Integers under Addition form Totally Ordered Group
Theorem
Let $\struct {\Z, +}$ denote the additive group of integers.
Let $\le$ be the usual ordering on $\Z$.
Then the ordered structure $\struct {\Z, +, \le}$ is a totally ordered group.
Proof
$\struct {\Z, +, \le}$ is an Ordered Structure
$(1)$
By Integer Addition is Closed, $\struct {\Z, +}$ is an algebraic structure.
$(2)$
$\le$ is an ordering on $\Z$.
Thus, $\struct {\Z, \le}$ is an ordered set.
$(3)$
By Ordering is Preserved on Integers by Addition and Integer Addition is Commutative, $\le$ is compatible with $+$.
Thus, $\struct {\Z, +, \le}$ is an ordered structure.
$\Box$
$\struct {\Z, +, \le}$ is a Totally Ordered Structure
By definition, the ordered structure $\struct {\Z, +, \le}$ is a totally ordered structure if and only if $\le$ is a total ordering.
This follows from Ordering on Integers is Total Ordering.
$\Box$
$\struct {\Z, +, \le}$ is a Totally Ordered Group
By definition, the totally ordered structure $\struct {\Z, +, \le}$ is a totally ordered group if and only if $\struct {\Z, +}$ is a group.
This follows from Integers under Addition form Abelian Group.
$\blacksquare$