Theorems · Theorem · functional analysis

cfc_inv

∀ {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]
  (f : R → R) (a : A),
  (∀ x ∈ spectrum R a, f x ≠ 0) →
    autoParam (ContinuousOn f (spectrum R a)) cfc_inv._auto_1 →
      autoParam (p a) cfc_inv._auto_3 → cfc (fun x => (f x)⁻¹) a = Ring.inverse (cfc f a)

Defined in

Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital

Cited by

3 results in Mathlib

Foundations

Depth 105 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemifieldStarRingMetricSpaceIsTopologicalSemiringContinuousStarTopologicalSpaceRingStarRingAlgebraContinuousFunctionalCalculusContinuousInv₀