PropositionStatement: AI-adaptedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-30

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 tangent-bundle projection and zero section are smooth

Statement

For a smooth manifold $M$ , the tangent-bundle projection $π : T M \to M$ and the zero section $0_{M} : M \to T M$ , $p \mapsto 0 \in T_{p} M$ , are smooth.

Facts & Assumptions

Given: A smooth manifold $M$ .

[L1]

The tangent bundle carries the smooth structure induced by bundle charts $\tilde{x}$ (Assuming countable choice, the tangent bundle has a canonical smooth 2n-manifold structure).

Proof

technique · direct

1.1

In a bundle chart $\tilde{x}$ , the projection is $(a, v) \mapsto a$ , which is smooth.

L1given

1.2

In the same chart, the zero section is $a \mapsto (a, 0)$ , which is smooth.

L1given

2.1

Therefore both $π$ and $0_{M}$ are smooth.

step 1.1step 1.2∎

Depends on

Assuming countable choice, the tangent bundle has a canonical smooth 2n-manifold structure

Used by

Nothing in the library uses this result yet.

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)
Nigel Hitchin, Differentiable Manifolds (standard reference, not scraped)