Sotomayor's Theorem for Discrete Dynamical Systems?

Question

One of my research projects has recently taken a leap forward using Sotomayor's Theorem, which provides sufficient conditions for a saddle-node bifurcation to occur in an n-dimensional ODE without first doing a center manifold reduction. (My reference is Perko's text, "Differential Equations and Dynamical Systems," which omits the proof.)

I would like to use an analogous theorem for maps, and seek either a reference or a method of proof. In particular, I conjecture that a map $F:\mathbb{R}^n\times\mathbb{R}\to\mathbb{R}^n$ undergoes a saddle-node bifurcation at $(x_*,\lambda_*)$ if the following five conditions hold:

1. $ F(x_*,\lambda_*)=x_* $
2. $ D_xF(x_*,\lambda_*) \text{ has eigenvalue 1 with left, right eigenvectors } w, \ v$
3. The other $n-1$ eigenvalues of $D_xF(x_*,\lambda_*)$ have modulus $\not=1$
4. $w\frac{\partial F}{\partial \lambda}(x_*,\lambda_*)\not=0$
5. $w(D_x^2F(x_*,\lambda_*)(v,v))\not=0$

Leads on a reference, proof method, or correction are much appreciated. Thanks!