TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-08-28

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.

A morphism factors uniquely through its image

Statement

Let $f : A \to B$ be a morphism in a category with kernels and cokernels. If $c : B \to C$ is a cokernel of $f$ and $i : im (f) \to B$ is a kernel of $c$ , then there exists a unique morphism $\hat{f} : A \to im (f)$ with

$i \circ \hat{f} = f .$

Facts & Assumptions

Given: A morphism $f : A \to B$ , a cokernel $c : B \to C$ of $f$ , and a kernel $i : im (f) \to B$ of $c$ .

[L1]

The image of $f$ is the kernel of a cokernel of $f$ (Image and coimage in a category with kernels and cokernels).

Proof

technique · direct

1.1L1given

Because $c$ is a cokernel of $f$ , one has $c \circ f = 0$ .

2.1L1step 1.1∎

The morphism $i$ is a kernel of $c$ , so step 1.1 gives a unique $\hat{f} : A \to im (f)$ with $i \hat{f} = f$ . That is the claimed factorization through the image.

Depends on

Image and coimage in a category with kernels and cokernels

Used by

Dependency tree · two levels

2 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

Gautam Tamme, Algebra II Lecture 9, §9.1 (standard reference, not scraped)