Alphabeta Math
CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-09-01
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  • 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.
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Outside a null set every translation makes a chosen value a transverse zero

Statement

Let f:M→Rn be smooth and let q∈Rn. Then outside a null subset of a∈Rn, the map p↦f(p)+a has q as a regular value.

Facts & Assumptions

Given: A smooth map f:M→Rn and a point q∈Rn.

[L1]

For the point submanifold {q}, generic translations are transverse to it outside a null set (Generic translations of a Euclidean-valued map are transverse).

[L2]

Transversality to a point is the regular-value condition (Transversality to a point is the regular-value condition).

Proof

technique · direct
1.1

Apply [L1] with Z={q}. Then outside a null subset of a∈Rn, the translated map fa(p)=f(p)+a is transverse to {q}.

L1given
2.1

By [L2], fa⋔{q} is equivalent to q being a regular value of fa.

L2step 1.1
3.1

Therefore outside a null set of translations, the chosen value becomes a transverse zero.

step 2.1∎

Depends on

Used by

Dependency tree · two levels

6 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