Theorems · Theorem · linear algebra

Module.Dual.eq_of_preReflection_mapsTo

∀ {R : Type u_1} {M : Type u_2} [inst : CommRing R] [inst_1 : AddCommGroup M] [inst_2 : Module R M] [CharZero R]
  [IsDomain R] [Module.IsTorsionFree R M] {x : M} {Φ : Set M},
  Φ.Finite →
    Submodule.span R Φ = ⊤ →
      ∀ {f g : Module.Dual R M},
        f x = 2 →
          Set.MapsTo (⇑(Module.preReflection x f)) Φ Φ → g x = 2 → Set.MapsTo (⇑(Module.preReflection x g)) Φ Φ → f = g

See also Module.Dual.eq_of_preReflection_mapsTo' for a variant of this lemma which applies when Φ does not span. This rather technical-looking lemma exists because it is exactly what is needed to establish various uniqueness results for root data / systems. One might regard this lemma as lying at the boundary of linear algebra and combinatorics since the finiteness assumption is the key.

Defined in

Mathlib.LinearAlgebra.Reflection

Cited by

2 results in Mathlib

Foundations

Depth 81 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingAddCommGroupModuleCharZeroIsDomainModule.IsTorsionFree