Square of Non-Zero Element of Ordered Integral Domain is Strictly Positive

Theorem
Let $\struct {D, +, \times}$ be an ordered integral domain.

Then:
 * $\forall x \in D: x \ne 0 \iff \map P {x \times x}$

where $\map P {x \times x}$ denotes that $x \times x$ has the positivity property.

That is, the square of any element of an ordered integral domain is positive that element is non-zero.

Proof
Let $x = 0$.

Then $x \times x = 0 \times 0 = 0$ by the properties of the ring zero.

Now suppose $x \ne 0$.

One of two cases applies:
 * $\map P x$


 * $\neg \map P x$

Let $\map P x$.

Then by definition of positivity:
 * $\map P {x \times x}$

Now suppose $\neg \map P x$.

Then by the trichotomy law of ordered integral domains:
 * $\map P {-x}$

Then again by definition:
 * $\map P {\paren {-x} \times \paren {-x} }$.

But by Product of Ring Negatives:
 * $\paren {-x} \times \paren {-x} = x \times x$

So again:
 * $\map P {x \times x}$

Hence the result.