Theorems · Theorem · nonassociative algebras

RootPairing.InvariantForm.exists_apply_eq_or

∀ {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [inst : CommRing R] [inst_1 : AddCommGroup M]
  [inst_2 : Module R M] [inst_3 : AddCommGroup N] [inst_4 : Module R N] {P : RootPairing ι R M N} [Finite ι]
  [CharZero R] [P.IsCrystallographic] [IsDomain R] [P.IsReduced] (B : P.InvariantForm) [P.IsIrreducible] [Nonempty ι],
  ∃ i j,
    ∀ (k : ι),
      (B.form (P.root k)) (P.root k) = (B.form (P.root i)) (P.root i) ∨
        (B.form (P.root k)) (P.root k) = (B.form (P.root j)) (P.root j)

A reduced irreducible finite crystallographic root system has roots of at most two different lengths.

Defined in

Mathlib.LinearAlgebra.RootSystem.Finite.Lemmas

Cited by

1 results in Mathlib

Foundations

Depth 97 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingAddCommGroupModuleAddCommGroupModuleFiniteCharZeroRootPairing.IsCrystallographicIsDomainRootPairing.IsReducedRootPairing.IsIrreducibleNonempty