awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity
Goal:
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst : RCLike 𝕜
inst_1 : NormedRing A
inst_2 : StarRing A
inst_3 : NormedAlgebra 𝕜 A
inst_4 : IsometricContinuousFunctionalCalculus 𝕜 A p
inst_5 : ContinuousStar A
inst_6 : TopologicalSpace X
s : Set 𝕜
hs : IsCompact s
f : C((↑s : Type u_2), 𝕜)
a : X → A
ha_cont : Continuous[inst_6, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] a
ha : ∀ (x : X), spectrum 𝕜 (a x) ⊆ s
ha' : ∀ (x : X), p (a x)
open : CompactSpace.{u_2} (↑s : Type u_2)
open₁ :
  ∀ (this : CompactSpace.{u_2} (↑s : Type u_2)),
    Continuous[inst_6, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun (x : X) ↦
      (cfcHomSuperset (p := p) ⋯ (s := s) ⋯ : (a : C((↑s : Type u_2), 𝕜)) → A) f
open₂ : CompactSpace.{u_2} (↑s : Type u_2)
open₃ open₄ :
  ∀ (this : CompactSpace.{u_2} (↑s : Type u_2)),
    Continuous[inst_6, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun (x : X) ↦
      (cfcHomSuperset (p := p) ⋯ (s := s) ⋯ : (a : C((↑s : Type u_2), 𝕜)) → A) f
⊢ Continuous[inst_6, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun (x : X) ↦
    (cfcHomSuperset (p := p) ⋯ (s := s) ⋯ : (a : C((↑s : Type u_2), 𝕜)) → A) f

Tactic:
open scoped ContinuousFunctionalCalculus in
  have : CompactSpace s := by rwa [isCompact_iff_compactSpace] at hs
  induction f using ContinuousMap.induction_on_of_compact with
  | const r =>
    have : ContinuousMap.const s r = algebraMap 𝕜 C(s, 𝕜) r := rfl
    simpa only [this, AlgHomClass.commutes] using continuous_const
  | id =>
    simp only [cfcHomSuperset_id]
    fun_prop
  | star_id =>
    simp only [map_star, cfcHomSuperset_id]
    fun_prop
  | add f g hf hg => simpa using hf.add hg
  | mul f g hf hg => simpa using hf.mul hg
  | frequently f hf =>
    apply continuous_of_uniform_approx_of_continuous
    rw [Metric.uniformity_basis_dist_le.forall_iff (by aesop)]
    intro ε hε
    simp only [Set.mem_setOf_eq, dist_eq_norm]
    obtain ⟨g, hg, g_cont⟩ := frequently_iff.mp hf (Metric.closedBall_mem_nhds f hε)
    simp only [Metric.mem_closedBall, dist_comm g, dist_eq_norm] at hg
    refine ⟨_, g_cont, fun x ↦ ?_⟩
    rw [← map_sub, cfcHomSuperset_apply]
    rw [isometry_cfcHom (R := 𝕜) _ (ha' x) |>.norm_map_of_map_zero (map_zero (cfcHom (ha' x)))]
    rw [ContinuousMap.norm_le _ hε.le] at hg ⊢
    aesop

Premises used: dist_eq_norm, map_sub, continuous_const, StarHomClass.map_star, id, Set.mem_setOf_eq, Metric.uniformity_basis_dist_le, cfcHomSuperset_apply, AlgHomClass.commutes, continuous_of_uniform_approx_of_continuous, cfcHomSuperset_id, Filter.HasBasis.forall_iff, Metric.mem_closedBall, ContinuousFunctionalCalculus
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity
Goal:
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst : RCLike 𝕜
inst_1 : NormedRing A
inst_2 : StarRing A
inst_3 : NormedAlgebra 𝕜 A
inst_4 : IsometricContinuousFunctionalCalculus 𝕜 A p
inst_5 : ContinuousStar A
s : Set 𝕜
hs : IsCompact s
f : 𝕜 → 𝕜
hf : ContinuousOn f s
⊢ Continuous[instTopologicalSpaceSubtype, PseudoMetricSpace.toUniformSpace.toTopologicalSpace]
    fun (x : { x : A // Membership.mem (γ := Set A) {a : A | p a ∧ spectrum 𝕜 a ⊆ s} x }) ↦
    (cfcHomSuperset (p := p) ⋯ (s := s) ⋯ : (a : C((↑s : Type u_2), 𝕜)) → A)
      { toFun := s.restrict (π := fun (a : 𝕜) ↦ 𝕜) f, continuous_toFun := ⋯ : C((↑s : Type u_2), 𝕜) }

