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.

Noether isomorphism theorems recovered from the nine lemma

Statement

The first and third isomorphism theorems in an abelian category can be recovered by placing the standard quotient diagrams into a 3×3 short-exact-column diagram and applying the nine lemma.

Facts & Assumptions

Given: The standard quotient diagrams attached to a subobject and to a chain of subobjects.

[L1]

The nine lemma reconstructs a missing short exact row from the surrounding short exact rows and columns (Nine lemma in an abelian category).

[L2]

The quotient objects and the first and third isomorphism theorems are already established in the abelian-category development (The quotient of an object by a subobject, First isomorphism theorem in an abelian category, Third isomorphism theorem in an abelian category, The quotient by the kernel followed by the image inclusion is the canonical epi-mono factorization).

Proof

technique · direct
1.1

For the first isomorphism theorem, insert the kernel, image, and cokernel factorization of a morphism into the standard 3×3 quotient diagram. The surrounding rows and columns are short exact by [L2], so [L1] forces the missing quotient row. That row is precisely the statement that the coimage and image quotients coincide.

L1L2givenconstruct
1.2

For the third isomorphism theorem, do the same with a chain of subobjects K≤H≤A. The canonical quotient maps provide the surrounding short exact rows and columns, and [L1] forces the remaining quotient row. By [L2], that row is exactly the third isomorphism theorem.

L1L2givenconstruct
2.1

Therefore the standard Noether isomorphism theorems are recoverable from the nine lemma.

step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

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