TheoremStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-28‡ 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.

AB5 implies AB4

Statement

Every abelian category satisfying AB5 also satisfies AB4.

Facts & Assumptions

Given: An abelian category satisfying AB5.

[L1]

AB5 is the directed-join distributivity law together with AB3 (The axioms AB5 and AB5*).

[L2]

AB4 is the assertion that small coproducts of monomorphisms are monic (The axioms AB4 and AB4*).

Proof

technique · direct

1.1

Let $(m_{i} : B_{i} ↣ A_{i})_{i \in I}$ be a small family of monomorphisms, and let $m : ∐_{i \in I} B_{i} \to ∐_{i \in I} A_{i}$ be the induced coproduct map, which exists by the AB3 part of [L1]. Let $k : K ↣ ∐_{i} B_{i}$ be its kernel. For each finite subset $F \subseteq I$ , let $S_{F} \leq ∐_{i} B_{i}$ be the finite partial sum of the summands with indices in $F$ . Because finite coproducts in an abelian category are biproducts, the restriction of $m$ to $S_{F}$ is a finite direct sum of monomorphisms and is therefore monic. Hence $K \land S_{F} = 0$ for every finite $F$ .

L1construct

2.1

The family $(S_{F})$ is directed and has join $∐_{i} B_{i}$ . Applying the AB5 identity [L1] with the fixed subobject $K$ gives $K = (⋁_{F} S_{F}) \land K = ⋁_{F} (S_{F} \land K) = 0.$ So the kernel of $m$ is zero, and therefore $m$ is monic. By [L2], this is exactly AB4.

L1L2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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

Charles A. Weibel, An Introduction to Homological Algebra, Exercise A.4.6 (standard reference, not scraped)