DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-07-25

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Absolute value in an ordered field

Definition

Let $F$ be an ordered field (Ordered field). The absolute value of $x \in F$ is

$∣ x ∣ : = {\begin{cases} x & if x \geq 0, \\ - x & if x < 0. \end{cases}$

Definition references

Remarks

The distance between $x$ and $y$ is $∣ x - y ∣$ .
The basic algebraic properties ( $∣ x ∣ \geq 0$ , $∣ x ∣ = 0 ⟺ x = 0$ , $∣ x y ∣ = ∣ x ∣ ∣ y ∣$ , and the characterisation $∣ x ∣ < c ⟺ - c < x < c$ ) are established in Basic properties of the absolute value; the triangle inequality is The triangle inequality.

Depends on

Ordered field

Dependency tree · 2 levels

Sources