Tactic:
continuous_cfcHomSuperset_left hs ⟨_, hf.restrict⟩ ((↑) : {a | p a ∧ spectrum 𝕜 a ⊆ s} → A)
        continuous_subtype_val (fun x ↦ x.2.2)

Premises used: continuous_subtype_val, ContinuousMap.mk, continuous_cfcHomSuperset_left, And.right, ContinuousOn.restrict, Subtype.val, Subtype.property
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity
Goal:
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst : RCLike 𝕜
inst_1 : NormedRing A
inst_2 : StarRing A
inst_3 : NormedAlgebra 𝕜 A
inst_4 : IsometricContinuousFunctionalCalculus 𝕜 A p
inst_5 : ContinuousStar A
s : Set 𝕜
hs : IsCompact s
f : 𝕜 → 𝕜
hf : ContinuousOn f s
h :
  ∀ (x : (↑{a : A | p a ∧ spectrum 𝕜 a ⊆ s} : Type u_3)),
    Set.restrict (π := fun (a : A) ↦ A) {a : A | p a ∧ spectrum 𝕜 a ⊆ s} (cfc (p := p) f) x =
      (cfcHomSuperset (p := p) ⋯ (s := s) ⋯ : (a : C((↑s : Type u_2), 𝕜)) → A)
        { toFun := s.restrict (π := fun (a : 𝕜) ↦ 𝕜) f, continuous_toFun := ⋯ : C((↑s : Type u_2), 𝕜) }
⊢ Continuous[instTopologicalSpaceSubtype, PseudoMetricSpace.toUniformSpace.toTopologicalSpace]
    (Set.restrict (π := fun (a : A) ↦ A) {a : A | p a ∧ spectrum 𝕜 a ⊆ s} (cfc (p := p) f))

Tactic:
convert!
      continuous_cfcHomSuperset_left hs ⟨_, hf.restrict⟩ ((↑) : {a | p a ∧ spectrum 𝕜 a ⊆ s} → A)
        continuous_subtype_val (fun x ↦ x.2.2) with
      x

Premises used: continuous_subtype_val, ContinuousMap.mk, continuous_cfcHomSuperset_left, And.right, ContinuousOn.restrict, Subtype.val, Subtype.property
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity
Goal:
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst : RCLike 𝕜
inst_1 : NormedRing A
inst_2 : StarRing A
inst_3 : NormedAlgebra 𝕜 A
inst_4 : IsometricContinuousFunctionalCalculus 𝕜 A p
inst_5 : ContinuousStar A
s : Set 𝕜
hs : IsCompact s
f : 𝕜 → 𝕜
hf : ContinuousOn f s
x : (↑{a : A | p a ∧ spectrum 𝕜 a ⊆ s} : Type u_3)
h :
  Eq.{u_3 + 1} (α := A)
    ((cfcHom (p := p) ⋯ : (a : C((↑(spectrum 𝕜 (↑x : A)) : Type u_2), 𝕜)) → A)
        { toFun := (spectrum 𝕜 (↑x : A)).restrict (π := fun (a : 𝕜) ↦ 𝕜) f, continuous_toFun := ⋯ :
          C((↑(spectrum 𝕜 (↑x : A)) : Type u_2), 𝕜) } :
      A)
    ((cfcHom (p := p) ⋯ : (a : C((↑(spectrum 𝕜 (↑x : A)) : Type u_2), 𝕜)) → A)
        ({ toFun := s.restrict (π := fun (a : 𝕜) ↦ 𝕜) f, continuous_toFun := ⋯ : C((↑s : Type u_2), 𝕜) }.comp
          { toFun := Subtype.map (p := Membership.mem (spectrum 𝕜 (↑x : A))) (q := Membership.mem s) id ⋯,
            continuous_toFun := ⋯ : C((↑(spectrum 𝕜 (↑x : A)) : Type u_2), (↑s : Type u_2)) }) :
      A)
⊢ Set.restrict (π := fun (a : A) ↦ A) {a : A | p a ∧ spectrum 𝕜 a ⊆ s} (cfc (p := p) f) x =
    (cfcHomSuperset (p := p) ⋯ (s := s) ⋯ : (a : C((↑s : Type u_2), 𝕜)) → A)
      { toFun := s.restrict (π := fun (a : 𝕜) ↦ 𝕜) f, continuous_toFun := ⋯ : C((↑s : Type u_2), 𝕜) }

Tactic:
rw [cfcHomSuperset_apply, Set.restrict_apply, cfc_apply _ _ x.2.1 (hf.mono x.2.2)]

