Theorems · Theorem · nonassociative algebras

RootPairing.Base.root_sub_root_mem_of_mem_of_mem

∀ {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [inst : CommRing R] [CharZero R] [IsDomain R]
  [inst_3 : AddCommGroup M] [inst_4 : Module R M] [inst_5 : AddCommGroup N] [inst_6 : Module R N]
  {P : RootPairing ι R M N} [Finite ι] [P.IsCrystallographic] {b : P.Base} (i j k : ι),
  i ≠ j →
    i ∈ b.support →
      j ∈ b.support →
        P.root k + P.root i - P.root j ∈ Set.range ⇑P.root →
          k ≠ j → P.root k + P.root i ∈ Set.range ⇑P.root → P.root k - P.root j ∈ Set.range ⇑P.root

This is Lemma 2.5 (a) from [Geck](Geck2017).

Defined in

Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Lemmas

Cited by

3 results in Mathlib

Foundations

Depth 96 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCharZeroIsDomainAddCommGroupModuleAddCommGroupModuleFiniteRootPairing.IsCrystallographic