Product forcing ($\mathbb{P}\times \mathbb{Q}$)

The conditions are pairs with coordinatewise extension: $(p,q)\le (p^{\prime },q^{\prime })$ exactly when $p\le p^{\prime }$ and $q\le q^{\prime }$. The projection of a product-generic filter is a pair of factor filters whose Cartesian product is the original filter. Each is ground-model generic, and mutual genericity for product forcing makes the second generic over the extension by the first.

If $H=G_0\times G_1$ is generic for product forcing over $M$, then $G_1$ is generic over $M[G_0]$. For a name $\dot{D}$ for a dense subset of the second factor, take $p_0\in G_0$ forcing density. Ground-model pairs with first coordinate incompatible with $p_0$, or with first coordinate below $p_0$ forcing the second into $\dot{D}$, form a dense product set. The existential clause of syntactic forcing and the membership clause for a canonical forcing name give the density witnesses. The product filter meets this set, cannot use the incompatible case, and hence its second projection meets ${\dot{D}}^{G_0}$.

For a generic filter $H$ in product forcing, let $G_0,G_1$ be its coordinate projections. They are filters in an ordered set. Given $p\in G_0$ and $q\in G_1$, choose separate pairs witnessing their membership and then a common strengthening in $H$. Upward closure puts $(p,q)$ in $H$, proving the displayed equality. Every ground-model dense factor set lifts to a dense product set, proving genericity of each projection over the ground model.