 Solution

Add the base-$p$ expansions of $x$ and $y$ column by column. Before carrying, the sum of all displayed digits is $d_p(x)+d_p(y)$. Each carry removes $p$ units from one column and adds one unit to the next, decreasing the total digit sum by $p-1$. After all carries the digits are those of $x+y$, so the subadditivity of the base-p digit sum gives