Theorems · Theorem · nonassociative algebras

RootPairing.RootPositiveForm.posForm_apply_root_root_le_zero_iff

∀ {ι : Type u_1} {R : Type u_2} {S : Type u_3} {M : Type u_4} {N : Type u_5} [inst : CommRing S]
  [inst_1 : LinearOrder S] [inst_2 : CommRing R] [inst_3 : Algebra S R] [inst_4 : AddCommGroup M] [inst_5 : Module R M]
  [inst_6 : AddCommGroup N] [inst_7 : Module R N] {P : RootPairing ι R M N} [inst_8 : P.IsValuedIn S]
  (B : RootPairing.RootPositiveForm S P) (i j : ι) [inst_9 : FaithfulSMul S R] [inst_10 : Module S M]
  [inst_11 : IsScalarTower S R M] [IsStrictOrderedRing S]
  (hi : optParam (P.root i ∈ Submodule.span S (Set.range ⇑P.root)) ⋯)
  (hj : optParam (P.root j ∈ Submodule.span S (Set.range ⇑P.root)) ⋯),
  (B.posForm ⟨P.root i, hi⟩) ⟨P.root j, hj⟩ ≤ 0 ↔ P.pairingIn S i j ≤ 0

Defined in

Mathlib.LinearAlgebra.RootSystem.RootPositive

Cited by

1 results in Mathlib

Foundations

Depth 46 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingLinearOrderCommRingAlgebraAddCommGroupModuleAddCommGroupModuleRootPairing.IsValuedInFaithfulSMulModuleIsScalarTowerIsStrictOrderedRing