Theorem: QuasispectrumRestricts.closedEmbedding_nonUnitalStarAlgHom
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✝²² : Semifield R
inst✝²¹ : StarRing R
inst✝²⁰ : MetricSpace R
inst✝¹⁹ : TopologicalSemiring R
inst✝¹⁸ : ContinuousStar R
inst✝¹⁷ : Field S
inst✝¹⁶ : StarRing S
inst✝¹⁵ : MetricSpace S
inst✝¹⁴ : TopologicalRing S
inst✝¹³ : ContinuousStar S
inst✝¹² : TopologicalSpace A
inst✝¹¹ : NonUnitalRing A
inst✝¹⁰ : StarRing A
inst✝⁹ : Module S A
inst✝⁸ : IsScalarTower S A A
inst✝⁷ : SMulCommClass S A A
inst✝⁶ : NonUnitalContinuousFunctionalCalculus S q
inst✝⁵ : Algebra R S
inst✝⁴ : Module R A
inst✝³ : IsScalarTower R S A
inst✝² : StarModule R S
inst✝¹ : ContinuousSMul R S
inst✝ : CompleteSpace R
a : A
φ : C(↑(σₙ S a), S)₀ →⋆ₙₐ[S] A
hφ : ClosedEmbedding ⇑φ
f : C(S, R)
h : QuasispectrumRestricts a ⇑f
halg : UniformEmbedding ⇑(algebraMap R S)
h_cpct : CompactSpace ↑(σₙ S a)
⊢ ClosedEmbedding ⇑(nonUnitalStarAlgHom φ h)
state_after: R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²² : Semifield R
inst✝²¹ : StarRing R
inst✝²⁰ : MetricSpace R
inst✝¹⁹ : TopologicalSemiring R
inst✝¹⁸ : ContinuousStar R
inst✝¹⁷ : Field S
inst✝¹⁶ : StarRing S
inst✝¹⁵ : MetricSpace S
inst✝¹⁴ : TopologicalRing S
inst✝¹³ : ContinuousStar S
inst✝¹² : TopologicalSpace A
inst✝¹¹ : NonUnitalRing A
inst✝¹⁰ : StarRing A
inst✝⁹ : Module S A
inst✝⁸ : IsScalarTower S A A
inst✝⁷ : SMulCommClass S A A
inst✝⁶ : NonUnitalContinuousFunctionalCalculus S q
inst✝⁵ : Algebra R S
inst✝⁴ : Module R A
inst✝³ : IsScalarTower R S A
inst✝² : StarModule R S
inst✝¹ : ContinuousSMul R S
inst✝ : CompleteSpace R
a : A
φ : C(↑(σₙ S a), S)₀ →⋆ₙₐ[S] A
hφ : ClosedEmbedding ⇑φ
f : C(S, R)
h : QuasispectrumRestricts a ⇑f
halg : UniformEmbedding ⇑(algebraMap R S)
h_cpct : CompactSpace ↑(σₙ S a)
this : CompactSpace ↑(σₙ R a)
⊢ ClosedEmbedding ⇑(nonUnitalStarAlgHom φ h)
tactic: have := h.compactSpace
state_before: R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²² : Semifield R
inst✝²¹ : StarRing R
inst✝²⁰ : MetricSpace R
inst✝¹⁹ : TopologicalSemiring R
inst✝¹⁸ : ContinuousStar R
inst✝¹⁷ : Field S
inst✝¹⁶ : StarRing S
inst✝¹⁵ : MetricSpace S
inst✝¹⁴ : TopologicalRing S
inst✝¹³ : ContinuousStar S
inst✝¹² : TopologicalSpace A
inst✝¹¹ : NonUnitalRing A
inst✝¹⁰ : StarRing A
inst✝⁹ : Module S A
inst✝⁸ : IsScalarTower S A A
inst✝⁷ : SMulCommClass S A A
inst✝⁶ : NonUnitalContinuousFunctionalCalculus S q
inst✝⁵ : Algebra R S
inst✝⁴ : Module R A
inst✝³ : IsScalarTower R S A
inst✝² : StarModule R S
