TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-29

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.

A zero arrow is detected by members

Statement

A morphism $h : R \to S$ in an abelian category is the zero morphism if and only if $h x \equiv 0$ for every member $x$ of $R$ .

Facts & Assumptions

Given: A morphism $h : R \to S$ .

[L1]

The zero member exists, and a member equivalent to zero is literally zero (Each object has a zero member and each member has a negative).

Proof

technique · direct

1.1L1algebra

If $h = 0$ , then for every member $x$ one has $h x = 0$ , hence $h x \equiv 0$ by [L1].

1.2L1assume-hyp

Conversely, assume $h x \equiv 0$ for every member $x$ of $R$ . Apply this to the identity member $1_{R} : R \to R$ . Then $h = h 1_{R} \equiv 0$ , so [L1] forces $h = 0$ .

2.1step 1.1step 1.2∎

Therefore the displayed member criterion detects exactly the zero morphism.

Depends on

Used by

FALSE: two morphisms that agree on every member are equal False statement

Dependency tree · two levels

6 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, Theorem VIII.4.3(iv) (standard reference, not scraped)