Theorems · Theorem · nonassociative algebras

RootPairing.posRootForm_posForm_pos_of_ne_zero

∀ {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [inst : Fintype ι] [inst_1 : AddCommGroup M]
  [inst_2 : AddCommGroup N] [inst_3 : CommRing R] [inst_4 : Module R M] [inst_5 : Module R N] (P : RootPairing ι R M N)
  (S : Type u_5) [inst_6 : CommRing S] [inst_7 : LinearOrder S] [inst_8 : IsStrictOrderedRing S] [IsDomain R]
  [inst_10 : Algebra S R] [inst_11 : FaithfulSMul S R] [inst_12 : P.IsValuedIn S] [inst_13 : Module S M]
  [inst_14 : IsScalarTower S R M] [inst_15 : Module S N] [IsScalarTower S R N] {x : ↥(P.rootSpan S)},
  x ≠ 0 → 0 < ((P.posRootForm S).posForm x) x

Defined in

Mathlib.LinearAlgebra.RootSystem.Finite.Nondegenerate

Cited by

3 results in Mathlib

Foundations

Depth 125 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FintypeAddCommGroupAddCommGroupCommRingModuleModuleCommRingLinearOrderIsStrictOrderedRingIsDomainAlgebraFaithfulSMulRootPairing.IsValuedInModuleIsScalarTowerModuleIsScalarTower