Intersection of Submonoids with Monoid Identity is Submonoid

Theorem
Let $\struct {S, \circ}$ be a monoid whose identity is $e_S$.

Let $I$ be an indexing set.

Let $\family {S_\alpha}_{\alpha \mathop \in I}$ be a family of submonoids of $S$.

For each $S_\alpha \in \family {S_\alpha}_{\alpha \mathop \in I}$, let $e_S \in S_\alpha$.

Let $\ds \bigcap_{\alpha \mathop \in I} S_\alpha$ denote the intersection of $\family {S_\alpha}$

Then $\ds \bigcap_{\alpha \mathop \in I} S_\alpha$ is a submonoid of $S$.

Proof
First we show that $\struct {\ds \bigcap_{\alpha \mathop \in I} S_\alpha, \circ}$ is a semigroup:

From the above we have that $\ds \bigcap_{\alpha \mathop \in I} S_\alpha$ is closed under $\circ$.

From Restriction of Associative Operation is Associative we have that $\circ$ is associative on $\ds \bigcap_{\alpha \mathop \in I} S_\alpha$.

Hence we have that $\struct {\ds \bigcap_{\alpha \mathop \in I} S_\alpha, \circ}$ is a semigroup.

Identity Element
We are given that:
 * $\forall \alpha \in I: e_S \in S_\alpha$

That is, for all $S_\alpha$, $e_S$ is the identity element of $S_\alpha$

Thus for all $S_\alpha$, $\struct {S_\alpha, \circ}$ is a monoid.