CorollaryStatement: AI-adaptedProof: 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.

Every vector subbundle has a smooth complement

Statement

Every smooth vector subbundle $S \subseteq E$ has a smooth complement in $E$ .

Facts & Assumptions

Given: A smooth vector subbundle $S \subseteq E$ .

[L1]

The bundle $E$ admits a smooth bundle metric (Every smooth vector bundle admits a smooth bundle metric).

[L2]

Orthogonal complements of subbundles are smooth subbundles (Orthogonal complements of subbundles are smooth subbundles).

Proof

technique · direct

1.1

Choose a smooth bundle metric on $E$ by [L1].

L1givenchoose

2.1

With that metric, [L2] gives a smooth subbundle $S^{⊥} \subseteq E$ , and fibrewise one has $E_{p} = S_{p} \oplus S_{p}^{⊥}$ . Thus $S^{⊥}$ is a smooth complement of $S$ .

L2step 1.1algebra∎

Depends on

Used by

Every short exact sequence of smooth vector bundles splits Corollary

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

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