Theorems · Theorem · nonassociative algebras

RootPairing.ncard_eq_finrank_of_linearIndepOn_of

∀ {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [Finite ι] [inst : AddCommGroup M]
  [inst_1 : AddCommGroup N] [inst_2 : CommRing R] [inst_3 : Module R M] [inst_4 : Module R N] (P : RootPairing ι R M N)
  [P.IsRootSystem] [Nontrivial R] {s : Set ι},
  LinearIndepOn R (⇑P.root) s →
    (∀ (i : ι), P.root i ∈ AddSubmonoid.closure (⇑P.root '' s) ∨ -P.root i ∈ AddSubmonoid.closure (⇑P.root '' s)) →
      s.ncard = Module.finrank R M

Defined in

Mathlib.LinearAlgebra.RootSystem.BaseExists

Cited by

1 results in Mathlib

Foundations

Depth 120 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FiniteAddCommGroupAddCommGroupCommRingModuleModuleRootPairing.IsRootSystemNontrivial