PropositionStatement: 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.

The canonical map to a quotient bundle is a smooth bundle map

Statement

If $S \subseteq E$ is a smooth vector subbundle, then the fibrewise quotient map $q : E \to E / S$ is a smooth vector bundle map over ${id}_{M}$ , and its kernel is $S$ .

Facts & Assumptions

Given: A smooth vector bundle $E \to M$ and a smooth subbundle $S \subseteq E$ .

[L1]

The quotient $E / S \to M$ is a smooth vector bundle (A vector bundle quotient by a subbundle is a smooth vector bundle).

Proof

technique · direct

1.1

In an adapted local frame $(e_{1}, \dots, e_{r})$ with $S$ spanned by the first $k$ vectors, the quotient bundle chart from [L1] identifies $q$ with the map $(p, u, w) \mapsto (p, w)$ , where $u \in R^{k}$ and $w \in R^{r - k}$ . Thus $q$ is smooth and fibrewise linear.

L1given

2.1

In the same coordinates, $q (p, u, w) = 0$ exactly when $w = 0$ , which means that the vector lies in the span of $e_{1}, \dots, e_{k}$ , namely in $S_{p}$ . Therefore $\ker q = S$ .

step 1.1algebra∎

Depends on

Used by

Dependency tree · two levels

7 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)