Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances

Goal state before:
pp: case continuous
𝕜 : Type u_1
A : Type u_2
inst✝⁹ : RCLike 𝕜
inst✝⁸ : NonUnitalNormedRing A
inst✝⁷ : StarRing A
inst✝⁶ : NormedSpace 𝕜 A
inst✝⁵ : IsScalarTower 𝕜 A A
inst✝⁴ : SMulCommClass 𝕜 A A
inst✝³ : StarModule 𝕜 A
p : A → Prop
p₁ : Unitization 𝕜 A → Prop
hp₁ : ∀ {x : A}, p₁ ↑x ↔ p x
inst✝² : ClosedEmbeddingContinuousFunctionalCalculus 𝕜 (Unitization 𝕜 A) p₁
inst✝¹ : CompleteSpace A
inst✝ : CStarRing A
a : A
ha : p a
ψ : C(↑(σₙ 𝕜 a), 𝕜)₀ →⋆ₙₐ[𝕜] A :=
  (↑(inrRangeEquiv 𝕜 A).symm).comp
    (codRestrict (cfcₙAux ⋯ a ha) (NonUnitalStarAlgHom.range (inrNonUnitalStarAlgHom 𝕜 A)) ⋯)
coe_ψ : ∀ (f : C(↑(σₙ 𝕜 a), 𝕜)₀), ↑(ψ f) = (cfcₙAux ⋯ a ha) f
⊢ Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ⇑ψ

pp: case injective
𝕜 : Type u_1
A : Type u_2
inst✝⁹ : RCLike 𝕜
inst✝⁸ : NonUnitalNormedRing A
inst✝⁷ : StarRing A
inst✝⁶ : NormedSpace 𝕜 A
inst✝⁵ : IsScalarTower 𝕜 A A
inst✝⁴ : SMulCommClass 𝕜 A A
inst✝³ : StarModule 𝕜 A
p : A → Prop
p₁ : Unitization 𝕜 A → Prop
hp₁ : ∀ {x : A}, p₁ ↑x ↔ p x
inst✝² : ClosedEmbeddingContinuousFunctionalCalculus 𝕜 (Unitization 𝕜 A) p₁
inst✝¹ : CompleteSpace A
inst✝ : CStarRing A
a : A
ha : p a
ψ : C(↑(σₙ 𝕜 a), 𝕜)₀ →⋆ₙₐ[𝕜] A :=
  (↑(inrRangeEquiv 𝕜 A).symm).comp
    (codRestrict (cfcₙAux ⋯ a ha) (NonUnitalStarAlgHom.range (inrNonUnitalStarAlgHom 𝕜 A)) ⋯)
coe_ψ : ∀ (f : C(↑(σₙ 𝕜 a), 𝕜)₀), ↑(ψ f) = (cfcₙAux ⋯ a ha) f
⊢ Function.Injective ⇑ψ

pp: case map_id
𝕜 : Type u_1
A : Type u_2
inst✝⁹ : RCLike 𝕜
inst✝⁸ : NonUnitalNormedRing A
inst✝⁷ : StarRing A
inst✝⁶ : NormedSpace 𝕜 A
inst✝⁵ : IsScalarTower 𝕜 A A
inst✝⁴ : SMulCommClass 𝕜 A A
inst✝³ : StarModule 𝕜 A
p : A → Prop
p₁ : Unitization 𝕜 A → Prop
hp₁ : ∀ {x : A}, p₁ ↑x ↔ p x
inst✝² : ClosedEmbeddingContinuousFunctionalCalculus 𝕜 (Unitization 𝕜 A) p₁
inst✝¹ : CompleteSpace A
inst✝ : CStarRing A
a : A
ha : p a
ψ : C(↑(σₙ 𝕜 a), 𝕜)₀ →⋆ₙₐ[𝕜] A :=
  (↑(inrRangeEquiv 𝕜 A).symm).comp
    (codRestrict (cfcₙAux ⋯ a ha) (NonUnitalStarAlgHom.range (inrNonUnitalStarAlgHom 𝕜 A)) ⋯)
coe_ψ : ∀ (f : C(↑(σₙ 𝕜 a), 𝕜)₀), ↑(ψ f) = (cfcₙAux ⋯ a ha) f
⊢ ψ { toContinuousMap := ContinuousMap.restrict (σₙ 𝕜 a) (ContinuousMap.id 𝕜), map_zero' := ⋯ } = a

