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

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: 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)
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)
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)
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)
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)
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_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)
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)
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)
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)
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)
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)
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)
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 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)
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)
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)
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)
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)
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 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 _]
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)
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: have := h_cpct a ((h a).mp ha).1
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)
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 ⋯) ⋯)
state_after: 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)
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
this : CompactSpace ↑(spectrum S a)
⊢ ClosedEmbedding ⇑(starAlgHom (cfcHom ⋯) ⋯)

tactic: exact ((h a).mp ha).2.closedEmbedding_starAlgHom
  (cfcHom_closedEmbedding ((h a).mp ha).1) halg
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)
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
this : CompactSpace ↑(spectrum S a)
⊢ ClosedEmbedding ⇑(starAlgHom (cfcHom ⋯) ⋯)
state_after: no goals

tactic: exact ((h a).mp ha).2.starAlgHom_id <| cfcHom_id ((h a).mp ha).1
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)
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
state_after: no goals

tactic: intro g
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)
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
state_after: 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)
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
g : C(↑(spectrum R a), R)
⊢ spectrum R ((starAlgHom (cfcHom ⋯) ⋯) g) = range ⇑g

tactic: rw [SpectrumRestricts.starAlgHom_apply]
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)
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
g : C(↑(spectrum R a), R)
⊢ spectrum R ((starAlgHom (cfcHom ⋯) ⋯) g) = range ⇑g
state_after: 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)
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
g : C(↑(spectrum R a), R)
⊢ spectrum R
      ((cfcHom ⋯)
        ({ toFun := ⇑(StarAlgHom.ofId R S), continuous_toFun := ⋯ }.comp
          (g.comp { toFun := Subtype.map ⇑f ⋯, continuous_toFun := ⋯ }))) =
    range ⇑g

tactic: simp only [← @spectrum.preimage_algebraMap (R := R) S, cfcHom_map_spectrum]
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)
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
g : C(↑(spectrum R a), R)
⊢ spectrum R
      ((cfcHom ⋯)
        ({ toFun := ⇑(StarAlgHom.ofId R S), continuous_toFun := ⋯ }.comp
          (g.comp { toFun := Subtype.map ⇑f ⋯, continuous_toFun := ⋯ }))) =
    range ⇑g
state_after: 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)
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
g : C(↑(spectrum R a), R)
⊢ ⇑(algebraMap R S) ⁻¹'
      range
        ⇑({ toFun := ⇑(StarAlgHom.ofId R S), continuous_toFun := ⋯ }.comp
            (g.comp { toFun := Subtype.map ⇑f ⋯, continuous_toFun := ⋯ })) =
    range ⇑g

tactic: ext x
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)
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
g : C(↑(spectrum R a), R)
⊢ ⇑(algebraMap R S) ⁻¹'
      range
        ⇑({ toFun := ⇑(StarAlgHom.ofId R S), continuous_toFun := ⋯ }.comp
            (g.comp { toFun := Subtype.map ⇑f ⋯, continuous_toFun := ⋯ })) =
    range ⇑g
state_after: case h
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)
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
g : C(↑(spectrum R a), R)
x : R
⊢ x ∈
      ⇑(algebraMap R S) ⁻¹'
        range
          ⇑({ toFun := ⇑(StarAlgHom.ofId R S), continuous_toFun := ⋯ }.comp
              (g.comp { toFun := Subtype.map ⇑f ⋯, continuous_toFun := ⋯ })) ↔
    x ∈ range ⇑g

tactic: constructor
state_before: case h
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)
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
g : C(↑(spectrum R a), R)
x : R
⊢ x ∈
      ⇑(algebraMap R S) ⁻¹'
        range
          ⇑({ toFun := ⇑(StarAlgHom.ofId R S), continuous_toFun := ⋯ }.comp
              (g.comp { toFun := Subtype.map ⇑f ⋯, continuous_toFun := ⋯ })) ↔
    x ∈ range ⇑g
state_after: case h.mp
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)
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
g : C(↑(spectrum R a), R)
x : R
⊢ x ∈
      ⇑(algebraMap R S) ⁻¹'
        range
          ⇑({ toFun := ⇑(StarAlgHom.ofId R S), continuous_toFun := ⋯ }.comp
              (g.comp { toFun := Subtype.map ⇑f ⋯, continuous_toFun := ⋯ })) →
    x ∈ range ⇑g

case h.mpr
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)
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
g : C(↑(spectrum R a), R)
x : R
⊢ x ∈ range ⇑g →
    x ∈
      ⇑(algebraMap R S) ⁻¹'
        range
          ⇑({ toFun := ⇑(StarAlgHom.ofId R S), continuous_toFun := ⋯ }.comp
              (g.comp { toFun := Subtype.map ⇑f ⋯, continuous_toFun := ⋯ }))

