PropositionStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-29

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.

Smoothness is local on the source

Statement

Let $M$ and $N$ be smooth manifolds, let $F : M \to N$ be continuous, and let $(U_{i})_{i \in I}$ be an open cover of $M$ . Then $F$ is smooth if and only if every restriction $F ∣_{U_{i}} : U_{i} \to N$ is smooth, where each $U_{i}$ carries the restricted smooth structure.

Facts & Assumptions

Given: A continuous map $F : M \to N$ and an open cover $(U_{i})_{i \in I}$ of $M$ .

[F1]

$F$ is smooth when its representative with respect to suitable smooth charts is smooth at every point, and this is independent of the chart pair ( $C^{r}$ and smooth maps between smooth manifolds, Chart independence of $C^{r}$ smoothness).

[F2]

An open subset $U \subseteq M$ carries the restricted smooth structure, whose charts are the restrictions $(V \cap U, φ ∣_{V \cap U})$ of smooth charts of $M$ (Smooth manifolds and their smooth charts, An open subset of a smooth manifold has a canonical restricted smooth structure).

Proof

technique · direct

1.1givenF1F2

Forward direction: suppose $F$ is smooth and fix $i \in I$ . For [given, F1, F2] $p \in U_{i}$ take a smooth chart $(V, ψ)$ of $N$ at $F (p)$ and a smooth chart $(W, φ)$ of $M$ at $p$ with $W \subseteq U_{i}$ and $F (W) \subseteq V$ . The representative of $F ∣_{U_{i}}$ with respect to $(W \cap U_{i}, φ ∣_{W \cap U_{i}})$ and $(V, ψ)$ is $ψ \circ F \circ φ^{- 1}$ restricted to $φ (W \cap U_{i})$ , which is smooth because [F1] makes $ψ \circ F \circ φ^{- 1}$ smooth and restricting to the open set $φ (W \cap U_{i})$ keeps every iterated coordinate derivative existing and continuous. By [F1], $F ∣_{U_{i}}$ is smooth at $p$ .

1.2givenF1F2choose

Reverse direction: suppose every $F ∣_{U_{i}}$ is smooth and let $p \in M$ . [given, F1, F2, choose] Choose $i \in I$ with $p \in U_{i}$ , then charts $(V, ψ)$ of $N$ at $F (p)$ and $(W, φ)$ of $M$ at $p$ with $W \subseteq U_{i}$ and $F (W) \subseteq V$ . The representative of $F$ with respect to $(W, φ)$ and $(V, ψ)$ equals the representative of $F ∣_{U_{i}}$ with respect to $(W \cap U_{i}, φ ∣_{W \cap U_{i}})$ and $(V, ψ)$ , which [F1] and the hypothesis make smooth, so $F$ is smooth at $p$ by [F1].

2.1givenstep 1.1step 1.2∎

Steps 1.1 and 1.2 prove the two directions at every point, so the [given, step 1.1, step 1.2] biconditional holds on all of $M$ .

Depends on

Used by

Dependency tree · two levels

15 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

Rob van der Vorst, Introduction to differentiable manifolds, §2, Theorem 2.15 (standard reference, not scraped)
Nigel Hitchin, Differentiable Manifolds, §2.4 (standard reference, not scraped)