Theorem: cfcₙHom_eq_cfcₙHom_of_cfcHom
File: Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/NonUnital.lean

Proof steps:
state_before: R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : Field R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : CompleteSpace R
inst✝⁷ : TopologicalRing R
inst✝⁶ : ContinuousStar R
inst✝⁵ : Ring A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Algebra R A
inst✝¹ : ContinuousFunctionalCalculus R p
h_cpct : ∀ (a : A), CompactSpace ↑(spectrum R a)
inst✝ : UniqueNonUnitalContinuousFunctionalCalculus R A
a : A
ha : p a
⊢ cfcₙHom ha = cfcₙHom_of_cfcHom R ha
state_after: R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : Field R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : CompleteSpace R
inst✝⁷ : TopologicalRing R
inst✝⁶ : ContinuousStar R
inst✝⁵ : Ring A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Algebra R A
inst✝¹ : ContinuousFunctionalCalculus R p
h_cpct : ∀ (a : A), CompactSpace ↑(spectrum R a)
inst✝ : UniqueNonUnitalContinuousFunctionalCalculus R A
a : A
ha : p a
h_cpct' : CompactSpace ↑(σₙ R a)
⊢ cfcₙHom ha = cfcₙHom_of_cfcHom R ha
tactic: have h_cpct' : CompactSpace (σₙ R a) := by
    specialize h_cpct a
    simp_rw [← isCompact_iff_compactSpace, quasispectrum_eq_spectrum_union_zero] at h_cpct ⊢
    exact h_cpct.union isCompact_singleton
state_before: R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : Field R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : CompleteSpace R
inst✝⁷ : TopologicalRing R
inst✝⁶ : ContinuousStar R
inst✝⁵ : Ring A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Algebra R A
inst✝¹ : ContinuousFunctionalCalculus R p
h_cpct : ∀ (a : A), CompactSpace ↑(spectrum R a)
inst✝ : UniqueNonUnitalContinuousFunctionalCalculus R A
a : A
ha : p a
h_cpct' : CompactSpace ↑(σₙ R a)
⊢ cfcₙHom ha = cfcₙHom_of_cfcHom R ha
state_after: case refine_1
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : Field R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : CompleteSpace R
inst✝⁷ : TopologicalRing R
inst✝⁶ : ContinuousStar R
inst✝⁵ : Ring A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Algebra R A
inst✝¹ : ContinuousFunctionalCalculus R p
h_cpct : ∀ (a : A), CompactSpace ↑(spectrum R a)
inst✝ : UniqueNonUnitalContinuousFunctionalCalculus R A
a : A
ha : p a
h_cpct' : CompactSpace ↑(σₙ R a)
⊢ ↑0 = 0

case refine_2
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : Field R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : CompleteSpace R
inst✝⁷ : TopologicalRing R
inst✝⁶ : ContinuousStar R
inst✝⁵ : Ring A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Algebra R A
inst✝¹ : ContinuousFunctionalCalculus R p
h_cpct : ∀ (a : A), CompactSpace ↑(spectrum R a)
inst✝ : UniqueNonUnitalContinuousFunctionalCalculus R A
a : A
ha : p a
h_cpct' : CompactSpace ↑(σₙ R a)
⊢ Continuous ⇑(cfcₙHom ha)

case refine_3
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : Field R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : CompleteSpace R
inst✝⁷ : TopologicalRing R
inst✝⁶ : ContinuousStar R
inst✝⁵ : Ring A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Algebra R A
inst✝¹ : ContinuousFunctionalCalculus R p
h_cpct : ∀ (a : A), CompactSpace ↑(spectrum R a)
inst✝ : UniqueNonUnitalContinuousFunctionalCalculus R A
a : A
ha : p a
h_cpct' : CompactSpace ↑(σₙ R a)
⊢ Continuous ⇑(cfcₙHom_of_cfcHom R ha)

case refine_4
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : Field R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : CompleteSpace R
inst✝⁷ : TopologicalRing R
inst✝⁶ : ContinuousStar R
inst✝⁵ : Ring A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Algebra R A
inst✝¹ : ContinuousFunctionalCalculus R p
h_cpct : ∀ (a : A), CompactSpace ↑(spectrum R a)
inst✝ : UniqueNonUnitalContinuousFunctionalCalculus R A
a : A
ha : p a
h_cpct' : CompactSpace ↑(σₙ R a)
⊢ (cfcₙHom ha) { toContinuousMap := ContinuousMap.restrict (σₙ R a) (ContinuousMap.id R), map_zero' := ?refine_1 } =
    (cfcₙHom_of_cfcHom R ha)
      { toContinuousMap := ContinuousMap.restrict (σₙ R a) (ContinuousMap.id R), map_zero' := ?refine_1 }
tactic: refine UniqueNonUnitalContinuousFunctionalCalculus.eq_of_continuous_of_map_id
      (σₙ R a) ?_ _ _ ?_ ?_ ?_