Order of Real Numbers is Dual of Order of their Negatives

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Theorem

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


Proof 1

Let $x > y$.

\(\displaystyle x\) \(>\) \(\displaystyle y\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle x + \paren {-x}\) \(>\) \(\displaystyle y + \paren {-x}\) Real Number Axioms: $\R O1$: compatibility with addition
\(\displaystyle \leadsto \ \ \) \(\displaystyle 0\) \(>\) \(\displaystyle y + \paren {-x}\) Real Number Axioms: $\R A4$: Inverses
\(\displaystyle \leadsto \ \ \) \(\displaystyle 0 + \paren {-y}\) \(>\) \(\displaystyle y + \paren {-x} + \paren {-y}\) Real Number Axioms: $\R O1$: compatibility with addition
\(\displaystyle \leadsto \ \ \) \(\displaystyle 0 + \paren {-y}\) \(>\) \(\displaystyle \paren {y + \paren {-y} } + \paren {-x}\) Real Number Axioms: $\R A1$: Associativity and $\R A2$: Commuativity
\(\displaystyle \leadsto \ \ \) \(\displaystyle 0 + \paren {-y}\) \(>\) \(\displaystyle 0 + \paren {-x}\) Real Number Axioms: $\R A4$: Inverses
\(\displaystyle \leadsto \ \ \) \(\displaystyle \paren {-y}\) \(>\) \(\displaystyle \paren {-x}\) Real Number Axioms: $\R A3$: Identity
\(\displaystyle \leadsto \ \ \) \(\displaystyle \paren {-x}\) \(<\) \(\displaystyle \paren {-y}\) Definition of Dual Ordering

$\Box$


Let $\paren {-x} < \paren {-y}$.

\(\displaystyle \paren {-x}\) \(<\) \(\displaystyle \paren {-y}\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \paren {-x} + x\) \(<\) \(\displaystyle \paren {-y} + x\) Real Number Axioms: $\R O1$: compatibility with addition
\(\displaystyle \leadsto \ \ \) \(\displaystyle 0\) \(<\) \(\displaystyle \paren {-y} + x\) Real Number Axioms: $\R A4$: Inverses
\(\displaystyle \leadsto \ \ \) \(\displaystyle 0 + y\) \(<\) \(\displaystyle \paren {-y} + x + y\) Real Number Axioms: $\R O1$: compatibility with addition
\(\displaystyle \leadsto \ \ \) \(\displaystyle 0 + y\) \(<\) \(\displaystyle \paren {\paren {-y} + y} + x\) Real Number Axioms: $\R A1$: Associativity and $\R A2$: Commuativity
\(\displaystyle \leadsto \ \ \) \(\displaystyle 0 + y\) \(<\) \(\displaystyle 0 + x\) Real Number Axioms: $\R A4$: Inverses
\(\displaystyle \leadsto \ \ \) \(\displaystyle y\) \(<\) \(\displaystyle x\) Real Number Axioms: $\R A3$: Identity
\(\displaystyle \leadsto \ \ \) \(\displaystyle x\) \(>\) \(\displaystyle y\) Definition of Dual Ordering

$\blacksquare$


Proof 2

\(\displaystyle x < y\) \(\iff\) \(\displaystyle y - x > 0\) Inequality iff Difference is Positive
\(\displaystyle \) \(\iff\) \(\displaystyle -x + y > 0\) Real Number Axioms: $\R A2$: Commutativity of Addition
\(\displaystyle \) \(\iff\) \(\displaystyle -x + -\left({ -y }\right) > 0\) Negative of Negative Real Number
\(\displaystyle \) \(\iff\) \(\displaystyle -x - \left({ -y }\right) > 0\) Definition of subtraction
\(\displaystyle \) \(\iff\) \(\displaystyle -y < -x\) Inequality iff Difference is Positive

$\blacksquare$


Sources