PropositionStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-29‡ 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.

Each object has a zero member and each member has a negative

Statement

For every object $A$ of an abelian category:

there is a zero member of $A$ ;
every member $x : X \to A$ has a negative member $- x : X \to A$ ;
for every member $x$ , one has $x \equiv 0$ if and only if $x = 0$ as a morphism.

Facts & Assumptions

Given: An abelian category and a member $x : X \to A$ .

[L1]

A zero object supplies zero morphisms between any two objects (A zero object supplies a unique compatible system of zero morphisms).

[L2]

An abelian category is additive, so every hom-set has negatives (Abelian category).

[L3]

Member equivalence is defined by comparison after epimorphisms (Equivalence of members).

Proof

technique · direct

1.1

By [L1], there is a zero morphism $0_{X, A} : X \to A$ , so $A$ has a zero member. By [L2], the additive inverse $- x : X \to A$ exists, so every member has a negative.

L1L2

1.2

Conversely, if $x \equiv 0$ , choose epimorphisms $u : W \to X$ and $v : W \to Y$ witnessing that equivalence. Then $x u = 0$ , and epicity of $u$ forces $x = 0$ .

L3assume-hypalgebra

2.1

If $x = 0$ , then $x \equiv 0$ is witnessed by the identity epic $1_{X} : X \to X$ , since $x 1_{X} = 01_{X}$ .

L3step 1.1

3.1

Step 1.1 proves claims 1 and 2, while steps 2.1 and 1.2 prove claim 3.

step 1.1step 2.1step 1.2∎

Depends on

Used by

The members of an object do not form a group Counterexample
Two morphisms agreeing on every member need not be equal Counterexample
A zero arrow is detected by members Theorem
Monicity by member cancellation Theorem
Monicity is detected by members Theorem
The subtraction surrogate Theorem

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

Saunders Mac Lane, Categories for the Working Mathematician, VIII.4 (standard reference, not scraped)