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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Interior product is a graded antiderivation

Statement

If $α \in {Alt}^{k} (V)$ , $β \in {Alt}^{ℓ} (V)$ , and $v \in V$ , then

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

Facts & Assumptions

Given: Alternating covectors $α, β$ of degrees $k, ℓ$ and a vector $v \in V$ .

[F1]

Interior product inserts $v$ into the first slot (Interior product on alternating covectors).

[F2]

The wedge product is the signed shuffle sum (The wedge product of alternating covectors).

Proof

technique · direct

1.1

Evaluate both sides on $(v_{2}, \dots, v_{k + ℓ})$ . By [F2], the terms in $(α \land β) (v, v_{2}, \dots, v_{k + ℓ})$ split into two groups: those where $v$ lands among the $k$ arguments sent to $α$ , and those where it lands among the $ℓ$ arguments sent to $β$ .

F2given

2.1

The first group is exactly $(ι_{v} α) \land β$ by [F1]. To move $v$ into the first slot of $β$ in the second group, it must cross the $k$ slots occupied by $α$ , which contributes the sign $(- 1)^{k}$ ; that group is therefore $(- 1)^{k} α \land ι_{v} β$ .

F1F2step 1.1algebra

3.1

Summing the two groups gives the claimed formula, so interior product is a graded antiderivation.

step 2.1∎

Depends on

Used by

Interior product on forms is a graded antiderivation Proposition

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)