PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31

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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.

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The inclusion of an embedded submanifold is a smooth embedding

Statement

If $S \subseteq M$ is an embedded submanifold equipped with the smooth structure from its restricted slice charts, then the inclusion $i : S ↪ M$ is a smooth embedding.

Facts & Assumptions

Given: An embedded submanifold $S \subseteq M$ with its slice-chart smooth structure.

[F1]

A smooth embedding is an injective immersion and a homeomorphism onto its image with the subspace topology (Smooth embeddings).

[L1]

The restricted slice charts form a smooth atlas on $S$ with the subspace topology (Slice-chart restrictions form a smooth atlas).

Proof

technique · direct

1.1

The inclusion $i : S ↪ M$ is injective and, by [L1], a homeomorphism of $S$ onto its image because the domain topology of $S$ is exactly the subspace topology.

L1given

1.2

In a restricted slice chart on $S$ and the ambient slice chart on $M$ , the coordinate representative of $i$ is $u \mapsto (u, 0)$ . Its differential is the coordinate inclusion, hence injective, so $i$ is an immersion.

2.1

Steps 1.1 and 1.2 verify the clauses in [F1], so $i$ is a smooth embedding.

F1step 1.1step 1.2∎

Depends on

Used by

The smooth structure of an embedded submanifold is unique Proposition

Dependency tree · two levels

7 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, Embedded Submanifolds (standard reference, not scraped)