TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck 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

The published module five lemma as an instance Example
Thom isomorphisms glue over two trivializing opens Lemma
Reduced and unreduced generalized cohomology theories correspond Proposition
Countable mayer vietoris open set principle Theorem
Integral complex projective bundle theorem Theorem
Mod-two real projective bundle theorem Theorem
The de Rham map is an isomorphism on a two-open union Theorem
Thom isomorphism for a trivial oriented bundle Theorem

Dependency tree · two levels

3 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

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)