Theorems · Theorem · functional analysis
cfc_real_eq_nnreal
∀ {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] [IsSemitopologicalRing A] [T2Space A]
[inst_8 : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint] [inst_9 : NonnegSpectrumClass ℝ A] {f : ℝ → ℝ} (a : A),
(∀ x ∈ spectrum ℝ a, 0 ≤ f x) →
autoParam (0 ≤ a) cfc_real_eq_nnreal._auto_1 → cfc f a = cfc (fun x => (f ↑x).toNNReal) a- Cited by
- 1 results in Mathlib
- Foundations
- Depth 199 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites22
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- 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
- NNRealstatement and proof · cited by 4,310
- StarRingstatement and proof · cited by 1,686
- T2Spacestatement and proof · cited by 1,351
- NNReal.toRealstatement and proof · cited by 1,260
- StarOrderedRingstatement and proof · cited by 587
- IsSelfAdjointstatement and proof · cited by 545
Cited by1
Results whose statement or proof uses this declaration.
- range_cfc_nnreal_eq_image_cfc_realproof · cited by 2