Alphabeta MathHow 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.
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 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.
The nine lemma forces the missing row in the standard diagram built from a section or retraction (Nine lemma in an abelian category).
The actual splitting conclusion is already recorded as the splitting lemma (Splitting lemma in an abelian category).
Proof
A section or retraction inserts the given short exact sequence into the usual diagram whose other two rows are visibly split exact. Applying [L1] makes the remaining row short exact as well.
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.
Hence the splitting lemma follows from the nine lemma.
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
- Peter Freyd, Abelian Categories, Lemma 2.68 (standard reference, not scraped)