TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-30‡ 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.

Five lemma in an abelian category

Statement

In a commutative diagram with exact rows

if $f_{1}, f_{2}, f_{4}, f_{5}$ are isomorphisms, then $f_{3}$ is an isomorphism.

Facts & Assumptions

Given: The commutative exact-row diagram in the statement.

[L1]

The sharp five lemma makes the middle map monic under one set of hypotheses and epic under the complementary one (Sharp five lemma in an abelian category).

[L2]

In an abelian category, a morphism that is both monic and epic is an isomorphism (An abelian category is balanced).

Proof

technique · direct

1.1

Because $f_{1}, f_{2}, f_{4}, f_{5}$ are isomorphisms, they are in particular monic and epic. The first half of [L1] therefore makes $f_{3}$ monic, and the second half of [L1] makes $f_{3}$ epic.

L1given

2.1

Applying [L2] to $f_{3}$ now shows that $f_{3}$ is an isomorphism.

L2step 1.1

3.1

Hence the five lemma holds in every abelian category.

step 2.1∎

Depends on

Used by

The published module five lemma as an instance Example

Dependency tree · two levels

7 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

Saunders Mac Lane, Categories for the Working Mathematician, Lemma VIII.4.4 (standard reference, not scraped)