LemmaStatement: 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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The commutator of vector-field derivations is again a derivation

Statement

Let $X$ and $Y$ be smooth vector fields on $M$ . Then the commutator $[X, Y] : C^{\infty} (M) \to C^{\infty} (M)$ defined by $[X, Y] f = X (Y f) - Y (X f)$ is an $R$ -linear derivation.

Facts & Assumptions

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

[L1]

Each smooth vector field acts on $C^{\infty} (M)$ as a derivation (A vector field acts as a derivation of smooth functions).

Proof

technique · direct

1.1L1given

By [L1], both $X$ and $Y$ are $R$ -linear derivations, so their commutator is automatically $R$ -linear. It remains to prove the Leibniz rule.

1.2L1given

Expand $X (Y (f g))$ using [L1] twice: $X (Y (f g)) = X (f Y g + g Y f) = X f Y g + f X (Y g) + X g Y f + g X (Y f) .$

1.3L1given

Similarly, $Y (X (f g)) = Y f X g + f Y (X g) + Y g X f + g Y (X f) .$

2.1step 1.2step 1.3algebra

Subtracting step 1.3 from step 1.2 cancels the mixed first-order products, leaving $[X, Y] (f g) = f [X, Y] g + g [X, Y] f .$

3.1step 1.1step 2.1∎

Therefore $[X, Y]$ is an $R$ -linear derivation of $C^{\infty} (M)$ .

Depends on

Used by

Nothing in the library uses this result yet.

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)