Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Restrict

Goal state before:
pp: R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝¹⁹ : Semifield R
inst✝¹⁸ : StarRing R
inst✝¹⁷ : MetricSpace R
inst✝¹⁶ : IsTopologicalSemiring R
inst✝¹⁵ : ContinuousStar R
inst✝¹⁴ : Semifield S
inst✝¹³ : StarRing S
inst✝¹² : MetricSpace S
inst✝¹¹ : IsTopologicalSemiring S
inst✝¹⁰ : ContinuousStar S
inst✝⁹ : Ring A
inst✝⁸ : StarRing A
inst✝⁷ : Algebra S A
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R A
inst✝⁴ : IsScalarTower R S A
inst✝³ : StarModule R S
inst✝² : ContinuousSMul R S
inst✝¹ : TopologicalSpace A
inst✝ : ContinuousFunctionalCalculus S A q
f : C(S, R)
halg : IsClosedEmbedding ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ SpectrumRestricts a ⇑f
a : A
ha : p a
⊢ ∃ φ,
    Continuous[_, inst✝¹] ⇑φ ∧
      Function.Injective ⇑φ ∧
        φ (ContinuousMap.restrict (spectrum R a) (ContinuousMap.id R)) = a ∧
          (∀ (f : C(↑(spectrum R a), R)), spectrum R (φ f) = range ⇑f) ∧ ∀ (f : C(↑(spectrum R a), R)), p (φ f)

Tactic:
refine ⟨((h a).mp ha).2.starAlgHom (cfcHom ((h a).mp ha).1 (R := S)),
      ?hom_continuous, ?hom_injective, ?hom_id, ?hom_map_spectrum, ?predicate_hom⟩

Goal state after:
case hom_continuous
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝¹⁹ : Semifield R
inst✝¹⁸ : StarRing R
inst✝¹⁷ : MetricSpace R
inst✝¹⁶ : IsTopologicalSemiring R
inst✝¹⁵ : ContinuousStar R
inst✝¹⁴ : Semifield S
inst✝¹³ : StarRing S
inst✝¹² : MetricSpace S
inst✝¹¹ : IsTopologicalSemiring S
inst✝¹⁰ : ContinuousStar S
inst✝⁹ : Ring A
inst✝⁸ : StarRing A
inst✝⁷ : Algebra S A
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R A
inst✝⁴ : IsScalarTower R S A
inst✝³ : StarModule R S
inst✝² : ContinuousSMul R S
inst✝¹ : TopologicalSpace A
inst✝ : ContinuousFunctionalCalculus S A q
f : C(S, R)
halg : IsClosedEmbedding ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ SpectrumRestricts a ⇑f
a : A
ha : p a
⊢ Continuous[_, inst✝¹] ⇑(starAlgHom (cfcHom ⋯) ⋯)

case hom_injective
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝¹⁹ : Semifield R
inst✝¹⁸ : StarRing R
inst✝¹⁷ : MetricSpace R
inst✝¹⁶ : IsTopologicalSemiring R
inst✝¹⁵ : ContinuousStar R
inst✝¹⁴ : Semifield S
inst✝¹³ : StarRing S
inst✝¹² : MetricSpace S
inst✝¹¹ : IsTopologicalSemiring S
inst✝¹⁰ : ContinuousStar S
inst✝⁹ : Ring A
inst✝⁸ : StarRing A
inst✝⁷ : Algebra S A
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R A
inst✝⁴ : IsScalarTower R S A
inst✝³ : StarModule R S
inst✝² : ContinuousSMul R S
inst✝¹ : TopologicalSpace A
inst✝ : ContinuousFunctionalCalculus S A q
f : C(S, R)
halg : IsClosedEmbedding ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ SpectrumRestricts a ⇑f
a : A
ha : p a
⊢ Function.Injective ⇑(starAlgHom (cfcHom ⋯) ⋯)

case hom_id
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝¹⁹ : Semifield R
inst✝¹⁸ : StarRing R
inst✝¹⁷ : MetricSpace R
inst✝¹⁶ : IsTopologicalSemiring R
inst✝¹⁵ : ContinuousStar R
inst✝¹⁴ : Semifield S
inst✝¹³ : StarRing S
inst✝¹² : MetricSpace S
inst✝¹¹ : IsTopologicalSemiring S
inst✝¹⁰ : ContinuousStar S
inst✝⁹ : Ring A
inst✝⁸ : StarRing A
inst✝⁷ : Algebra S A
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R A
inst✝⁴ : IsScalarTower R S A
inst✝³ : StarModule R S
inst✝² : ContinuousSMul R S
inst✝¹ : TopologicalSpace A
inst✝ : ContinuousFunctionalCalculus S A q
f : C(S, R)
halg : IsClosedEmbedding ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ SpectrumRestricts a ⇑f
a : A
ha : p a
⊢ (starAlgHom (cfcHom ⋯) ⋯) (ContinuousMap.restrict (spectrum R a) (ContinuousMap.id R)) = a

case hom_map_spectrum
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝¹⁹ : Semifield R
inst✝¹⁸ : StarRing R
inst✝¹⁷ : MetricSpace R
inst✝¹⁶ : IsTopologicalSemiring R
inst✝¹⁵ : ContinuousStar R
inst✝¹⁴ : Semifield S
inst✝¹³ : StarRing S
inst✝¹² : MetricSpace S
inst✝¹¹ : IsTopologicalSemiring S
inst✝¹⁰ : ContinuousStar S
inst✝⁹ : Ring A
inst✝⁸ : StarRing A
inst✝⁷ : Algebra S A
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R A
inst✝⁴ : IsScalarTower R S A
inst✝³ : StarModule R S
inst✝² : ContinuousSMul R S
inst✝¹ : TopologicalSpace A
inst✝ : ContinuousFunctionalCalculus S A q
f : C(S, R)
halg : IsClosedEmbedding ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ SpectrumRestricts a ⇑f
a : A
ha : p a
⊢ ∀ (f_1 : C(↑(spectrum R a), R)), spectrum R ((starAlgHom (cfcHom ⋯) ⋯) f_1) = range ⇑f_1

case predicate_hom
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝¹⁹ : Semifield R
inst✝¹⁸ : StarRing R
inst✝¹⁷ : MetricSpace R
inst✝¹⁶ : IsTopologicalSemiring R
inst✝¹⁵ : ContinuousStar R
inst✝¹⁴ : Semifield S
inst✝¹³ : StarRing S
inst✝¹² : MetricSpace S
inst✝¹¹ : IsTopologicalSemiring S
inst✝¹⁰ : ContinuousStar S
inst✝⁹ : Ring A
inst✝⁸ : StarRing A
inst✝⁷ : Algebra S A
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R A
inst✝⁴ : IsScalarTower R S A
inst✝³ : StarModule R S
inst✝² : ContinuousSMul R S
inst✝¹ : TopologicalSpace A
inst✝ : ContinuousFunctionalCalculus S A q
f : C(S, R)
halg : IsClosedEmbedding ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ SpectrumRestricts a ⇑f
a : A
ha : p a
⊢ ∀ (f_1 : C(↑(spectrum R a), R)), p ((starAlgHom (cfcHom ⋯) ⋯) f_1)