Order of Real Numbers is Dual of Order of their Negatives

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Theorem

$\forall x, y \in \R: x > y \iff \paren {-x}< \paren {-y}$


Proof 1

Let $x > y$.

\(\ds x\) \(>\) \(\ds y\)
\(\ds \leadsto \ \ \) \(\ds x + \paren {-x}\) \(>\) \(\ds y + \paren {-x}\) Real Number Axioms: $\R \text O 1$: compatibility with addition
\(\ds \leadsto \ \ \) \(\ds 0\) \(>\) \(\ds y + \paren {-x}\) Real Number Axioms: $\R \text A 4$: Inverses
\(\ds \leadsto \ \ \) \(\ds 0 + \paren {-y}\) \(>\) \(\ds y + \paren {-x} + \paren {-y}\) Real Number Axioms: $\R \text O 1$: compatibility with addition
\(\ds \leadsto \ \ \) \(\ds 0 + \paren {-y}\) \(>\) \(\ds \paren {y + \paren {-y} } + \paren {-x}\) Real Number Axioms: $\R \text A 1$: Associativity and $\R \text A 2$: Commuativity
\(\ds \leadsto \ \ \) \(\ds 0 + \paren {-y}\) \(>\) \(\ds 0 + \paren {-x}\) Real Number Axioms: $\R \text A 4$: Inverses
\(\ds \leadsto \ \ \) \(\ds \paren {-y}\) \(>\) \(\ds \paren {-x}\) Real Number Axioms: $\R \text A 3$: Identity
\(\ds \leadsto \ \ \) \(\ds \paren {-x}\) \(<\) \(\ds \paren {-y}\) Definition of Dual Ordering

$\Box$


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

\(\ds \paren {-x}\) \(<\) \(\ds \paren {-y}\)
\(\ds \leadsto \ \ \) \(\ds \paren {-x} + x\) \(<\) \(\ds \paren {-y} + x\) Real Number Axioms: $\R \text O 1$: compatibility with addition
\(\ds \leadsto \ \ \) \(\ds 0\) \(<\) \(\ds \paren {-y} + x\) Real Number Axioms: $\R \text A 4$: Inverses
\(\ds \leadsto \ \ \) \(\ds 0 + y\) \(<\) \(\ds \paren {-y} + x + y\) Real Number Axioms: $\R \text O 1$: compatibility with addition
\(\ds \leadsto \ \ \) \(\ds 0 + y\) \(<\) \(\ds \paren {\paren {-y} + y} + x\) Real Number Axioms: $\R \text A 1$: Associativity and $\R \text A 2$: Commuativity
\(\ds \leadsto \ \ \) \(\ds 0 + y\) \(<\) \(\ds 0 + x\) Real Number Axioms: $\R \text A 4$: Inverses
\(\ds \leadsto \ \ \) \(\ds y\) \(<\) \(\ds x\) Real Number Axioms: $\R \text A 3$: Identity
\(\ds \leadsto \ \ \) \(\ds x\) \(>\) \(\ds y\) Definition of Dual Ordering

$\blacksquare$


Proof 2

\(\ds x\) \(<\) \(\ds y\)
\(\ds \leadstoandfrom \ \ \) \(\ds y - x\) \(>\) \(\ds 0\) Inequality iff Difference is Positive
\(\ds \leadstoandfrom \ \ \) \(\ds \) \(>\) \(\ds 0\) Real Number Axioms: $\R \text A 2 $: Commutativity of Addition
\(\ds \leadstoandfrom \ \ \) \(\ds -x + -\paren {-y}\) \(>\) \(\ds 0\) Negative of Negative Real Number
\(\ds \leadstoandfrom \ \ \) \(\ds -x - \paren {-y}\) \(>\) \(\ds 0\) Definition of Real Subtraction
\(\ds \leadstoandfrom \ \ \) \(\ds -y\) \(<\) \(\ds -x\) Inequality iff Difference is Positive

$\blacksquare$


Sources