Theorems · Theorem · nonassociative algebras

RootPairing.Base.exists_eq_sum_and_forall_sum_mem_of_isPos

∀ {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} {b : P.Base}
  [inst_5 : CharZero R] [Finite ι] [IsDomain R] [P.IsCrystallographic] {i : ι},
  b.IsPos i →
    ∃ n f,
      Set.range f ⊆ ↑b.support ∧
        P.root i = ∑ m, P.root (f m) ∧ ∀ (m : Fin n), ∑ m' ≤ m, P.root (f m') ∈ Set.range ⇑P.root

This lemma is included mostly for comparison with the informal literature. Usually RootPairing.Base.IsPos.induction_on_add will be more useful.

Defined in

Mathlib.LinearAlgebra.RootSystem.Base

Cited by

0 results in Mathlib

Foundations

Depth 96 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingAddCommGroupModuleAddCommGroupModuleCharZeroFiniteIsDomainRootPairing.IsCrystallographic