TheoremStatement: AI-adaptedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-30

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 chain rule for differentials of smooth maps

Statement

If $F : M \to N$ and $G : N \to P$ are smooth, then $d (G \circ F)_{p} = d G_{F (p)} \circ d F_{p}$ for every $p \in M$ .

Facts & Assumptions

Given: Smooth maps $F : M \to N$ and $G : N \to P$ and a point $p \in M$ .

[F1]

The differential is defined by pullback of target germs (The differential of a smooth map).

Proof

technique · direct

1.1

Let $v \in T_{p} M$ and let $[h]$ be a smooth germ at $G (F (p))$ . Then $d (G \circ F)_{p} (v) ([h]) = v ([h \circ G \circ F])$ by [F1].

F1given

2.1

Also $(d G_{F (p)} \circ d F_{p}) (v) ([h]) = d F_{p} (v) ([h \circ G]) = v ([h \circ G \circ F])$ by two uses of [F1].

F1step 1.1

3.1

The two linear maps agree on every $v$ and every $[h]$ , so they are equal.

step 1.1step 2.1∎

Depends on

The differential of a smooth map

Used by

The differential of a diffeomorphism is an isomorphism Corollary
Cotangent pullback is contravariantly functorial Proposition
The tangent-bundle construction is functorial Proposition

Dependency tree · two levels

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