PropositionStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04

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.

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Leibniz rules for the Lie bracket with function multiples

Statement

For smooth vector fields $X, Y$ and smooth functions $f, g$ on $M$ ,

$[X, f Y] = f [X, Y] + (X f) Y$

and

$[f X, g Y] = f g [X, Y] + f (X g) Y - g (Y f) X .$

Facts & Assumptions

Given: Smooth vector fields $X, Y$ and smooth functions $f, g, h$ .

[L1]

Smooth vector fields act as derivations on smooth functions (A vector field acts as a derivation of smooth functions).

Proof

technique · direct

1.1

For any test function $h$ , $[X, f Y] (h) = X (f Y h) - f Y (X h) = f [X, Y] (h) + (X f) Y h$ by one application of the Leibniz rule from [L1]. Since this holds for every $h$ , one has $[X, f Y] = f [X, Y] + (X f) Y$ .

L1given

2.1

Apply step 1.1 with $f X$ in place of $X$ and use [L1] once more: $[f X, g Y] = g [f X, Y] + (f X g) Y = g (f [X, Y] - (Y f) X) + f (X g) Y .$ Collecting terms gives $[f X, g Y] = f g [X, Y] + f (X g) Y - g (Y f) X .$

L1step 1.1algebra

3.1

Therefore the Lie bracket satisfies the displayed Leibniz rules with function multiples.

step 1.1step 2.1∎

Depends on

Used by

FALSE: the Lie bracket is Cⁱnfty-linear in each vector-field entry False statement
The Lie derivative of a vector field equals the Lie bracket Theorem

Dependency tree · two levels

5 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

Will J. Merry, Differential Geometry (standard reference, not scraped)
Nigel Hitchin, Differentiable Manifolds (standard reference, not scraped)