Theorems · Theorem · nonassociative algebras

RootPairing.setOfPred_root_add_zsmul_eq_Icc_of_linearIndependent

∀ {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [Finite ι] [inst : CommRing R] [CharZero R] [IsDomain R]
  [inst_3 : AddCommGroup M] [inst_4 : Module R M] [inst_5 : AddCommGroup N] [inst_6 : Module R N]
  {P : RootPairing ι R M N} [P.IsCrystallographic] {i j : ι},
  LinearIndependent R ![P.root i, P.root j] →
    ∃ q ≤ 0, ∃ p ≥ 0, {z | P.root j + z • P.root i ∈ Set.range ⇑P.root} = Set.Icc q p

Note that it is often more convenient to use RootPairing.root_add_zsmul_mem_range_iff than to invoke this lemma directly.

Defined in

Mathlib.LinearAlgebra.RootSystem.Chain

Cited by

5 results in Mathlib

Foundations

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

Assumes

FiniteCommRingCharZeroIsDomainAddCommGroupModuleAddCommGroupModuleRootPairing.IsCrystallographic