Theorems · Theorem · functional analysis
algebraMap_le_iff_le_spectrum
∀ {R : Type u_1} {A : Type u_2} {p : A → Prop} [inst : CommRing R] [inst_1 : PartialOrder R] [inst_2 : StarRing R]
[inst_3 : MetricSpace R] [inst_4 : IsTopologicalRing R] [inst_5 : ContinuousStar R] [ContinuousSqrt R]
[StarOrderedRing R] [inst_8 : TopologicalSpace A] [inst_9 : Ring A] [inst_10 : StarRing A] [inst_11 : PartialOrder A]
[StarOrderedRing A] [inst_13 : Algebra R A] [instCFC : ContinuousFunctionalCalculus R A p] [NonnegSpectrumClass R A]
{r : R} {a : A},
autoParam (p a) algebraMap_le_iff_le_spectrum._auto_1 → ((algebraMap R A) r ≤ a ↔ ∀ x ∈ spectrum R a, r ≤ x)- Cited by
- 3 results in Mathlib
- Foundations
- Depth 103 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites21
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · cited by 62,936
- Setstatement · cited by 53,352
- TopologicalSpacestatement and proof · cited by 24,529
- CommRingstatement and proof · cited by 17,173
- Algebrastatement and proof · cited by 11,388
- RingHomstatement · cited by 10,189
- Ringstatement and proof · cited by 7,463
- PartialOrderstatement and proof · cited by 6,410
- Algebra.algebraMapstatement and proof · cited by 4,706
- StarRingstatement and proof · cited by 1,686
- MetricSpacestatement and proof · cited by 1,684
- StarOrderedRingstatement and proof · cited by 587
Cited by3
Results whose statement or proof uses this declaration.
- CStarAlgebra.isUnit_of_leproof · cited by 3
- CFC.exists_pos_algebraMap_le_iffproof · cited by 1
- CFC.one_le_iffproof · cited by 1