Existing in Mathlib · Order theory
Jordan–Hölder Theorem
In any JordanHolderLattice, two composition series with the same head and the same last have equivalent adjacent steps after a bijection reindexes the step positions.
- Jordan–Hölder lattices
- composition series
- maximal steps
- equivalence by reindexing
- lattice induction
- isomorphism of adjacent pairs
Exact theorem
Exact Mathlib statement
theorem CompositionSeries.jordan_holder {X : Type u} [Lattice X] [JordanHolderLattice X] (s₁ s₂ : CompositionSeries X) (hb : s₁.head = s₂.head) (ht : s₁.last = s₂.last) : CompositionSeries.Equivalent s₁ s₂The theorem at a glance
Jordan–Hölder theorem at a glance
A visual guide to the theorem's hypotheses, structure, and conclusion; the exact statement gives the formal detail.
Jordan–Hölder theorem at a glance
Detailed visual description
The poster states the abstract JordanHolderLattice theorem above two composition-chain rows whose paired endpoint stones mark the same head and the same last. Five adjacent-step emblems on each row are joined by a crossed one-to-one matching, emphasizing equivalence after reindexing rather than equality in the displayed order. Four source-faithful route lines summarize the induction: remove the second series' last step, replace the first series by an equivalent one through the same penultimate point, match the shortened prefixes, and append the common final interval.
Statement structure
From hypotheses to conclusion
This map summarizes the statement's structure; the exact Mathlib declaration remains authoritative.
Equivalent composition series with equal endpoints
Detailed visual description
An upper green composition route and a lower blue composition route each contain five differently ordered adjacent-step medallions between distinct paired endpoints. The visible label “same head” explains the left bracket around the two equal head circles, and “same last” explains the corresponding right bracket around the two equal last circles. Exactly five crossing gold threads pair recurring shapes one to one; the visible “f” names this position bijection and “Iso” names the relation between matched adjacent pairs. No other prose or mathematical label appears.
Why it matters
A mathematical landmark
Jordan–Hölder is the structural uniqueness theorem for composition series, and Mathlib states it at a notably reusable order-theoretic level. The source file documents groups and modules as motivating settings for the abstraction, while this exact declaration cleanly exposes the core conclusion: matching endpoints determine equivalence only after adjacent steps are reindexed and compared through Iso.
ProofAtlas record
What has been checked
Mathlib is the source of the theorem; the local Lean replay and page review are separate.
Claim boundary
No new theorem is claimed
This page indexes Mathlib's abstract order-theoretic Jordan–Hölder declaration. For two CompositionSeries in a JordanHolderLattice, equality of their heads and equality of their lasts imply Equivalent, meaning that a bijection of step positions matches adjacent pairs through the lattice's Iso relation. The conclusion is not equality, an order-preserving identification, or a unique matching. The declaration itself does not mention groups, modules, subgroups, factor groups, bottom-to-top series, or an additional finite-length assumption.
- Equivalent is not literal equality: it supplies a bijection between step-index types and Iso-related adjacent pairs.
- The matching bijection need not preserve the displayed order of the two series and is not asserted to be unique.
- The hypotheses identify each series' own head and last; they do not say those endpoints are the bottom and top of the ambient lattice.
- The theorem adds no separate finite-length hypothesis; the finite nonempty series structure is already packaged by CompositionSeries.
- Groups and modules are motivating contexts documented by the source file, but they are not named in this declaration's binder surface, and this page does not introduce factor groups.
- ProofAtlas is indexing an existing Mathlib theorem, and generated explanations and visuals are not proof evidence.
Source and local evidence
Where the theorem comes from
- Existing declaration
CompositionSeries.jordan_holderin mathlib- Relationship
- The checked artifact is the upstream declaration itself
- Local Lean evidence
artifact.library.mathlib.jordan-holder-theorem.v001