Lexicographic Order is Ordering

Theorem
Let $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ be ordered sets.

Let $\preccurlyeq_l$ be the lexicographic order on $S_1 \times S_2$''':
 * $\tuple {x_1, x_2} \preccurlyeq_l \tuple {y_1, y_2} \iff \paren {x_1 \prec_1 y_1} \lor \paren {x_1 = y_1 \land x_2 \preccurlyeq_2 y_2}$

Then $\preccurlyeq_l$ is an ordering on $S_1 \times S_2$.

Proof
In the following, $\tuple {x_1, x_2}, \tuple {y_1, y_2}, \tuple {z_1, z_2} \in S_1 \times S_2$.

Checking in turn each of the criteria for an ordering:

Reflexivity
From Equality of Ordered Pairs:
 * $x_1 = x_2 \land y_1 = y_2 \iff \tuple {x_1, x_2} = \tuple {y_1, y_2}$

Thus:
 * $\tuple {x_1, x_2} = \tuple {x_1, x_2}$

and so:
 * $\tuple {x_1, x_2} \preccurlyeq_l \tuple {x_1, x_2}$

by definition of lexicographic order on $S_1 \times S_2$.

So $\preccurlyeq_l$ has been shown to be reflexive.

Transitivity
Let:
 * $\tuple {x_1, x_2} \preccurlyeq_l \tuple {y_1, y_2}$

and:
 * $\tuple {y_1, y_2} \preccurlyeq_l \tuple {z_1, z_2}$

$(1): \quad$ Let $x_1 = y_1 = z_1$.

Then by definition of lexicographic order on $S_1 \times S_2$:


 * $x_2 \preccurlyeq_2 y_2$

and:
 * $y_2 \preccurlyeq_2 z_2$

As $\preccurlyeq_2$ is an ordering, it is transitive.

Thus:
 * $x_2 \preceq z_2$

and it follows by definition of lexicographic order on $S_1 \times S_2$ that:
 * $\tuple {x_1, x_2} \preccurlyeq_l \tuple {z_1, z_2}$

$(2): \quad$ Let $x_1 = y_1$ and $y_1 \ne z_1$.

Then by definition of lexicographic order on $S_1 \times S_2$:
 * $y_1 \prec z_1$

But as $x_1 = y_1$:
 * $x_1 \prec z_1$

and so by definition of lexicographic order on $S_1 \times S_2$:
 * $\tuple {x_1, x_2} \preccurlyeq_l \tuple {z_1, z_2}$

$(3): \quad$ Let $x_1 \ne y_1$ and $y_1 = z_1$.

Then by definition of lexicographic order on $S_1 \times S_2$:
 * $x_1 \prec y_1$

But as $y_1 = z_1$:
 * $x_1 \prec z_1$

and so by definition of lexicographic order on $S_1 \times S_2$:
 * $\tuple {x_1, x_2} \preccurlyeq_l \tuple {z_1, z_2}$

$(4): \quad$ Let $x_1 \ne y_1$ and $y_1 \ne z_1$.

Then by definition of lexicographic order on $S_1 \times S_2$:
 * $x_1 \prec y_1$

and:
 * $y_1 \prec z_1$

As $\preccurlyeq_1$ is an ordering, it is transitive.
 * $x_1 \prec z_1$

So by definition of lexicographic order on $S_1 \times S_2$:
 * $\tuple {x_1, x_2} \preccurlyeq_l \tuple {z_1, z_2}$

Thus in all cases it can be seen that:
 * $\tuple {x_1, x_2} \preccurlyeq_l \tuple {z_1, z_2}$

So $\preccurlyeq_l$ has been shown to be transitive.

Antisymmetry
Suppose that:
 * $\tuple {x_1, x_2} \preccurlyeq_l \tuple {y_1, y_2}$

and:
 * $\tuple {y_1, y_2} \preccurlyeq_l \tuple {x_1, x_2}$

Suppose $x_1 \ne y_1$.

Then by definition of lexicographic order on $S_1 \times S_2$:
 * $x_1 \preccurlyeq_1 y_1$

and:
 * $y_1 \preccurlyeq_1 x_1$

But $\preccurlyeq_1$ is an ordering and so $x_1 = y_1$.

From that contradiction it follows that $x_1 = y_1$.

Then by definition of lexicographic order on $S_1 \times S_2$:
 * $x_2 \preccurlyeq_2 y_2$

and:
 * $y_2 \preccurlyeq_2 x_2$

As $\preccurlyeq_2$ is an ordering, it is antisymmetric.

Therefore:
 * $x_2 = y_2$

and so:
 * $\tuple {x_1, x_2} = \tuple {y_1, y_2}$

So $\preccurlyeq_l$ has been shown to be antisymmetric.

$\preccurlyeq_l$ has been shown to be reflexive, transitive and antisymmetric.

Hence by definition it is an ordering.