awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital
Goal:
R : Type u_1
A : Type u_2
p : A → Prop
inst : Semifield R
inst_1 : StarRing R
inst_2 : MetricSpace R
inst_3 : IsTopologicalSemiring R
inst_4 : ContinuousStar R
inst_5 : Ring A
inst_6 : StarRing A
inst_7 : TopologicalSpace A
inst_8 : Algebra R A
inst_9 : ClosedEmbeddingContinuousFunctionalCalculus R A p
inst_10 : CompleteSpace R
a : A
ha : p a
f : C((↑(spectrum R a) : Type u_1), (↑(σₙ R a) : Type u_1)) :=
  { toFun := Set.inclusion (s := spectrum R a) (t := σₙ R a) ⋯, continuous_toFun := ⋯ }
⊢ Eq (α := Set (↑(σₙ R a) : Type u_1))
    (range (ι := (↑(spectrum R a) : Type u_1)) (⇑f : (a_1 : (↑(spectrum R a) : Type u_1)) → (↑(σₙ R a) : Type u_1)) ∪
      range (ι := Unit) (⇑0 : (a_1 : Unit) → (↑(σₙ R a) : Type u_1)))
    univ

Tactic:
simp only [← Subtype.val_injective.image_injective.eq_iff, f, ContinuousMap.coe_mk,
        ContinuousMap.coe_zero, range_zero, image_union, image_singleton,
        quasispectrum.coe_zero, ← range_comp, val_comp_inclusion, image_univ, Subtype.range_coe,
        quasispectrum_eq_spectrum_union_zero]

Premises used: Set.range_comp, Set.image_univ, Set.val_comp_inclusion, Set.image_singleton, quasispectrum.coe_zero, Set.range_zero, Subtype.val_injective, Function.Injective.eq_iff, ContinuousMap.coe_zero, Subtype.range_coe, ContinuousMap.coe_mk, quasispectrum_eq_spectrum_union_zero, Function.Injective.image_injective, Set.image_union
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital
Goal:
R : Type u_1
A : Type u_2
p : A → Prop
inst : Semifield R
inst_1 : StarRing R
inst_2 : MetricSpace R
inst_3 : IsTopologicalSemiring R
inst_4 : ContinuousStar R
inst_5 : Ring A
inst_6 : StarRing A
inst_7 : TopologicalSpace A
inst_8 : Algebra R A
inst_9 : ClosedEmbeddingContinuousFunctionalCalculus R A p
inst_10 : CompleteSpace R
a : A
ha : p a
f : C((↑(spectrum R a) : Type u_1), (↑(σₙ R a) : Type u_1)) :=
  { toFun := Set.inclusion (s := spectrum R a) (t := σₙ R a) ⋯, continuous_toFun := ⋯ }
⊢ Eq.{u_1 + 1} (α := UniformSpace C((↑(σₙ R a) : Type u_1), R)) inferInstance
    (UniformSpace.comap (β := C((↑(spectrum R a) : Type u_1), R)) (fun (x : C((↑(σₙ R a) : Type u_1), R)) ↦ x.comp f)
        inferInstance ⊓
      UniformSpace.comap (β := C(Unit, R)) (fun (x : C((↑(σₙ R a) : Type u_1), R)) ↦ x.comp 0) inferInstance)

Tactic:
have := uniformSpace_eq_inf_precomp_of_cover (β := R) f (0 : C(Unit, σₙ R a))
    (map_continuous f).isProperMap (map_continuous 0).isProperMap <| by
      simp only [← Subtype.val_injective.image_injective.eq_iff, f, ContinuousMap.coe_mk,
        ContinuousMap.coe_zero, range_zero, image_union, image_singleton,
        quasispectrum.coe_zero, ← range_comp, val_comp_inclusion, image_univ, Subtype.range_coe,
        quasispectrum_eq_spectrum_union_zero]
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital
Goal:
R : Type u_1
A : Type u_2
p : A → Prop
inst : Semifield R
inst_1 : StarRing R
inst_2 : MetricSpace R
inst_3 : IsTopologicalSemiring R
inst_4 : ContinuousStar R
inst_5 : Ring A
inst_6 : StarRing A
inst_7 : TopologicalSpace A
inst_8 : Algebra R A
inst_9 : ClosedEmbeddingContinuousFunctionalCalculus R A p
inst_10 : CompleteSpace R
a : A
ha : p a
f : C((↑(spectrum R a) : Type u_1), (↑(σₙ R a) : Type u_1)) :=
  { toFun := Set.inclusion (s := spectrum R a) (t := σₙ R a) ⋯, continuous_toFun := ⋯ }
this :
  Eq.{u_1 + 1} (α := UniformSpace C((↑(σₙ R a) : Type u_1), R)) inferInstance
    (UniformSpace.comap (β := C((↑(spectrum R a) : Type u_1), R)) (fun (x : C((↑(σₙ R a) : Type u_1), R)) ↦ x.comp f)
        inferInstance ⊓
      UniformSpace.comap (β := C(Unit, R)) (fun (x : C((↑(σₙ R a) : Type u_1), R)) ↦ x.comp 0) inferInstance)
⊢ ∀ (U : Set (C(Unit, R) × C(Unit, R))),
    Membership.mem (γ := Filter (C(Unit, R) × C(Unit, R))) (𝓤 C(Unit, R)) U → (0, 0) ∈ U

