Theorems · Theorem · nonassociative algebras

RootPairing.Base.exists_mem_span_pairingIn_ne_zero_and_pairwise_ne

∀ {ι : Type u_1} {M : Type u_3} {N : Type u_4} [inst : AddCommGroup M] [inst_1 : AddCommGroup N] [Finite ι]
  {K : Type u_6} [inst_3 : Field K] [CharZero K] [inst_5 : Module K M] [inst_6 : Module K N] {P : RootPairing ι K M N}
  [P.IsRootSystem] [inst_8 : P.IsCrystallographic] (b : P.Base),
  ∃ d ∈ Submodule.span K (Set.range fun i j => ↑(P.pairingIn ℤ j ↑i)),
    (∀ (i : ι), d i ≠ 0) ∧ Pairwise (Function.onFun (fun x1 x2 => x1 ≠ x2) d)

Defined in

Mathlib.LinearAlgebra.RootSystem.CartanMatrix

Cited by

0 results in Mathlib

Foundations

Depth 133 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

AddCommGroupAddCommGroupFiniteFieldCharZeroModuleModuleRootPairing.IsRootSystemRootPairing.IsCrystallographic