Theorems · Theorem · nonassociative algebras

RootPairing.span_root_image_eq_top_of_forall_orthogonal

∀ {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) [NeZero 2]
  [P.IsIrreducible] (s : Set ι),
  s.Nonempty →
    (∀ (j : ι), P.root j ∉ Submodule.span R (⇑P.root '' s) → ∀ i ∈ s, P.IsOrthogonal j i) →
      Submodule.span R (⇑P.root '' s) = ⊤

Defined in

Mathlib.LinearAlgebra.RootSystem.Irreducible

Cited by

1 results in Mathlib

Foundations

Depth 68 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingAddCommGroupModuleAddCommGroupModuleNeZeroRootPairing.IsIrreducible