User:Dfeuer/sandbox

Reflexive Closure and Reflexive Reduction

 * User:Dfeuer/Reflexive Closure of Transitive Relation is Transitive
 * User:Dfeuer/Reflexive Closure of Antisymmetric Relation is Antisymmetric

Lexicographic Orderings

 * User:Dfeuer/Definition:Lexicographic Ordering on Product
 * User:Dfeuer/Definition:Lexicographic Ordering of Finite Sequences


 * User:Dfeuer/Well-Founded Relation Determines Minimal Elements


 * Product of Positive Element and Element Greater than One


 * User:Dfeuer/Strictly Positive Power of Element Greater than One Not Less than Element

Properties of Ordered Groups

 * User:Dfeuer/OG1
 * User:Dfeuer/OG2
 * User:Dfeuer/OG3
 * User:Dfeuer/OG4
 * User:Dfeuer/Operating on Ordered Group Relationships
 * User:Dfeuer/OG5

Properties of Ordered Rings

 * Union of Relations Compatible with Operation is Compatible
 * Intersection of Relations Compatible with Operation is Compatible
 * Inverse of Relation Compatible with Operation is Compatible
 * Definition:Relation Conversely Compatible with Operation
 * Definition:Relation Strongly Compatible with Operation
 * User:Dfeuer/Operating on Transitive Relationships Compatible with Operation
 * User:Dfeuer/Operating Repeatedly on Transitive Relationship Compatible with Operation


 * Complement of Relation Compatible with Group is Compatible
 * Relation Compatible with Group is Strongly Compatible
 * Relation Conversely Compatible with Group is Strongly Compatible


 * User:Dfeuer/OR1
 * User:Dfeuer/OR2
 * User:Dfeuer/OR3
 * User:Dfeuer/OR4
 * User:Dfeuer/OR5
 * User:Dfeuer/OR6
 * User:Dfeuer/OR7
 * User:Dfeuer/OR8
 * User:Dfeuer/OR9
 * User:Dfeuer/OR10
 * User:Dfeuer/OR11

scrap:

Lemma
Let $S$ be an ordered set.

Let $F$ be a totally ordered field.

Let $f,g \colon S \to F$.

Let $h\colon S \to F$ with $h(x) = f(x)g(x)$.

Suppose that for some $q \in F$ and some $w \in S$ with $q > 0$, $x \succ w \implies f(x) \ge q$.

Suppose that $g$ is increasing and its image is unbounded above.

Then the image of $h$ is unbounded above.

Useful links
Axiom of Foundation at NLab