LemmaStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-08-02

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.

The intersection of a nonempty family of normal subgroups is normal

Let $G$ be a group and let $N$ be a nonempty family of normal subgroups of $G$ . Then

$K : = ⋂_{N \in N} N$

is a normal subgroup of $G$ .

Given: A group $G$ and a nonempty family $N$ of normal subgroups of $G$ .

[L1]

The intersection of a nonempty family of subgroups of a group is a subgroup (The intersection of a nonempty family of subgroups of $G$ is a subgroup of $G$ ).

[L2]

technique · direct

1.1L1

By [L1], the set $K = ⋂_{N \in N} N$ is a subgroup of $G$ .

1.2givenL2

Fix $g \in G$ and $x \in K$ . For every $N \in N$ , one has $x \in N$ and $N ⊴ G$ , so $g x g^{- 1} \in N$ by [L2]. Hence $g x g^{- 1} \in K$ .

2.1step 1.1step 1.2L2∎

Thus $g K g^{- 1} \subseteq K$ for every $g \in G$ , and [L2] gives $K ⊴ G$ .

Depends on