Tactic:
have : ∀ U ∈ 𝓤 (C(Unit, R)), (0, 0) ∈ U := fun U hU ↦ refl_mem_uniformity hU
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital
Goal:
R : Type u_1
A : Type u_2
p : A → Prop
inst : Semifield R
inst_1 : StarRing R
inst_2 : MetricSpace R
inst_3 : IsTopologicalSemiring R
inst_4 : ContinuousStar R
inst_5 : Ring A
inst_6 : StarRing A
inst_7 : TopologicalSpace A
inst_8 : Algebra R A
inst_9 : ClosedEmbeddingContinuousFunctionalCalculus R A p
inst_10 : CompleteSpace R
a : A
ha : p a
f : C((↑(spectrum R a) : Type u_1), (↑(σₙ R a) : Type u_1)) :=
  { toFun := Set.inclusion (s := spectrum R a) (t := σₙ R a) ⋯, continuous_toFun := ⋯ }
this_1 :
  Eq.{u_1 + 1} (α := UniformSpace C((↑(σₙ R a) : Type u_1), R)) inferInstance
    (UniformSpace.comap (β := C((↑(spectrum R a) : Type u_1), R)) (fun (x : C((↑(σₙ R a) : Type u_1), R)) ↦ x.comp f)
        inferInstance ⊓
      UniformSpace.comap (β := C(Unit, R)) (fun (x : C((↑(σₙ R a) : Type u_1), R)) ↦ x.comp 0) inferInstance)
this :
  ∀ (U : Set (C(Unit, R) × C(Unit, R))),
    Membership.mem (γ := Filter (C(Unit, R) × C(Unit, R))) (𝓤 C(Unit, R)) U → (0, 0) ∈ U
⊢ Eq.{u_1 + 1} (α := Filter (C((↑(σₙ R a) : Type u_1), R)₀ × C((↑(σₙ R a) : Type u_1), R)₀))
    (Filter.comap (β := C(Unit, R) × C(Unit, R))
      (fun (x : C((↑(σₙ R a) : Type u_1), R)₀ × C((↑(σₙ R a) : Type u_1), R)₀) ↦ (0, 0)) (𝓤 C(Unit, R)))
    ⊤

