Order Embedding between Quotient Fields is Unique

Theorem
Let $$\left({R_1, +_1, \circ_1, \le_1}\right)$$ and $$\left({S, +_2, \circ_2, \le_2}\right)$$ be totally ordered integral domains.

Let $$K, L$$ be totally ordered quotient fields of $$\left({R_1, +_1, \circ_1, \le_1}\right)$$ and $$\left({S, +_2, \circ_2, \le_2}\right)$$ respectively.

Let $$\phi: R \to S$$ be a order monomorphism.

Then there is one and only one order monomorphism $$\psi: K \to L$$ extending $$\phi$$.

Also:


 * $$\forall x \in R, y \in R^*: \psi \left({\frac x y}\right) = \frac {\phi \left({x}\right)} {\phi \left({y}\right)}$$

If $$\phi: R \to S$$ is an order isomorphism, then so is $$\psi$$.

Proof
By Quotient Field is Unique, all we need to show is:


 * $$\forall x_1, x_2 \in R, y_1, y_2 \in R_+^*: \frac {x_1} {y_1} \le \frac {x_2} {y_2} \iff \frac {\phi \left({x_1}\right)} {\phi \left({y_1}\right)} \le \frac {\phi \left({x_2}\right)} {\phi \left({y_2}\right)}$$


 * Let $$x_1 / y_1 \le x_2 / y_2$$, where $$y_1, y_2 \in R_+^*$$.

As $$y_1, y_2 \in R_+^*$$, it follows that $$0 < y_1 \circ_1 y_2$$ and $$0 < 1 / \left({y_1 \circ_1 y_2}\right)$$.

We also have:
 * $$0 < \phi \left({y_1}\right) \circ_2 \phi \left({y_2}\right) = \phi \left({y_1 \circ_1 y_2}\right)$$

Therefore:


 * $$x_1 \circ_1 y_2 = \frac {x_1} {y_1} \circ_1 \left({y_1 \circ y_2}\right) \le \frac {x_2} {y_2} \circ_1 \left({y_1 \circ_1 y_2}\right) = x_2 \circ_1 y_1$$

Conversely, let $$x_1 \circ_1 y_2 \le x_2 \circ_1 y_1$$. Then:


 * $$\frac {x_1} {y_1} = x_1 \circ_1 y_2 \circ_1 \left({\frac 1 {y_1 \circ_1 y_2}}\right) \le x_2 \circ_1 y_1 \circ_1 \left({\frac 1 {y_1 \circ_1 y_2}}\right) = \frac {x_2} {y_2}$$

That is, we have:


 * $$\frac {x_1} {y_1} \le \frac {x_2} {y_2} \iff x_1 \circ_1 y_2 \le x_2 \circ_1 y_1$$

Similarly:


 * $$\frac {\phi \left({x_1}\right)} {\phi \left({y_1}\right)} \le \frac {\phi \left({x_2}\right)} {\phi \left({y_2}\right)} \iff \phi \left({x_1}\right) \circ_2 \phi \left({y_2}\right) \le \phi \left({x_2}\right) \circ_2 \phi \left({y_1}\right)$$

Now $$\phi: R \to S$$ is an order monomorphism.

Therefore:
 * $$x_1 \circ_1 y_2 \le x_2 \circ_1 y_1 \iff \phi \left({x_1 \circ_1 y_2}\right) \le \phi \left({x_2 \circ_1 y_1}\right)$$

The result follows.