TheoremStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-08-30‡ rests on unproved material (inherited)

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.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

Weak four lemma with the exactness hypotheses named

Statement

In the four-term commutative diagram of the four lemma, the two conclusions already follow from exactness at the four middle nodes that are actually used:

exactness at $X$ , $Y$ , $X^{'}$ , and $Y^{'}$ , together with $α, γ$ epic and $δ$ monic, implies $β$ epic;
exactness at $X$ , $Y$ , $X^{'}$ , and $Y^{'}$ , together with $β, δ$ monic and $α$ epic, implies $γ$ monic.

Facts & Assumptions

Given: The four-term commutative diagram underlying the four lemma.

[L1]

Monicity is equivalent to cancellation on members (Monicity by member cancellation).

[L2]

Epicity is equivalent to the member-lifting property (Epimorphy is detected by members).

[L3]

Exactness at a node is equivalent to the member-lifting condition (Exactness is detected by members).

[L4]

The common-refinement construction for member equivalence puts finitely many witness equalities on one epic domain, where hom-set subtraction is defined (Equivalence of members, Member equivalence is transitive, Abelian category).

Proof

technique · direct

1.1

Write the top row as $W \overset{i}{\to} X \overset{j}{\to} Y \overset{p}{\to} Z$ and the bottom row as $W^{'} \overset{i^{'}}{\to} X^{'} \overset{j^{'}}{\to} Y^{'} \overset{p^{'}}{\to} Z^{'}$ . Assume that exactness holds at $X$ , $Y$ , $X^{'}$ , and $Y^{'}$ , and that $α$ and $γ$ are epic while $δ$ is monic. Let $x^{'}$ be a member of $X^{'}$ . By epicity of $γ$ and [L2], choose a member $y$ of $Y$ with $γ y \equiv j^{'} x^{'}$ . Then $δ p y = p^{'} γ y \equiv p^{'} j^{'} x^{'} \equiv 0,$ so monicity of $δ$ and [L1] give $p y \equiv 0$ . Exactness at $Y$ gives a member $x_{0}$ of $X$ with $j x_{0} \equiv y$ by [L3].

L1L2L3assume-hypconstruct

1.2

Assume instead that exactness holds at $X$ , $Y$ , $X^{'}$ , and $Y^{'}$ , that $β$ and $δ$ are monic, and that $α$ is epic. Let $y$ and $y^{'}$ be members of $Y$ with $γ y \equiv γ y^{'}$ . Then $δ p y = p^{'} γ y \equiv p^{'} γ y^{'} = δ p y^{'},$ so [L1] gives $p y \equiv p y^{'}$ . By [L4], replace $y$ and $y^{'}$ by representatives on one common epic refinement of the witnesses for both equalities and define $t : = y - y^{'}$ . Then $γ t = p t = 0$ and $y = y^{'} + t$ on that domain. Exactness at $Y$ gives a member $x$ of $X$ with $j x \equiv t$ by [L3].

L1L3L4assume-hypchooseconstructalgebra

2.1

Now $j^{'} β x_{0} = γ j x_{0} \equiv γ y \equiv j^{'} x^{'} .$ By [L4], replace $β x_{0}$ and $x^{'}$ by representatives on a common epic domain and define $z^{'} : = x^{'} - β x_{0}$ . Then $j^{'} z^{'} = 0$ and $x^{'} = β x_{0} + z^{'}$ on that domain. Exactness at $X^{'}$ gives a member $w^{'}$ of $W^{'}$ with $i^{'} w^{'} \equiv z^{'}$ by [L3], and epicity of $α$ gives a member $w$ of $W$ with $α w \equiv w^{'}$ by [L2]. Therefore $β (x_{0} + i w) \equiv β x_{0} + i^{'} α w \equiv β x_{0} + i^{'} w^{'} \equiv β x_{0} + z^{'} \equiv x^{'} .$ So [L2] makes $β$ epic.

L2L3L4step 1.1chooseconstructalgebra

2.2

From step 1.2 we get $j^{'} β x = γ j x \equiv γ t \equiv 0.$ Exactness at $X^{'}$ gives a member $w^{'}$ of $W^{'}$ with $i^{'} w^{'} \equiv β x$ by [L3]. Since $α$ is epic, choose $w$ in $W$ with $α w \equiv w^{'}$ by [L2]. Then $β (x - i w) \equiv β x - i^{'} α w \equiv β x - i^{'} w^{'} \equiv 0.$ Because $β$ is monic, [L1] yields $x \equiv i w$ , and therefore $t \equiv j x \equiv j i w \equiv 0.$ So $y \equiv y^{'}$ , and [L1] makes $γ$ monic.

L1L2L3step 1.2algebra

3.1

Hence the weak four lemma follows from the named middle-node exactness hypotheses alone.

step 1.2step 2.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

20 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

Charles A. Weibel, An Introduction to Homological Algebra, Exercise 1.3.3 (standard reference, not scraped)
Saunders Mac Lane, Homology, Chapter XII, Section 3 (standard reference, not scraped)