Premises used: Set.restrict_apply, cfcHomSuperset_apply, And.right, And.left, cfc_apply, ContinuousOn.mono, Subtype.property
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity
Goal:
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst : RCLike 𝕜
inst_1 : NormedRing A
inst_2 : StarRing A
inst_3 : NormedAlgebra 𝕜 A
inst_4 : IsometricContinuousFunctionalCalculus 𝕜 A p
inst_5 : ContinuousStar A
s : Set 𝕜
hs : IsCompact s
f : 𝕜 → 𝕜
hf : ContinuousOn f s
x : (↑{a : A | p a ∧ spectrum 𝕜 a ⊆ s} : Type u_3)
⊢ Eq.{u_3 + 1} (α := A)
    ((cfcHom (p := p) ⋯ : (a : C((↑(spectrum 𝕜 (↑x : A)) : Type u_2), 𝕜)) → A)
        { toFun := (spectrum 𝕜 (↑x : A)).restrict (π := fun (a : 𝕜) ↦ 𝕜) f, continuous_toFun := ⋯ :
          C((↑(spectrum 𝕜 (↑x : A)) : Type u_2), 𝕜) } :
      A)
    ((cfcHom (p := p) ⋯ : (a : C((↑(spectrum 𝕜 (↑x : A)) : Type u_2), 𝕜)) → A)
        ({ toFun := s.restrict (π := fun (a : 𝕜) ↦ 𝕜) f, continuous_toFun := ⋯ : C((↑s : Type u_2), 𝕜) }.comp
          { toFun := Subtype.map (p := Membership.mem (spectrum 𝕜 (↑x : A))) (q := Membership.mem s) id ⋯,
            continuous_toFun := ⋯ : C((↑(spectrum 𝕜 (↑x : A)) : Type u_2), (↑s : Type u_2)) }) :
      A)

Tactic:
congr!
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity
Goal:
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst : RCLike 𝕜
inst_1 : NormedRing A
inst_2 : StarRing A
inst_3 : NormedAlgebra 𝕜 A
inst_4 : IsometricContinuousFunctionalCalculus 𝕜 A p
inst_5 : ContinuousStar A
s : Set 𝕜
hs : IsCompact s
⊢ ContinuousOn (Y := A)
    (fun (fa : Prod.{u_2, u_3} (𝕜 →ᵤ[{s}] 𝕜) A) ↦
      cfc.{u_2, u_3} (R := 𝕜) (p := p) ((toFun {s} : (a : 𝕜 →ᵤ[{s}] 𝕜) → 𝕜 → 𝕜) fa.1) fa.2)
    (SProd.sprod (α := Set (𝕜 →ᵤ[{s}] 𝕜)) (β := Set A)
      {f : 𝕜 →ᵤ[{s}] 𝕜 | ContinuousOn (Y := 𝕜) ((toFun {s} : (a : 𝕜 →ᵤ[{s}] 𝕜) → 𝕜 → 𝕜) f) s}
      {a : A | p a ∧ spectrum 𝕜 a ⊆ s})

Tactic:
continuousOn_prod_of_continuousOn_lipschitzOnWith _ 1
    (fun f hf ↦ continuousOn_cfc A hs ((toFun {s}) f) hf)
    (fun a ⟨_, ha'⟩ ↦ lipschitzOnWith_cfc_fun_of_subset a ha')

Premises used: cfc, setOf, lipschitzOnWith_cfc_fun_of_subset, _private.0.continuousOn_cfc_setProd.match_1, continuousOn_prod_of_continuousOn_lipschitzOnWith, Prod.mk, UniformOnFun.toFun, Prod.fst, LipschitzOnWith, ContinuousOn, continuousOn_cfc, Singleton.singleton, OfNat.ofNat, Prod.snd, DFunLike.coe
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity
Goal:
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst : RCLike 𝕜
inst_1 : NormedRing A
inst_2 : StarRing A
inst_3 : NormedAlgebra 𝕜 A
inst_4 : IsometricContinuousFunctionalCalculus 𝕜 A p
inst_5 : ContinuousStar A
s : Set 𝕜
hs : IsCompact s
f : 𝕜 → 𝕜
a : X → A
a₀ : A
l : Filter X
ha_tendsto : Tendsto a l (𝓝 a₀)
ha : ∀ᶠ (x : X) in l, spectrum 𝕜 (a x) ⊆ s
ha' : ∀ᶠ (x : X) in l, p (a x)
ha₀ : spectrum 𝕜 a₀ ⊆ s
ha₀' : p a₀
hf : ContinuousOn f s
apply : Tendsto a l (𝓝[{a : A | p a ∧ spectrum 𝕜 a ⊆ s}] a₀)
⊢ Tendsto (fun (x : X) ↦ cfc (p := p) f (a x)) l (𝓝 (cfc (p := p) f a₀))

