Matrix Entrywise Addition is Associative

Theorem
Let $\map \MM {m, n}$ be a $m \times n$ matrix space over one of the standard number systems.

For $\mathbf A, \mathbf B \in \map \MM {m, n}$, let $\mathbf A + \mathbf B$ be defined as the matrix entrywise sum of $\mathbf A$ and $\mathbf B$.

The operation $+$ is associative on $\map \MM {m, n}$.

That is:
 * $\paren {\mathbf A + \mathbf B} + \mathbf C = \mathbf A + \paren {\mathbf B + \mathbf C}$

for all $\mathbf A$, $\mathbf B$ and $\mathbf C$ in $\map \MM {m, n}$.

Also see

 * Matrix Entrywise Addition is Commutative