Product on Left with Idempotent Element under Left Self-Distributive Operation is Idempotent

Theorem
Let $\struct {S, \circ}$ be an algebraic structure.

Let $\circ$ be left self-distributive.

Let $a \in S$ be an idempotent element of $\struct {S, \circ}$.

Then for all $b \in S$, $b \circ a$ is an idempotent element of $\struct {S, \circ}$

Proof
Let $a \in S$ be an idempotent element of $\struct {S, \circ}$.

We have:

The result follows by definition of idempotent element.