Theorems · Theorem · functional analysis
nnnorm_cfc_nnreal_le_iff
∀ {A : Type u_1} [inst : NormedRing A] [inst_1 : StarRing A] [inst_2 : NormedAlgebra ℝ A] [inst_3 : PartialOrder A]
[inst_4 : StarOrderedRing A] [inst_5 : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint]
[inst_6 : NonnegSpectrumClass ℝ A] (f : NNReal → NNReal) (a : A) (c : NNReal),
autoParam (ContinuousOn f (spectrum NNReal a)) nnnorm_cfc_nnreal_le_iff._auto_1 →
autoParam (0 ≤ a) nnnorm_cfc_nnreal_le_iff._auto_3 → (‖cfc f a‖₊ ≤ c ↔ ∀ x ∈ spectrum NNReal a, f x ≤ c)- Cited by
- 0 results in Mathlib
- Foundations
- Depth 201 from the axioms · uses propext, Classical.choice, Quot.sound
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Cites18
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
- PartialOrderstatement and proof · cited by 6,410
- NNRealstatement and proof · cited by 4,310
- LE.le.transproof · cited by 3,151
- StarRingstatement and proof · cited by 1,686
- ContinuousOnstatement and proof · cited by 1,411
- NormedAlgebrastatement and proof · cited by 1,165
- NNNorm.nnnormstatement and proof · cited by 952
- NormedRingstatement and proof · cited by 924
- StarOrderedRingstatement and proof · cited by 587
- IsSelfAdjointstatement and proof · cited by 545
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