Tactic:
apply continuousOn_cfc A hs f |>.continuousWithinAt ⟨ha₀', ha₀⟩ |>.tendsto.comp

Premises used: And.intro, Filter.Tendsto.comp, continuousOn_cfc, ContinuousWithinAt.tendsto, ContinuousOn.continuousWithinAt
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity
Goal:
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst : RCLike 𝕜
inst_1 : NormedRing A
inst_2 : StarRing A
inst_3 : NormedAlgebra 𝕜 A
inst_4 : IsometricContinuousFunctionalCalculus 𝕜 A p
inst_5 : ContinuousStar A
s : Set 𝕜
hs : IsCompact s
f : 𝕜 → 𝕜
a : X → A
a₀ : A
l : Filter X
ha_tendsto : Tendsto a l (𝓝 a₀)
ha : ∀ᶠ (x : X) in l, spectrum 𝕜 (a x) ⊆ s
ha' : ∀ᶠ (x : X) in l, p (a x)
ha₀ : spectrum 𝕜 a₀ ⊆ s
ha₀' : p a₀
hf : ContinuousOn f s
rw : Tendsto a l (𝓝 a₀) ∧ ∀ᶠ (n : X) in l, Membership.mem (γ := Set A) {a : A | p a ∧ spectrum 𝕜 a ⊆ s} (a n)
⊢ Tendsto a l (𝓝[{a : A | p a ∧ spectrum 𝕜 a ⊆ s}] a₀)

Tactic:
rw [tendsto_nhdsWithin_iff]

Premises used: tendsto_nhdsWithin_iff
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity
Goal:
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst : RCLike 𝕜
inst_1 : NormedRing A
inst_2 : StarRing A
inst_3 : NormedAlgebra 𝕜 A
inst_4 : IsometricContinuousFunctionalCalculus 𝕜 A p
inst_5 : ContinuousStar A
s : Set 𝕜
hs : IsCompact s
f : 𝕜 → 𝕜
a : X → A
a₀ : A
l : Filter X
ha_tendsto : Tendsto a l (𝓝 a₀)
ha : ∀ᶠ (x : X) in l, spectrum 𝕜 (a x) ⊆ s
ha' : ∀ᶠ (x : X) in l, p (a x)
ha₀ : spectrum 𝕜 a₀ ⊆ s
ha₀' : p a₀
hf : ContinuousOn f s
⊢ Tendsto a l (𝓝 a₀) ∧ ∀ᶠ (n : X) in l, Membership.mem (γ := Set A) {a : A | p a ∧ spectrum 𝕜 a ⊆ s} (a n)

Tactic:
⟨ha_tendsto, ha'.and ha⟩

Premises used: Filter.Eventually.and, And.intro
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity
Goal:
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst : RCLike 𝕜
inst_1 : NormedRing A
inst_2 : StarRing A
inst_3 : NormedAlgebra 𝕜 A
inst_4 : IsometricContinuousFunctionalCalculus 𝕜 A p
inst_5 : ContinuousStar A
s : Set 𝕜
hs : IsCompact s
f : 𝕜 → 𝕜
a : X → A
a₀ : A
l : Filter X
ha_tendsto : Tendsto a l (𝓝 a₀)
ha : ∀ᶠ (x : X) in l, spectrum 𝕜 (a x) ⊆ s
ha' : ∀ᶠ (x : X) in l, p (a x)
ha₀ : spectrum 𝕜 a₀ ⊆ s
ha₀' : p a₀
hf : ContinuousOn f s
⊢ Tendsto a l (𝓝 a₀) ∧ ∀ᶠ (n : X) in l, Membership.mem (γ := Set A) {a : A | p a ∧ spectrum 𝕜 a ⊆ s} (a n)

Tactic:
exact ⟨ha_tendsto, ha'.and ha⟩
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity
Goal:
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst : RCLike 𝕜
inst_1 : NormedRing A
inst_2 : StarRing A
inst_3 : NormedAlgebra 𝕜 A
inst_4 : IsometricContinuousFunctionalCalculus 𝕜 A p
inst_5 : ContinuousStar A
inst_6 : TopologicalSpace X
s : Set 𝕜
hs : IsCompact s
f : 𝕜 → 𝕜
a : X → A
x₀ : X
ha_cont : ContinuousAt a x₀
ha : ∀ᶠ (x : X) in 𝓝 x₀, spectrum 𝕜 (a x) ⊆ s
ha' : ∀ᶠ (x : X) in 𝓝 x₀, p (a x)
hf : ContinuousOn f s
⊢ ContinuousAt.{u_1, u_3} (Y := A) (fun (x : X) ↦ cfc (p := p) f (a x)) x₀

