Inverse for Rational Multiplication

Theorem
Each element $x$ of the set of non-zero rational numbers $\Q^*$ has an inverse element $\frac 1 x$ under the operation of rational number multiplication:
 * $\displaystyle \forall x \in \Q^*: \exists \frac 1 x \in \Q^*: x \times \frac 1 x = 1 = \frac 1 x \times x$

Proof
From the definition, the field $\left({\Q, +, \times}\right)$ of rational numbers is the quotient field of the integral domain $\left({\Z, +, \times}\right)$ of integers.

From Rational Multiplication Identity is One, the identity for $\left({\Q, \times}\right)$ is $1 = \dfrac 1 1 = \dfrac p p$ where $p \in \Z$ and $p \ne 0$.

From the definition of product inverses in quotient field, we have that:
 * $\left({\dfrac p q}\right)^{-1} = \dfrac q p$

which can be demonstrated directly:

Note that this is defined only when $p \ne 0$ and $q \ne 0$.

Now let $x = \dfrac p q \in \Q$.

We define $\dfrac 1 x$ as $1 \times \left({\dfrac p q}\right)^{-1}$.

Hence: