TheoremStatement: AI-adaptedProof: 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 for a morphism of long exact sequences

Statement

Let $\dots \to A_{n - 1} \to A_{n} \to A_{n + 1} \to A_{n + 2} \to A_{n + 3} \to \dots$ and $\dots \to B_{n - 1} \to B_{n} \to B_{n + 1} \to B_{n + 2} \to B_{n + 3} \to \dots$ be long exact sequences in an abelian category, together with a morphism of these sequences. If the four comparison maps at $A_{n - 1}, A_{n}, A_{n + 2}, A_{n + 3}$ are isomorphisms, then the comparison map $A_{n + 1} \to B_{n + 1}$ is an isomorphism.

Facts & Assumptions

Given: The morphism of long exact sequences in the statement.

[L1]

Every five-term exact window satisfies the sharp five lemma (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

Extract the five-term window $A_{n - 1} \to A_{n} \to A_{n + 1} \to A_{n + 2} \to A_{n + 3}$ and the corresponding window in the $B$ -sequence. Exactness of the long sequences makes both rows exact.

L1given

2.1

Because the four surrounding comparison maps are isomorphisms, they satisfy both halves of the hypotheses of [L1]. Hence the middle comparison map $A_{n + 1} \to B_{n + 1}$ is both monic and epic.

L1step 1.1

3.1

Therefore that middle comparison map is an isomorphism by [L2].

L2step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

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, VIII.4 (standard reference, not scraped)