LemmaStatement: AI-adaptedProof: AI-generatedprecheck passaudited 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.

Pullback of a target germ by a smooth map is a well-defined source germ

Statement

If $F : M \to N$ is smooth, $p \in M$ , and $[g] \in C_{F (p)}^{\infty} (N)$ , then the germ of $g \circ F$ at $p$ depends only on the germ $[g]$ at $F (p)$ .

Facts & Assumptions

Given: A smooth map $F : M \to N$ , a point $p \in M$ , and a germ $[g]$ at $F (p)$ .

[F1]

Equal germs agree on some neighbourhood of the base point (The germ of a smooth function at a point).

[F2]

The differential uses the pullback germ $[g \circ F]$ at $p$ (The differential of a smooth map).

Proof

technique · direct

1.1

If $[g] = [h]$ at $F (p)$ , then $g$ and $h$ agree on some neighbourhood $V$ of $F (p)$ . By continuity of $F$ , the inverse image of $V$ is a neighbourhood of $p$ on which $g \circ F = h \circ F$ .

F1given

2.1

Therefore $g \circ F$ and $h \circ F$ define the same germ at $p$ , so the pullback germ in [F2] is well defined.

F2step 1.1

3.1

Hence the target germ determines a unique source germ under pullback by $F$ .

step 2.1∎

Depends on

Used by

The differential sends derivations to derivations and is linear Lemma

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

Dependency tree · two levels

4 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)