Theorems · Theorem · functional analysis

CStarAlgebra.isStrictlyPositive_iff_isStrictlyPositive_sqrt_and_eq_sqrt_mul_sqrt

∀ {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] {a : A},
  IsStrictlyPositive a ↔ IsStrictlyPositive (CFC.sqrt a) ∧ a = CFC.sqrt a * CFC.sqrt a

Defined in: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic
Cited by: 0 results in Mathlib
Foundations: Depth 221 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: PartialOrder Ring StarRing TopologicalSpace StarOrderedRing Algebra ContinuousFunctionalCalculus NonnegSpectrumClass IsSemitopologicalRing T2Space

Mathlib docs Source Loogle LeanSearch

Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

Cites19

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

Realstatement and proof · cited by 25,697
TopologicalSpacestatement and proof · cited by 24,529
Algebrastatement and proof · cited by 11,388
Ringstatement and proof · cited by 7,463
PartialOrderstatement and proof · cited by 6,410
StarRingstatement and proof · cited by 1,686
IsUnitproof · cited by 1,602
T2Spacestatement and proof · cited by 1,351
Star.starproof · cited by 1,082
StarOrderedRingstatement and proof · cited by 587
IsSelfAdjointstatement and proof · cited by 545
spectrumproof · cited by 510

Cited by0

Results whose statement or proof uses this declaration.

Nothing cites this yet.