TheoremStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-28‡ rests on unproved material (inherited)

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.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

Objects of finite length form an abelian subcategory

Statement

In an abelian category, the full subcategory whose objects have finite length is an abelian subcategory.

Facts & Assumptions

Given: An abelian category $A$ .

[L1]

Finite length is the property defined in Object of finite length.

[L2]

Finite length is stable under passing to subobjects and quotients, and is additive across a subobject (Length is additive along a subobject).

[L3]

A full subcategory is abelian precisely when it is closed under kernels, cokernels, and finite biproducts computed in the ambient abelian category (Abelian subcategory and exact embedding).

Proof

technique · direct

1.1

Let $f : X \to Y$ be a morphism between finite-length objects. Since $\ker (f) \leq X$ and $im (f) \leq Y$ , [L2] makes both $\ker (f)$ and $im (f)$ finite length. The quotient $Y / im (f)$ is then finite length by [L2], so the cokernel of $f$ is finite length as well.

L1L2construct

1.2

If $X$ and $Y$ have finite length, then the inclusion $X ↣ X \oplus Y$ has quotient $Y$ . Applying [L2] to that inclusion shows that $X \oplus Y$ has finite length. So the finite-length objects are closed under finite biproducts.

L2construct

2.1

Steps 1.1 and 1.2 are exactly the kernel, cokernel, and finite-biproduct closures required by [L3]. Therefore the full subcategory of finite-length objects is an abelian subcategory.

L3step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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

Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, and Victor Ostrik, Tensor Categories, Section 1.5 (standard reference, not scraped)