Multiplication of Piecewise functions

I have two separate Piecewise functions wit the same domain ranges. Is it possible to get a third PWF that is the product of the first two?

p1 = Piecewise[{{1, x < 1}}, 2];
p2 = Piecewise[{{2, x > 2}}, 1];
p3 = PiecewiseExpand[p1*p2]