Theorems · Theorem · nonassociative algebras

RootPairing.coroot_mem_or_neg_mem_closure_of_root

∀ {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [Finite ι] [inst : AddCommGroup M]
  [inst_1 : AddCommGroup N] [inst_2 : Field R] [CharZero R] [inst_4 : Module R M] [inst_5 : Module R N]
  (P : RootPairing ι R M N) [P.IsRootSystem] [P.IsCrystallographic] [P.IsReduced] (s : Set ι),
  LinearIndepOn R (⇑P.root) s →
    (∀ (i : ι), P.root i ∈ AddSubmonoid.closure (⇑P.root '' s) ∨ -P.root i ∈ AddSubmonoid.closure (⇑P.root '' s)) →
      ∀ (i : ι),
        P.coroot i ∈ AddSubmonoid.closure (⇑P.coroot '' s) ∨ -P.coroot i ∈ AddSubmonoid.closure (⇑P.coroot '' s)

This is really just an auxiliary result en route to RootPairing.Base.mk'.

Defined in

Mathlib.LinearAlgebra.RootSystem.BaseExists

Cited by

0 results in Mathlib

Foundations

Depth 133 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FiniteAddCommGroupAddCommGroupFieldCharZeroModuleModuleRootPairing.IsRootSystemRootPairing.IsCrystallographicRootPairing.IsReduced