Alphabeta Math
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.

An exact functor transports every diagram lemma

Statement

Let F:A→B be an exact functor between abelian categories. Then F carries every instance of the short five lemma, snake lemma, four lemma, sharp five lemma, and nine lemma in A to the corresponding valid instance in B. For the snake lemma, the connecting morphism is carried to the connecting morphism under the canonical kernel and cokernel comparison isomorphisms.

Facts & Assumptions

Given: An exact functor F:A→B.

[L1]

Exactness is equivalent to preserving kernels and cokernels, and one-sided exactness preserves monomorphisms and epimorphisms (An additive functor is exact exactly when it preserves kernels and cokernels, A left exact functor preserves monomorphisms and a right exact functor preserves epimorphisms).

[L2]

The connecting morphism is characterized uniquely by a pullback-pushout square, and the named diagram lemmas have already been proved in any abelian category (The connecting morphism exists and is unique, 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, The diagram lemmas hold in the opposite category).

Proof

technique · direct
1.1

By [L1], the functor F preserves short exact sequences, kernels, cokernels, monomorphisms, and epimorphisms. Therefore applying F to any diagram that satisfies the hypotheses of one of the listed lemmas again produces a diagram satisfying the same type of hypotheses in B.

L1given
2.1

For the short five, four, sharp five, and nine lemmas, the conclusions are therefore immediate from the corresponding theorem in B, namely [L2].

L1L2step 1.1
2.2

For the snake lemma, F preserves the pullback, pushout, kernel, and cokernel data used in the construction of δ. The resulting morphism in B satisfies the same universal-property square, so uniqueness in [L2] identifies it with the connecting morphism of the image diagram.

L1L2step 1.1
3.1

Hence every diagram lemma on this page is transported by an exact functor, with the connecting morphism respected under the canonical comparisons.

step 2.1step 2.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

34 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