Tactic:
Filter.comap_const_of_mem this

Premises used: Filter.comap_const_of_mem
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital
Goal:
R : Type u_1
A : Type u_2
p : A → Prop
inst : Semifield R
inst_1 : StarRing R
inst_2 : MetricSpace R
inst_3 : IsTopologicalSemiring R
inst_4 : ContinuousStar R
inst_5 : Ring A
inst_6 : StarRing A
inst_7 : TopologicalSpace A
inst_8 : Algebra R A
inst_9 : ClosedEmbeddingContinuousFunctionalCalculus R A p
inst_10 : CompleteSpace R
a : A
ha : p a
f : C((↑(spectrum R a) : Type u_1), (↑(σₙ R a) : Type u_1)) :=
  { toFun := Set.inclusion (s := spectrum R a) (t := σₙ R a) ⋯, continuous_toFun := ⋯ }
this_1 :
  Eq.{u_1 + 1} (α := UniformSpace C((↑(σₙ R a) : Type u_1), R)) inferInstance
    (UniformSpace.comap (β := C((↑(spectrum R a) : Type u_1), R)) (fun (x : C((↑(σₙ R a) : Type u_1), R)) ↦ x.comp f)
        inferInstance ⊓
      UniformSpace.comap (β := C(Unit, R)) (fun (x : C((↑(σₙ R a) : Type u_1), R)) ↦ x.comp 0) inferInstance)
this :
  ∀ (U : Set (C(Unit, R) × C(Unit, R))),
    Membership.mem (γ := Filter (C(Unit, R) × C(Unit, R))) (𝓤 C(Unit, R)) U → (0, 0) ∈ U
h :
  ∀ (u v : C((↑(σₙ R a) : Type u_1), R)₀),
    Eq (α := C(Unit, R))
      (Function.comp (β := C((↑(σₙ R a) : Type u_1), R) × C((↑(σₙ R a) : Type u_1), R))
          (Prod.map (fun (x : C((↑(σₙ R a) : Type u_1), R)) ↦ x.comp 0) fun (x : C((↑(σₙ R a) : Type u_1), R)) ↦
            x.comp 0)
          (fun (x : C((↑(σₙ R a) : Type u_1), R)₀ × C((↑(σₙ R a) : Type u_1), R)₀) ↦
            ((↑x.1 : C((↑(σₙ R a) : Type u_1), R)), (↑x.2 : C((↑(σₙ R a) : Type u_1), R))))
          (u, v)).1
      0
h₁ :
  ∀ (u v : C((↑(σₙ R a) : Type u_1), R)₀),
    Eq (α := C(Unit, R))
      (Function.comp (β := C((↑(σₙ R a) : Type u_1), R) × C((↑(σₙ R a) : Type u_1), R))
          (Prod.map (fun (x : C((↑(σₙ R a) : Type u_1), R)) ↦ x.comp 0) fun (x : C((↑(σₙ R a) : Type u_1), R)) ↦
            x.comp 0)
          (fun (x : C((↑(σₙ R a) : Type u_1), R)₀ × C((↑(σₙ R a) : Type u_1), R)₀) ↦
            ((↑x.1 : C((↑(σₙ R a) : Type u_1), R)), (↑x.2 : C((↑(σₙ R a) : Type u_1), R))))
          (u, v)).2
      0
⊢ Eq.{u_1 + 1} (α := Filter (C((↑(σₙ R a) : Type u_1), R)₀ × C((↑(σₙ R a) : Type u_1), R)₀))
    (Filter.comap (β := C(Unit, R) × C(Unit, R))
      (Function.comp (β := C((↑(σₙ R a) : Type u_1), R) × C((↑(σₙ R a) : Type u_1), R))
        (Prod.map (fun (x : C((↑(σₙ R a) : Type u_1), R)) ↦ x.comp 0) fun (x : C((↑(σₙ R a) : Type u_1), R)) ↦ x.comp 0)
        fun (x : C((↑(σₙ R a) : Type u_1), R)₀ × C((↑(σₙ R a) : Type u_1), R)₀) ↦
        ((↑x.1 : C((↑(σₙ R a) : Type u_1), R)), (↑x.2 : C((↑(σₙ R a) : Type u_1), R))))
      (𝓤 C(Unit, R)))
    ⊤

