CorollaryStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-27

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 biproduct of morphisms is the diagonal matrix

Statement

For morphisms $f : A \to A^{'}$ and $g : B \to B^{'}$ , the induced morphism

$f \oplus g : A \oplus B \to A^{'} \oplus B^{'}$

has matrix

$(\begin{array}{cc} f & 0 \\ 0 & g \end{array}) .$

Facts & Assumptions

Given: Morphisms $f : A \to A^{'}$ and $g : B \to B^{'}$ in an additive category.

[L1]

Morphisms between finite biproducts are determined by their matrix entries (Morphisms between finite biproducts correspond to matrices).

[L2]

Composition of those morphisms is ordinary matrix multiplication (Composition of morphisms between finite biproducts is matrix multiplication).

Proof

technique · direct

1.1

By definition, $f \oplus g$ is the unique morphism whose composites with the two injections and two projections act as $f$ and $g$ on the matching summands and as zero on the off-diagonal summands.

2.1

Therefore its four matrix entries are $f$ , $0$ , $0$ , and $g$ in the expected positions. The multiplication rule of [L2] is exactly what makes this diagonal description functorial under further direct sums and composition.

L1L2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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.2 (standard reference, not scraped)