Theorem: SpectrumRestricts.cfc
File: Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Restrict.lean

Proof steps:
state_before: R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²⁰ : Semifield R
inst✝¹⁹ : StarRing R
inst✝¹⁸ : MetricSpace R
inst✝¹⁷ : TopologicalSemiring R
inst✝¹⁶ : ContinuousStar R
inst✝¹⁵ : Semifield S
inst✝¹⁴ : StarRing S
inst✝¹³ : MetricSpace S
inst✝¹² : TopologicalSemiring S
inst✝¹¹ : ContinuousStar S
inst✝¹⁰ : TopologicalSpace A
inst✝⁹ : Ring A
inst✝⁸ : StarRing A
inst✝⁷ : Algebra S A
inst✝⁶ : ContinuousFunctionalCalculus S q
inst✝⁵ : Algebra R S
inst✝⁴ : Algebra R A
inst✝³ : IsScalarTower R S A
inst✝² : StarModule R S
inst✝¹ : ContinuousSMul R S
inst✝ : CompleteSpace R
f : C(S, R)
halg : UniformEmbedding ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ SpectrumRestricts a ⇑f
h_cpct : ∀ (a : A), q a → CompactSpace ↑(spectrum S a)
a : A
ha : p a
⊢ ∃ φ,
    ClosedEmbedding ⇑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)
state_after: case hom_closedEmbedding
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²⁰ : Semifield R
inst✝¹⁹ : StarRing R
inst✝¹⁸ : MetricSpace R
inst✝¹⁷ : TopologicalSemiring R
inst✝¹⁶ : ContinuousStar R
inst✝¹⁵ : Semifield S
inst✝¹⁴ : StarRing S
inst✝¹³ : MetricSpace S
inst✝¹² : TopologicalSemiring S
inst✝¹¹ : ContinuousStar S
inst✝¹⁰ : TopologicalSpace A
inst✝⁹ : Ring A
inst✝⁸ : StarRing A
inst✝⁷ : Algebra S A
inst✝⁶ : ContinuousFunctionalCalculus S q
inst✝⁵ : Algebra R S
inst✝⁴ : Algebra R A
inst✝³ : IsScalarTower R S A
inst✝² : StarModule R S
inst✝¹ : ContinuousSMul R S
inst✝ : CompleteSpace R
f : C(S, R)
halg : UniformEmbedding ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ SpectrumRestricts a ⇑f
h_cpct : ∀ (a : A), q a → CompactSpace ↑(spectrum S a)
a : A
ha : p a
⊢ ClosedEmbedding ⇑(starAlgHom (cfcHom ⋯) ⋯)

case hom_id
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²⁰ : Semifield R
inst✝¹⁹ : StarRing R
inst✝¹⁸ : MetricSpace R
inst✝¹⁷ : TopologicalSemiring R
inst✝¹⁶ : ContinuousStar R
inst✝¹⁵ : Semifield S
inst✝¹⁴ : StarRing S
inst✝¹³ : MetricSpace S
inst✝¹² : TopologicalSemiring S
inst✝¹¹ : ContinuousStar S
inst✝¹⁰ : TopologicalSpace A
inst✝⁹ : Ring A
inst✝⁸ : StarRing A
inst✝⁷ : Algebra S A
inst✝⁶ : ContinuousFunctionalCalculus S q
inst✝⁵ : Algebra R S
inst✝⁴ : Algebra R A
inst✝³ : IsScalarTower R S A
inst✝² : StarModule R S
inst✝¹ : ContinuousSMul R S
inst✝ : CompleteSpace R
f : C(S, R)
halg : UniformEmbedding ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ SpectrumRestricts a ⇑f
h_cpct : ∀ (a : A), q a → CompactSpace ↑(spectrum S a)
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✝²⁰ : Semifield R
inst✝¹⁹ : StarRing R
inst✝¹⁸ : MetricSpace R
inst✝¹⁷ : TopologicalSemiring R
inst✝¹⁶ : ContinuousStar R
inst✝¹⁵ : Semifield S
inst✝¹⁴ : StarRing S
inst✝¹³ : MetricSpace S
inst✝¹² : TopologicalSemiring S
inst✝¹¹ : ContinuousStar S
inst✝¹⁰ : TopologicalSpace A
inst✝⁹ : Ring A
inst✝⁸ : StarRing A
inst✝⁷ : Algebra S A
inst✝⁶ : ContinuousFunctionalCalculus S q
inst✝⁵ : Algebra R S
inst✝⁴ : Algebra R A
