LemmaStatement: Literature-sourcedProof: Literature-sourcedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-27

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.

Vanishing sets reverse inclusions

Statement

Let $R$ be a commutative ring and let $I \subseteq J ⊴ R$ be ideals. Then $V (J) \subseteq V (I)$ .

Facts & Assumptions

Given: A commutative ring $R$ and ideals $I \subseteq J ⊴ R$ .

[L1]

$V (K)$ is the set of prime ideals containing the ideal $K$ (The prime spectrum and vanishing sets).

Proof

technique · direct

1.1

Let $p \in V (J)$ . By [L1], this means $J \subseteq p$ . Since $I \subseteq J$ , one also has $I \subseteq p$ .

L1given

2.1

The containment $I \subseteq p$ says exactly that $p \in V (I)$ by [L1]. Therefore every element of $V (J)$ lies in $V (I)$ .

L1step 1.1

3.1

Hence $V (J) \subseteq V (I)$ .

step 2.1∎

Depends on

The prime spectrum and vanishing sets

Used by

Nothing in the library uses this result yet.

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

J. S. Milne, A Primer of Commutative Algebra, v4.03, §14 The spectrum of a ring (standard reference, not scraped)
The Stacks Project, Section 10.17: The spectrum of a ring (standard reference, not scraped)