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.

Sharp five lemma in an abelian category

Statement

In a commutative diagram with exact rows

the following hold:

if $f_{1}$ is epic and $f_{2}, f_{4}$ are monic, then $f_{3}$ is monic;
if $f_{2}, f_{4}$ are epic and $f_{5}$ is monic, then $f_{3}$ is epic.

Facts & Assumptions

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

[L1]

The four lemma gives the monic and epic conclusions on any four-column window with exact rows (Four lemma in an abelian category).

Proof

technique · direct

1.1

For the monic clause, apply [L1] to the left four columns $A_{1} \to A_{2} \to A_{3} \to A_{4},$ $B_{1} \to B_{2} \to B_{3} \to B_{4} .$ The hypotheses there are exactly that $f_{1}$ is epic and that $f_{2}$ and $f_{4}$ are monic, so the four lemma gives that $f_{3}$ is monic.

L1assume-hyp

1.2

For the epic clause, apply [L1] to the right four columns $A_{2} \to A_{3} \to A_{4} \to A_{5},$ $B_{2} \to B_{3} \to B_{4} \to B_{5} .$ The hypotheses there are exactly that $f_{2}$ and $f_{4}$ are epic and that $f_{5}$ is monic, so the four lemma gives that $f_{3}$ is epic.

L1assume-hyp

2.1

Therefore the sharp five lemma holds in both halves.

step 1.1step 1.2∎

Depends on

Four lemma in an abelian category

Used by

FALSE: the five lemma needs only that the two middle maps are monic False statement
Why the five lemma asks for isomorphisms in the middle Remark
An exact functor transports every diagram lemma Theorem
Five lemma for a morphism of long exact sequences Theorem
Five lemma in an abelian category Theorem
The diagram lemmas hold in the opposite category Theorem

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

The Stacks Project, Section 12.5, Lemma 12.5.20 (standard reference, not scraped)
Charles A. Weibel, An Introduction to Homological Algebra, Exercise 1.3.3 (standard reference, not scraped)