inst✝³ : IsScalarTower R S A
inst✝² : StarModule R S
inst✝¹ : ContinuousSMul R S
inst✝ : CompleteSpace R
f : C(S, R)
halg : UniformEmbedding ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ SpectrumRestricts a ⇑f
h_cpct : ∀ (a : A), q a → CompactSpace ↑(spectrum S a)
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✝²⁰ : Semifield R
inst✝¹⁹ : StarRing R
inst✝¹⁸ : MetricSpace R
inst✝¹⁷ : TopologicalSemiring R
inst✝¹⁶ : ContinuousStar R
inst✝¹⁵ : Semifield S
inst✝¹⁴ : StarRing S
inst✝¹³ : MetricSpace S
inst✝¹² : TopologicalSemiring S
inst✝¹¹ : ContinuousStar S
inst✝¹⁰ : TopologicalSpace A
inst✝⁹ : Ring A
inst✝⁸ : StarRing A
inst✝⁷ : Algebra S A
inst✝⁶ : ContinuousFunctionalCalculus S q
inst✝⁵ : Algebra R S
inst✝⁴ : Algebra R A
inst✝³ : IsScalarTower R S A
inst✝² : StarModule R S
inst✝¹ : ContinuousSMul R S
inst✝ : CompleteSpace R
f : C(S, R)
halg : UniformEmbedding ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ SpectrumRestricts a ⇑f
h_cpct : ∀ (a : A), q a → CompactSpace ↑(spectrum S a)
a : A
ha : p a
⊢ ∀ (f_1 : C(↑(spectrum R a), R)), p ((starAlgHom (cfcHom ⋯) ⋯) f_1)
tactic: refine ⟨((h a).mp ha).2.starAlgHom (cfcHom ((h a).mp ha).1 (R := S)),
      ?hom_closedEmbedding, ?hom_id, ?hom_map_spectrum, ?predicate_hom⟩
state_before: case hom_closedEmbedding
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²⁰ : Semifield R
inst✝¹⁹ : StarRing R
inst✝¹⁸ : MetricSpace R
inst✝¹⁷ : TopologicalSemiring R
inst✝¹⁶ : ContinuousStar R
inst✝¹⁵ : Semifield S
inst✝¹⁴ : StarRing S
inst✝¹³ : MetricSpace S
inst✝¹² : TopologicalSemiring S
inst✝¹¹ : ContinuousStar S
inst✝¹⁰ : TopologicalSpace A
inst✝⁹ : Ring A
inst✝⁸ : StarRing A
inst✝⁷ : Algebra S A
inst✝⁶ : ContinuousFunctionalCalculus S q
inst✝⁵ : Algebra R S
inst✝⁴ : Algebra R A
inst✝³ : IsScalarTower R S A
inst✝² : StarModule R S
inst✝¹ : ContinuousSMul R S
inst✝ : CompleteSpace R
f : C(S, R)
halg : UniformEmbedding ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ SpectrumRestricts a ⇑f
h_cpct : ∀ (a : A), q a → CompactSpace ↑(spectrum S a)
a : A
ha : p a
⊢ ClosedEmbedding ⇑(starAlgHom (cfcHom ⋯) ⋯)

case hom_id
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²⁰ : Semifield R
inst✝¹⁹ : StarRing R
inst✝¹⁸ : MetricSpace R
inst✝¹⁷ : TopologicalSemiring R
inst✝¹⁶ : ContinuousStar R
inst✝¹⁵ : Semifield S
inst✝¹⁴ : StarRing S
inst✝¹³ : MetricSpace S
inst✝¹² : TopologicalSemiring S
inst✝¹¹ : ContinuousStar S
inst✝¹⁰ : TopologicalSpace A
inst✝⁹ : Ring A
inst✝⁸ : StarRing A
inst✝⁷ : Algebra S A
inst✝⁶ : ContinuousFunctionalCalculus S q
inst✝⁵ : Algebra R S
inst✝⁴ : Algebra R A
inst✝³ : IsScalarTower R S A
inst✝² : StarModule R S
inst✝¹ : ContinuousSMul R S
inst✝ : CompleteSpace R
f : C(S, R)
halg : UniformEmbedding ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ SpectrumRestricts a ⇑f
h_cpct : ∀ (a : A), q a → CompactSpace ↑(spectrum S a)
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✝²⁰ : Semifield R
inst✝¹⁹ : StarRing R
inst✝¹⁸ : MetricSpace R
inst✝¹⁷ : TopologicalSemiring R
inst✝¹⁶ : ContinuousStar R
inst✝¹⁵ : Semifield S
