Theorems · Theorem · nonassociative algebras

RootPairing.exists_ge_zero_eq_rootForm

∀ {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] [P.IsValuedIn S] [inst_13 : Fintype ι],
  ∀ x ∈ Submodule.span S (Set.range ⇑P.root), ∃ s ≥ 0, (algebraMap S R) s = (P.RootForm x) x

Defined in

Mathlib.LinearAlgebra.RootSystem.Finite.CanonicalBilinear

Cited by

1 results in Mathlib

Foundations

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

Assumes

CommRingAddCommGroupModuleAddCommGroupModuleCommRingLinearOrderIsStrictOrderedRingAlgebraFaithfulSMulModuleIsScalarTowerRootPairing.IsValuedInFintype