LemmaStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 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.

Chart independence of $C^{r}$ smoothness

Statement

Let $M$ and $N$ be smooth manifolds, let $F : M \to N$ be continuous at $p \in M$ , and let $r \in N_{0} \cup {\infty}$ . Let $(U, φ)$ , $(U^{'}, φ^{'})$ be smooth charts of $M$ at $p$ and $(V, ψ)$ , $(V^{'}, ψ^{'})$ smooth charts of $N$ at $F (p)$ , with $F (U) \subseteq V$ and $F (U^{'}) \subseteq V^{'}$ . If the representative $ψ \circ F \circ φ^{- 1}$ is of class $C^{r}$ on a neighbourhood of $φ (p)$ , then the representative $ψ^{'} \circ F \circ φ^{' - 1}$ is of class $C^{r}$ on a neighbourhood of $φ^{'} (p)$ . Testing one chart pair therefore agrees with testing any other.

Facts & Assumptions

Given: The manifolds, map, point, smoothness class $r$ , and the four charts of the Statement, with $ψ \circ F \circ φ^{- 1}$ of class $C^{r}$ near $φ (p)$ .

[F1]

Any two charts of a smooth manifold are smoothly compatible: their domains are disjoint or both transition maps are smooth (Smoothly compatible charts and the smoothness of Euclidean transition maps, Smooth manifolds and their smooth charts).

[L1]

If $u : W \to W^{'}$ is smooth and $g : W^{'} \to W^{''}$ is $C^{r}$ , then $g \circ u$ is $C^{r}$ ; and if $h : W \to W^{'}$ is $C^{r}$ and $v : W^{'} \to W^{''}$ is smooth, then $v \circ h$ is $C^{r}$ (Compatibility of smooth atlases is an equivalence relation, and smooth Euclidean maps compose).

Proof

technique · direct

1.1

The overlaps $U \cap U^{'}$ and $V \cap V^{'}$ contain $p$ and $F (p)$ , so both are [given, F1] nonempty; by [F1] the transitions $φ \circ φ^{' - 1}$ on $φ^{'} (U \cap U^{'})$ and $ψ^{'} \circ ψ^{- 1}$ on $ψ (V \cap V^{'})$ are smooth. Because $F$ is continuous at $p$ and $F (p) \in V \cap V^{'}$ , there is an open neighbourhood $U_{p} \subseteq U \cap U^{'}$ of $p$ with $F (U_{p}) \subseteq V \cap V^{'}$ .

givenF1

2.1

On $φ^{'} (U_{p})$ the new representative factors as

givenF1L1step 1.1

$ψ^{'} \circ F \circ φ^{' - 1} = (ψ^{'} \circ ψ^{- 1}) \circ (ψ \circ F \circ φ^{- 1}) \circ (φ \circ φ^{' - 1}) .$

First [L1] composes the smooth $φ \circ φ^{' - 1}$ after the $C^{r}$ map $ψ \circ F \circ φ^{- 1}$ and keeps $C^{r}$ ; then [L1] composes the smooth $ψ^{'} \circ ψ^{- 1}$ after that and keeps $C^{r}$ . The middle factor is $C^{r}$ on the image of $φ^{'} (U_{p})$ under $φ \circ φ^{' - 1}$ , which is an open neighbourhood of $φ (p)$ inside the set where the given representative is $C^{r}$ ; hence the composite is $C^{r}$ on $φ^{'} (U_{p})$ . [given, F1, L1, step 1.1]

3.1

The set $φ^{'} (U_{p})$ is an open neighbourhood of $φ^{'} (p)$ , so the [given, step 2.1] representative $ψ^{'} \circ F \circ φ^{' - 1}$ is $C^{r}$ near $φ^{'} (p)$ . The reverse implication is the same argument with the chart pairs interchanged.

givenstep 2.1∎

Depends on

Used by

Identity maps and composites of smooth maps are smooth Proposition
Smooth maps are continuous Proposition
Smoothness is local on the source Proposition

Cited to discharge well-definedness by Cʳ and smooth maps between smooth manifolds.

Dependency tree · two levels

17 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

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

Chart independence of CrCr smoothness