pp: case map_spec
𝕜 : Type u_1
A : Type u_2
inst✝⁹ : RCLike 𝕜
inst✝⁸ : NonUnitalNormedRing A
inst✝⁷ : StarRing A
inst✝⁶ : NormedSpace 𝕜 A
inst✝⁵ : IsScalarTower 𝕜 A A
inst✝⁴ : SMulCommClass 𝕜 A A
inst✝³ : StarModule 𝕜 A
p : A → Prop
p₁ : Unitization 𝕜 A → Prop
hp₁ : ∀ {x : A}, p₁ ↑x ↔ p x
inst✝² : ClosedEmbeddingContinuousFunctionalCalculus 𝕜 (Unitization 𝕜 A) p₁
inst✝¹ : CompleteSpace A
inst✝ : CStarRing A
a : A
ha : p a
ψ : C(↑(σₙ 𝕜 a), 𝕜)₀ →⋆ₙₐ[𝕜] A :=
  (↑(inrRangeEquiv 𝕜 A).symm).comp
    (codRestrict (cfcₙAux ⋯ a ha) (NonUnitalStarAlgHom.range (inrNonUnitalStarAlgHom 𝕜 A)) ⋯)
coe_ψ : ∀ (f : C(↑(σₙ 𝕜 a), 𝕜)₀), ↑(ψ f) = (cfcₙAux ⋯ a ha) f
f : C(↑(σₙ 𝕜 a), 𝕜)₀
⊢ σₙ 𝕜 (ψ f) = Set.range ⇑f

pp: case isStarNormal
𝕜 : Type u_1
A : Type u_2
inst✝⁹ : RCLike 𝕜
inst✝⁸ : NonUnitalNormedRing A
inst✝⁷ : StarRing A
inst✝⁶ : NormedSpace 𝕜 A
inst✝⁵ : IsScalarTower 𝕜 A A
inst✝⁴ : SMulCommClass 𝕜 A A
inst✝³ : StarModule 𝕜 A
p : A → Prop
p₁ : Unitization 𝕜 A → Prop
hp₁ : ∀ {x : A}, p₁ ↑x ↔ p x
inst✝² : ClosedEmbeddingContinuousFunctionalCalculus 𝕜 (Unitization 𝕜 A) p₁
inst✝¹ : CompleteSpace A
inst✝ : CStarRing A
a : A
ha : p a
ψ : C(↑(σₙ 𝕜 a), 𝕜)₀ →⋆ₙₐ[𝕜] A :=
  (↑(inrRangeEquiv 𝕜 A).symm).comp
    (codRestrict (cfcₙAux ⋯ a ha) (NonUnitalStarAlgHom.range (inrNonUnitalStarAlgHom 𝕜 A)) ⋯)
coe_ψ : ∀ (f : C(↑(σₙ 𝕜 a), 𝕜)₀), ↑(ψ f) = (cfcₙAux ⋯ a ha) f
f : C(↑(σₙ 𝕜 a), 𝕜)₀
⊢ p (ψ f)

Tactic:
case continuous =>
      rw [isometry_inr (𝕜 := 𝕜) |>.isEmbedding.continuous_iff]
      have := continuous_cfcₙAux hp₁ a ha
      simp only [coe_comp, NonUnitalStarAlgHom.coe_coe, Function.comp_def,
        inrRangeEquiv_symm_apply, coe_codRestrict, ψ]
      fun_prop

Goal state after:
case injective
𝕜 : Type u_1
A : Type u_2
inst✝⁹ : RCLike 𝕜
inst✝⁸ : NonUnitalNormedRing A
inst✝⁷ : StarRing A
inst✝⁶ : NormedSpace 𝕜 A
inst✝⁵ : IsScalarTower 𝕜 A A
inst✝⁴ : SMulCommClass 𝕜 A A
inst✝³ : StarModule 𝕜 A
p : A → Prop
p₁ : Unitization 𝕜 A → Prop
hp₁ : ∀ {x : A}, p₁ ↑x ↔ p x
inst✝² : ClosedEmbeddingContinuousFunctionalCalculus 𝕜 (Unitization 𝕜 A) p₁
inst✝¹ : CompleteSpace A
inst✝ : CStarRing A
a : A
ha : p a
ψ : C(↑(σₙ 𝕜 a), 𝕜)₀ →⋆ₙₐ[𝕜] A :=
  (↑(inrRangeEquiv 𝕜 A).symm).comp
    (codRestrict (cfcₙAux ⋯ a ha) (NonUnitalStarAlgHom.range (inrNonUnitalStarAlgHom 𝕜 A)) ⋯)
coe_ψ : ∀ (f : C(↑(σₙ 𝕜 a), 𝕜)₀), ↑(ψ f) = (cfcₙAux ⋯ a ha) f
⊢ Function.Injective ⇑ψ

