Theorems · Theorem · nonassociative algebras

RootPairing.exists_coroot_ne

∀ {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] [IsDomain R] [IsDomain S] [inst_9 : Algebra S R] [inst_10 : FaithfulSMul S R]
  [inst_11 : P.IsValuedIn S] [inst_12 : Module S M] [inst_13 : IsScalarTower S R M] [inst_14 : Module S N]
  [IsScalarTower S R N] [P.IsAnisotropic] {x : ↥(P.rootSpan S)}, x ≠ 0 → ∃ i, (P.coroot'In S i) x ≠ 0

Defined in

Mathlib.LinearAlgebra.RootSystem.Finite.Nondegenerate

Cited by

1 results in Mathlib

Foundations

Depth 124 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FintypeAddCommGroupAddCommGroupCommRingModuleModuleCommRingIsDomainIsDomainAlgebraFaithfulSMulRootPairing.IsValuedInModuleIsScalarTowerModuleIsScalarTowerRootPairing.IsAnisotropic