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.