Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Restrict

Goal state before:
pp: case neg.inl
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²⁵ : Semifield R
inst✝²⁴ : StarRing R
inst✝²³ : MetricSpace R
inst✝²² : IsTopologicalSemiring R
inst✝²¹ : ContinuousStar R
inst✝²⁰ : Field S
inst✝¹⁹ : StarRing S
inst✝¹⁸ : MetricSpace S
inst✝¹⁷ : IsTopologicalRing S
inst✝¹⁶ : ContinuousStar S
inst✝¹⁵ : NonUnitalRing A
inst✝¹⁴ : StarRing A
inst✝¹³ : Module S A
inst✝¹² : IsScalarTower S A A
inst✝¹¹ : SMulCommClass S A A
inst✝¹⁰ : Algebra R S
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R S A
inst✝⁷ : StarModule R S
inst✝⁶ : ContinuousSMul R S
inst✝⁵ : TopologicalSpace A
inst✝⁴ : NonUnitalContinuousFunctionalCalculus S A q
inst✝³ : IsScalarTower R A A
inst✝² : SMulCommClass R A A
inst✝¹ : NonUnitalContinuousFunctionalCalculus R A p
inst✝ : UniqueHom R A
f : C(S, R)
halg : IsClosedEmbedding ⇑(algebraMap R S)
a : A
hpa : p a
hqa : q a
h : QuasispectrumRestricts a ⇑f
g : R → R
hg : ¬ContinuousOn g (σₙ R a)
⊢ cfcₙ g a = cfcₙ (fun x ↦ (algebraMap R S) (g (f x))) a

pp: case neg.inr
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²⁵ : Semifield R
inst✝²⁴ : StarRing R
inst✝²³ : MetricSpace R
inst✝²² : IsTopologicalSemiring R
inst✝²¹ : ContinuousStar R
inst✝²⁰ : Field S
inst✝¹⁹ : StarRing S
inst✝¹⁸ : MetricSpace S
inst✝¹⁷ : IsTopologicalRing S
inst✝¹⁶ : ContinuousStar S
inst✝¹⁵ : NonUnitalRing A
inst✝¹⁴ : StarRing A
inst✝¹³ : Module S A
inst✝¹² : IsScalarTower S A A
inst✝¹¹ : SMulCommClass S A A
inst✝¹⁰ : Algebra R S
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R S A
inst✝⁷ : StarModule R S
inst✝⁶ : ContinuousSMul R S
inst✝⁵ : TopologicalSpace A
inst✝⁴ : NonUnitalContinuousFunctionalCalculus S A q
inst✝³ : IsScalarTower R A A
inst✝² : SMulCommClass R A A
inst✝¹ : NonUnitalContinuousFunctionalCalculus R A p
inst✝ : UniqueHom R A
f : C(S, R)
halg : IsClosedEmbedding ⇑(algebraMap R S)
a : A
hpa : p a
hqa : q a
h : QuasispectrumRestricts a ⇑f
g : R → R
hg : ¬g 0 = 0
⊢ cfcₙ g a = cfcₙ (fun x ↦ (algebraMap R S) (g (f x))) a

Tactic:
· have : ¬ ContinuousOn (fun x ↦ algebraMap R S (g (f x)) : S → S) (σₙ S a) := by
        refine fun hg' ↦ hg ?_
        rw [halg.isEmbedding.continuousOn_iff]
        simpa [halg.isEmbedding.continuousOn_iff, Function.comp_def, h.left_inv _] using
          hg'.comp halg.isEmbedding.continuous.continuousOn
          (fun _ : R ↦ quasispectrum.algebraMap_mem S)
      rw [cfcₙ_apply_of_not_continuousOn a hg, cfcₙ_apply_of_not_continuousOn a this]

Goal state after:
case neg.inr
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²⁵ : Semifield R
inst✝²⁴ : StarRing R
inst✝²³ : MetricSpace R
inst✝²² : IsTopologicalSemiring R
inst✝²¹ : ContinuousStar R
inst✝²⁰ : Field S
inst✝¹⁹ : StarRing S
inst✝¹⁸ : MetricSpace S
inst✝¹⁷ : IsTopologicalRing S
inst✝¹⁶ : ContinuousStar S
inst✝¹⁵ : NonUnitalRing A
inst✝¹⁴ : StarRing A
inst✝¹³ : Module S A
inst✝¹² : IsScalarTower S A A
inst✝¹¹ : SMulCommClass S A A
inst✝¹⁰ : Algebra R S
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R S A
inst✝⁷ : StarModule R S
inst✝⁶ : ContinuousSMul R S
inst✝⁵ : TopologicalSpace A
inst✝⁴ : NonUnitalContinuousFunctionalCalculus S A q
inst✝³ : IsScalarTower R A A
inst✝² : SMulCommClass R A A
inst✝¹ : NonUnitalContinuousFunctionalCalculus R A p
inst✝ : UniqueHom R A
f : C(S, R)
halg : IsClosedEmbedding ⇑(algebraMap R S)
a : A
hpa : p a
hqa : q a
h : QuasispectrumRestricts a ⇑f
g : R → R
hg : ¬g 0 = 0
⊢ cfcₙ g a = cfcₙ (fun x ↦ (algebraMap R S) (g (f x))) a