Theorems · Theorem · nonassociative algebras

RootPairing.eq_baseOf_iff

∀ {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] (s : Set ι) (f : M →+ ℚ),
  (∀ i ∈ s, f (P.root i) = 1) →
    (∀ (i : ι), f (P.root i) ≠ 0) →
      (s = IsAddIndecomposable.baseOf (⇑P.root) f ↔
        LinearIndepOn R (⇑P.root) s ∧
          ∀ (i : ι), P.root i ∈ AddSubmonoid.closure (⇑P.root '' s) ∨ -P.root i ∈ AddSubmonoid.closure (⇑P.root '' s))

Defined in

Mathlib.LinearAlgebra.RootSystem.BaseExists

Cited by

0 results in Mathlib

Foundations

Depth 130 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FiniteAddCommGroupAddCommGroupFieldCharZeroModuleModuleRootPairing.IsRootSystemRootPairing.IsCrystallographic