TheoremStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-28

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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.

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First isomorphism theorem in an abelian category

Statement

For every morphism $f : A \to B$ in an abelian category, there is a canonical isomorphism

$A / \ker (f) ≅ im (f) .$

Facts & Assumptions

Given: An abelian category and a morphism $f : A \to B$ .

[L1]

The quotient by a subobject is the cokernel of a representing monomorphism (The quotient of an object by a subobject, The quotient by a subobject is independent of the chosen representing monomorphism).

[L2]

The coimage is the cokernel of the kernel, and the image is the kernel of the cokernel (Image and coimage in a category with kernels and cokernels).

[L3]

The canonical morphism $coim (f) \to im (f)$ exists (The canonical morphism from the coimage to the image exists and is unique).

[L4]

In an abelian category that canonical morphism is an isomorphism (Abelian category).

Proof

technique · direct

1.1

By [L1] and [L2], the quotient $A / \ker (f)$ is exactly the coimage of $f$ .

L1L2

2.1

The map from step 1.1 to $im (f)$ is the canonical coimage-to-image morphism of [L3], and [L4] makes it an isomorphism. Therefore $A / \ker (f) ≅ im (f)$ canonically.

L3L4step 1.1∎

Depends on

Used by

Dependency tree · two levels

14 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)
Junhan Tan, The Freyd-Mitchell Embedding Theorem, Fact 2.8 (standard reference, not scraped)