Power Set of Monoid under Induced Operation is Monoid

Theorem
Let $\struct {G, \circ}$ be a monoid with identity $e$.

Let $\struct {\powerset G, \circ_\PP}$ be the algebraic structure consisting of the power set of $G$ and the operation induced on $\powerset S$ by $\circ$.

Then $\struct {\powerset G, \circ_\PP}$ is a monoid with identity $\set e$.

Proof
By definition of a monoid, $\struct {G, \circ}$ is a semigroup.

By Power Set of Semigroup under Induced Operation is Semigroup, $\struct {\powerset G, \circ_\PP}$ is a semigroup.

By Subset Product by Identity Singleton, $\set e$ is an identity for $\struct {\powerset G, \circ_\PP}$.

Thus $\struct {\powerset G, \circ_\PP}$ is a monoid with identity $\set e$.