Theorems · Theorem · functional analysis

cfc_comp_polynomial

∀ {R : Type u_1} {A : Type u_2} {p : A → Prop} [inst : CommSemiring 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] [instCFC : ContinuousFunctionalCalculus R A p]
  [ContinuousMap.UniqueHom R A] (q : Polynomial R) (f : R → R) (a : A),
  autoParam (ContinuousOn f ((fun x => Polynomial.eval x q) '' spectrum R a)) cfc_comp_polynomial._auto_1 →
    autoParam (p a) cfc_comp_polynomial._auto_3 →
      cfc (fun x => f (Polynomial.eval x q)) a = cfc f ((Polynomial.aeval a) q)

Defined in

Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital

Cited by

0 results in Mathlib

Foundations

Depth 113 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringStarRingMetricSpaceIsTopologicalSemiringContinuousStarTopologicalSpaceRingStarRingAlgebraContinuousFunctionalCalculusContinuousMap.UniqueHom