Tactic:
convert! Filter.comap_const_of_mem this with ⟨u, v⟩

Premises used: Filter.comap_const_of_mem
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital
Goal:
R : Type u_1
A : Type u_2
p : A → Prop
inst : Semifield R
inst_1 : StarRing R
inst_2 : MetricSpace R
inst_3 : IsTopologicalSemiring R
inst_4 : ContinuousStar R
inst_5 : Ring A
inst_6 : StarRing A
inst_7 : TopologicalSpace A
inst_8 : Algebra R A
inst_9 : ClosedEmbeddingContinuousFunctionalCalculus R A p
inst_10 : CompleteSpace R
a : A
ha : p a
f : C((↑(spectrum R a) : Type u_1), (↑(σₙ R a) : Type u_1)) :=
  { toFun := Set.inclusion (s := spectrum R a) (t := σₙ R a) ⋯, continuous_toFun := ⋯ }
this_1 :
  Eq.{u_1 + 1} (α := UniformSpace C((↑(σₙ R a) : Type u_1), R)) inferInstance
    (UniformSpace.comap (β := C((↑(spectrum R a) : Type u_1), R)) (fun (x : C((↑(σₙ R a) : Type u_1), R)) ↦ x.comp f)
        inferInstance ⊓
      UniformSpace.comap (β := C(Unit, R)) (fun (x : C((↑(σₙ R a) : Type u_1), R)) ↦ x.comp 0) inferInstance)
this :
  ∀ (U : Set (C(Unit, R) × C(Unit, R))),
    Membership.mem (γ := Filter (C(Unit, R) × C(Unit, R))) (𝓤 C(Unit, R)) U → (0, 0) ∈ U
u v : C((↑(σₙ R a) : Type u_1), R)₀
h :
  Eq (α := R)
    (((Function.comp (β := C((↑(σₙ R a) : Type u_1), R) × C((↑(σₙ R a) : Type u_1), R))
              (Prod.map (fun (x : C((↑(σₙ R a) : Type u_1), R)) ↦ x.comp 0) fun (x : C((↑(σₙ R a) : Type u_1), R)) ↦
                x.comp 0)
              (fun (x : C((↑(σₙ R a) : Type u_1), R)₀ × C((↑(σₙ R a) : Type u_1), R)₀) ↦
                ((↑x.1 : C((↑(σₙ R a) : Type u_1), R)), (↑x.2 : C((↑(σₙ R a) : Type u_1), R))))
              (u, v)).1 :
          (a : Unit) → R)
        PUnit.unit :
      R)
    ((0 : (a : Unit) → R) PUnit.unit : R)
⊢ Eq (α := C(Unit, R))
    (Function.comp (β := C((↑(σₙ R a) : Type u_1), R) × C((↑(σₙ R a) : Type u_1), R))
        (Prod.map (fun (x : C((↑(σₙ R a) : Type u_1), R)) ↦ x.comp 0) fun (x : C((↑(σₙ R a) : Type u_1), R)) ↦ x.comp 0)
        (fun (x : C((↑(σₙ R a) : Type u_1), R)₀ × C((↑(σₙ R a) : Type u_1), R)₀) ↦
          ((↑x.1 : C((↑(σₙ R a) : Type u_1), R)), (↑x.2 : C((↑(σₙ R a) : Type u_1), R))))
        (u, v)).1
    0

Tactic:
ext ⟨x, rfl⟩

Premises used: rfl
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital
Goal:
R : Type u_1
A : Type u_2
p : A → Prop
inst : Semifield R
inst_1 : StarRing R
inst_2 : MetricSpace R
inst_3 : IsTopologicalSemiring R
inst_4 : ContinuousStar R
inst_5 : Ring A
inst_6 : StarRing A
inst_7 : TopologicalSpace A
inst_8 : Algebra R A
inst_9 : ClosedEmbeddingContinuousFunctionalCalculus R A p
inst_10 : CompleteSpace R
a : A
ha : p a
f : C((↑(spectrum R a) : Type u_1), (↑(σₙ R a) : Type u_1)) :=
  { toFun := Set.inclusion (s := spectrum R a) (t := σₙ R a) ⋯, continuous_toFun := ⋯ }
this_1 :
  Eq.{u_1 + 1} (α := UniformSpace C((↑(σₙ R a) : Type u_1), R)) inferInstance
    (UniformSpace.comap (β := C((↑(spectrum R a) : Type u_1), R)) (fun (x : C((↑(σₙ R a) : Type u_1), R)) ↦ x.comp f)
        inferInstance ⊓
      UniformSpace.comap (β := C(Unit, R)) (fun (x : C((↑(σₙ R a) : Type u_1), R)) ↦ x.comp 0) inferInstance)
this :
  ∀ (U : Set (C(Unit, R) × C(Unit, R))),
    Membership.mem (γ := Filter (C(Unit, R) × C(Unit, R))) (𝓤 C(Unit, R)) U → (0, 0) ∈ U
u v : C((↑(σₙ R a) : Type u_1), R)₀
h :
  Eq (α := R)
    (((Function.comp (β := C((↑(σₙ R a) : Type u_1), R) × C((↑(σₙ R a) : Type u_1), R))
              (Prod.map (fun (x : C((↑(σₙ R a) : Type u_1), R)) ↦ x.comp 0) fun (x : C((↑(σₙ R a) : Type u_1), R)) ↦
                x.comp 0)
              (fun (x : C((↑(σₙ R a) : Type u_1), R)₀ × C((↑(σₙ R a) : Type u_1), R)₀) ↦
                ((↑x.1 : C((↑(σₙ R a) : Type u_1), R)), (↑x.2 : C((↑(σₙ R a) : Type u_1), R))))
              (u, v)).2 :
          (a : Unit) → R)
        PUnit.unit :
      R)
    ((0 : (a : Unit) → R) PUnit.unit : R)
⊢ Eq (α := C(Unit, R))
    (Function.comp (β := C((↑(σₙ R a) : Type u_1), R) × C((↑(σₙ R a) : Type u_1), R))
        (Prod.map (fun (x : C((↑(σₙ R a) : Type u_1), R)) ↦ x.comp 0) fun (x : C((↑(σₙ R a) : Type u_1), R)) ↦ x.comp 0)
        (fun (x : C((↑(σₙ R a) : Type u_1), R)₀ × C((↑(σₙ R a) : Type u_1), R)₀) ↦
          ((↑x.1 : C((↑(σₙ R a) : Type u_1), R)), (↑x.2 : C((↑(σₙ R a) : Type u_1), R))))
        (u, v)).2
    0

