ExampleConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck 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.

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The trivial homomorphism $G\to H$ has kernel $G$ and image $\{e_H\}$

Example

The trivial homomorphism $G\to H$ has kernel $G$ and image $\{e_H\}$ .

Facts & Assumptions

Given: Groups $G,H$ and $t:G\to H$ defined by $t(g)=e_H$ .

[L1]

A group homomorphism preserves products (Monoid homomorphism and group homomorphism).

[L2]

Kernels and images are defined by inverse images and value sets (The kernel and image of a group homomorphism).

[L3]

Every group has an identity element (Group and abelian group).

Verification

technique · direct

1.1

For $g,h\in G$ , $t(gh)=e_H=e_He_H=t(g)t(h)$ , so $t$ is a homomorphism.

L1L2L3givenalgebra

2.1

Every $g\in G$ satisfies $t(g)=e_H$ , while the only value of $t$ is $e_H$ .

step 1.1L1L2L3givenalgebra

3.1

Thus $\ker t=G$ and $\operatorname{im}t=\{e_H\}$ .

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 13 results over 11 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Ernst, An Inquiry-Based Approach to Abstract Algebra, Homomorphisms (standard reference, not scraped)

The trivial homomorphism G→HG\to HG→H has kernel GGG and image {eH}\{e_H\}{eH​}