Theorem: RCLike.uniqueContinuousFunctionalCalculus_of_compactSpace_spectrum
File: Mathlib/Topology/ContinuousFunction/UniqueCFC.lean

tactic: convert h using 1
state_before: 𝕜 : Type u_1
A : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : TopologicalSpace A
inst✝³ : T2Space A
inst✝² : Ring A
inst✝¹ : StarRing A
inst✝ : Algebra 𝕜 A
h✝ : ∀ (a : A), CompactSpace ↑(spectrum 𝕜 a)
s : Set 𝕜
hs : CompactSpace ↑s
φ ψ : C(↑s, 𝕜) →⋆ₐ[𝕜] A
hφ : Continuous ⇑φ
hψ : Continuous ⇑ψ
h : φ (ContinuousMap.restrict s (ContinuousMap.id 𝕜)) = ψ (ContinuousMap.restrict s (ContinuousMap.id 𝕜))
⊢ φ ((Polynomial.toContinuousMapOnAlgHom s) Polynomial.X) = ψ ((Polynomial.toContinuousMapOnAlgHom s) Polynomial.X)
state_after: case h.e'_2
𝕜 : Type u_1
A : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : TopologicalSpace A
inst✝³ : T2Space A
inst✝² : Ring A
inst✝¹ : StarRing A
inst✝ : Algebra 𝕜 A
h✝ : ∀ (a : A), CompactSpace ↑(spectrum 𝕜 a)
s : Set 𝕜
hs : CompactSpace ↑s
φ ψ : C(↑s, 𝕜) →⋆ₐ[𝕜] A
hφ : Continuous ⇑φ
hψ : Continuous ⇑ψ
h : φ (ContinuousMap.restrict s (ContinuousMap.id 𝕜)) = ψ (ContinuousMap.restrict s (ContinuousMap.id 𝕜))
⊢ φ ((Polynomial.toContinuousMapOnAlgHom s) Polynomial.X) = φ (ContinuousMap.restrict s (ContinuousMap.id 𝕜))

case h.e'_3
𝕜 : Type u_1
A : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : TopologicalSpace A
inst✝³ : T2Space A
inst✝² : Ring A
inst✝¹ : StarRing A
inst✝ : Algebra 𝕜 A
h✝ : ∀ (a : A), CompactSpace ↑(spectrum 𝕜 a)
s : Set 𝕜
hs : CompactSpace ↑s
φ ψ : C(↑s, 𝕜) →⋆ₐ[𝕜] A
hφ : Continuous ⇑φ
hψ : Continuous ⇑ψ
h : φ (ContinuousMap.restrict s (ContinuousMap.id 𝕜)) = ψ (ContinuousMap.restrict s (ContinuousMap.id 𝕜))
⊢ ψ ((Polynomial.toContinuousMapOnAlgHom s) Polynomial.X) = ψ (ContinuousMap.restrict s (ContinuousMap.id 𝕜))

tactic: all_goals exact congr_arg _ (by ext; simp)
state_before: case h.e'_2
𝕜 : Type u_1
A : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : TopologicalSpace A
inst✝³ : T2Space A
inst✝² : Ring A
inst✝¹ : StarRing A
inst✝ : Algebra 𝕜 A
h✝ : ∀ (a : A), CompactSpace ↑(spectrum 𝕜 a)
s : Set 𝕜
hs : CompactSpace ↑s
φ ψ : C(↑s, 𝕜) →⋆ₐ[𝕜] A
hφ : Continuous ⇑φ
hψ : Continuous ⇑ψ
h : φ (ContinuousMap.restrict s (ContinuousMap.id 𝕜)) = ψ (ContinuousMap.restrict s (ContinuousMap.id 𝕜))
⊢ φ ((Polynomial.toContinuousMapOnAlgHom s) Polynomial.X) = φ (ContinuousMap.restrict s (ContinuousMap.id 𝕜))

