Alphabeta Math
CorollaryStatement: 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.

Inverse image preserves meets and direct image preserves joins

Statement

Let f:A→A′ be a morphism in an abelian category. Then for subobjects C1,C2≤A′ and B1,B2≤A one has

f∗(C1∧C2)=f∗C1∧f∗C2,f∗(B1∨B2)=f∗B1∨f∗B2.

Facts & Assumptions

Given: A morphism f:A→A′ and the displayed subobjects.

[L1]

The subobject maps f∗ and f∗ form a Galois connection (Direct and inverse image of subobjects form a Galois connection).

[L2]

Subobjects form lattices, so meets and joins are characterized by their order universal properties (The subobjects of an object in an abelian category form a lattice).

Proof

technique · direct
1.1

For any subobject B≤A, B≤f∗(C1∧C2)  ⟺  f∗B≤C1∧C2  ⟺  f∗B≤C1 and f∗B≤C2  ⟺  B≤f∗C1 and B≤f∗C2. By [L2], this says that f∗(C1∧C2) is the meet of f∗C1 and f∗C2.

L1L2algebra
1.2

For any subobject C≤A′, f∗(B1∨B2)≤C  ⟺  B1∨B2≤f∗C  ⟺  B1≤f∗C and B2≤f∗C  ⟺  f∗B1≤C and f∗B2≤C. Again [L2] identifies this with the universal property of the join f∗B1∨f∗B2.

L1L2algebra
2.1

Steps 1.1 and 1.2 are exactly the two displayed identities.

step 1.1step 1.2∎

Depends on

Used by

Dependency tree · two levels

10 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