inst✝¹⁴ : StarRing S
inst✝¹³ : MetricSpace S
inst✝¹² : TopologicalSemiring S
inst✝¹¹ : ContinuousStar S
inst✝¹⁰ : TopologicalSpace A
inst✝⁹ : Ring A
inst✝⁸ : StarRing A
inst✝⁷ : Algebra S A
inst✝⁶ : ContinuousFunctionalCalculus S q
inst✝⁵ : Algebra R S
inst✝⁴ : Algebra R A
inst✝³ : IsScalarTower R S A
inst✝² : StarModule R S
inst✝¹ : ContinuousSMul R S
inst✝ : CompleteSpace R
f : C(S, R)
halg : UniformEmbedding ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ SpectrumRestricts a ⇑f
h_cpct : ∀ (a : A), q a → CompactSpace ↑(spectrum S a)
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✝²⁰ : Semifield R
inst✝¹⁹ : StarRing R
inst✝¹⁸ : MetricSpace R
inst✝¹⁷ : TopologicalSemiring R
inst✝¹⁶ : ContinuousStar R
inst✝¹⁵ : Semifield S
inst✝¹⁴ : StarRing S
inst✝¹³ : MetricSpace S
inst✝¹² : TopologicalSemiring S
inst✝¹¹ : ContinuousStar S
inst✝¹⁰ : TopologicalSpace A
inst✝⁹ : Ring A
inst✝⁸ : StarRing A
inst✝⁷ : Algebra S A
inst✝⁶ : ContinuousFunctionalCalculus S q
inst✝⁵ : Algebra R S
inst✝⁴ : Algebra R A
inst✝³ : IsScalarTower R S A
inst✝² : StarModule R S
inst✝¹ : ContinuousSMul R S
inst✝ : CompleteSpace R
f : C(S, R)
halg : UniformEmbedding ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ SpectrumRestricts a ⇑f
h_cpct : ∀ (a : A), q a → CompactSpace ↑(spectrum S a)
a : A
ha : p a
⊢ ∀ (f_1 : C(↑(spectrum R a), R)), p ((starAlgHom (cfcHom ⋯) ⋯) f_1)
state_after: case hom_id
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²⁰ : Semifield R
inst✝¹⁹ : StarRing R
inst✝¹⁸ : MetricSpace R
inst✝¹⁷ : TopologicalSemiring R
inst✝¹⁶ : ContinuousStar R
inst✝¹⁵ : Semifield S
inst✝¹⁴ : StarRing S
inst✝¹³ : MetricSpace S
inst✝¹² : TopologicalSemiring S
inst✝¹¹ : ContinuousStar S
inst✝¹⁰ : TopologicalSpace A
inst✝⁹ : Ring A
inst✝⁸ : StarRing A
inst✝⁷ : Algebra S A
inst✝⁶ : ContinuousFunctionalCalculus S q
inst✝⁵ : Algebra R S
inst✝⁴ : Algebra R A
inst✝³ : IsScalarTower R S A
inst✝² : StarModule R S
inst✝¹ : ContinuousSMul R S
inst✝ : CompleteSpace R
f : C(S, R)
halg : UniformEmbedding ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ SpectrumRestricts a ⇑f
h_cpct : ∀ (a : A), q a → CompactSpace ↑(spectrum S a)
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✝²⁰ : Semifield R
inst✝¹⁹ : StarRing R
inst✝¹⁸ : MetricSpace R
inst✝¹⁷ : TopologicalSemiring R
inst✝¹⁶ : ContinuousStar R
inst✝¹⁵ : Semifield S
inst✝¹⁴ : StarRing S
inst✝¹³ : MetricSpace S
inst✝¹² : TopologicalSemiring S
inst✝¹¹ : ContinuousStar S
inst✝¹⁰ : TopologicalSpace A
inst✝⁹ : Ring A
inst✝⁸ : StarRing A
inst✝⁷ : Algebra S A
inst✝⁶ : ContinuousFunctionalCalculus S q
inst✝⁵ : Algebra R S
inst✝⁴ : Algebra R A
inst✝³ : IsScalarTower R S A
inst✝² : StarModule R S
inst✝¹ : ContinuousSMul R S
inst✝ : CompleteSpace R
f : C(S, R)
halg : UniformEmbedding ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ SpectrumRestricts a ⇑f
h_cpct : ∀ (a : A), q a → CompactSpace ↑(spectrum S a)
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✝²⁰ : Semifield R
inst✝¹⁹ : StarRing R
inst✝¹⁸ : MetricSpace R
inst✝¹⁷ : TopologicalSemiring R