tactic: rintro ⟨y, hy⟩
state_before: case h.mp
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)
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
g : C(↑(spectrum R a), R)
x : R
⊢ x ∈
      ⇑(algebraMap R S) ⁻¹'
        range
          ⇑({ toFun := ⇑(StarAlgHom.ofId R S), continuous_toFun := ⋯ }.comp
              (g.comp { toFun := Subtype.map ⇑f ⋯, continuous_toFun := ⋯ })) →
    x ∈ range ⇑g
state_after: case h.mp.intro
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)
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
g : C(↑(spectrum R a), R)
x : R
y : ↑(spectrum S a)
hy :
  ({ toFun := ⇑(StarAlgHom.ofId R S), continuous_toFun := ⋯ }.comp
        (g.comp { toFun := Subtype.map ⇑f ⋯, continuous_toFun := ⋯ }))
      y =
    (algebraMap R S) x
⊢ x ∈ range ⇑g

tactic: have := congr_arg f hy
state_before: case h.mp.intro
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)
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
g : C(↑(spectrum R a), R)
x : R
y : ↑(spectrum S a)
hy :
  ({ toFun := ⇑(StarAlgHom.ofId R S), continuous_toFun := ⋯ }.comp
        (g.comp { toFun := Subtype.map ⇑f ⋯, continuous_toFun := ⋯ }))
      y =
    (algebraMap R S) x
⊢ x ∈ range ⇑g
state_after: case h.mp.intro
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)
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
g : C(↑(spectrum R a), R)
x : R
y : ↑(spectrum S a)
hy :
  ({ toFun := ⇑(StarAlgHom.ofId R S), continuous_toFun := ⋯ }.comp
        (g.comp { toFun := Subtype.map ⇑f ⋯, continuous_toFun := ⋯ }))
      y =
    (algebraMap R S) x
this :
  f
      (({ toFun := ⇑(StarAlgHom.ofId R S), continuous_toFun := ⋯ }.comp
          (g.comp { toFun := Subtype.map ⇑f ⋯, continuous_toFun := ⋯ }))
        y) =
    f ((algebraMap R S) x)
⊢ x ∈ range ⇑g

tactic: simp only [ContinuousMap.coe_mk, ContinuousMap.comp_apply, StarAlgHom.ofId_apply] at this
state_before: case h.mp.intro
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)
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
g : C(↑(spectrum R a), R)
x : R
y : ↑(spectrum S a)
hy :
  ({ toFun := ⇑(StarAlgHom.ofId R S), continuous_toFun := ⋯ }.comp
        (g.comp { toFun := Subtype.map ⇑f ⋯, continuous_toFun := ⋯ }))
      y =
    (algebraMap R S) x
this :
  f
      (({ toFun := ⇑(StarAlgHom.ofId R S), continuous_toFun := ⋯ }.comp
          (g.comp { toFun := Subtype.map ⇑f ⋯, continuous_toFun := ⋯ }))
        y) =
    f ((algebraMap R S) x)
⊢ x ∈ range ⇑g
state_after: case h.mp.intro
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)
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
g : C(↑(spectrum R a), R)
x : R
y : ↑(spectrum S a)
hy :
  ({ toFun := ⇑(StarAlgHom.ofId R S), continuous_toFun := ⋯ }.comp
        (g.comp { toFun := Subtype.map ⇑f ⋯, continuous_toFun := ⋯ }))
      y =
    (algebraMap R S) x
this : f ((algebraMap R S) (g (Subtype.map ⇑f ⋯ y))) = f ((algebraMap R S) x)
⊢ x ∈ range ⇑g

tactic: rw [((h a).mp ha).2.left_inv _, ((h a).mp ha).2.left_inv _] at this
state_before: case h.mp.intro
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)
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
g : C(↑(spectrum R a), R)
x : R
y : ↑(spectrum S a)
hy :
  ({ toFun := ⇑(StarAlgHom.ofId R S), continuous_toFun := ⋯ }.comp
        (g.comp { toFun := Subtype.map ⇑f ⋯, continuous_toFun := ⋯ }))
      y =
    (algebraMap R S) x
this : f ((algebraMap R S) (g (Subtype.map ⇑f ⋯ y))) = f ((algebraMap R S) x)
⊢ x ∈ range ⇑g
state_after: case h.mp.intro
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)
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
g : C(↑(spectrum R a), R)
x : R
y : ↑(spectrum S a)
hy :
  ({ toFun := ⇑(StarAlgHom.ofId R S), continuous_toFun := ⋯ }.comp
        (g.comp { toFun := Subtype.map ⇑f ⋯, continuous_toFun := ⋯ }))
      y =
    (algebraMap R S) x
this : g (Subtype.map ⇑f ⋯ y) = x
⊢ x ∈ range ⇑g

