Real Number is Greater than Zero iff its Negative is Less than Zero

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Theorem

$\forall x \in \R: x > 0 \iff \left({-x}\right) < 0$


Proof

Let $x > 0$.

\(\displaystyle x\) \(>\) \(\displaystyle 0\)
\(\displaystyle \implies \ \ \) \(\displaystyle x + \left({-x}\right)\) \(>\) \(\displaystyle 0 + \left({-x}\right)\) Real Number Axioms: $\R O2$: compatibility with addition
\(\displaystyle \implies \ \ \) \(\displaystyle 0\) \(>\) \(\displaystyle 0 + \left({-x}\right)\) Real Number Axioms: $\R A4$: Inverses
\(\displaystyle \implies \ \ \) \(\displaystyle 0\) \(>\) \(\displaystyle \left({-x}\right)\) Real Number Axioms: $\R A3$: Identity

$\Box$


Let $\left({-x}\right) < 0$.

\(\displaystyle \left({-x}\right)\) \(<\) \(\displaystyle 0\)
\(\displaystyle \implies \ \ \) \(\displaystyle x + \left({-x}\right)\) \(<\) \(\displaystyle x + 0\) Real Number Axioms: $\R O2$: compatibility with addition
\(\displaystyle \implies \ \ \) \(\displaystyle 0\) \(<\) \(\displaystyle x + 0\) Real Number Axioms: $\R A4$: Inverses
\(\displaystyle \implies \ \ \) \(\displaystyle 0\) \(<\) \(\displaystyle x\) Real Number Axioms: $\R A3$: Identity

$\blacksquare$


Sources