Left neutral element is unique idempotent with left inverse in semigroup

Statement
Suppose $$(S,*)$$ is a semigroup (in other words, $$*$$ is an associative binary operation $$S \times S \to S$$) and $$e$$ is a left neutral element for the magma, i.e., we have:

$$e * a = a = \ \forall \ a \in S$$

Suppose $$u$$ is an idempotent element of $$S$$ (i.e., $$u * u = u$$) with a left inverse $$v$$ with respect to $$e$$, so $$v * u = e$$. Then, $$u = e$$.

Related facts

 * Left neutral element is unique in semigroup if every element has a left inverse

Proof
Given: Semigroup $$(S,*)$$. Element $$e \in S$$ such that $$e * a = a \ \forall a \in S$$, and for all $$a \in S$$. Idempotent $$u \in S$$ and element $$v \in S$$ such that $$v * u = e$$.

To prove: $$u = e$$

Proof: Consider the product $$v * u * u$$. Parenthesized as $$(v * u) * u$$, it simplifies to $$e * u = u$$. Parenthesized to $$v * (u * u)$$, it simplifies to $$v * u = e$$. Thus, $$u = e$$.