ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31

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.

The graph of a bundle map as a subbundle of a Whitney sum

Example

If $Φ : E \to F$ is a smooth vector bundle map over ${id}_{M}$ , then its graph

$Γ_{Φ} : = {(e, Φ (e)) : e \in E} \subseteq E \oplus F$

is a smooth vector subbundle of the Whitney sum.

Facts & Assumptions

Given: A smooth bundle map $Φ : E \to F$ over one base $M$ .

[L1]

The Whitney sum $E \oplus F$ is a smooth vector bundle (Whitney sums are smooth vector bundles).

[L2]

Constant-rank images of bundle maps over one base are smooth subbundles (Constant-rank kernels and images of bundle maps over one base are subbundles).

Verification

technique · direct

1.1

Define $G : E \to E \oplus F$ by $G (e) = (e, Φ (e))$ . This is a smooth bundle map over ${id}_{M}$ , and each fibre map $G_{p} (v) = (v, Φ_{p} (v))$ is injective, hence has constant rank $rank E$ .

L1givenconstruct

2.1

The image of $G$ is exactly the graph $Γ_{Φ}$ . Therefore [L2] implies that $Γ_{Φ}$ is a smooth vector subbundle of $E \oplus F$ .

L2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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

John M. Lee, Introduction to Smooth Manifolds (standard reference, not scraped)
Will J. Merry, Differential Geometry (standard reference, not scraped)