case map_id
𝕜 : Type u_1
A : Type u_2
inst✝⁹ : RCLike 𝕜
inst✝⁸ : NonUnitalNormedRing A
inst✝⁷ : StarRing A
inst✝⁶ : NormedSpace 𝕜 A
inst✝⁵ : IsScalarTower 𝕜 A A
inst✝⁴ : SMulCommClass 𝕜 A A
inst✝³ : StarModule 𝕜 A
p : A → Prop
p₁ : Unitization 𝕜 A → Prop
hp₁ : ∀ {x : A}, p₁ ↑x ↔ p x
inst✝² : ClosedEmbeddingContinuousFunctionalCalculus 𝕜 (Unitization 𝕜 A) p₁
inst✝¹ : CompleteSpace A
inst✝ : CStarRing A
a : A
ha : p a
ψ : C(↑(σₙ 𝕜 a), 𝕜)₀ →⋆ₙₐ[𝕜] A :=
  (↑(inrRangeEquiv 𝕜 A).symm).comp
    (codRestrict (cfcₙAux ⋯ a ha) (NonUnitalStarAlgHom.range (inrNonUnitalStarAlgHom 𝕜 A)) ⋯)
coe_ψ : ∀ (f : C(↑(σₙ 𝕜 a), 𝕜)₀), ↑(ψ f) = (cfcₙAux ⋯ a ha) f
⊢ ψ { toContinuousMap := ContinuousMap.restrict (σₙ 𝕜 a) (ContinuousMap.id 𝕜), map_zero' := ⋯ } = a

case map_spec
𝕜 : Type u_1
A : Type u_2
inst✝⁹ : RCLike 𝕜
inst✝⁸ : NonUnitalNormedRing A
inst✝⁷ : StarRing A
inst✝⁶ : NormedSpace 𝕜 A
inst✝⁵ : IsScalarTower 𝕜 A A
inst✝⁴ : SMulCommClass 𝕜 A A
inst✝³ : StarModule 𝕜 A
p : A → Prop
p₁ : Unitization 𝕜 A → Prop
hp₁ : ∀ {x : A}, p₁ ↑x ↔ p x
inst✝² : ClosedEmbeddingContinuousFunctionalCalculus 𝕜 (Unitization 𝕜 A) p₁
inst✝¹ : CompleteSpace A
inst✝ : CStarRing A
a : A
ha : p a
ψ : C(↑(σₙ 𝕜 a), 𝕜)₀ →⋆ₙₐ[𝕜] A :=
  (↑(inrRangeEquiv 𝕜 A).symm).comp
    (codRestrict (cfcₙAux ⋯ a ha) (NonUnitalStarAlgHom.range (inrNonUnitalStarAlgHom 𝕜 A)) ⋯)
coe_ψ : ∀ (f : C(↑(σₙ 𝕜 a), 𝕜)₀), ↑(ψ f) = (cfcₙAux ⋯ a ha) f
f : C(↑(σₙ 𝕜 a), 𝕜)₀
⊢ σₙ 𝕜 (ψ f) = Set.range ⇑f

case isStarNormal
𝕜 : Type u_1
A : Type u_2
inst✝⁹ : RCLike 𝕜
inst✝⁸ : NonUnitalNormedRing A
inst✝⁷ : StarRing A
inst✝⁶ : NormedSpace 𝕜 A
inst✝⁵ : IsScalarTower 𝕜 A A
inst✝⁴ : SMulCommClass 𝕜 A A
inst✝³ : StarModule 𝕜 A
p : A → Prop
p₁ : Unitization 𝕜 A → Prop
hp₁ : ∀ {x : A}, p₁ ↑x ↔ p x
inst✝² : ClosedEmbeddingContinuousFunctionalCalculus 𝕜 (Unitization 𝕜 A) p₁
inst✝¹ : CompleteSpace A
inst✝ : CStarRing A
a : A
ha : p a
ψ : C(↑(σₙ 𝕜 a), 𝕜)₀ →⋆ₙₐ[𝕜] A :=
  (↑(inrRangeEquiv 𝕜 A).symm).comp
    (codRestrict (cfcₙAux ⋯ a ha) (NonUnitalStarAlgHom.range (inrNonUnitalStarAlgHom 𝕜 A)) ⋯)
coe_ψ : ∀ (f : C(↑(σₙ 𝕜 a), 𝕜)₀), ↑(ψ f) = (cfcₙAux ⋯ a ha) f
f : C(↑(σₙ 𝕜 a), 𝕜)₀
⊢ p (ψ f)