TheoremStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 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.

Zassenhaus butterfly lemma in an abelian category

Statement

Let $A_{0} \leq A_{1}$ and $B_{0} \leq B_{1}$ be subobjects of an object $X$ in an abelian category. Then there is a canonical isomorphism

$\frac{A_{0} \lor (A_{1} \land B_{1})}{A_{0} \lor (A_{1} \land B_{0})} ≅ \frac{(A_{1} \land B_{1}) \lor B_{0}}{(A_{0} \land B_{1}) \lor B_{0}} .$

Facts & Assumptions

Given: Subobjects $A_{0} \leq A_{1}$ and $B_{0} \leq B_{1}$ of an object $X$ .

[L1]

The subobject lattice of $X$ is modular (The subobject lattice of an abelian category is modular).

[L2]

For subobjects $U, V$ of a common object, one has $(U \lor V) / V ≅ U / (U \land V)$ (Second isomorphism theorem in an abelian category).

[L3]

Nested quotients satisfy the third isomorphism theorem (Third isomorphism theorem in an abelian category).

Proof

technique · direct

1.1

Put $M : = A_{1} \land B_{1}$ , $U : = A_{1} \land B_{0}$ , and $V : = A_{0} \land B_{1}$ . Since $U \leq M$ , the second isomorphism theorem [L2] applied to $A_{0}$ and $M$ inside $A_{0} \lor M$ gives $\frac{A_{0} \lor M}{A_{0} \lor U} ≅ \frac{M}{M \land (A_{0} \lor U)} .$ By modularity [L1] inside the interval below $M$ , $M \land (A_{0} \lor U) = (M \land A_{0}) \lor U = V \lor U .$ So the left quotient is canonically isomorphic to $M / (V \lor U)$ .

L1L2construct

1.2

Similarly, since $V \leq M$ , the second isomorphism theorem [L2] applied to $M$ and $B_{0}$ inside $M \lor B_{0}$ gives $\frac{M \lor B_{0}}{V \lor B_{0}} ≅ \frac{M}{M \land (V \lor B_{0})} .$ Again modularity yields $M \land (V \lor B_{0}) = V \lor (M \land B_{0}) = V \lor U .$ So the right quotient is also canonically isomorphic to $M / (V \lor U)$ .

L1L2construct

2.1

The two quotients in steps 1.1 and 1.2 are canonically isomorphic to the same quotient of $M$ , hence to each other. The nested-quotient compatibility of [L3] identifies these isomorphisms with the displayed butterfly quotient comparison.

L3step 1.1step 1.2∎

Depends on

Used by

Dependency tree · two levels

15 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

Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, and Victor Ostrik, Tensor Categories, Section 1.5 (standard reference, not scraped)