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.

The splitting lemma follows from the nine lemma

Statement

If a short exact sequence in an abelian category admits a section or a retraction, then the splitting conclusion can be recovered by applying the nine lemma to the induced 3×3 diagram.

Facts & Assumptions

Given: A short exact sequence together with either a section of its right-hand map or a retraction of its left-hand map.

[L1]

The nine lemma forces the missing row in the standard 3×3 diagram built from a section or retraction (Nine lemma in an abelian category).

[L2]

The actual splitting conclusion is already recorded as the splitting lemma (Splitting lemma in an abelian category).

Proof

technique · direct
1.1

A section or retraction inserts the given short exact sequence into the usual 3×3 diagram whose other two rows are visibly split exact. Applying [L1] makes the remaining row short exact as well.

L1givenconstruct
2.1

The data in that recovered short exact row are exactly the biproduct data named in [L2]. So the nine-lemma route reproduces the splitting lemma statement.

L2step 1.1
3.1

Hence the splitting lemma follows from the nine lemma.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

16 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