-- Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Restrict.lean
lemma cfcₙHom_eq_restrict (f : C(S, R)) (halg : IsUniformEmbedding (algebraMap R S)) {a : A}
    (hpa : p a) (hqa : q a) (h : QuasispectrumRestricts a f) :
    cfcₙHom hpa = h.nonUnitalStarAlgHom (cfcₙHom hqa)

Proof state:
case hφ₁
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²⁶ : Semifield R
inst✝²⁵ : StarRing R
inst✝²⁴ : MetricSpace R
inst✝²³ : IsTopologicalSemiring R
inst✝²² : ContinuousStar R
inst✝²¹ : Field S
inst✝²⁰ : StarRing S
inst✝¹⁹ : MetricSpace S
inst✝¹⁸ : IsTopologicalRing S
inst✝¹⁷ : ContinuousStar S
inst✝¹⁶ : NonUnitalRing A
inst✝¹⁵ : StarRing A
inst✝¹⁴ : Module S A
inst✝¹³ : IsScalarTower S A A
inst✝¹² : SMulCommClass S A A
inst✝¹¹ : Algebra R S
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R S A
inst✝⁸ : StarModule R S
inst✝⁷ : ContinuousSMul R S
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalContinuousFunctionalCalculus S q
inst✝⁴ : CompleteSpace R
inst✝³ : IsScalarTower R A A
inst✝² : SMulCommClass R A A
inst✝¹ : NonUnitalContinuousFunctionalCalculus R p
inst✝ : UniqueHom R A
f : C(S, R)
halg : IsUniformEmbedding ⇑(algebraMap R S)
a : A
hpa : p a
hqa : q a
h : QuasispectrumRestricts a ⇑f
⊢ Continuous ⇑(nonUnitalStarAlgHom (cfcₙHom hqa) h)

Tactic:
exact h.isClosedEmbedding_nonUnitalStarAlgHom (cfcₙHom_isClosedEmbedding hqa) halg |>.continuous

Result:
no goals

Proof state:
case hφ₂
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²⁶ : Semifield R
inst✝²⁵ : StarRing R
inst✝²⁴ : MetricSpace R
inst✝²³ : IsTopologicalSemiring R
inst✝²² : ContinuousStar R
inst✝²¹ : Field S
inst✝²⁰ : StarRing S
inst✝¹⁹ : MetricSpace S
inst✝¹⁸ : IsTopologicalRing S
inst✝¹⁷ : ContinuousStar S
inst✝¹⁶ : NonUnitalRing A
inst✝¹⁵ : StarRing A
inst✝¹⁴ : Module S A
inst✝¹³ : IsScalarTower S A A
inst✝¹² : SMulCommClass S A A
inst✝¹¹ : Algebra R S
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R S A
inst✝⁸ : StarModule R S
inst✝⁷ : ContinuousSMul R S
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalContinuousFunctionalCalculus S q
inst✝⁴ : CompleteSpace R
inst✝³ : IsScalarTower R A A
inst✝² : SMulCommClass R A A
inst✝¹ : NonUnitalContinuousFunctionalCalculus R p
inst✝ : UniqueHom R A
f : C(S, R)
halg : IsUniformEmbedding ⇑(algebraMap R S)
a : A
hpa : p a
hqa : q a
h : QuasispectrumRestricts a ⇑f
⊢ (nonUnitalStarAlgHom (cfcₙHom hqa) h)
      { toContinuousMap := ContinuousMap.restrict (σₙ R a) (ContinuousMap.id R), map_zero' := ⋯ } =
    a

Tactic:
exact h.nonUnitalStarAlgHom_id (cfcₙHom_id hqa)

