Left Operation is Distributive over Idempotent Operation

Theorem
Let $\struct {S, \circ, \leftarrow}$ be an algebraic structure where:
 * $\leftarrow$ is the left operation
 * $\circ$ is any arbitrary binary operation.

Then:
 * $\leftarrow$ is distributive over $\circ$


 * $\circ$ is idempotent.
 * $\circ$ is idempotent.

Proof
From Left Operation is Right Distributive over All Operations:
 * $\forall a, b, c \in S: \paren {a \circ b} \leftarrow c = \paren {a \leftarrow c} \circ \paren {b \leftarrow c}$

for all binary operations $\circ$.

It remains to show that $\leftarrow$ is left distributive over $\circ$ $\circ$ is idempotent.

Necessary Condition
Let $\circ$ be idempotent.

Then:

Thus $\leftarrow$ is left distributive over $\circ$.

Sufficient Condition
Let $\leftarrow$ be left distributive over $\circ$.

Let $a \in S$ be arbitrary.

Then:

Hence $\circ$ is idempotent.