Alphabeta Math
ExampleConstruction: AI-adaptedVerification: AI-adaptedprecheck 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 trivial homomorphism G→H has kernel G and image {eH}

Example

The trivial homomorphism G→H has kernel G and image {eH}.

Facts & Assumptions

Given: Groups G,H and t:G→H defined by t(g)=eH.

[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∈G, t(gh)=eH=eHeH=t(g)t(h), so t is a homomorphism.

L1L2L3givenalgebra
2.1

Every g∈G satisfies t(g)=eH, while the only value of t is eH.

step 1.1L1L2L3givenalgebra
3.1

Thus ker⁡t=G and im⁡t={eH}.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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