Tactic:
ext ⟨x, rfl⟩

Premises used: rfl
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital
Goal:
R : Type u_1
A : Type u_2
p : A → Prop
inst : Semifield R
inst_1 : StarRing R
inst_2 : MetricSpace R
inst_3 : IsTopologicalSemiring R
inst_4 : ContinuousStar R
inst_5 : Ring A
inst_6 : StarRing A
inst_7 : TopologicalSpace A
inst_8 : Algebra R A
inst_9 : ClosedEmbeddingContinuousFunctionalCalculus R A p
inst_10 : CompleteSpace R
a : A
ha : p a
f : C((↑(spectrum R a) : Type u_1), (↑(σₙ R a) : Type u_1)) :=
  { toFun := Set.inclusion (s := spectrum R a) (t := σₙ R a) ⋯, continuous_toFun := ⋯ }
this_1 :
  Eq.{u_1 + 1} (α := UniformSpace C((↑(σₙ R a) : Type u_1), R)) inferInstance
    (UniformSpace.comap (β := C((↑(spectrum R a) : Type u_1), R)) (fun (x : C((↑(σₙ R a) : Type u_1), R)) ↦ x.comp f)
        inferInstance ⊓
      UniformSpace.comap (β := C(Unit, R)) (fun (x : C((↑(σₙ R a) : Type u_1), R)) ↦ x.comp 0) inferInstance)
this :
  ∀ (U : Set (C(Unit, R) × C(Unit, R))),
    Membership.mem (γ := Filter (C(Unit, R) × C(Unit, R))) (𝓤 C(Unit, R)) U → (0, 0) ∈ U
u v : C((↑(σₙ R a) : Type u_1), R)₀
⊢ Eq (α := R)
    (((Function.comp (β := C((↑(σₙ R a) : Type u_1), R) × C((↑(σₙ R a) : Type u_1), R))
              (Prod.map (fun (x : C((↑(σₙ R a) : Type u_1), R)) ↦ x.comp 0) fun (x : C((↑(σₙ R a) : Type u_1), R)) ↦
                x.comp 0)
              (fun (x : C((↑(σₙ R a) : Type u_1), R)₀ × C((↑(σₙ R a) : Type u_1), R)₀) ↦
                ((↑x.1 : C((↑(σₙ R a) : Type u_1), R)), (↑x.2 : C((↑(σₙ R a) : Type u_1), R))))
              (u, v)).1 :
          (a : Unit) → R)
        PUnit.unit :
      R)
    ((0 : (a : Unit) → R) PUnit.unit : R)

