CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-09-01

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 submersion is transverse to every embedded submanifold

Statement

If $F : M \to N$ is a submersion, then $F ⋔ Z$ for every embedded submanifold $Z \subseteq N$ .

Facts & Assumptions

Given: A submersion $F : M \to N$ and an embedded submanifold $Z \subseteq N$ .

[F1]

A submersion has surjective differential at every point (Immersions, submersions, and constant-rank maps).

[F2]

Transversality to $Z$ means $d F_{p} (T_{p} M) + T_{F (p)} Z = T_{F (p)} N$ at each point of $F^{- 1} (Z)$ (A smooth map transverse to an embedded submanifold).

Proof

technique · direct

1.1

Let $p \in F^{- 1} (Z)$ . By [F1], $d F_{p} (T_{p} M) = T_{F (p)} N$ .

F1given

2.1

Therefore $d F_{p} (T_{p} M) + T_{F (p)} Z = T_{F (p)} N$ , so [F2] gives $F ⋔ Z$ at $p$ .

F2step 1.1

3.1

Since $p$ was arbitrary, $F$ is transverse to every embedded submanifold.

step 2.1∎

Depends on

Used by

Generic translations of a Euclidean-valued map are transverse Corollary

Dependency tree · two levels

5 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, 2nd ed. (standard reference, not scraped)