Theorems · Theorem · functional analysis

cfcHom_comp

∀ {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] {a : A} (ha : p a)
  [ContinuousMap.UniqueHom R A] (f : C(↑(spectrum R a), R)) (f' : C(↑(spectrum R a), ↑(spectrum R ((cfcHom ha) f)))),
  (∀ (x : ↑(spectrum R a)), f x = ↑(f' x)) →
    ∀ (g : C(↑(spectrum R ((cfcHom ha) f)), R)), (cfcHom ha) (g.comp f') = (cfcHom ⋯) g

Defined in

Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital

Cited by

1 results in Mathlib

Foundations

Depth 97 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringStarRingMetricSpaceIsTopologicalSemiringContinuousStarTopologicalSpaceRingStarRingAlgebraContinuousFunctionalCalculusContinuousMap.UniqueHom