TheoremStatement: AI-adaptedProof: AI-generatedprecheck passaudited 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.

Whitney sums are smooth vector bundles

Statement

If $E \to M$ and $F \to M$ are smooth vector bundles of ranks $r$ and $s$ , then $E \oplus F \to M$ is a smooth vector bundle of rank $r + s$ .

Facts & Assumptions

Given: Smooth vector bundles $E \to M$ and $F \to M$ .

[L1]

On a common trivializing neighborhood, vector bundle charts identify $E$ and $F$ with $U \times R^{r}$ and $U \times R^{s}$ (Vector bundle charts and transition functions).

Proof

technique · direct

1.1

On a common trivializing neighborhood $U$ , use [L1] to identify $E ∣_{U} \oplus F ∣_{U}$ with $U \times (R^{r} \oplus R^{s}) ≅ U \times R^{r + s}$ . This gives a local trivialization of the Whitney sum.

L1givenconstruct

2.1

If the transition matrices for $E$ and $F$ are $g_{β α}$ and $h_{β α}$ , then the transition matrix for $E \oplus F$ is the block diagonal matrix $diag (g_{β α}, h_{β α})$ , which is smooth on overlaps. Therefore these local trivializations define a smooth rank- $(r + s)$ bundle.

L1step 1.1algebra∎

Depends on

Used by

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

Dependency tree · two levels

16 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)