Theorems · Theorem · nonassociative algebras

RootPairing.root_add_root_mem_of_pairingIn_neg

∀ {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.pairingIn ℤ i j < 0 → P.root i ≠ -P.root j → P.root i + P.root j ∈ Set.range ⇑P.root

If two roots make an obtuse angle then their sum is a root (provided it is not zero). See RootPairing.pairingIn_le_zero_of_root_add_mem for a partial converse.

Defined in

Mathlib.LinearAlgebra.RootSystem.Finite.Lemmas

Cited by

2 results in Mathlib

Foundations

Depth 95 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingAddCommGroupModuleAddCommGroupModuleFiniteCharZeroRootPairing.IsCrystallographicIsDomain