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.
If is a unit, then
Statement
If is a commutative ring, , , and is a unit, then
Facts & Assumptions
Given: as in the statement.
The adjugate identity is (For every positive-sized square matrix over a commutative ring, ).
A positive-sized square matrix whose determinant is a unit is invertible (A positive-sized square matrix over a commutative ring is invertible if and only if its determinant is a unit).
A two-sided inverse in a monoid is unique (In a monoid, a left inverse and a right inverse of the same element are equal; hence an invertible element has exactly one inverse, and it is two-sided).
Proof
Put . Scaling [L1] by gives .
By [L2], exists. Both it and are two-sided inverses of , so [L3] makes them equal.
Depends on
For every positive-sized square matrix over a commutative ring,
A positive-sized square matrix over a commutative ring is invertible if and only if its determinant is a unit
In a monoid, a left inverse and a right inverse of the same element are equal; hence an invertible element has exactly one inverse, and it is two-sided