Tactic:
map_zero u

Premises used: map_zero
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital
Goal:
R : Type u_1
A : Type u_2
p : A → Prop
inst : Semifield R
inst_1 : StarRing R
inst_2 : MetricSpace R
inst_3 : IsTopologicalSemiring R
inst_4 : ContinuousStar R
inst_5 : Ring A
inst_6 : StarRing A
inst_7 : TopologicalSpace A
inst_8 : Algebra R A
inst_9 : ClosedEmbeddingContinuousFunctionalCalculus R A p
inst_10 : CompleteSpace R
a : A
ha : p a
f : C((↑(spectrum R a) : Type u_1), (↑(σₙ R a) : Type u_1)) :=
  { toFun := Set.inclusion (s := spectrum R a) (t := σₙ R a) ⋯, continuous_toFun := ⋯ }
this_1 :
  Eq.{u_1 + 1} (α := UniformSpace C((↑(σₙ R a) : Type u_1), R)) inferInstance
    (UniformSpace.comap (β := C((↑(spectrum R a) : Type u_1), R)) (fun (x : C((↑(σₙ R a) : Type u_1), R)) ↦ x.comp f)
        inferInstance ⊓
      UniformSpace.comap (β := C(Unit, R)) (fun (x : C((↑(σₙ R a) : Type u_1), R)) ↦ x.comp 0) inferInstance)
this :
  ∀ (U : Set (C(Unit, R) × C(Unit, R))),
    Membership.mem (γ := Filter (C(Unit, R) × C(Unit, R))) (𝓤 C(Unit, R)) U → (0, 0) ∈ U
u v : C((↑(σₙ R a) : Type u_1), R)₀
⊢ Eq (α := R)
    (((Function.comp (β := C((↑(σₙ R a) : Type u_1), R) × C((↑(σₙ R a) : Type u_1), R))
              (Prod.map (fun (x : C((↑(σₙ R a) : Type u_1), R)) ↦ x.comp 0) fun (x : C((↑(σₙ R a) : Type u_1), R)) ↦
                x.comp 0)
              (fun (x : C((↑(σₙ R a) : Type u_1), R)₀ × C((↑(σₙ R a) : Type u_1), R)₀) ↦
                ((↑x.1 : C((↑(σₙ R a) : Type u_1), R)), (↑x.2 : C((↑(σₙ R a) : Type u_1), R))))
              (u, v)).1 :
          (a : Unit) → R)
        PUnit.unit :
      R)
    ((0 : (a : Unit) → R) PUnit.unit : R)

Tactic:
exact map_zero u

Premises used: map_zero
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital
Goal:
R : Type u_1
A : Type u_2
p : A → Prop
inst : Semifield R
inst_1 : StarRing R
inst_2 : MetricSpace R
inst_3 : IsTopologicalSemiring R
inst_4 : ContinuousStar R
inst_5 : Ring A
inst_6 : StarRing A
inst_7 : TopologicalSpace A
inst_8 : Algebra R A
inst_9 : ClosedEmbeddingContinuousFunctionalCalculus R A p
inst_10 : CompleteSpace R
a : A
ha : p a
f : C((↑(spectrum R a) : Type u_1), (↑(σₙ R a) : Type u_1)) :=
  { toFun := Set.inclusion (s := spectrum R a) (t := σₙ R a) ⋯, continuous_toFun := ⋯ }
this_1 :
  Eq.{u_1 + 1} (α := UniformSpace C((↑(σₙ R a) : Type u_1), R)) inferInstance
    (UniformSpace.comap (β := C((↑(spectrum R a) : Type u_1), R)) (fun (x : C((↑(σₙ R a) : Type u_1), R)) ↦ x.comp f)
        inferInstance ⊓
      UniformSpace.comap (β := C(Unit, R)) (fun (x : C((↑(σₙ R a) : Type u_1), R)) ↦ x.comp 0) inferInstance)
this :
  ∀ (U : Set (C(Unit, R) × C(Unit, R))),
    Membership.mem (γ := Filter (C(Unit, R) × C(Unit, R))) (𝓤 C(Unit, R)) U → (0, 0) ∈ U
u v : C((↑(σₙ R a) : Type u_1), R)₀
⊢ Eq (α := R)
    (((Function.comp (β := C((↑(σₙ R a) : Type u_1), R) × C((↑(σₙ R a) : Type u_1), R))
              (Prod.map (fun (x : C((↑(σₙ R a) : Type u_1), R)) ↦ x.comp 0) fun (x : C((↑(σₙ R a) : Type u_1), R)) ↦
                x.comp 0)
              (fun (x : C((↑(σₙ R a) : Type u_1), R)₀ × C((↑(σₙ R a) : Type u_1), R)₀) ↦
                ((↑x.1 : C((↑(σₙ R a) : Type u_1), R)), (↑x.2 : C((↑(σₙ R a) : Type u_1), R))))
              (u, v)).2 :
          (a : Unit) → R)
        PUnit.unit :
      R)
    ((0 : (a : Unit) → R) PUnit.unit : R)

