-- Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Instances.lean
theorem RCLike.nonUnitalContinuousFunctionalCalculus :
    NonUnitalContinuousFunctionalCalculus 𝕜 (p : A → Prop) where
  predicate_zero

Proof state:
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 : ContinuousFunctionalCalculus 𝕜 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 : ContinuousFunctionalCalculus 𝕜 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 : ContinuousFunctionalCalculus 𝕜 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 map_id => exact inr_injective (R := 𝕜) <| coe_ψ _ ▸ cfcₙAux_id hp₁ a ha

Result:
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 : ContinuousFunctionalCalculus 𝕜 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 : ContinuousFunctionalCalculus 𝕜 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)

Proof state:
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 : ContinuousFunctionalCalculus 𝕜 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
⊢ IsClosedEmbedding ⇑ψ

Tactic:
apply isometry_inr (𝕜 := 𝕜) (A := A) |>.isClosedEmbedding |>.of_comp_iff.mp

Result:
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 : ContinuousFunctionalCalculus 𝕜 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
⊢ IsClosedEmbedding (Unitization.inr ∘ ⇑ψ)

Proof state:
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 : ContinuousFunctionalCalculus 𝕜 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:
exact hp₁.mp <| coe_ψ _ ▸ cfcHom_predicate (R := 𝕜) (hp₁.mpr ha) _

Result:
no goals

Proof state:
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 : ContinuousFunctionalCalculus 𝕜 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 isStarNormal => exact hp₁.mp <| coe_ψ _ ▸ cfcHom_predicate (R := 𝕜) (hp₁.mpr ha) _

Result:
no goals

Proof state:
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 : ContinuousFunctionalCalculus 𝕜 p₁
inst✝¹ : CompleteSpace A
inst✝ : CStarRing A
⊢ p 0

Tactic:
rw [← hp₁, Unitization.inr_zero 𝕜]

Result:
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 : ContinuousFunctionalCalculus 𝕜 p₁
inst✝¹ : CompleteSpace A
inst✝ : CStarRing A
⊢ p₁ 0