Theorems · Theorem · functional analysis

cfc_sub

∀ {R : Type u_1} {A : Type u_2} {p : A → Prop} [inst : CommRing R] [inst_1 : StarRing R] [inst_2 : MetricSpace R]
  [inst_3 : IsTopologicalRing 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] (f g : R → R) (a : A),
  autoParam (ContinuousOn f (spectrum R a)) _auto_630✝ →
    autoParam (ContinuousOn g (spectrum R a)) _auto_632✝ → cfc (fun x => f x - g x) a = cfc f a - cfc g a

Defined in

Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital

Cited by

5 results in Mathlib

Foundations

Depth 101 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingStarRingMetricSpaceIsTopologicalRingContinuousStarTopologicalSpaceRingStarRingAlgebraContinuousFunctionalCalculus