Negative of Absolute Value/Corollary 1

Corollary to Negative of Absolute Value

Let $x, y \in \R$ be a real numbers.

Let $\size x$ be the absolute value of $x$.

Then:

$\size x < y \iff -y < x < y$

Proof

Necessary Condition

Let $\size x < y$.

Then from Negative of Absolute Value:

$x \le \size x$

and:

$\size x \ge -x$

So $x < y$ and $-x < y$, and so $x > -y$ from Ordering of Inverses in Ordered Monoid.

It follows that $-y < x < y$.

$\Box$

Sufficient Condition

Let $-y < x < y$.

Then $x < y$ and $-x < y$.

For all $x$:

$\size x = x$

or:

$\size x = -x$

Thus it follows that $\size x < y$.

$\blacksquare$