Theorems · Theorem · functional analysis
cfc_comp
∀ {R : Type u_1} {A : Type u_2} {p : A → Prop} [inst : CommSemiring 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] [instCFC : ContinuousFunctionalCalculus R A p]
[ContinuousMap.UniqueHom R A] (g f : R → R) (a : A),
autoParam (p a) cfc_comp._auto_1 →
autoParam (ContinuousOn g (f '' spectrum R a)) cfc_comp._auto_3 →
autoParam (ContinuousOn f (spectrum R a)) cfc_comp._auto_5 → cfc (g ∘ f) a = cfc g (cfc f a)- Cited by
- 5 results in Mathlib
- Foundations
- Depth 101 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites33
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- Setproof · cited by 53,352
- TopologicalSpacestatement and proof · cited by 24,529
- Algebrastatement and proof · cited by 11,388
- CommSemiringstatement and proof · cited by 10,911
- Ringstatement and proof · cited by 7,463
- Set.Elemproof · cited by 7,166
- Set.imagestatement and proof · cited by 5,609
- ContinuousMapproof · cited by 2,491
- Set.extproof · cited by 2,266
- StarRingstatement and proof · cited by 1,686
- MetricSpacestatement and proof · cited by 1,684
Cited by5
Results whose statement or proof uses this declaration.
- cfc_comp'proof · cited by 21
- CFC.rpow_rpowproof · cited by 6
- CFC.cfc_rpowproof · cited by 1
- CFC.rpow_rpow_of_exponent_nonnegproof · cited by 1
- CFC.log_smulproof · cited by 0