Induced Structure Identity

Theorem
Let $\struct {T, \circ}$ be an algebraic structure, and let $S$ be a set.

Let $\struct {T^S, \oplus}$ be the structure on $T^S$ induced by $\circ$.

Let $e$ be an identity for $\circ$.

Then the constant mapping $f_e: S \to T$ defined as:


 * $\forall x \in S: \map {f_e} x = e$

is the identity for the pointwise operation $\oplus$ induced on $T^S$ by $\circ$.

Proof
Let $\struct {T, \circ}$ be an algebraic structure with an identity $e$.

Let $f \in T^S$.

Then: