Alphabeta Math
CorollaryStatement: AI-adaptedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 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.

A morphism of short exact sequences with invertible outer maps is invertible

Statement

In a morphism of short exact sequences in an abelian category, if the left and right vertical maps are isomorphisms, then the middle vertical map is an isomorphism.

Facts & Assumptions

Given: A morphism of short exact sequences whose outer vertical maps are isomorphisms.

[L1]

The short five lemma makes the middle map both monic and epic (Short five lemma in an abelian category).

[L2]

Every morphism that is both monic and epic in an abelian category is an isomorphism (An abelian category is balanced).

Proof

technique · direct
1.1

Because the outer maps are isomorphisms, they are monic and epic. Therefore [L1] shows that the middle map is monic and epic.

L1given
2.1

Applying [L2] to that middle map shows that it is an isomorphism.

L2step 1.1
3.1

Hence a morphism of short exact sequences with invertible outer maps has invertible middle map as well.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 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