TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-13

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 Five Lemma for modules

Statement

In a commutative diagram with exact rows

the middle map $c$ is injective if $a$ is surjective and $b, d$ are injective, and it is surjective if $b, d$ are surjective and $e$ is injective. In particular, if $a, b, d, e$ are isomorphisms, then $c$ is an isomorphism.

Facts & Assumptions

Given: The commutative diagram in the statement, with exact rows.

Diagram: $u ⁣ : A \to B$ , $v ⁣ : B \to C$ , $w ⁣ : C \to D$ , $x ⁣ : D \to E$ , $u^{'} ⁣ : A^{'} \to B^{'}$ , $v^{'} ⁣ : B^{'} \to C^{'}$ , $w^{'} ⁣ : C^{'} \to D^{'}$ , $x^{'} ⁣ : D^{'} \to E^{'}$ , $a ⁣ : A \to A^{'}$ , $b ⁣ : B \to B^{'}$ , $c ⁣ : C \to C^{'}$ , $d ⁣ : D \to D^{'}$ , $e ⁣ : E \to E^{'}$ .

[C1]

$b \circ u = u^{'} \circ a$ (given).

[C2]

$c \circ v = v^{'} \circ b$ (given).

[C3]

$d \circ w = w^{'} \circ c$ (given).

[C4]

$e \circ x = x^{'} \circ d$ (given).

[L1]

In such a diagram, $a$ surjective with $b, d$ injective implies $c$ injective, while $b, d$ surjective with $e$ injective implies $c$ surjective (The injective and surjective Four Lemmas).

Proof

technique · direct

1.1

Under the first set of hypotheses, the injective Four Lemma [L1] applied to the diagram [C1] to [C4] gives that $c$ is injective.

C1C2C3C4L1

1.2

Under the second set of hypotheses, the surjective Four Lemma [L1] applied to the same diagram gives that $c$ is surjective.

C1C2C3C4L1

2.1

If $a, b, d, e$ are isomorphisms, then $a, b, d$ meet the first hypotheses and $b, d, e$ meet the second; steps 1.1 and 1.2 make $c$ both injective and surjective, hence an isomorphism.

step 1.1step 1.2∎

Depends on

The injective and surjective Four Lemmas

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 7 results over 6 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

A. Kleshchev, Lectures on Abstract Algebra for Graduate Students, sections 3.6, 3.14, and 3.15 (standard reference, not scraped)
The Stacks Project, Algebra (standard reference, not scraped)
P. Hekmati, Homological Algebra, section 3.1 (standard reference, not scraped)