In the solution manual, the solution of problem 3.41 starts with the description of "implicit function theorem" which I write below:
Suppose $F:\mathbb{R}^n\times\mathbb{R}^m\to\mathbb{R}$ (where I think it should be $F:\mathbb{R}^n\times\mathbb{R}^m\to\mathbb{R}^{m \text{ or }n}$ please clarify this also whether it should be $n$ or $m$ because this point is also critical in my view in understanding the solution) satisfies
$F(\bar{\mathbb{u}},\bar{\mathbb{v}})=0$
$F$ is continuously differentiable function and $D_{\mathbb{v}}F(\mathbb{u,v})$ is nonsingular in a neighborhood of $(\bar{\mathbb{u}},\bar{\mathbb{v}})$.
Then there exists a continuously differentiable function $\phi : \mathbb{R}^n \to \mathbb{R}^m$, that satisfies $\bar{\mathbb{v}} = \phi(\bar{\mathbb{u}})$ and $$F(\mathbb{u},\phi(\mathbb{u})) = 0$$ in the neighborhood of $\bar{\mathbb{u}}$. After this description the solution manual says:
Applying this to $\mathbb{u}=\mathbb{y},\mathbb{v}=\mathbb{x}$ and $F(\mathbb{u,v})=\triangledown f(\mathbb{x})-\mathbb{y}$, we see ....
Now, I do not understand how $F(\mathbb{u,v})=F(\mathbb{y,x})=\triangledown f(\mathbb{x})-\mathbb{y}$ is a right description of a function when $n \neq m$ since $F(\mathbb{y,x})$ has either dimensions $n$ or $m$ and $\triangledown f(\mathbb{x})$ and $\mathbb{y}$ have different dimensions if $n \neq m$ and therefore their subtraction is not a valid thing. Please clarify how to understand this in solution manual. Thanks in advance.