Tactic:
ha_cont.tendsto.cfc hs f ha ha' ha.self_of_nhds ha'.self_of_nhds

Premises used: Filter.Tendsto.cfc, Filter.Eventually.self_of_nhds, ContinuousAt, ContinuousAt.tendsto
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity
Goal:
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst : RCLike 𝕜
inst_1 : NormedRing A
inst_2 : StarRing A
inst_3 : NormedAlgebra 𝕜 A
inst_4 : IsometricContinuousFunctionalCalculus 𝕜 A p
inst_5 : ContinuousStar A
inst_6 : TopologicalSpace X
s : Set 𝕜
hs : IsCompact s
f : 𝕜 → 𝕜
a : X → A
x₀ : X
t : Set X
hx₀ : x₀ ∈ t
ha_cont : ContinuousWithinAt a t x₀
ha : ∀ᶠ (x : X) in 𝓝[t] x₀, spectrum 𝕜 (a x) ⊆ s
ha' : ∀ᶠ (x : X) in 𝓝[t] x₀, p (a x)
hf : ContinuousOn f s
⊢ ContinuousWithinAt.{u_1, u_3} (Y := A) (fun (x : X) ↦ cfc (p := p) f (a x)) t x₀

Tactic:
ha_cont.tendsto.cfc hs f ha ha' (ha.self_of_nhdsWithin hx₀) (ha'.self_of_nhdsWithin hx₀)

Premises used: Filter.Tendsto.cfc, Filter.Eventually.self_of_nhdsWithin, ContinuousWithinAt, ContinuousWithinAt.tendsto
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity
Goal:
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst : RCLike 𝕜
inst_1 : NormedRing A
inst_2 : StarRing A
inst_3 : NormedAlgebra 𝕜 A
inst_4 : IsometricContinuousFunctionalCalculus 𝕜 A p
inst_5 : ContinuousStar A
inst_6 : TopologicalSpace X
s : X → Set 𝕜
f : 𝕜 → 𝕜
a : X → A
t : Set X
hs : ∀ x ∈ t, IsCompact (s x)
ha_cont : ContinuousOn a t
ha : ∀ x₀ ∈ t, ∀ᶠ (x : X) in 𝓝[t] x₀, spectrum 𝕜 (a x) ⊆ s x₀
ha' : ∀ x ∈ t, p (a x)
hf : ∀ x ∈ t, ContinuousOn f (s x)
⊢ ∀ x ∈ t, ContinuousWithinAt a t x

Tactic:
rw [ContinuousOn] at ha_cont ⊢

Premises used: ContinuousOn
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity
Goal:
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst : RCLike 𝕜
inst_1 : NormedRing A
inst_2 : StarRing A
inst_3 : NormedAlgebra 𝕜 A
inst_4 : IsometricContinuousFunctionalCalculus 𝕜 A p
inst_5 : ContinuousStar A
inst_6 : TopologicalSpace X
s : X → Set 𝕜
f : 𝕜 → 𝕜
a : X → A
t : Set X
hs : ∀ x ∈ t, IsCompact (s x)
ha_cont : ContinuousOn a t
ha : ∀ x₀ ∈ t, ∀ᶠ (x : X) in 𝓝[t] x₀, spectrum 𝕜 (a x) ⊆ s x₀
ha' : ∀ x ∈ t, p (a x)
hf : ∀ x ∈ t, ContinuousOn f (s x)
ha_cont₁ : ∀ x ∈ t, ContinuousWithinAt a t x
⊢ ∀ x₀ ∈ t, ∀ᶠ (x : X) in 𝓝[t] x₀, spectrum 𝕜 (a x) ⊆ s x₀

