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.

F-relatedness is equivalent to the derivation intertwining law

Statement

Let $F : M \to N$ be smooth, let $X$ be a smooth vector field on $M$ , and let $Y$ be a smooth vector field on $N$ . Then $X$ and $Y$ are $F$ -related if and only if

$X (f \circ F) = (Y f) \circ F$

for every $f \in C^{\infty} (N)$ .

Facts & Assumptions

Given: A smooth map $F : M \to N$ and smooth vector fields $X$ on $M$ and $Y$ on $N$ .

[L1]

The differential satisfies $d F_{p} (v) (f) = v (f \circ F)$ for $v \in T_{p} M$ and $f \in C^{\infty} (N)$ (The differential of a smooth map).

[L2]

The action of a vector field on functions is defined pointwise by $(X h) (p) = X_{p} (h)$ (The action of a vector field on smooth functions).

[L3]

For a point inside an open set there is a smooth bump function equal to $1$ on a neighbourhood of that point and supported in the open set (A manifold bump for a compact set inside an open set).

Proof

technique · direct

1.1L1L2given

Assume $X$ and $Y$ are $F$ -related. For any $p \in M$ and $f \in C^{\infty} (N)$ , [L1] and [L2] give $(X (f \circ F)) (p) = X_{p} (f \circ F) = d F_{p} (X_{p}) (f) = Y_{F (p)} (f) = ((Y f) \circ F) (p)$ .

1.2L1L2L3given

Conversely, assume $X (f \circ F) = (Y f) \circ F$ for every smooth $f$ on $N$ . Fix $p \in M$ , and let $[g] \in C_{F (p)}^{\infty} (N)$ be a smooth germ. Choose an open neighbourhood $W$ of $F (p)$ on which $g$ is represented by a smooth function, and use [L3] to choose $χ : N \to [0, 1]$ that is $1$ on a neighbourhood of $F (p)$ and has support contained in $W$ . Let $h$ be the global smooth function obtained by extending $χ g$ by $0$ outside $W$ . Then $[h] = [g]$ at $F (p)$ . The hypothesis applied to $h$ gives $X_{p} (h \circ F) = Y_{F (p)} (h)$ , and [L1] identifies the left-hand side with $d F_{p} (X_{p}) ([h])$ . Thus $d F_{p} (X_{p}) ([g]) = Y_{F (p)} ([g])$ for every germ $[g]$ , so $d F_{p} (X_{p}) = Y_{F (p)}$ .

2.1step 1.1step 1.2∎

Therefore $X$ and $Y$ are $F$ -related exactly when the derivation intertwining law holds for every smooth target function.

F-relatedness is equivalent to the derivation intertwining law

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources