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.

Roots in a finite product of groups

PDF p. 13: Divisibility of the Picard group

Statement

Let A₁, …, Aᵣ be finitely many abelian groups, with r ≥ 0, and n a positive integer. If multiplication by n is surjective on each Aᵢ, then it is surjective on their product. The empty product is the zero group.

Facts & Assumptions

Given: The groups and n as stated.

Goal: Prove surjectivity on the product.

Proof

1.1givenchoose

Given an element $(a_{i})$ of the product, choose for each of the finitely many indices an element $x_{i}$ with $n x_{i} = a_{i}$ ; this uses only finite choice. If $r = 0$ , take the unique element of the empty product.

2.1step 1.1algebra∎

Addition in the product is coordinatewise, so $n (x_{i}) = (n x_{i}) = (a_{i})$ . Hence every product element has a preimage.

Used by · 1

Divisibility of the Picard group of an affine curve