Tactic:
rw [ContinuousOn] at ha_cont ⊢

Premises used: ContinuousOn
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity
Goal:
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst : RCLike 𝕜
inst_1 : NormedRing A
inst_2 : StarRing A
inst_3 : NormedAlgebra 𝕜 A
inst_4 : IsometricContinuousFunctionalCalculus 𝕜 A p
inst_5 : ContinuousStar A
inst_6 : TopologicalSpace X
s : X → Set 𝕜
f : 𝕜 → 𝕜
a : X → A
t : Set X
hs : ∀ x ∈ t, IsCompact (s x)
ha_cont : ContinuousOn a t
ha : ∀ x₀ ∈ t, ∀ᶠ (x : X) in 𝓝[t] x₀, spectrum 𝕜 (a x) ⊆ s x₀
ha' : ∀ x ∈ t, p (a x)
hf : ∀ x ∈ t, ContinuousOn f (s x)
ha_cont₁ : ∀ x ∈ t, ContinuousWithinAt a t x
ha₁ : ∀ x₀ ∈ t, ∀ᶠ (x : X) in 𝓝[t] x₀, spectrum 𝕜 (a x) ⊆ s x₀
⊢ ∀ x ∈ t, p (a x)

Tactic:
rw [ContinuousOn] at ha_cont ⊢

Premises used: ContinuousOn
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity
Goal:
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst : RCLike 𝕜
inst_1 : NormedRing A
inst_2 : StarRing A
inst_3 : NormedAlgebra 𝕜 A
inst_4 : IsometricContinuousFunctionalCalculus 𝕜 A p
inst_5 : ContinuousStar A
inst_6 : TopologicalSpace X
s : X → Set 𝕜
f : 𝕜 → 𝕜
a : X → A
t : Set X
hs : ∀ x ∈ t, IsCompact (s x)
ha_cont : ContinuousOn a t
ha : ∀ x₀ ∈ t, ∀ᶠ (x : X) in 𝓝[t] x₀, spectrum 𝕜 (a x) ⊆ s x₀
ha' : ∀ x ∈ t, p (a x)
hf : ∀ x ∈ t, ContinuousOn f (s x)
ha_cont₁ : ∀ x ∈ t, ContinuousWithinAt a t x
ha₁ : ∀ x₀ ∈ t, ∀ᶠ (x : X) in 𝓝[t] x₀, spectrum 𝕜 (a x) ⊆ s x₀
ha'₁ : ∀ x ∈ t, p (a x)
⊢ ∀ x ∈ t, ContinuousOn f (s x)

Tactic:
rw [ContinuousOn] at ha_cont ⊢

Premises used: ContinuousOn
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity
Goal:
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst : RCLike 𝕜
inst_1 : NormedRing A
inst_2 : StarRing A
inst_3 : NormedAlgebra 𝕜 A
inst_4 : IsometricContinuousFunctionalCalculus 𝕜 A p
inst_5 : ContinuousStar A
inst_6 : TopologicalSpace X
s : X → Set 𝕜
f : 𝕜 → 𝕜
a : X → A
t : Set X
hs : ∀ x ∈ t, IsCompact (s x)
ha_cont : ∀ x ∈ t, ContinuousWithinAt a t x
ha : ∀ x₀ ∈ t, ∀ᶠ (x : X) in 𝓝[t] x₀, spectrum 𝕜 (a x) ⊆ s x₀
ha' : ∀ x ∈ t, p (a x)
hf : ∀ x ∈ t, ContinuousOn f (s x)
⊢ ∀ x ∈ t, ContinuousWithinAt.{u_1, u_3} (Y := A) (fun (x : X) ↦ cfc (p := p) f (a x)) t x

Tactic:
fun x hx ↦ (ha_cont x hx).cfc (hs x hx) f hx ?_ ?_ (hf x hx)

Premises used: ContinuousWithinAt.cfc
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity
Goal:
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst : RCLike 𝕜
inst_1 : NormedRing A
inst_2 : StarRing A
inst_3 : NormedAlgebra 𝕜 A
inst_4 : IsometricContinuousFunctionalCalculus 𝕜 A p
inst_5 : ContinuousStar A
inst_6 : TopologicalSpace X
s : X → Set 𝕜
f : 𝕜 → 𝕜
a : X → A
t : Set X
hs : ∀ x ∈ t, IsCompact (s x)
ha_cont : ∀ x ∈ t, ContinuousWithinAt a t x
ha : ∀ x₀ ∈ t, ∀ᶠ (x : X) in 𝓝[t] x₀, spectrum 𝕜 (a x) ⊆ s x₀
ha' : ∀ x ∈ t, p (a x)
hf : ∀ x ∈ t, ContinuousOn f (s x)
refine_1 : ∀ (x : X) (hx : x ∈ t), ∀ᶠ (x_1 : X) in 𝓝[t] x, spectrum 𝕜 (a x_1) ⊆ s x
refine_2 : ∀ (x : X) (hx : x ∈ t), ∀ᶠ (x : X) in 𝓝[t] x, p (a x)
⊢ ∀ x ∈ t, ContinuousWithinAt.{u_1, u_3} (Y := A) (fun (x : X) ↦ cfc (p := p) f (a x)) t x

