbf x \in \mathbb{R}^{n}$ there exists one and only one $\mathbf y \in \mathbb{R}^{m}$ that satisfies the identity. This implies the existence of an implicit function: $\mathbf y = \mathbf g(\mathbf x)$ on all solutions of the system in the previously mentioned neighborhood of $(\mathbf x_0, \mathbf y_0)$.

Now the problem: this is not a proof, this is just the outline of a proof. Does the Kronecker-Rouché-Capelli theorem really say this? Where have I been to loose? How do I deal with the fact that $(||(−_0,−_0)||)$ is a function of $\bf y$?

I'd like to be helped in making this proof more rigorous or to be told that it is all wrong. If you could point me to an already existing proof based on Kronecker-Rouché-Capelli's theorem it would be more than enough.

Answer 1

Since I think this question may be relevant for someone else I will answer myself referring to peek-a-boo's comment.

Krc is indeed a special case of IFT, but using IFT here is overkill. But the converse is what you're asking for. Information about derivatives is "infinitesimal behavior" but the IFT gives you local behavior (i.e on an entire open neighborhood). So the IFT is an infinitesimal to local theorem, and this can never be achieved by simple linear algebra only. The linear algebra certainly motivates the IFT (and I've written several answers regarding this intuitive motivation), but once you get down to the task of proving the general statement, it is obvious that linear algebra alone is insufficient.