User:Dfeuer/Product Order is Order

Theorem
Let $\left({S_i,\preceq_i}\right)$ be an ordered set for each $i \in I$.

And let $\preceq$ be the User:Dfeuer/Definition:Product Order on $S = \displaystyle \prod_{i \mathop\in I} S_i$.

Then $\left({S, \preceq}\right)$ is an ordered set.

Proof
$\preceq$ is reflexive:

Let $x \in S$.

Since $\preceq_i$ is reflexive for each $i \in I$, $x_i \preceq_i x_i$ for each $i \in i$.

Thus by the definition of the product order, $x \preceq x$.

$\preceq$ is transitive:

Suppose that $x \preceq y$ and $y \preceq z$.

Then by the definition of the product order, for each $i \in I$, $x_i \preceq_i y_i$ and $y_i \preceq_i z_i$.

Since each $\preceq_i$ is transitive,

for each $i \in I$, $x_i \preceq_i z_i$.

Thus, by the definition of the product order, $x \preceq z$.

$\preceq$ is antisymmetric:

Suppose that $x \preceq y$ and $y \preceq x$.

By the definition of the product order, for each $i \in I$, $x_i \preceq_i y_i$ and $y_i \preceq_i x_i$.

Since each $\preceq_i$ is anti-symmetric, for each $i \in I$, $x_i = y_i$.

By the definition of cartesian product, $x = y$.