LemmaStatement: AI-generatedProof: AI-generated

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.

Integer multiplication on rational vector spaces

PDF p. 6: Finite-coefficient comparison

Statement

For a rational vector space $V$ and a positive integer $n$ , multiplication by $n$ is an automorphism of the underlying abelian group. A rational vector space is a module over the field $Q$ .

Facts & Assumptions

Given: A rational vector space $V$ and an integer $n > 0$ .

Goal: Show that $n : V \to V$ is an automorphism of abelian groups.

Proof

1.1givenalgebra

The scalar $1 / n$ belongs to $Q$ . Multiplication by $1 / n$ is additive and is both a left and right inverse to multiplication by $n$ , by the module axioms.

2.1step 1.1∎

Thus multiplication by $n$ is an automorphism, as claimed.

Used by · 1

Integer multiplication on rational complexes