PropositionStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06

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.

Lie derivatives are natural for related vector fields

Statement

If $F : M \to N$ is smooth and $X$ and $Y$ are $F$ -related, then $L_{X} (F^{*} ω) = F^{*} (L_{Y} ω) .$

Facts & Assumptions

Given: The manifolds, forms, vector fields, maps, and coordinates explicitly named in the statement.

[F1]

The preceding result states that For every smooth map $F : M \to N$ and every form $ω$ on $N$ , $d (F^{*} ω) = F^{*} (d ω) .$ (The exterior derivative commutes with pullback).

[F2]

For every vector field $Z$ and differential form $η$ , $L_{Z} η = d (ι_{Z} η) + ι_{Z} (d η)$ (Cartan's magic formula).

Proof

technique · direct

1.1F1given

Relatedness and the definition of pullback give $ι_{X} (F^{*} ω) = F^{*} (ι_{Y} ω)$ ; [F1] also gives $d (F^{*} ω) = F^{*} (d ω)$ .

2.1F1F2step 1.1∎

Apply [F2] to both sides of the first identity and use step 1.1: $L_{X} (F^{*} ω) = d (F^{*} (ι_{Y} ω)) + F^{*} (ι_{Y} d ω) = F^{*} (L_{Y} ω) .$

Depends on

Used by

Nothing in the library uses this result yet.

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, 2nd ed. (standard reference, not scraped)