LemmaStatement: AI-adaptedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-30

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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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A derivation annihilates constant germs

Statement

Every derivation at $p$ sends every constant germ to $0$ .

Facts & Assumptions

Given: A derivation $v \in T_{p} M$ .

[F1]

A derivation is linear and satisfies the Leibniz rule on germs (Derivations at a point and the tangent space).

Proof

technique · direct

1.1

Let $1$ denote the constant germ with value $1$ . Applying the Leibniz rule from [F1] to $1 \cdot 1$ gives $v (1) = 1 v (1) + 1 v (1) = 2 v (1)$ , so $v (1) = 0$ .

F1given

2.1

Every constant germ is $c \cdot 1$ for some $c \in R$ , so linearity from [F1] gives $v (c \cdot 1) = c v (1) = 0$ .

F1step 1.1

3.1

Thus every constant germ is annihilated by $v$ .

step 2.1∎

Depends on

Derivations at a point and the tangent space

Used by

Coordinate derivations form a basis of the tangent space Theorem

Dependency tree · two levels

3 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 (standard reference, not scraped)
Will J. Merry, Differential Geometry (standard reference, not scraped)
Nigel Hitchin, Differentiable Manifolds (standard reference, not scraped)