equivalence in absolute values

• November 5th 2012, 03:04 PM
psolaki
equivalence in absolute values
Is the following equivalence true:

$|x+y|=|x|+|y| \Longleftrightarrow (x+y\geq 0\Longrightarrow x+y =|x| +|y|)\wedge(x+y<0\Longrightarrow x+y=-(|x|+|y|)$
because i can only prove that:

$|x+y|=|x|+|y| \Longrightarrow (x+y\geq 0\Longrightarrow x+y =|x| +|y|)\vee(x+y<0\Longrightarrow x+y=-(|x|+|y|)$