Theorems · Theorem · functional analysis

CFC.sqrt_map_prod

∀ {A : Type u_1} [inst : PartialOrder A] [inst_1 : NonUnitalRing A] [inst_2 : TopologicalSpace A] [inst_3 : StarRing A]
  [inst_4 : Module ℝ A] [inst_5 : SMulCommClass ℝ A A] [inst_6 : IsScalarTower ℝ A A] [inst_7 : StarOrderedRing A]
  [inst_8 : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint] [inst_9 : NonnegSpectrumClass ℝ A]
  [IsSemitopologicalRing A] [T2Space A] {B : Type u_2} [inst_12 : PartialOrder B] [inst_13 : NonUnitalRing B]
  [inst_14 : TopologicalSpace B] [inst_15 : StarRing B] [inst_16 : Module ℝ B] [inst_17 : SMulCommClass ℝ B B]
  [inst_18 : IsScalarTower ℝ B B] [inst_19 : StarOrderedRing B]
  [inst_20 : NonUnitalContinuousFunctionalCalculus ℝ B IsSelfAdjoint]
  [inst_21 : NonUnitalContinuousFunctionalCalculus ℝ (A × B) IsSelfAdjoint] [IsSemitopologicalRing B] [T2Space B]
  [inst_24 : NonnegSpectrumClass ℝ B] [inst_25 : NonnegSpectrumClass ℝ (A × B)] {a : A} {b : B},
  autoParam (0 ≤ a) CFC.sqrt_map_prod._auto_1 →
    autoParam (0 ≤ b) CFC.sqrt_map_prod._auto_3 → CFC.sqrt (a, b) = (CFC.sqrt a, CFC.sqrt b)

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

PartialOrderNonUnitalRingTopologicalSpaceStarRingModuleSMulCommClassIsScalarTowerStarOrderedRingNonUnitalContinuousFunctionalCalculusNonnegSpectrumClassIsSemitopologicalRingT2SpacePartialOrderNonUnitalRingTopologicalSpaceStarRingModuleSMulCommClassIsScalarTowerStarOrderedRingNonUnitalContinuousFunctionalCalculusNonUnitalContinuousFunctionalCalculusIsSemitopologicalRingT2SpaceNonnegSpectrumClassNonnegSpectrumClass