Matrix Entrywise Addition over Ring is Closed/Proof 1

Proof
Let $\mathbf A = \sqbrk a_{m n}$ and $\mathbf B = \sqbrk b_{m n}$ be elements of $\map {\MM_R} {m, n}$.

Let $\sqbrk c_{m n} = \sqbrk a_{m n} + \sqbrk b_{m n}$.

By definition of matrix entrywise addition:
 * $\forall i \in \closedint 1 m, j \in \closedint 1 n: a_{i j} + b_{i j} = c_{i j}$

By ring axiom $\text A 0$, $R$ is closed under addition.

Hence:
 * $\forall i \in \closedint 1 m, j \in \closedint 1 n: c_{i j} \in R$

From the definition of matrix entrywise addition, $\sqbrk c_{m n}$ has the same order as both $\sqbrk a_{m n}$ and $\sqbrk b_{m n}$.

Thus it follows that:
 * $\sqbrk c_{m n} \in \map {\MM_R} {m, n}$

Thus $\struct {\map {\MM_R} {m, n}, +}$, as it is defined, is closed.