Element Commutes with Product of Commuting Elements

Theorem
Let $\circ$ be a binary operation on a set $S$.

Let $\circ$ be associative.

Let $x, y, z \in S$.


 * If $x$ commutes with both $y$ and $z$, then $x$ commutes with $y \circ z$.


 * If $x$ and $y$ both commute with $z$, then $x \circ y$ commutes with $z$.

General Theorem
Let $\circ$ be a binary operation on a set $S$.

Let $\circ$ be associative.

Let $\left \langle {a_k} \right \rangle_{1 \le k \le n}$ be a sequence of terms of $S$.

Let $b \in S$.

If $b$ commutes with $a_k$ for each $k \in \left[{1 \,.\,.\, n}\right]$, then $b$ commutes with $a_1 \circ \cdots \circ a_n$.

Proof
The following are demonstrated by associativity of $\circ$ and the defined commutativity relations.


 * If $x$ commutes with both $y$ and $z$, then $x$ commutes with $y \circ z$:


 * If $x$ and $y$ both commute with $z$, then $x \circ y$ commutes with $z$:

The truth of the general theorem can be proved by induction.