Identity of Algebraic Structure is Preserved in Substructure

Theorem

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

Let $\struct {T, \circ}$ be a algebraic substructure of $\struct {S, \circ}$.

That is, let $T \subseteq S$.

Let $e \in T$.

Then $e$ is an identity of $\struct {T, \circ}$.

Proof

Let $x \in T$.

By the definition of subset, $x \in S$.

Since $e$ is an identity of $\struct {S, \circ}$:

$e \circ x = x \circ e = x$

Since this holds for all $x \in T$, $e$ is an identity of $\struct {T, \circ}$.

$\blacksquare$