Theorems · Theorem · nonassociative algebras

RootPairing.IsG2.pairingIn_mem_zero_one_three

∀ {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)
  [inst_5 : P.IsCrystallographic] [P.IsReduced] [P.IsIrreducible] [Finite ι] [CharZero R] [IsDomain R] [P.IsG2]
  (i j : ι), P.root i ≠ P.root j → P.root i ≠ -P.root j → P.pairingIn ℤ i j ∈ {-3, -1, 0, 1, 3}

Defined in

Mathlib.LinearAlgebra.RootSystem.Finite.G2

Cited by

1 results in Mathlib

Foundations

Depth 101 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingAddCommGroupModuleAddCommGroupModuleRootPairing.IsCrystallographicRootPairing.IsReducedRootPairing.IsIrreducibleFiniteCharZeroIsDomainRootPairing.IsG2