PropositionStatement: 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 morphism carries members to members and preserves equivalence

Statement

Let $f : A \to B$ be a morphism.

If $x : X \to A$ is a member of $A$ , then $f x : X \to B$ is a member of $B$ .
If $x \equiv y$ as members of $A$ , then $f x \equiv f y$ as members of $B$ .

Facts & Assumptions

Given: A morphism $f : A \to B$ and members $x : X \to A$ , $y : Y \to A$ .

[L1]

Member equivalence means equality after precomposition by one common pair of epimorphisms (Equivalence of members).

Proof

technique · direct

1.1given

The composite $f x : X \to B$ is again a morphism into $B$ , so it is a member of $B$ .

1.2L1algebra

If $x \equiv y$ , choose epimorphisms $u : W \to X$ and $v : W \to Y$ with $x u = y v$ by [L1]. Postcomposing with $f$ gives $f x u = f y v$ , and the same epimorphisms witness $f x \equiv f y$ .

2.1step 1.1step 1.2∎

Therefore every morphism carries members to members and preserves their equivalence.

A morphism carries members to members and preserves equivalence

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources