Minimum Function on a Metric

1. The problem statement, all variables and given/known data

Let (X, d1) and (Y,d2) be metric space. Define a function d:(X x Y)x(X x Y) to R by d((x1,y,1), (x2,y2))=min{d(x1,x2),d(y1,y2)}. Is d a metric on X x Y? Explain

2. Relevant equations


3. The attempt at a solution

Is it enough to say that the min for d((x1,y1),d(x2,y2)) is the distance function between two points and since the distance function is the minimum distance between 2 points and a metric, then d is a metric?

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