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 $T_{F (p)} N$ .

The differential sends derivations to derivations and is linear

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources