Theorems · Theorem · nonassociative algebras

RootPairing.pairingIn_le_zero_iff

∀ {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) (S : Type u_5)
  [inst_5 : CommRing S] [inst_6 : LinearOrder S] [IsStrictOrderedRing S] [inst_8 : Algebra S R] [FaithfulSMul S R]
  [inst_10 : Module S M] [IsScalarTower S R M] [inst_12 : P.IsValuedIn S] {i j : ι} [Finite ι] [NeZero 2] [IsDomain R]
  [Module.IsTorsionFree R M], P.pairingIn S i j ≤ 0 ↔ P.pairingIn S j i ≤ 0

Defined in

Mathlib.LinearAlgebra.RootSystem.Finite.CanonicalBilinear

Cited by

0 results in Mathlib

Foundations

Depth 64 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingAddCommGroupModuleAddCommGroupModuleCommRingLinearOrderIsStrictOrderedRingAlgebraFaithfulSMulModuleIsScalarTowerRootPairing.IsValuedInFiniteNeZeroIsDomainModule.IsTorsionFree