proof of basic theorem about ordered groups
Consider . Since can be written as a pairwise disjoint union, exactly one of the following conditions must hold:
By definition of the ordering relation, if the first condition holds. If the second condition holds, then . If the third condition holds, then we must have for some . Taking inverses, this means that , so , or equivalently . Hence, one of the following three conditions must hold:
Suppose that , so . Then
By the defining property of , we have . Also,
hence , so
By property 3, implies and likewise implies . Then, by property 2, we conclude .
By the hypothesis, . By the defining property, . Since , we have . In other words, .
By definition, means that . Since and , this is equivalent to stating that .
|Title||proof of basic theorem about ordered groups|
|Date of creation||2013-03-22 14:54:46|
|Last modified on||2013-03-22 14:54:46|
|Last modified by||rspuzio (6075)|