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.
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- 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 pushout of a monomorphism is a monomorphism
Statement
In an abelian category, the pushout of a monomorphism is a monomorphism.
Facts & Assumptions
Given: An abelian category and a pushout square whose left leg is monic.
The opposite of an abelian category is abelian (The opposite of an abelian category is abelian).
In an abelian category, pullbacks of epimorphisms are epimorphisms (The pullback of an epimorphism is an epimorphism).
Proof
Passing to the opposite category turns the given pushout square into a pullback square, and the given monomorphism into an epimorphism. By [L1], the opposite category is still abelian.
Apply [L2] in the opposite category. The opposite of the resulting epimorphism is exactly the pushout leg in the original square, so that leg is monic.