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

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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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Interior product on forms is a graded antiderivation

Statement

If $X$ is a smooth vector field and $α \in Ω^{k} (M)$ , $β \in Ω^{ℓ} (M)$ , then

$ι_{X} (α \land β) = ι_{X} α \land β + (- 1)^{k} α \land ι_{X} β .$

Facts & Assumptions

Given: A smooth vector field $X$ and forms $α, β$ of degrees $k, ℓ$ .

[F1]

Interior product of a form is defined pointwise from the fibrewise interior product (Interior product of a form by a vector field).

[L1]

Fibrewise interior product is a graded antiderivation (Interior product is a graded antiderivation).

Proof

technique · direct

1.1F1given

At each point $p \in M$ , [F1] identifies $(ι_{X} (α \land β))_{p}$ with $ι_{X_{p}} (α_{p} \land β_{p})$ .

2.1F1L1step 1.1

Applying [L1] in the vector space $T_{p} M$ gives $ι_{X_{p}} (α_{p} \land β_{p}) = ι_{X_{p}} α_{p} \land β_{p} + (- 1)^{k} α_{p} \land ι_{X_{p}} β_{p} .$ Using [F1] again identifies this with the fibre at $p$ of the claimed form identity.

3.1step 2.1∎

Since the two forms agree at every point, the displayed identity holds on $M$ .

Depends on

Used by

Cartan commutator identities Proposition
Cartan's magic formula Theorem

Dependency tree · two levels

8 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)