TheoremStatement: AI-adaptedProof: AI-generatedprecheck passaudited 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.

The diagram lemmas hold in the opposite category

Statement

If $A$ is abelian, then every diagram lemma proved on this page remains valid in $A^{o p}$ , and each dual statement is one of the named lemmas on the same page.

Facts & Assumptions

Given: An abelian category $A$ .

[L1]

The opposite of an abelian category is abelian (The opposite of an abelian category is abelian).

[L2]

The snake, four, sharp five, nine, and sharp nine lemmas have already been proved in an arbitrary abelian category (Snake lemma in an abelian category, Four lemma in an abelian category, Sharp five lemma in an abelian category, Nine lemma in an abelian category, Sharp nine lemma).

Proof

technique · direct

1.1

By [L1], the opposite category $A^{o p}$ is abelian, so each theorem listed in [L2] applies there as stated.

L1L2given

2.1

Interpreting those statements back in $A$ swaps kernels with cokernels, monic with epic, pullbacks with pushouts, and top-row exactness with bottom-row exactness. Those are exactly the dual formulations already named on this page.

L1L2step 1.1

3.1

Therefore every diagram lemma on this page is closed under passage to the opposite category.

step 2.1∎

Depends on

Used by

An exact functor transports every diagram lemma Theorem

Dependency tree · two levels

21 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 (standard reference, not scraped)