tactic: exact ⟨_, this⟩
state_before: case h.mp.intro
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)
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
g : C(↑(spectrum R a), R)
x : R
y : ↑(spectrum S a)
hy :
  ({ toFun := ⇑(StarAlgHom.ofId R S), continuous_toFun := ⋯ }.comp
        (g.comp { toFun := Subtype.map ⇑f ⋯, continuous_toFun := ⋯ }))
      y =
    (algebraMap R S) x
this : g (Subtype.map ⇑f ⋯ y) = x
⊢ x ∈ range ⇑g
state_after: no goals

tactic: rintro ⟨y, rfl⟩
state_before: case h.mpr
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)
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
g : C(↑(spectrum R a), R)
x : R
⊢ x ∈ range ⇑g →
    x ∈
      ⇑(algebraMap R S) ⁻¹'
        range
          ⇑({ toFun := ⇑(StarAlgHom.ofId R S), continuous_toFun := ⋯ }.comp
              (g.comp { toFun := Subtype.map ⇑f ⋯, continuous_toFun := ⋯ }))
state_after: case h.mpr.intro
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)
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
g : C(↑(spectrum R a), R)
y : ↑(spectrum R a)
⊢ g y ∈
    ⇑(algebraMap R S) ⁻¹'
      range
        ⇑({ toFun := ⇑(StarAlgHom.ofId R S), continuous_toFun := ⋯ }.comp
            (g.comp { toFun := Subtype.map ⇑f ⋯, continuous_toFun := ⋯ }))

tactic: rw [Set.mem_preimage]
state_before: case h.mpr.intro
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)
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
g : C(↑(spectrum R a), R)
y : ↑(spectrum R a)
⊢ g y ∈
    ⇑(algebraMap R S) ⁻¹'
      range
        ⇑({ toFun := ⇑(StarAlgHom.ofId R S), continuous_toFun := ⋯ }.comp
            (g.comp { toFun := Subtype.map ⇑f ⋯, continuous_toFun := ⋯ }))
state_after: case h.mpr.intro
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)
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
g : C(↑(spectrum R a), R)
y : ↑(spectrum R a)
⊢ (algebraMap R S) (g y) ∈
    range
      ⇑({ toFun := ⇑(StarAlgHom.ofId R S), continuous_toFun := ⋯ }.comp
          (g.comp { toFun := Subtype.map ⇑f ⋯, continuous_toFun := ⋯ }))

tactic: refine ⟨⟨algebraMap R S y, spectrum.algebraMap_mem S y.prop⟩, ?_⟩
state_before: case h.mpr.intro
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)
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
g : C(↑(spectrum R a), R)
y : ↑(spectrum R a)
⊢ (algebraMap R S) (g y) ∈
    range
      ⇑({ toFun := ⇑(StarAlgHom.ofId R S), continuous_toFun := ⋯ }.comp
          (g.comp { toFun := Subtype.map ⇑f ⋯, continuous_toFun := ⋯ }))
state_after: case h.mpr.intro
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)
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
g : C(↑(spectrum R a), R)
y : ↑(spectrum R a)
⊢ ({ toFun := ⇑(StarAlgHom.ofId R S), continuous_toFun := ⋯ }.comp
        (g.comp { toFun := Subtype.map ⇑f ⋯, continuous_toFun := ⋯ }))
      ⟨(algebraMap R S) ↑y, ⋯⟩ =
    (algebraMap R S) (g y)

tactic: simp only [ContinuousMap.coe_mk, ContinuousMap.comp_apply, StarAlgHom.ofId_apply]
state_before: case h.mpr.intro
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)
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
g : C(↑(spectrum R a), R)
y : ↑(spectrum R a)
⊢ ({ toFun := ⇑(StarAlgHom.ofId R S), continuous_toFun := ⋯ }.comp
        (g.comp { toFun := Subtype.map ⇑f ⋯, continuous_toFun := ⋯ }))
      ⟨(algebraMap R S) ↑y, ⋯⟩ =
    (algebraMap R S) (g y)
state_after: case h.mpr.intro
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)
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
g : C(↑(spectrum R a), R)
y : ↑(spectrum R a)
⊢ (algebraMap R S) (g (Subtype.map ⇑f ⋯ ⟨(algebraMap R S) ↑y, ⋯⟩)) = (algebraMap R S) (g y)

tactic: congr
state_before: case h.mpr.intro
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)
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
g : C(↑(spectrum R a), R)
y : ↑(spectrum R a)
⊢ (algebraMap R S) (g (Subtype.map ⇑f ⋯ ⟨(algebraMap R S) ↑y, ⋯⟩)) = (algebraMap R S) (g y)
state_after: case h.mpr.intro.h.e_6.h.h.e_6.h
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)
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
g : C(↑(spectrum R a), R)
y : ↑(spectrum R a)
⊢ Subtype.map ⇑f ⋯ ⟨(algebraMap R S) ↑y, ⋯⟩ = y