Result:
no goals

Proof state:
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²⁶ : Semifield R
inst✝²⁵ : StarRing R
inst✝²⁴ : MetricSpace R
inst✝²³ : IsTopologicalSemiring R
inst✝²² : ContinuousStar R
inst✝²¹ : Field S
inst✝²⁰ : StarRing S
inst✝¹⁹ : MetricSpace S
inst✝¹⁸ : IsTopologicalRing S
inst✝¹⁷ : ContinuousStar S
inst✝¹⁶ : NonUnitalRing A
inst✝¹⁵ : StarRing A
inst✝¹⁴ : Module S A
inst✝¹³ : IsScalarTower S A A
inst✝¹² : SMulCommClass S A A
inst✝¹¹ : Algebra R S
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R S A
inst✝⁸ : StarModule R S
inst✝⁷ : ContinuousSMul R S
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalContinuousFunctionalCalculus S q
inst✝⁴ : CompleteSpace R
inst✝³ : IsScalarTower R A A
inst✝² : SMulCommClass R A A
inst✝¹ : NonUnitalContinuousFunctionalCalculus R p
inst✝ : UniqueHom R A
f : C(S, R)
halg : IsUniformEmbedding ⇑(algebraMap R S)
a : A
hpa : p a
hqa : q a
h : QuasispectrumRestricts a ⇑f
⊢ cfcₙHom hpa = nonUnitalStarAlgHom (cfcₙHom hqa) h

Tactic:
apply cfcₙHom_eq_of_continuous_of_map_id

Result:
case hφ₁
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²⁶ : Semifield R
inst✝²⁵ : StarRing R
inst✝²⁴ : MetricSpace R
inst✝²³ : IsTopologicalSemiring R
inst✝²² : ContinuousStar R
inst✝²¹ : Field S
inst✝²⁰ : StarRing S
inst✝¹⁹ : MetricSpace S
inst✝¹⁸ : IsTopologicalRing S
inst✝¹⁷ : ContinuousStar S
inst✝¹⁶ : NonUnitalRing A
inst✝¹⁵ : StarRing A
inst✝¹⁴ : Module S A
inst✝¹³ : IsScalarTower S A A
inst✝¹² : SMulCommClass S A A
inst✝¹¹ : Algebra R S
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R S A
inst✝⁸ : StarModule R S
inst✝⁷ : ContinuousSMul R S
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalContinuousFunctionalCalculus S q
inst✝⁴ : CompleteSpace R
inst✝³ : IsScalarTower R A A
inst✝² : SMulCommClass R A A
inst✝¹ : NonUnitalContinuousFunctionalCalculus R p
inst✝ : UniqueHom R A
f : C(S, R)
halg : IsUniformEmbedding ⇑(algebraMap R S)
a : A
hpa : p a
hqa : q a
h : QuasispectrumRestricts a ⇑f
⊢ Continuous ⇑(nonUnitalStarAlgHom (cfcₙHom hqa) h)

case hφ₂
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²⁶ : Semifield R
inst✝²⁵ : StarRing R
inst✝²⁴ : MetricSpace R
inst✝²³ : IsTopologicalSemiring R
inst✝²² : ContinuousStar R
inst✝²¹ : Field S
inst✝²⁰ : StarRing S
inst✝¹⁹ : MetricSpace S
inst✝¹⁸ : IsTopologicalRing S
inst✝¹⁷ : ContinuousStar S
inst✝¹⁶ : NonUnitalRing A
inst✝¹⁵ : StarRing A
inst✝¹⁴ : Module S A
inst✝¹³ : IsScalarTower S A A
inst✝¹² : SMulCommClass S A A
inst✝¹¹ : Algebra R S
inst✝¹⁰ : Module R A
inst✝⁹ : IsScalarTower R S A
inst✝⁸ : StarModule R S
inst✝⁷ : ContinuousSMul R S
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalContinuousFunctionalCalculus S q
inst✝⁴ : CompleteSpace R
inst✝³ : IsScalarTower R A A
inst✝² : SMulCommClass R A A
inst✝¹ : NonUnitalContinuousFunctionalCalculus R p
inst✝ : UniqueHom R A
f : C(S, R)
halg : IsUniformEmbedding ⇑(algebraMap R S)
a : A
hpa : p a
hqa : q a
h : QuasispectrumRestricts a ⇑f
⊢ (nonUnitalStarAlgHom (cfcₙHom hqa) h)
      { toContinuousMap := ContinuousMap.restrict (σₙ R a) (ContinuousMap.id R), map_zero' := ⋯ } =
    a