Theorems · Theorem · functional analysis

cfc_tsub

∀ {A : Type u_1} [inst : TopologicalSpace A] [inst_1 : Ring A] [inst_2 : PartialOrder A] [inst_3 : StarRing A]
  [inst_4 : StarOrderedRing A] [inst_5 : Algebra ℝ A] [IsTopologicalRing A] [T2Space A]
  [inst_8 : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint] [inst_9 : NonnegSpectrumClass ℝ A] (f g : NNReal → NNReal)
  (a : A),
  (∀ x ∈ spectrum NNReal a, g x ≤ f x) →
    autoParam (0 ≤ a) cfc_tsub._auto_1 →
      autoParam (ContinuousOn f (spectrum NNReal a)) cfc_tsub._auto_3 →
        autoParam (ContinuousOn g (spectrum NNReal a)) cfc_tsub._auto_5 → cfc (fun x => f x - g x) a = cfc f a - cfc g a

Defined in

Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order

Cited by

1 results in Mathlib

Foundations

Depth 199 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceRingPartialOrderStarRingStarOrderedRingAlgebraIsTopologicalRingT2SpaceContinuousFunctionalCalculusNonnegSpectrumClass