ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 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.

Members of an abelian group correspond to its subgroups

Example

In $A b$ , a member $x : X \to A$ corresponds exactly to the subgroup $im (x) \leq A$ . Two members are equivalent exactly when they have the same image subgroup. Thus the member-calculus bijection of Members modulo equivalence correspond to subobjects becomes the usual identification of subgroup data with image subgroups.

Facts & Assumptions

Given: An abelian group $A$ and a homomorphism $x : X \to A$ .

[L1]

The category $A b$ is abelian (Abelian groups form an abelian category).

[L2]

Member-equivalence classes correspond to subobjects (Members modulo equivalence correspond to subobjects).

Verification

technique · direct

1.1L1L2

In $A b$ , subobjects of $A$ are exactly subgroup inclusions into $A$ , so [L2] identifies the class of $x$ with the subgroup represented by its image inclusion.

2.1step 1.1∎

Therefore the abstract member-subobject correspondence is, in this category, the ordinary rule $x \mapsto im (x)$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

14 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)