inst✝¹ : ContinuousSMul R S
inst✝ : CompleteSpace R
a : A
φ : C(↑(σₙ S a), S)₀ →⋆ₙₐ[S] A
hφ : ClosedEmbedding ⇑φ
f : C(S, R)
h : QuasispectrumRestricts a ⇑f
halg : UniformEmbedding ⇑(algebraMap R S)
h_cpct : CompactSpace ↑(σₙ S a)
this : CompactSpace ↑(σₙ R a)
⊢ ClosedEmbedding ⇑(nonUnitalStarAlgHom φ h)
state_after: R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²² : Semifield R
inst✝²¹ : StarRing R
inst✝²⁰ : MetricSpace R
inst✝¹⁹ : TopologicalSemiring R
inst✝¹⁸ : ContinuousStar R
inst✝¹⁷ : Field S
inst✝¹⁶ : StarRing S
inst✝¹⁵ : MetricSpace S
inst✝¹⁴ : TopologicalRing S
inst✝¹³ : ContinuousStar S
inst✝¹² : TopologicalSpace A
inst✝¹¹ : NonUnitalRing A
inst✝¹⁰ : StarRing A
inst✝⁹ : Module S A
inst✝⁸ : IsScalarTower S A A
inst✝⁷ : SMulCommClass S A A
inst✝⁶ : NonUnitalContinuousFunctionalCalculus S q
inst✝⁵ : Algebra R S
inst✝⁴ : Module R A
inst✝³ : IsScalarTower R S A
inst✝² : StarModule R S
inst✝¹ : ContinuousSMul R S
inst✝ : CompleteSpace R
a : A
φ : C(↑(σₙ S a), S)₀ →⋆ₙₐ[S] A
hφ : ClosedEmbedding ⇑φ
f : C(S, R)
h : QuasispectrumRestricts a ⇑f
halg : UniformEmbedding ⇑(algebraMap R S)
h_cpct : CompactSpace ↑(σₙ S a)
this✝ : CompactSpace ↑(σₙ R a)
this : h.homeomorph.symm 0 = 0
⊢ ClosedEmbedding ⇑(nonUnitalStarAlgHom φ h)
tactic: have : h.homeomorph.symm 0 = 0 := Subtype.ext (map_zero <| algebraMap _ _)
state_before: R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²² : Semifield R
inst✝²¹ : StarRing R
inst✝²⁰ : MetricSpace R
inst✝¹⁹ : TopologicalSemiring R
inst✝¹⁸ : ContinuousStar R
inst✝¹⁷ : Field S
inst✝¹⁶ : StarRing S
inst✝¹⁵ : MetricSpace S
inst✝¹⁴ : TopologicalRing S
inst✝¹³ : ContinuousStar S
inst✝¹² : TopologicalSpace A
inst✝¹¹ : NonUnitalRing A
inst✝¹⁰ : StarRing A
inst✝⁹ : Module S A
inst✝⁸ : IsScalarTower S A A
inst✝⁷ : SMulCommClass S A A
inst✝⁶ : NonUnitalContinuousFunctionalCalculus S q
inst✝⁵ : Algebra R S
inst✝⁴ : Module R A
inst✝³ : IsScalarTower R S A
inst✝² : StarModule R S
inst✝¹ : ContinuousSMul R S
inst✝ : CompleteSpace R
a : A
φ : C(↑(σₙ S a), S)₀ →⋆ₙₐ[S] A
hφ : ClosedEmbedding ⇑φ
f : C(S, R)
h : QuasispectrumRestricts a ⇑f
halg : UniformEmbedding ⇑(algebraMap R S)
h_cpct : CompactSpace ↑(σₙ S a)
this✝ : CompactSpace ↑(σₙ R a)
this : h.homeomorph.symm 0 = 0
⊢ ClosedEmbedding ⇑(nonUnitalStarAlgHom φ h)
state_after: no goals
tactic: refine hφ.comp <| UniformEmbedding.toClosedEmbedding <| .comp
    (ContinuousMapZero.uniformEmbedding_comp _ halg)
    (UniformEquiv.arrowCongrLeft₀ h.homeomorph.symm this |>.uniformEmbedding)