tactic: exact Subtype.ext (((h a).mp ha).2.left_inv y)
state_before: case h.mpr.intro.h.e_6.h.h.e_6.h
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)
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
g : C(↑(spectrum R a), R)
y : ↑(spectrum R a)
⊢ Subtype.map ⇑f ⋯ ⟨(algebraMap R S) ↑y, ⋯⟩ = y
state_after: no goals

tactic: intro g
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)
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: 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)
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
g : C(↑(spectrum R a), R)
⊢ p ((starAlgHom (cfcHom ⋯) ⋯) g)

tactic: rw [h]
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)
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
g : C(↑(spectrum R a), R)
⊢ p ((starAlgHom (cfcHom ⋯) ⋯) g)
state_after: 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)
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
g : C(↑(spectrum R a), R)
⊢ q ((starAlgHom (cfcHom ⋯) ⋯) g) ∧ SpectrumRestricts ((starAlgHom (cfcHom ⋯) ⋯) g) ⇑f

tactic: refine ⟨cfcHom_predicate _ _, ?_⟩
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)
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
g : C(↑(spectrum R a), R)
⊢ q ((starAlgHom (cfcHom ⋯) ⋯) g) ∧ SpectrumRestricts ((starAlgHom (cfcHom ⋯) ⋯) g) ⇑f
state_after: 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)
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
g : C(↑(spectrum R a), R)
⊢ SpectrumRestricts ((starAlgHom (cfcHom ⋯) ⋯) g) ⇑f

tactic: refine .of_rightInvOn (((h a).mp ha).2.left_inv) fun s hs ↦ ?_
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)
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
g : C(↑(spectrum R a), R)
⊢ SpectrumRestricts ((starAlgHom (cfcHom ⋯) ⋯) g) ⇑f
state_after: 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)
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
g : C(↑(spectrum R a), R)
s : S
hs : s ∈ spectrum S ((starAlgHom (cfcHom ⋯) ⋯) g)
⊢ (algebraMap R S) (f s) = s

tactic: rw [SpectrumRestricts.starAlgHom_apply, cfcHom_map_spectrum] at hs
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)
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
g : C(↑(spectrum R a), R)
s : S
hs : s ∈ spectrum S ((starAlgHom (cfcHom ⋯) ⋯) g)
⊢ (algebraMap R S) (f s) = s
state_after: 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)
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
g : C(↑(spectrum R a), R)
s : S
hs :
  s ∈
    range
      ⇑({ toFun := ⇑(StarAlgHom.ofId R S), continuous_toFun := ⋯ }.comp
          (g.comp { toFun := Subtype.map ⇑f ⋯, continuous_toFun := ⋯ }))
⊢ (algebraMap R S) (f s) = s

tactic: obtain ⟨r, rfl⟩ := hs
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)
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
g : C(↑(spectrum R a), R)
s : S
hs :
  s ∈
    range
      ⇑({ toFun := ⇑(StarAlgHom.ofId R S), continuous_toFun := ⋯ }.comp
          (g.comp { toFun := Subtype.map ⇑f ⋯, continuous_toFun := ⋯ }))
⊢ (algebraMap R S) (f s) = s
state_after: case intro
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)
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
g : C(↑(spectrum R a), R)
r : ↑(spectrum S a)
⊢ (algebraMap R S)
      (f
        (({ toFun := ⇑(StarAlgHom.ofId R S), continuous_toFun := ⋯ }.comp
            (g.comp { toFun := Subtype.map ⇑f ⋯, continuous_toFun := ⋯ }))
          r)) =
    ({ toFun := ⇑(StarAlgHom.ofId R S), continuous_toFun := ⋯ }.comp
        (g.comp { toFun := Subtype.map ⇑f ⋯, continuous_toFun := ⋯ }))
      r

tactic: simp [((h a).mp ha).2.left_inv _]
state_before: case intro
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)
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
g : C(↑(spectrum R a), R)
r : ↑(spectrum S a)
⊢ (algebraMap R S)
      (f
        (({ toFun := ⇑(StarAlgHom.ofId R S), continuous_toFun := ⋯ }.comp
            (g.comp { toFun := Subtype.map ⇑f ⋯, continuous_toFun := ⋯ }))
          r)) =
    ({ toFun := ⇑(StarAlgHom.ofId R S), continuous_toFun := ⋯ }.comp
        (g.comp { toFun := Subtype.map ⇑f ⋯, continuous_toFun := ⋯ }))
      r
state_after: no goals