Rainbow numbers of $[m]\times [n]$ for $x_1+x_2=x_3$

Consider the set $[m]\times [n]=\{(i,j):1\le i\le m,1\le j\le n\}$ and the equation $x_1+x_2=x_3$, namely $eq$. The \emph{rainbow number of $[m]\times [n]$ for $eq$}, denoted $\text{rb}([m]\times [n],eq)$, is the smallest number of colors such that for every surjective $\text{rb}([m]\times [n],eq)$-coloring of $[m]\times [n]$ there must exist a solution to $eq$, with component-wise addition, where every element of the solution set is assigned a distinct color. This paper determines that $\text{rb}([m]\times [n],eq)=m+n+1$ for all values of $m$ and $n$ that a greater than or equal to $2$.