Alphabeta MathHow 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.
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These labels describe origin, not correctness: citations and verification chips remain separate evidence.
The third isomorphism theorem in abelian groups matches the categorical statement
Example
For nested subgroups of an abelian group, the quotient is canonically isomorphic to . This is exactly the categorical third isomorphism theorem specialized to .
Facts & Assumptions
Given: Subgroups of an abelian group.
The categorical third isomorphism theorem holds in every abelian category (Third isomorphism theorem in an abelian category).
The ordinary third isomorphism theorem holds for modules, hence for abelian groups (Third isomorphism theorem for modules).
Verification
Since abelian groups form an abelian category, [L1] applies to the inclusions .
The resulting isomorphism is the familiar quotient-group map described by [L2], so the categorical statement reproduces the ordinary one without change.
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
- Saunders Mac Lane, Categories for the Working Mathematician, VIII.3 (standard reference, not scraped)