Tactic:
refine fun x hx ↦ (ha_cont x hx).cfc (hs x hx) f hx ?_ ?_ (hf x hx)

Premises used: ContinuousWithinAt.cfc
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity
Goal:
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst : RCLike 𝕜
inst_1 : NormedRing A
inst_2 : StarRing A
inst_3 : NormedAlgebra 𝕜 A
inst_4 : IsometricContinuousFunctionalCalculus 𝕜 A p
inst_5 : ContinuousStar A
inst_6 : TopologicalSpace X
s : X → Set 𝕜
f : 𝕜 → 𝕜
a : X → A
t : Set X
hs : ∀ x ∈ t, IsCompact (s x)
ha_cont : ∀ x ∈ t, ContinuousWithinAt a t x
ha : ∀ x₀ ∈ t, ∀ᶠ (x : X) in 𝓝[t] x₀, spectrum 𝕜 (a x) ⊆ s x₀
ha' : ∀ x ∈ t, p (a x)
hf : ∀ x ∈ t, ContinuousOn f (s x)
x : X
hx : x ∈ t
h : ∀ (x_1 : X) (hx : spectrum 𝕜 (a x_1) ⊆ s x) (hxt : x_1 ∈ t), spectrum 𝕜 (a x_1) ⊆ s x
⊢ ∀ᶠ (x_1 : X) in 𝓝[t] x, spectrum 𝕜 (a x_1) ⊆ s x

Tactic:
filter_upwards [ha x hx, self_mem_nhdsWithin] with x hx hxt

Premises used: self_mem_nhdsWithin
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity
Goal:
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst : RCLike 𝕜
inst_1 : NormedRing A
inst_2 : StarRing A
inst_3 : NormedAlgebra 𝕜 A
inst_4 : IsometricContinuousFunctionalCalculus 𝕜 A p
inst_5 : ContinuousStar A
inst_6 : TopologicalSpace X
s : X → Set 𝕜
f : 𝕜 → 𝕜
a : X → A
t : Set X
hs : ∀ x ∈ t, IsCompact (s x)
ha_cont : ∀ x ∈ t, ContinuousWithinAt a t x
ha : ∀ x₀ ∈ t, ∀ᶠ (x : X) in 𝓝[t] x₀, spectrum 𝕜 (a x) ⊆ s x₀
ha' : ∀ x ∈ t, p (a x)
hf : ∀ x ∈ t, ContinuousOn f (s x)
x : X
hx : x ∈ t
h : ∀ (x_1 : X) (hx : spectrum 𝕜 (a x_1) ⊆ s x) (hxt : x_1 ∈ t), p (a x_1)
⊢ ∀ᶠ (x : X) in 𝓝[t] x, p (a x)

Tactic:
filter_upwards [ha x hx, self_mem_nhdsWithin] with x hx hxt

Premises used: self_mem_nhdsWithin
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity
Goal:
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst : RCLike 𝕜
inst_1 : NormedRing A
inst_2 : StarRing A
inst_3 : NormedAlgebra 𝕜 A
inst_4 : IsometricContinuousFunctionalCalculus 𝕜 A p
inst_5 : ContinuousStar A
inst_6 : TopologicalSpace X
s : X → Set 𝕜
f : 𝕜 → 𝕜
a : X → A
t : Set X
hs : ∀ x ∈ t, IsCompact (s x)
ha_cont : ∀ x ∈ t, ContinuousWithinAt a t x
ha : ∀ x₀ ∈ t, ∀ᶠ (x : X) in 𝓝[t] x₀, spectrum 𝕜 (a x) ⊆ s x₀
ha' : ∀ x ∈ t, p (a x)
hf : ∀ x ∈ t, ContinuousOn f (s x)
x_1 : X
hx_1 : x_1 ∈ t
x : X
hx : spectrum 𝕜 (a x) ⊆ s x_1
hxt : x ∈ t
⊢ spectrum 𝕜 (a x) ⊆ s x_1

