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.
- 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.
Two isomorphism components of a morphism of triangles force the third
Statement
In a morphism of distinguished triangles, if any two components are isomorphisms, then the third is an isomorphism.
Facts & Assumptions
Given: A morphism of distinguished triangles with two isomorphism components.
Proof
Rotate source and target triangles simultaneously until the two known components are adjacent to the unknown one.
The rotated morphism has the same component isomorphisms up to translation, so the triangulated five lemma makes the remaining component invertible; undoing the rotation proves the claim.
Depends on
Used by
Dependency tree · two levels
7 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
- The Stacks Project, Lemma 13.4.3 (standard reference, not scraped)