Theorems · Theorem · nonassociative algebras

RootPairing.baseOf_root_eq_baseOf_coroot

∀ {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 : Field R] [CharZero R] [inst_4 : Module R M] [inst_5 : Module R N]
  (P : RootPairing ι R M N) [P.IsRootSystem] [P.IsCrystallographic] [P.IsReduced] (f : M →+ ℚ),
  (∀ (i : ι), f (P.root i) ≠ 0) →
    ∀ (g : N →+ ℚ),
      (∀ (i : ι), g (P.coroot i) ≠ 0) →
        (∀ (i : ι), 0 < f (P.root i) ↔ 0 < g (P.coroot i)) →
          IsAddIndecomposable.baseOf (⇑P.root) f = IsAddIndecomposable.baseOf (⇑P.coroot) g

Defined in: Mathlib.LinearAlgebra.RootSystem.BaseExists
Cited by: 1 results in Mathlib
Foundations: Depth 132 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: Finite AddCommGroup AddCommGroup Field CharZero Module Module RootPairing.IsRootSystem RootPairing.IsCrystallographic RootPairing.IsReduced

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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
AddCommGroupstatement and proof · cited by 12,871
Fieldstatement and proof · cited by 7,404
AddMonoidHomstatement and proof · cited by 3,230
Finitestatement and proof · cited by 3,029
Function.Embeddingstatement · cited by 988
CharZerostatement and proof · cited by 932
RootPairingstatement and proof · cited by 710
RootPairing.rootstatement and proof · cited by 326
RootPairing.IsCrystallographicstatement and proof · cited by 160

Cited by1

Results whose statement or proof uses this declaration.

RootPairing.coroot_mem_or_neg_mem_closure_of_rootproof · cited by 0