If $x$ and $y$ are integers then prove that $(-x)+(-y)=-(x+y)$

Question

If $x$ and $y$ are integers then prove that $(-x)+(-y)=-(x+y)$

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My attempt:

I did google, and also it may be theorem itself. If we take common $'-'$ then itself proved itself in one line. Although, I don't its formal proof.

Can you explain it, please?

Answer 1

$-(x+y)=(-1)(x+y)=(-1)(x)+(-1)(y)=(-x)+(-y)$

Answer 2

The argument will depend on what is assumed.

At the most basic level, $-z $ denotes the (unique) number such that $-z+z=0$. So the equality $(-x)+(-y)=-(x+y) $ can (should, possibly) be read as saying the $(-x)+(-y) $ is the additive inverse of $x+y $; that is,$-(x+y) $. So, here, one needs to check that \begin{align} [(-x)+(-y)]+(x+y)&=(-x)+(-y)+(y+x)=(-x)+[(-y)+y ]+x\\ \ \\ &=(-x)+0+x=(-x)+x=0 \end{align}

In terms of axioms and properties, this comes before of the equality $-z=(-1)z $ (which usually requires a proof on its own).