Theorems · Theorem · functional analysis
isUnit_cfc_iff
∀ {R : Type u_1} {A : Type u_2} {p : A → Prop} [inst : Semifield R] [inst_1 : StarRing R] [inst_2 : MetricSpace R]
[inst_3 : IsTopologicalSemiring R] [inst_4 : ContinuousStar R] [inst_5 : TopologicalSpace A] [inst_6 : Ring A]
[inst_7 : StarRing A] [inst_8 : Algebra R A] [inst_9 : ContinuousFunctionalCalculus R A p] (f : R → R) (a : A),
autoParam (ContinuousOn f (spectrum R a)) isUnit_cfc_iff._auto_1 →
autoParam (p a) isUnit_cfc_iff._auto_3 → (IsUnit (cfc f a) ↔ ∀ x ∈ spectrum R a, f x ≠ 0)- Cited by
- 2 results in Mathlib
- Foundations
- Depth 102 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites16
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement and proof · cited by 53,352
- TopologicalSpacestatement and proof · cited by 24,529
- Algebrastatement and proof · cited by 11,388
- Ringstatement and proof · cited by 7,463
- StarRingstatement and proof · cited by 1,686
- MetricSpacestatement and proof · cited by 1,684
- IsUnitstatement · cited by 1,602
- ContinuousOnstatement and proof · cited by 1,411
- ContinuousStarstatement and proof · cited by 543
- spectrumstatement and proof · cited by 510
- IsTopologicalSemiringstatement and proof · cited by 442
- Semifieldstatement and proof · cited by 439
Cited by2
Results whose statement or proof uses this declaration.
- CFC.isUnit_rpow_iffproof · cited by 2
- isUnit_cfcproof · cited by 0