CorollaryStatement: AI-adaptedProof: AI-adaptedprecheck passaudited 2026-08-02

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.

Complex de Moivre formula for every integer exponent

Statement

Let $j : Z \to Q$ be the integer embedding of The integers embed in the rationals, let $ι_{R} : Q \to R$ be the ordered-field embedding of The unique embedding of ℚ into an ordered field, and put $κ_{R} : = ι_{R} \circ j$ . For every integer $m$ and real $θ$ , $(\cos θ + i \sin θ)^{m} = \cos (κ_{R} (m) θ) + i \sin (κ_{R} (m) θ) .$ The conventions and prerequisite facts used below are recorded in Euler's formula: $\exp (i θ) = \cos θ + i \sin θ$ for every real $θ$ , $\exp (z + w) = \exp z \exp w$ , and the complex exponential extends the real exponential, $C = R [x] / (x^{2} + 1)$ is a field, every element is uniquely $a + b i$ , and every nonzero element has inverse $(a - b i) / (a^{2} + b^{2})$ , Integer powers in the complex field.

Facts & Assumptions

Given: An integer $m$ and real $θ$ .

Proof

technique · direct

1.1

Euler's formula identifies the base with $\exp (i θ)$ .

given

1.2

Repeated addition handles nonnegative powers by the exponential addition law; inverses handle negative powers.

given

2.1

Euler's formula at $κ_{R} (m) θ$ gives the displayed result.

given∎

Depends on

Used by

The n-th roots of a complex number and the n distinct roots of unity for every n≥1 Theorem

Dependency tree · two levels

37 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

J. Lebl, Basic Analysis I: Complex Numbers and the Complex Exponential (standard reference, not scraped)