Theorem: elementalStarAlgebra.bijective_characterSpaceToSpectrum
File: Mathlib/Analysis/NormedSpace/Star/ContinuousFunctionalCalculus.lean

tactic: refine ⟨fun φ ψ h => starAlgHomClass_ext ℂ ?_ ?_ ?_, ?_⟩
state_before: A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
⊢ Function.Bijective (characterSpaceToSpectrum a)
state_after: case refine_1
A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
φ ψ : ↑(characterSpace ℂ ↥(elementalStarAlgebra ℂ a))
h : characterSpaceToSpectrum a φ = characterSpaceToSpectrum a ψ
⊢ Continuous ⇑φ

case refine_2
A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
φ ψ : ↑(characterSpace ℂ ↥(elementalStarAlgebra ℂ a))
h : characterSpaceToSpectrum a φ = characterSpaceToSpectrum a ψ
⊢ Continuous ⇑ψ

case refine_3
A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
φ ψ : ↑(characterSpace ℂ ↥(elementalStarAlgebra ℂ a))
h : characterSpaceToSpectrum a φ = characterSpaceToSpectrum a ψ
⊢ φ ⟨a, ⋯⟩ = ψ ⟨a, ⋯⟩

case refine_4
A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
⊢ Function.Surjective (characterSpaceToSpectrum a)

tactic: exact (map_continuous φ)
state_before: case refine_1
A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
φ ψ : ↑(characterSpace ℂ ↥(elementalStarAlgebra ℂ a))
h : characterSpaceToSpectrum a φ = characterSpaceToSpectrum a ψ
⊢ Continuous ⇑φ
state_after: no goals

tactic: exact (map_continuous ψ)
state_before: case refine_2
A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
φ ψ : ↑(characterSpace ℂ ↥(elementalStarAlgebra ℂ a))
h : characterSpaceToSpectrum a φ = characterSpaceToSpectrum a ψ
⊢ Continuous ⇑ψ
state_after: no goals

tactic: simpa only [elementalStarAlgebra.characterSpaceToSpectrum, Subtype.mk_eq_mk,
  ContinuousMap.coe_mk] using h
state_before: case refine_3
A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
φ ψ : ↑(characterSpace ℂ ↥(elementalStarAlgebra ℂ a))
h : characterSpaceToSpectrum a φ = characterSpaceToSpectrum a ψ
⊢ φ ⟨a, ⋯⟩ = ψ ⟨a, ⋯⟩
state_after: no goals

tactic: rintro ⟨z, hz⟩
state_before: case refine_4
A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
⊢ Function.Surjective (characterSpaceToSpectrum a)
state_after: case refine_4.mk
A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
z : ℂ
hz : z ∈ spectrum ℂ a
⊢ ∃ a_1, characterSpaceToSpectrum a a_1 = ⟨z, hz⟩

tactic: have hz' := (StarSubalgebra.spectrum_eq (elementalStarAlgebra.isClosed ℂ a)
  ⟨a, self_mem ℂ a⟩).symm.subst hz
state_before: case refine_4.mk
A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
z : ℂ
hz : z ∈ spectrum ℂ a
⊢ ∃ a_1, characterSpaceToSpectrum a a_1 = ⟨z, hz⟩
state_after: case refine_4.mk
A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
z : ℂ
hz : z ∈ spectrum ℂ a
hz' : z ∈ spectrum ℂ ⟨a, ⋯⟩
⊢ ∃ a_1, characterSpaceToSpectrum a a_1 = ⟨z, hz⟩

tactic: rw [CharacterSpace.mem_spectrum_iff_exists] at hz'
state_before: case refine_4.mk
A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
z : ℂ
hz : z ∈ spectrum ℂ a
hz' : z ∈ spectrum ℂ ⟨a, ⋯⟩
⊢ ∃ a_1, characterSpaceToSpectrum a a_1 = ⟨z, hz⟩
state_after: case refine_4.mk
A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
z : ℂ
hz : z ∈ spectrum ℂ a
hz' : ∃ f, f ⟨a, ⋯⟩ = z
⊢ ∃ a_1, characterSpaceToSpectrum a a_1 = ⟨z, hz⟩

tactic: obtain ⟨φ, rfl⟩ := hz'
state_before: case refine_4.mk
A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
z : ℂ
hz : z ∈ spectrum ℂ a
hz' : ∃ f, f ⟨a, ⋯⟩ = z
⊢ ∃ a_1, characterSpaceToSpectrum a a_1 = ⟨z, hz⟩
state_after: case refine_4.mk.intro
A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
φ : ↑(characterSpace ℂ ↥(elementalStarAlgebra ℂ a))
hz : φ ⟨a, ⋯⟩ ∈ spectrum ℂ a
⊢ ∃ a_1, characterSpaceToSpectrum a a_1 = ⟨φ ⟨a, ⋯⟩, hz⟩

tactic: exact ⟨φ, rfl⟩
state_before: case refine_4.mk.intro
A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
φ : ↑(characterSpace ℂ ↥(elementalStarAlgebra ℂ a))
hz : φ ⟨a, ⋯⟩ ∈ spectrum ℂ a
⊢ ∃ a_1, characterSpaceToSpectrum a a_1 = ⟨φ ⟨a, ⋯⟩, hz⟩
state_after: no goals