The Segre-Veronese map is a closed embedding

Apply ${\nu }_{m,a}$ and ${\nu }_{n,b}$ to the two factors. Their target coordinates are respectively all degree-$a$ and degree-$b$ monomials.

Applying Segre to those two images produces exactly the products $x^{\alpha }{y}^{\beta }$, in the fixed product ordering. It is therefore the fixed-bidegree map of the statement.

Let $X$ and $Y$ be the two Veronese images. The Veronese lemma makes them closed projective subvarieties with regular inverse maps. The construction in cor-projective-variety-product-exists, applied to $X,Y$, realizes their Segre image as a closed subset of the target projective space, with regular projections recovering its two factors. Compose these projections with the Veronese inverses. By the product universal property they give a regular map from this closed image to ${\mathbf{P}}^m\times {\mathbf{P}}^n$, inverse to the map in step 2.1. That forward map is a morphism by the multihomogeneous theorem, since a nonzero coordinate $x_i$ and a nonzero coordinate $y_j$ give the nonzero monomial $x_i^a{y}_j^b$. Thus the displayed map is an isomorphism onto a closed subvariety, as asserted. The argument includes a degree equal to one and a factor ${\mathbf{P}}^0$.