Theorems · Theorem · functional analysis

CFC.rpow_eq_rpow_prod

∀ {A : Type u_1} [inst : PartialOrder A] [inst_1 : Ring A] [inst_2 : StarRing A] [inst_3 : TopologicalSpace A]
  [inst_4 : StarOrderedRing A] [inst_5 : Algebra ℝ A] [inst_6 : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint]
  [inst_7 : NonnegSpectrumClass ℝ A] [IsSemitopologicalRing A] [T2Space A] {B : Type u_2} [inst_10 : PartialOrder B]
  [inst_11 : Ring B] [inst_12 : StarRing B] [inst_13 : TopologicalSpace B] [inst_14 : StarOrderedRing B]
  [inst_15 : Algebra ℝ B] [inst_16 : ContinuousFunctionalCalculus ℝ B IsSelfAdjoint]
  [inst_17 : ContinuousFunctionalCalculus ℝ (A × B) IsSelfAdjoint] [IsSemitopologicalRing B] [T2Space B]
  [inst_20 : StarOrderedRing (A × B)] [inst_21 : NonnegSpectrumClass ℝ B] [inst_22 : NonnegSpectrumClass ℝ (A × B)]
  {a : A} {b : B} {x : ℝ},
  IsUnit a →
    IsUnit b →
      autoParam (0 ≤ a) CFC.rpow_eq_rpow_prod._auto_1 →
        autoParam (0 ≤ b) CFC.rpow_eq_rpow_prod._auto_3 → CFC.rpow (a, b) x = (a, b) ^ x

Defined in

Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic

Cited by

0 results in Mathlib

Foundations

Depth 210 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

PartialOrderRingStarRingTopologicalSpaceStarOrderedRingAlgebraContinuousFunctionalCalculusNonnegSpectrumClassIsSemitopologicalRingT2SpacePartialOrderRingStarRingTopologicalSpaceStarOrderedRingAlgebraContinuousFunctionalCalculusContinuousFunctionalCalculusIsSemitopologicalRingT2SpaceStarOrderedRingNonnegSpectrumClassNonnegSpectrumClass