Q:

Properties of sum of two metrics

I try to show that the sum of two metrics is again a metric for the product space $X\times Y$
$$d((x_1,y_1),(x_2,y_2))=d_X(x_1,y_1)+d_Y(x_2,y_2)$$
I showed the triangle inequality but fail at

Showing identity of indiscernibles. I see that this can only be true if $$d_X(x_1,y_1)=d_Y(x_2,y_2)=0$$ because of positivity. But how can I conclude that $(x_1,x_2)=(x_2,y_2)$?

Because $d((1,1),(2,2))=d_X(1,1)+d_Y(2,2)=0$ but obviously $(1,1)\neq (2,2)$

I also don't know how to show symmetry. Symmetry must imply that $d_X(x_1,y_1)-d_X(x_2,y_2)=d_Y(x_1,y_1)-d_Y(x_2,y_2)$, but this is only true for $d_X=d_Y$ or?

A:

Since $d_X(x_1,x_2)=d_Y(y_1,y_2)=0$, $x_1=x_2$ and $y_1=y_2$. So, $(x_1,x_2)=(y_1,y_2)$.
$d_X(x_1,x_2)+d_Y(y_1,y_2)=d_X(x_2,x_1)+d_Y(y_2,y_1)$, since $d_X$ and $d_Y$ are distances.