Tactic:
map_zero v

Premises used: map_zero
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital
Goal:
R : Type u_1
A : Type u_2
p : A → Prop
inst : Semifield R
inst_1 : StarRing R
inst_2 : MetricSpace R
inst_3 : IsTopologicalSemiring R
inst_4 : ContinuousStar R
inst_5 : Ring A
inst_6 : StarRing A
inst_7 : TopologicalSpace A
inst_8 : Algebra R A
inst_9 : ClosedEmbeddingContinuousFunctionalCalculus R A p
inst_10 : CompleteSpace R
a : A
ha : p a
f : C((↑(spectrum R a) : Type u_1), (↑(σₙ R a) : Type u_1)) :=
  { toFun := Set.inclusion (s := spectrum R a) (t := σₙ R a) ⋯, continuous_toFun := ⋯ }
this_1 :
  Eq.{u_1 + 1} (α := UniformSpace C((↑(σₙ R a) : Type u_1), R)) inferInstance
    (UniformSpace.comap (β := C((↑(spectrum R a) : Type u_1), R)) (fun (x : C((↑(σₙ R a) : Type u_1), R)) ↦ x.comp f)
        inferInstance ⊓
      UniformSpace.comap (β := C(Unit, R)) (fun (x : C((↑(σₙ R a) : Type u_1), R)) ↦ x.comp 0) inferInstance)
this :
  ∀ (U : Set (C(Unit, R) × C(Unit, R))),
    Membership.mem (γ := Filter (C(Unit, R) × C(Unit, R))) (𝓤 C(Unit, R)) U → (0, 0) ∈ U
u v : C((↑(σₙ R a) : Type u_1), R)₀
⊢ Eq (α := R)
    (((Function.comp (β := C((↑(σₙ R a) : Type u_1), R) × C((↑(σₙ R a) : Type u_1), R))
              (Prod.map (fun (x : C((↑(σₙ R a) : Type u_1), R)) ↦ x.comp 0) fun (x : C((↑(σₙ R a) : Type u_1), R)) ↦
                x.comp 0)
              (fun (x : C((↑(σₙ R a) : Type u_1), R)₀ × C((↑(σₙ R a) : Type u_1), R)₀) ↦
                ((↑x.1 : C((↑(σₙ R a) : Type u_1), R)), (↑x.2 : C((↑(σₙ R a) : Type u_1), R))))
              (u, v)).2 :
          (a : Unit) → R)
        PUnit.unit :
      R)
    ((0 : (a : Unit) → R) PUnit.unit : R)

Tactic:
exact map_zero v

Premises used: map_zero
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital
Goal:
R : Type u_1
A : Type u_2
p : A → Prop
inst : Semifield R
inst_1 : StarRing R
inst_2 : MetricSpace R
inst_3 : IsTopologicalSemiring R
inst_4 : ContinuousStar R
inst_5 : Ring A
inst_6 : StarRing A
inst_7 : TopologicalSpace A
inst_8 : Algebra R A
inst_9 : ContinuousFunctionalCalculus R A p
a : A
⊢ CompactSpace.{u_1} (↑(spectrum R a) : Type u_1)

Tactic:
have h_cpct : CompactSpace (spectrum R a) := inferInstance
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital
Goal:
R : Type u_1
A : Type u_2
p : A → Prop
inst : Semifield R
inst_1 : StarRing R
inst_2 : MetricSpace R
inst_3 : IsTopologicalSemiring R
inst_4 : ContinuousStar R
inst_5 : Ring A
inst_6 : StarRing A
inst_7 : TopologicalSpace A
inst_8 : Algebra R A
inst_9 : ContinuousFunctionalCalculus R A p
a : A
h_cpct : CompactSpace.{u_1} (↑(spectrum R a) : Type u_1)
simp : ∀ (h_cpct : IsCompact (spectrum R a)), IsCompact (spectrum R a ∪ {0})
⊢ CompactSpace.{u_1} (↑(σₙ R a) : Type u_1)

Tactic:
simp only [← isCompact_iff_compactSpace, quasispectrum_eq_spectrum_union_zero] at h_cpct ⊢

Premises used: isCompact_iff_compactSpace, quasispectrum_eq_spectrum_union_zero
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital
Goal:
R : Type u_1
A : Type u_2
p : A → Prop
inst : Semifield R
inst_1 : StarRing R
inst_2 : MetricSpace R
inst_3 : IsTopologicalSemiring R
inst_4 : ContinuousStar R
inst_5 : Ring A
inst_6 : StarRing A
inst_7 : TopologicalSpace A
inst_8 : Algebra R A
inst_9 : ContinuousFunctionalCalculus R A p
a : A
h_cpct : IsCompact (spectrum R a)
⊢ IsCompact (spectrum R a ∪ {0})

