LemmaStatement: 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 differential sends derivations to derivations and is linear

Statement

For a smooth map $F : M \to N$ and a point $p \in M$ , the map $d F_{p} : T_{p} M \to T_{F (p)} N$ is well defined and linear.

Facts & Assumptions

Given: A smooth map $F : M \to N$ and a point $p \in M$ .

[F1]

The differential is defined by $d F_{p} (v) ([g]) = v ([g \circ F])$ (The differential of a smooth map).

[F2]

Pulling back a target germ by $F$ gives a well-defined source germ (Pullback of a target germ by a smooth map is a well-defined source germ).

[F3]

Derivations are linear and satisfy the Leibniz rule (Derivations at a point and the tangent space).

Proof

technique · direct

1.1F1F2given

By [F2], the formula of [F1] is well defined on target germs.

2.1F1F3step 1.1

If $v \in T_{p} M$ , then $d F_{p} (v)$ is linear because $v$ is, and it satisfies the Leibniz rule because $v ([g h] \circ F) = v ((g \circ F) (h \circ F))$ and [F3] applies.

3.1F1step 2.1∎

The assignment $v \mapsto d F_{p} (v)$ is linear because the defining formula of [F1] is linear in $v$ . Therefore $d F_{p}$ is a well-defined linear map into $T_{F (p)} N$ .

Depends on

Used by

Cited to discharge well-definedness by The differential of a smooth map.

Dependency tree · two levels

6 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)