User:Ascii/ProofWiki Sampling Notes for Theorems/Number Theory


 * 1) Set of Integers Bounded Below by Integer has Smallest Element
 * Let $\Z$ be the set of integers.
 * Let $\le$ be the ordering on the integers.
 * Let $\O \subset S \subseteq \Z$ such that $S$ is bounded below in $\struct {\Z, \le}$.
 * Then $S$ has a smallest element.
 * 1) Integers Bounded Above by Integer has Greatest Element
 * Let $\Z$ be the set of integers.
 * Let $\le$ be the ordering on the integers.
 * Let $\O \subset S \subseteq \Z$ such that $S$ is bounded above in $\struct {\Z, \le}$.
 * Then $S$ has a greatest element.
 * 1) Division Theorem
 * $\forall a, b \in \Z, b \ne 0: \exists! q, r \in \Z: a = q b + r, 0 \le r < \left\lvert{b}\right\rvert$
 * 1) Integer Divisor Results/One Divides all Integers
 * $\forall n \in \Z: 1 \divides n \land -1 \divides n$
 * 1) Integer Divisor Results/Integer Divides Itself
 * $\forall n \in \Z: n \divides n$
 * 1) Integer Divisor Results/Integer Divides its Negative
 * $\forall n \in \Z: n \divides (-n)$
 * 1) Integer Divisor Results/Integer Divides its Absolute Value
 * $\forall n \in \Z: n \divides \left \lvert {n}\right \rvert$
 * $\forall n \in \Z: \left \lvert {n}\right \rvert \divides n$
 * 1) Integer Divisor Results/Integer Divides Zero
 * $\forall n \in \Z: n \divides 0$
 * 1) Zero Divides Zero
 * $\forall n \in \Z: 0 \divides n \implies n = 0$
 * 1) Divisor Relation is Transitive
 * $\forall x, y, z \in \Z: x \divides y \land y \divides z \implies x \divides z$
 * 1) Divisor Relation is Antisymmetric
 * $\forall a, b \in \Z_{>0}: a \divides b \land b \divides a \implies a = b$
 * 1) Divisor Relation on Positive Integers is Partial Ordering
 * The divisor relation is a partial ordering of $\Z_{>0}$.
 * 1) Integer Divisor Results/Divisors of Negative Values
 * $\forall m, n \in \Z: m \mathrel \backslash n \iff -m \mathrel \backslash n \iff m \mathrel \backslash -n \iff -m \mathrel \backslash -n$
 * 1) Existence of Greatest Common Divisor
 * $\forall a, b \in \Z: a \ne 0 \lor b \ne 0$, there exists a largest $d \in \Z_{>0}$ such that $d \mathrel \backslash a$ and $d \mathrel \backslash b$.
 * That is, the greatest common divisor of $a$ and $b$ always exists.
 * 1) Greatest Common Divisor is at least 1
 * The greatest common divisor of $a$ and $b$ is at least $1$: $\forall a, b \in \Z_{\ne 0}: \gcd \set {a, b} \ge 1$