 One-Cohen-real preservation of GCH

Over a Generalized continuum hypothesis ground model, adding one Cohen real preserves every infinite power $2^{\lambda }={\lambda }^{+}$. For the countable forcing, nice subset forcing names for $\lambda$ number at most $(2^{{\aleph }_0}{)}^{\lambda }=2^{\lambda }$ in the ground model. Old subsets give the matching lower bound and the countable chain condition for forcing preserves cardinals. Starting from the constructible universe, the new real is nonconstructible while the constructible levels remain unchanged.