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.

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

Related vector fields have related Lie brackets

Statement

If smooth vector fields $X_{1}, X_{2}$ on $M$ are respectively $F$ -related to smooth vector fields $Y_{1}, Y_{2}$ on $N$ , then $[X_{1}, X_{2}]$ is $F$ -related to $[Y_{1}, Y_{2}]$ .

Facts & Assumptions

Given: A smooth map $F : M \to N$ , vector fields $X_{1}, X_{2}$ on $M$ , and vector fields $Y_{1}, Y_{2}$ on $N$ with each $X_{i}$ $F$ -related to $Y_{i}$ .

[L1]

$F$ -relatedness is equivalent to the intertwining identity on smooth functions (F-relatedness is equivalent to the derivation intertwining law).

[L2]

The commutator of vector-field derivations is again a derivation (The commutator of vector-field derivations is again a derivation).

[L3]

Every derivation comes from a unique smooth vector field (Derivations of smooth functions are exactly smooth vector fields).

Proof

technique · direct

1.1

For every $f \in C^{\infty} (N)$ , [L1] gives $X_{2} (f \circ F) = (Y_{2} f) \circ F and X_{1} (f \circ F) = (Y_{1} f) \circ F .$

L1given

2.1

Apply $X_{1}$ to the first identity and $X_{2}$ to the second. Using [L1] again on the resulting target functions yields $X_{1} X_{2} (f \circ F) = (Y_{1} Y_{2} f) \circ F, X_{2} X_{1} (f \circ F) = (Y_{2} Y_{1} f) \circ F .$

L1step 1.1

3.1

Subtracting the identities in step 2.1 gives $[X_{1}, X_{2}] (f \circ F) = ([Y_{1}, Y_{2}] f) \circ F$ for every smooth $f$ on $N$ . By [L2], [L3], and [L1], this is exactly the statement that $[X_{1}, X_{2}]$ and $[Y_{1}, Y_{2}]$ are $F$ -related.

L1L2L3step 2.1

4.1

Therefore related vector fields have related Lie brackets.

step 3.1∎

Depends on

Used by

Diffeomorphism pushforward preserves Lie brackets Corollary

Dependency tree · two levels

13 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)
Will J. Merry, Differential Geometry (standard reference, not scraped)