inst✝¹⁶ : ContinuousStar R
inst✝¹⁵ : Semifield S
inst✝¹⁴ : StarRing S
inst✝¹³ : MetricSpace S
inst✝¹² : TopologicalSemiring S
inst✝¹¹ : ContinuousStar S
inst✝¹⁰ : TopologicalSpace A
inst✝⁹ : Ring A
inst✝⁸ : StarRing A
inst✝⁷ : Algebra S A
inst✝⁶ : ContinuousFunctionalCalculus S q
inst✝⁵ : Algebra R S
inst✝⁴ : Algebra R A
inst✝³ : IsScalarTower R S A
inst✝² : StarModule R S
inst✝¹ : ContinuousSMul R S
inst✝ : CompleteSpace R
f : C(S, R)
halg : UniformEmbedding ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ SpectrumRestricts a ⇑f
h_cpct : ∀ (a : A), q a → CompactSpace ↑(spectrum S a)
a : A
ha : p a
⊢ ∀ (f_1 : C(↑(spectrum R a), R)), p ((starAlgHom (cfcHom ⋯) ⋯) f_1)
tactic: case hom_closedEmbedding =>
      have := h_cpct a ((h a).mp ha).1
      exact ((h a).mp ha).2.closedEmbedding_starAlgHom
        (cfcHom_closedEmbedding ((h a).mp ha).1) halg
state_before: case hom_id
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²⁰ : Semifield R
inst✝¹⁹ : StarRing R
inst✝¹⁸ : MetricSpace R
inst✝¹⁷ : TopologicalSemiring R
inst✝¹⁶ : ContinuousStar R
inst✝¹⁵ : Semifield S
inst✝¹⁴ : StarRing S
inst✝¹³ : MetricSpace S
inst✝¹² : TopologicalSemiring S
inst✝¹¹ : ContinuousStar S
inst✝¹⁰ : TopologicalSpace A
inst✝⁹ : Ring A
inst✝⁸ : StarRing A
inst✝⁷ : Algebra S A
inst✝⁶ : ContinuousFunctionalCalculus S q
inst✝⁵ : Algebra R S
inst✝⁴ : Algebra R A
inst✝³ : IsScalarTower R S A
inst✝² : StarModule R S
inst✝¹ : ContinuousSMul R S
inst✝ : CompleteSpace R
f : C(S, R)
halg : UniformEmbedding ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ SpectrumRestricts a ⇑f
h_cpct : ∀ (a : A), q a → CompactSpace ↑(spectrum S a)
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✝²⁰ : Semifield R
inst✝¹⁹ : StarRing R
inst✝¹⁸ : MetricSpace R
inst✝¹⁷ : TopologicalSemiring R
inst✝¹⁶ : ContinuousStar R
inst✝¹⁵ : Semifield S
inst✝¹⁴ : StarRing S
inst✝¹³ : MetricSpace S
inst✝¹² : TopologicalSemiring S
inst✝¹¹ : ContinuousStar S
inst✝¹⁰ : TopologicalSpace A
inst✝⁹ : Ring A
inst✝⁸ : StarRing A
inst✝⁷ : Algebra S A
inst✝⁶ : ContinuousFunctionalCalculus S q
inst✝⁵ : Algebra R S
inst✝⁴ : Algebra R A
inst✝³ : IsScalarTower R S A
inst✝² : StarModule R S
inst✝¹ : ContinuousSMul R S
inst✝ : CompleteSpace R
f : C(S, R)
halg : UniformEmbedding ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ SpectrumRestricts a ⇑f
h_cpct : ∀ (a : A), q a → CompactSpace ↑(spectrum S a)
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✝²⁰ : Semifield R
inst✝¹⁹ : StarRing R
inst✝¹⁸ : MetricSpace R
inst✝¹⁷ : TopologicalSemiring R
inst✝¹⁶ : ContinuousStar R
inst✝¹⁵ : Semifield S
inst✝¹⁴ : StarRing S
inst✝¹³ : MetricSpace S
inst✝¹² : TopologicalSemiring S
inst✝¹¹ : ContinuousStar S
inst✝¹⁰ : TopologicalSpace A
inst✝⁹ : Ring A
inst✝⁸ : StarRing A
inst✝⁷ : Algebra S A
inst✝⁶ : ContinuousFunctionalCalculus S q
inst✝⁵ : Algebra R S
inst✝⁴ : Algebra R A
inst✝³ : IsScalarTower R S A
inst✝² : StarModule R S
inst✝¹ : ContinuousSMul R S
inst✝ : CompleteSpace R
f : C(S, R)
halg : UniformEmbedding ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ SpectrumRestricts a ⇑f
