PropositionStatement: Literature-sourcedProof: AI-generatedprecheck 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.

A smooth map with everywhere smooth local inverses is a local diffeomorphism

Statement

Let $F : M \to N$ be a smooth map of smooth manifolds. Assume that for every $p \in M$ there are open neighbourhoods $U_{p} \subseteq M$ of $p$ and $V_{p} \subseteq N$ of $F (p)$ together with a smooth map $G_{p} : V_{p} \to U_{p}$ such that

$G_{p} \circ F ∣_{U_{p}} = {id}_{U_{p}}, F ∣_{U_{p}} \circ G_{p} = {id}_{V_{p}} .$

Then $F$ is a local diffeomorphism.

Facts & Assumptions

Given: A smooth map $F : M \to N$ satisfying the local inverse hypothesis of the Statement.

[F1]

A diffeomorphism is a bijective smooth map with smooth inverse, and a local diffeomorphism is a map that restricts near every point to a diffeomorphism onto an open subset of the target (Diffeomorphisms and local diffeomorphisms of manifolds).

[F2]

Identity maps and composites of smooth maps are smooth (Identity maps and composites of smooth maps are smooth).

Proof

technique · direct

1.1givenchoose

Fix $p \in M$ and choose the neighbourhoods $U_{p}$ , $V_{p}$ , and the smooth map [given, choose] $G_{p}$ from the hypothesis. The identities $G_{p} \circ F ∣_{U_{p}} = {id}_{U_{p}}$ and $F ∣_{U_{p}} \circ G_{p} = {id}_{V_{p}}$ show that the restriction $F ∣_{U_{p}} : U_{p} \to V_{p}$ is bijective with inverse $G_{p} : V_{p} \to U_{p}$ .

2.1F1F2step 1.1

The restriction $F ∣_{U_{p}}$ is smooth because it is the same map as $F$ with [F1, F2, step 1.1] a smaller domain, and $G_{p}$ is smooth by hypothesis. Therefore step 1.1 makes $F ∣_{U_{p}} : U_{p} \to V_{p}$ a diffeomorphism by [F1].

3.1F1step 2.1∎

Since $V_{p}$ is open in $N$ by hypothesis, step 2.1 exhibits $p$ in an open [F1, step 2.1] neighbourhood on which $F$ is a diffeomorphism onto an open subset of $N$ . By [F1] this is exactly the local-diffeomorphism condition.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 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 (standard reference, not scraped)
Nigel Hitchin, Differentiable Manifolds, §2.4 (standard reference, not scraped)