Theorems · Theorem · nonassociative algebras

RootPairing.pairingIn_pairingIn_mem_set_of_isCrystal_of_isRed

∀ {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] [inst_7 : P.IsCrystallographic] (i j : ι) [IsDomain R] [P.IsReduced],
  (P.pairingIn ℤ i j, P.pairingIn ℤ j i) ∈
    {(0, 0), (1, 1), (-1, -1), (1, 2), (2, 1), (-1, -2), (-2, -1), (1, 3), (3, 1), (-1, -3), (-3, -1), (2, 2), (-2, -2)}

Defined in

Mathlib.LinearAlgebra.RootSystem.Finite.Lemmas

Cited by

6 results in Mathlib

Foundations

Depth 94 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingAddCommGroupModuleAddCommGroupModuleFiniteCharZeroRootPairing.IsCrystallographicIsDomainRootPairing.IsReduced