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.

Symmetric nine lemma

Statement

In a commutative 3×3 diagram, suppose the middle row and middle column are short exact. If any three of the remaining four rows and columns are short exact, then the fourth is short exact.

Facts & Assumptions

Given: The commutative 3×3 diagram in the statement.

[L1]

The sharp nine lemma recovers a missing outer row from the two rows below it and the three columns (Sharp nine lemma).

[L2]

Passing to the opposite category preserves abelianity and reverses exact sequences (The opposite of an abelian category is abelian).

[L3]

Transposing the indexing of a commutative 3×3 diagram exchanges rows with columns while preserving commutativity and exactness.

Proof

technique · direct
1.1

If the missing exact line is the top row, [L1] applies directly. If it is the bottom row, apply [L1] in the opposite category and redraw the reversed exact sequences from top to bottom; [L2] then transports the result back.

L1L2assume-case topassume-case bottom
2.1

If the missing exact line is the left or right column, transpose the diagram using [L3]. The missing column becomes an outer row, so step 1.1 applies to the transposed diagram and transports back.

L1L2L3step 1.1cases
3.1

Thus any one of the four outer rows and columns is forced by the other three together with the short exact middle row and middle column.

step 1.1step 2.1cases-exhaustive∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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