case h.e'_3
𝕜 : Type u_1
A : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : TopologicalSpace A
inst✝³ : T2Space A
inst✝² : Ring A
inst✝¹ : StarRing A
inst✝ : Algebra 𝕜 A
h✝ : ∀ (a : A), CompactSpace ↑(spectrum 𝕜 a)
s : Set 𝕜
hs : CompactSpace ↑s
φ ψ : C(↑s, 𝕜) →⋆ₐ[𝕜] A
hφ : Continuous ⇑φ
hψ : Continuous ⇑ψ
h : φ (ContinuousMap.restrict s (ContinuousMap.id 𝕜)) = ψ (ContinuousMap.restrict s (ContinuousMap.id 𝕜))
⊢ ψ ((Polynomial.toContinuousMapOnAlgHom s) Polynomial.X) = ψ (ContinuousMap.restrict s (ContinuousMap.id 𝕜))
state_after: no goals

tactic: exact congr_arg _ (by ext; simp)
state_before: case h.e'_3
𝕜 : Type u_1
A : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : TopologicalSpace A
inst✝³ : T2Space A
inst✝² : Ring A
inst✝¹ : StarRing A
inst✝ : Algebra 𝕜 A
h✝ : ∀ (a : A), CompactSpace ↑(spectrum 𝕜 a)
s : Set 𝕜
hs : CompactSpace ↑s
φ ψ : C(↑s, 𝕜) →⋆ₐ[𝕜] A
hφ : Continuous ⇑φ
hψ : Continuous ⇑ψ
h : φ (ContinuousMap.restrict s (ContinuousMap.id 𝕜)) = ψ (ContinuousMap.restrict s (ContinuousMap.id 𝕜))
⊢ ψ ((Polynomial.toContinuousMapOnAlgHom s) Polynomial.X) = ψ (ContinuousMap.restrict s (ContinuousMap.id 𝕜))
state_after: no goals

tactic: ext
state_before: 𝕜 : Type u_1
A : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : TopologicalSpace A
inst✝³ : T2Space A
inst✝² : Ring A
inst✝¹ : StarRing A
inst✝ : Algebra 𝕜 A
h✝ : ∀ (a : A), CompactSpace ↑(spectrum 𝕜 a)
s : Set 𝕜
hs : CompactSpace ↑s
φ ψ : C(↑s, 𝕜) →⋆ₐ[𝕜] A
hφ : Continuous ⇑φ
hψ : Continuous ⇑ψ
h : φ (ContinuousMap.restrict s (ContinuousMap.id 𝕜)) = ψ (ContinuousMap.restrict s (ContinuousMap.id 𝕜))
⊢ (Polynomial.toContinuousMapOnAlgHom s) Polynomial.X = ContinuousMap.restrict s (ContinuousMap.id 𝕜)
state_after: case h
𝕜 : Type u_1
A : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : TopologicalSpace A
inst✝³ : T2Space A
inst✝² : Ring A
inst✝¹ : StarRing A
inst✝ : Algebra 𝕜 A
h✝ : ∀ (a : A), CompactSpace ↑(spectrum 𝕜 a)
s : Set 𝕜
hs : CompactSpace ↑s
φ ψ : C(↑s, 𝕜) →⋆ₐ[𝕜] A
hφ : Continuous ⇑φ
hψ : Continuous ⇑ψ
h : φ (ContinuousMap.restrict s (ContinuousMap.id 𝕜)) = ψ (ContinuousMap.restrict s (ContinuousMap.id 𝕜))
a✝ : ↑s
⊢ ((Polynomial.toContinuousMapOnAlgHom s) Polynomial.X) a✝ = (ContinuousMap.restrict s (ContinuousMap.id 𝕜)) a✝

tactic: simp
state_before: case h
𝕜 : Type u_1
A : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : TopologicalSpace A
inst✝³ : T2Space A
inst✝² : Ring A
inst✝¹ : StarRing A
inst✝ : Algebra 𝕜 A
h✝ : ∀ (a : A), CompactSpace ↑(spectrum 𝕜 a)
s : Set 𝕜
hs : CompactSpace ↑s
φ ψ : C(↑s, 𝕜) →⋆ₐ[𝕜] A
hφ : Continuous ⇑φ
hψ : Continuous ⇑ψ
h : φ (ContinuousMap.restrict s (ContinuousMap.id 𝕜)) = ψ (ContinuousMap.restrict s (ContinuousMap.id 𝕜))
a✝ : ↑s
⊢ ((Polynomial.toContinuousMapOnAlgHom s) Polynomial.X) a✝ = (ContinuousMap.restrict s (ContinuousMap.id 𝕜)) a✝
state_after: no goals