h_cpct : ∀ (a : A), q a → CompactSpace ↑(spectrum S a)
a : A
ha : p a
⊢ ∀ (f_1 : C(↑(spectrum R a), R)), p ((starAlgHom (cfcHom ⋯) ⋯) f_1)
state_after: case hom_map_spectrum
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²⁰ : Semifield R
inst✝¹⁹ : StarRing R
inst✝¹⁸ : MetricSpace R
inst✝¹⁷ : TopologicalSemiring R
inst✝¹⁶ : ContinuousStar R
inst✝¹⁵ : Semifield S
inst✝¹⁴ : StarRing S
inst✝¹³ : MetricSpace S
inst✝¹² : TopologicalSemiring S
inst✝¹¹ : ContinuousStar S
inst✝¹⁰ : TopologicalSpace A
inst✝⁹ : Ring A
inst✝⁸ : StarRing A
inst✝⁷ : Algebra S A
inst✝⁶ : ContinuousFunctionalCalculus S q
inst✝⁵ : Algebra R S
inst✝⁴ : Algebra R A
inst✝³ : IsScalarTower R S A
inst✝² : StarModule R S
inst✝¹ : ContinuousSMul R S
inst✝ : CompleteSpace R
f : C(S, R)
halg : UniformEmbedding ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ SpectrumRestricts a ⇑f
h_cpct : ∀ (a : A), q a → CompactSpace ↑(spectrum S a)
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✝²⁰ : Semifield R
inst✝¹⁹ : StarRing R
inst✝¹⁸ : MetricSpace R
inst✝¹⁷ : TopologicalSemiring R
inst✝¹⁶ : ContinuousStar R
inst✝¹⁵ : Semifield S
inst✝¹⁴ : StarRing S
inst✝¹³ : MetricSpace S
inst✝¹² : TopologicalSemiring S
inst✝¹¹ : ContinuousStar S
inst✝¹⁰ : TopologicalSpace A
inst✝⁹ : Ring A
inst✝⁸ : StarRing A
inst✝⁷ : Algebra S A
inst✝⁶ : ContinuousFunctionalCalculus S q
inst✝⁵ : Algebra R S
inst✝⁴ : Algebra R A
inst✝³ : IsScalarTower R S A
inst✝² : StarModule R S
inst✝¹ : ContinuousSMul R S
inst✝ : CompleteSpace R
f : C(S, R)
halg : UniformEmbedding ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ SpectrumRestricts a ⇑f
h_cpct : ∀ (a : A), q a → CompactSpace ↑(spectrum S a)
a : A
ha : p a
⊢ ∀ (f_1 : C(↑(spectrum R a), R)), p ((starAlgHom (cfcHom ⋯) ⋯) f_1)
tactic: case hom_id => exact ((h a).mp ha).2.starAlgHom_id <| cfcHom_id ((h a).mp ha).1
state_before: case predicate_hom
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²⁰ : Semifield R
inst✝¹⁹ : StarRing R
inst✝¹⁸ : MetricSpace R
inst✝¹⁷ : TopologicalSemiring R
inst✝¹⁶ : ContinuousStar R
inst✝¹⁵ : Semifield S
inst✝¹⁴ : StarRing S
inst✝¹³ : MetricSpace S
inst✝¹² : TopologicalSemiring S
inst✝¹¹ : ContinuousStar S
inst✝¹⁰ : TopologicalSpace A
inst✝⁹ : Ring A
inst✝⁸ : StarRing A
inst✝⁷ : Algebra S A
inst✝⁶ : ContinuousFunctionalCalculus S q
inst✝⁵ : Algebra R S
inst✝⁴ : Algebra R A
inst✝³ : IsScalarTower R S A
inst✝² : StarModule R S
inst✝¹ : ContinuousSMul R S
inst✝ : CompleteSpace R
f : C(S, R)
halg : UniformEmbedding ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ SpectrumRestricts a ⇑f
h_cpct : ∀ (a : A), q a → CompactSpace ↑(spectrum S a)
a : A
ha : p a
⊢ ∀ (f_1 : C(↑(spectrum R a), R)), p ((starAlgHom (cfcHom ⋯) ⋯) f_1)
state_after: no goals
tactic: case predicate_hom =>
      intro g
      rw [h]
      refine ⟨cfcHom_predicate _ _, ?_⟩
      refine .of_rightInvOn (((h a).mp ha).2.left_inv) fun s hs ↦ ?_
      rw [SpectrumRestricts.starAlgHom_apply, cfcHom_map_spectrum] at hs
      obtain ⟨r, rfl⟩ := hs
      simp [((h a).mp ha).2.left_inv _]