Theorems · Theorem · nonassociative algebras

RootPairing.root_add_zsmul_mem_range_iff

∀ {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [inst : Finite ι] [inst_1 : CommRing R]
  [inst_2 : CharZero R] [inst_3 : IsDomain R] [inst_4 : AddCommGroup M] [inst_5 : Module R M] [inst_6 : AddCommGroup N]
  [inst_7 : Module R N] {P : RootPairing ι R M N} [inst_8 : P.IsCrystallographic] {i j : ι},
  LinearIndependent R ![P.root i, P.root j] →
    ∀ {z : ℤ},
      P.root j + z • P.root i ∈ Set.range ⇑P.root ↔
        z ∈ Set.Icc (-↑(RootPairing.chainBotCoeff i j)) ↑(RootPairing.chainTopCoeff i j)

Defined in: Mathlib.LinearAlgebra.RootSystem.Chain
Cited by: 4 results in Mathlib
Foundations: Depth 101 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: Finite CommRing CharZero IsDomain AddCommGroup Module AddCommGroup Module RootPairing.IsCrystallographic

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Cites23

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

DFunLike.coestatement and proof · cited by 62,936
Setstatement · cited by 53,352
Modulestatement and proof · cited by 20,661
CommRingstatement and proof · cited by 17,173
AddCommGroupstatement and proof · cited by 12,871
Set.rangestatement and proof · cited by 4,705
Finitestatement and proof · cited by 3,029
IsDomainstatement and proof · cited by 2,196
Set.Iccstatement · cited by 1,702
Function.Embeddingstatement · cited by 988
CharZerostatement and proof · cited by 932
Matrix.vecConsstatement and proof · cited by 852

Cited by4

Results whose statement or proof uses this declaration.

RootPairing.chainBotCoeff_sub_chainTopCoeffproof · cited by 4
RootPairing.setOfPred_root_add_zsmul_mem_eq_Iccproof · cited by 2
RootPairing.chainCoeff_chainTopIdx_auxproof · cited by 2
RootPairing.root_sub_zsmul_mem_range_iffproof · cited by 1