PropositionStatement: 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.

Transversality is invariant under diffeomorphic change of source and target

Statement

Let $Φ : M^{'} \to M$ and $Ψ : N \to N^{'}$ be diffeomorphisms.

If $F : M \to N$ and $G : P \to N$ are transverse, then $Ψ \circ F \circ Φ$ and $Ψ \circ G$ are transverse.
If $Z \subseteq N$ is an embedded submanifold and $F ⋔ Z$ , then $Ψ \circ F \circ Φ ⋔ Ψ (Z)$ .

Facts & Assumptions

Given: Diffeomorphisms $Φ$ and $Ψ$ as above.

[F1]

Transversality is defined by spanning conditions on differential images and target tangent spaces (Transverse smooth maps, A smooth map transverse to an embedded submanifold).

[L1]

Differentials obey the chain rule, and the differential of a diffeomorphism is an isomorphism (The chain rule for differentials of smooth maps, The differential of a diffeomorphism is an isomorphism).

Proof

technique · direct

1.1

By [L1], the differentials $d Φ$ , $d Ψ$ , and their inverses are linear isomorphisms. Applying the chain rule to the composite maps multiplies the original differentials by these isomorphisms on the source and target sides.

L1given

2.1

A linear isomorphism preserves the property that a sum of subspaces is the whole target space. Therefore the spanning conditions in [F1] hold for the original maps exactly when they hold for the conjugated maps.

F1step 1.1algebra

3.1

Hence both notions of transversality are invariant under diffeomorphic changes of source and target.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

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