Theorems · Theorem · functional analysis

cfc_comp_zpow

∀ {R : Type u_1} {A : Type u_2} {p : A → Prop} [inst : Semifield R] [inst_1 : StarRing R] [inst_2 : MetricSpace R]
  [inst_3 : IsTopologicalSemiring R] [inst_4 : ContinuousStar R] [inst_5 : TopologicalSpace A] [inst_6 : Ring A]
  [inst_7 : StarRing A] [inst_8 : Algebra R A] [inst_9 : ContinuousFunctionalCalculus R A p] [ContinuousInv₀ R]
  [ContinuousMap.UniqueHom R A] (f : R → R) (n : ℤ) (a : Aˣ),
  autoParam (ContinuousOn f ((fun x => x ^ n) '' spectrum R ↑a)) cfc_comp_zpow._auto_1 →
    autoParam (p ↑a) cfc_comp_zpow._auto_3 → cfc (fun x => f (x ^ n)) ↑a = cfc f ↑(a ^ n)

Defined in

Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital

Cited by

0 results in Mathlib

Foundations

Depth 108 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemifieldStarRingMetricSpaceIsTopologicalSemiringContinuousStarTopologicalSpaceRingStarRingAlgebraContinuousFunctionalCalculusContinuousInv₀ContinuousMap.UniqueHom