Tactic:
h_cpct |>.union isCompact_singleton

Premises used: isCompact_singleton, IsCompact.union
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital
Goal:
R : Type u_1
A : Type u_2
p : A → Prop
inst : Semifield R
inst_1 : StarRing R
inst_2 : MetricSpace R
inst_3 : IsTopologicalSemiring R
inst_4 : ContinuousStar R
inst_5 : Ring A
inst_6 : StarRing A
inst_7 : TopologicalSpace A
inst_8 : Algebra R A
inst_9 : ContinuousFunctionalCalculus R A p
a : A
h_cpct : IsCompact (spectrum R a)
⊢ IsCompact (spectrum R a ∪ {0})

Tactic:
exact h_cpct |>.union isCompact_singleton

Premises used: isCompact_singleton, IsCompact.union
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital
Goal:
R : Type u_1
A : Type u_2
p : A → Prop
inst : Semifield R
inst_1 : StarRing R
inst_2 : MetricSpace R
inst_3 : IsTopologicalSemiring R
inst_4 : ContinuousStar R
inst_5 : Ring A
inst_6 : StarRing A
inst_7 : TopologicalSpace A
inst_8 : Algebra R A
inst_9 : ContinuousFunctionalCalculus R A p
inst_10 : ContinuousMapZero.UniqueHom R A
a : A
ha : p a
⊢ (cfcₙHom_of_cfcHom R (p := p) ha : (a : C((↑(σₙ R a) : Type u_1), R)₀) → A)
      (ContinuousMapZero.id (σₙ R a) : C((↑(σₙ R a) : Type u_1), R)₀) =
    a

Tactic:
simpa only [cfcₙHom_id ha] using cfcHom_id ha

Premises used: cfcₙHom_id, cfcHom_id
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital
Goal:
R : Type u_1
A : Type u_2
p : A → Prop
inst : Semifield R
inst_1 : StarRing R
inst_2 : MetricSpace R
inst_3 : IsTopologicalSemiring R
inst_4 : ContinuousStar R
inst_5 : Ring A
inst_6 : StarRing A
inst_7 : TopologicalSpace A
inst_8 : Algebra R A
inst_9 : ClosedEmbeddingContinuousFunctionalCalculus R A p
inst_10 : ContinuousMapZero.UniqueHom R A
inst_11 : CompleteSpace R
a : A
ha : p a
rw :
  IsClosedEmbedding.{u_1, u_2} (X := C((↑(σₙ R a) : Type u_1), R)₀) (Y := A)
    (⇑(cfcₙHom_of_cfcHom R (p := p) ha) : (a : C((↑(σₙ R a) : Type u_1), R)₀) → A)
⊢ IsClosedEmbedding.{u_1, u_2} (X := C((↑(σₙ R a) : Type u_1), R)₀) (Y := A)
    (⇑(cfcₙHom (p := p) ha) : (a : C((↑(σₙ R a) : Type u_1), R)₀) → A)

Tactic:
rw [cfcₙHom_eq_cfcₙHom_of_cfcHom (R := R) ha]

Premises used: cfcₙHom_eq_cfcₙHom_of_cfcHom
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital
Goal:
R : Type u_1
A : Type u_2
p : A → Prop
inst : Semifield R
inst_1 : StarRing R
inst_2 : MetricSpace R
inst_3 : IsTopologicalSemiring R
inst_4 : ContinuousStar R
inst_5 : Ring A
inst_6 : StarRing A
inst_7 : TopologicalSpace A
inst_8 : Algebra R A
inst_9 : ClosedEmbeddingContinuousFunctionalCalculus R A p
inst_10 : ContinuousMapZero.UniqueHom R A
inst_11 : CompleteSpace R
a : A
ha : p a
⊢ IsClosedEmbedding.{u_1, u_2} (X := C((↑(σₙ R a) : Type u_1), R)₀) (Y := A)
    (⇑(cfcₙHom_of_cfcHom R (p := p) ha) : (a : C((↑(σₙ R a) : Type u_1), R)₀) → A)

Tactic:
isClosedEmbedding_cfcₙHom_of_cfcHom ha

Premises used: isClosedEmbedding_cfcₙHom_of_cfcHom