TheoremStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 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.

The join of two subobjects is their least upper bound

Statement

Let $B$ and $C$ be subobjects of an object $A$ in an abelian category. Then the subobject $B \lor C$ of The join of two subobjects in an abelian category is the least upper bound of $B$ and $C$ in the subobject order of $A$ .

Facts & Assumptions

Given: Monomorphisms $b : B ↣ A$ and $c : C ↣ A$ representing the two subobjects.

[L1]

The join $B \lor C$ is the image of the induced map $[b, c] : B \oplus C \to A$ (The join of two subobjects in an abelian category).

[L2]

The image of a morphism is the least subobject through which that morphism factors (The image is the least subobject through which a morphism factors).

[L3]

A subobject inequality is exactly factorization of representatives (Subobjects and quotient objects form oppositely oriented partially ordered collections).

Proof

technique · direct

1.1

Let $j : J ↣ A$ be the image inclusion of $[b, c]$ . Since $[b, c] ι_{B} = b$ and $[b, c] ι_{C} = c$ for the biproduct injections, the factorization of $[b, c]$ through $j$ makes both $b$ and $c$ factor through $j$ . Thus $B \leq J$ and $C \leq J$ , so $J$ is an upper bound of the two subobjects.

L1L2L3

1.2

Let $n : N ↣ A$ be any common upper bound. Then $b = n u$ and $c = n v$ for suitable $u : B \to N$ and $v : C \to N$ . By the universal property of $B \oplus C$ , the induced map satisfies $[b, c] = n [u, v]$ . Now [L2] says that the image inclusion $j$ factors through every monomorphism through which $[b, c]$ factors, so $J \leq N$ .

L2L3construct

2.1

Step 1.1 gives that $J$ is an upper bound, and step 1.2 gives that it lies below every upper bound. By [L3], this is exactly the least-upper-bound claim.

L3step 1.1step 1.2∎

Depends on

Used by

The subobjects of an object in an abelian category form a lattice Theorem

Cited to discharge well-definedness by The join of two subobjects in an abelian category.

Dependency tree · two levels

11 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

Daniel Murfet, Abelian Categories, Section 4.2 (standard reference, not scraped)