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.

An injection identifies its domain with its image

Statement

If $f : G \to H$ is an injective homomorphism of abelian groups, its corestriction $G \to im f$ is an isomorphism.

Facts & Assumptions

Given: The injective group homomorphism in the statement.

Goal: Construct the asserted isomorphism.

Proof

1.1given

The corestriction is a homomorphism and is surjective by the definition of image. It remains injective because the original map is injective.

2.1step 1.1algebra∎

Its set-theoretic inverse preserves addition: if $f (a)$ and $f (b)$ are image elements, their sum is $f (a + b)$ . Thus the inverse sends their sum to the sum of their inverse images. The corestriction is consequently an isomorphism.

Used by · 1

Divisibility of an image with torsion kernel