TheoremStatement: 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.

$\ker (\exp) = 2 π i Z$ , and $\exp z = \exp w$ exactly when $z - w \in 2 π i Z$

Given: $z = x + i y$ and $w = u + i v$ .

technique · direct

1.1given

Cartesian form shows $\exp z = 1$ forces $e^{x} = 1$ and $(\cos y, \sin y) = (1, 0)$ .

1.2given

Strict monotonicity gives $x = 0$ , while the trigonometric period theorem gives $y \in 2 π Z$ .

2.1given∎

The addition law turns equality of two exponential values into membership of $z - w$ in the kernel, and the converse is immediate.

Depends on