Tactic:
hx
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity
Goal:
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst : RCLike 𝕜
inst_1 : NormedRing A
inst_2 : StarRing A
inst_3 : NormedAlgebra 𝕜 A
inst_4 : IsometricContinuousFunctionalCalculus 𝕜 A p
inst_5 : ContinuousStar A
inst_6 : TopologicalSpace X
s : X → Set 𝕜
f : 𝕜 → 𝕜
a : X → A
t : Set X
hs : ∀ x ∈ t, IsCompact (s x)
ha_cont : ∀ x ∈ t, ContinuousWithinAt a t x
ha : ∀ x₀ ∈ t, ∀ᶠ (x : X) in 𝓝[t] x₀, spectrum 𝕜 (a x) ⊆ s x₀
ha' : ∀ x ∈ t, p (a x)
hf : ∀ x ∈ t, ContinuousOn f (s x)
x_1 : X
hx_1 : x_1 ∈ t
x : X
hx : spectrum 𝕜 (a x) ⊆ s x_1
hxt : x ∈ t
⊢ p (a x)

Tactic:
ha' x hxt
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity
Goal:
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst : RCLike 𝕜
inst_1 : NormedRing A
inst_2 : StarRing A
inst_3 : NormedAlgebra 𝕜 A
inst_4 : IsometricContinuousFunctionalCalculus 𝕜 A p
inst_5 : ContinuousStar A
inst_6 : TopologicalSpace X
s : X → Set 𝕜
f : 𝕜 → 𝕜
a : X → A
t : Set X
hs : ∀ x ∈ t, IsCompact (s x)
ha_cont : ∀ x ∈ t, ContinuousWithinAt a t x
ha : ∀ x₀ ∈ t, ∀ᶠ (x : X) in 𝓝[t] x₀, spectrum 𝕜 (a x) ⊆ s x₀
ha' : ∀ x ∈ t, p (a x)
hf : ∀ x ∈ t, ContinuousOn f (s x)
x_1 : X
hx_1 : x_1 ∈ t
x : X
hx : spectrum 𝕜 (a x) ⊆ s x_1
hxt : x ∈ t
⊢ spectrum 𝕜 (a x) ⊆ s x_1

Tactic:
exacts [hx, ha' x hxt]
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity
Goal:
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst : RCLike 𝕜
inst_1 : NormedRing A
inst_2 : StarRing A
inst_3 : NormedAlgebra 𝕜 A
inst_4 : IsometricContinuousFunctionalCalculus 𝕜 A p
inst_5 : ContinuousStar A
inst_6 : TopologicalSpace X
s : Set 𝕜
hs : IsCompact s
f : 𝕜 → 𝕜
a : X → A
t : Set X
ha_cont : ContinuousOn a t
ha : ∀ x ∈ t, spectrum 𝕜 (a x) ⊆ s
ha' : ∀ x ∈ t, p (a x)
hf : ContinuousOn f s
⊢ ContinuousOn.{u_1, u_3} (Y := A) (fun (x : X) ↦ cfc (p := p) f (a x)) t

Tactic:
ContinuousOn.cfc _ (fun _ _ ↦ hs) ha_cont (fun _ _ ↦ ?_) ha'

Premises used: ContinuousOn.cfc
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity
Goal:
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst : RCLike 𝕜
inst_1 : NormedRing A
inst_2 : StarRing A
inst_3 : NormedAlgebra 𝕜 A
inst_4 : IsometricContinuousFunctionalCalculus 𝕜 A p
inst_5 : ContinuousStar A
inst_6 : TopologicalSpace X
s : Set 𝕜
hs : IsCompact s
f : 𝕜 → 𝕜
a : X → A
t : Set X
ha_cont : ContinuousOn a t
ha : ∀ x ∈ t, spectrum 𝕜 (a x) ⊆ s
ha' : ∀ x ∈ t, p (a x)
hf : ContinuousOn f s
refine : ∀ x ∈ t, ∀ᶠ (x : X) in 𝓝[t] x, spectrum 𝕜 (a x) ⊆ s
⊢ ContinuousOn.{u_1, u_3} (Y := A) (fun (x : X) ↦ cfc (p := p) f (a x)) t

Tactic:
refine ContinuousOn.cfc _ (fun _ _ ↦ hs) ha_cont (fun _ _ ↦ ?_) ha'

Premises used: ContinuousOn.cfc
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity
Goal:
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst : RCLike 𝕜
inst_1 : NormedRing A
inst_2 : StarRing A
inst_3 : NormedAlgebra 𝕜 A
inst_4 : IsometricContinuousFunctionalCalculus 𝕜 A p
inst_5 : ContinuousStar A
inst_6 : TopologicalSpace X
s : Set 𝕜
hs : IsCompact s
f : 𝕜 → 𝕜
a : X → A
t : Set X
ha_cont : ContinuousOn a t
ha : ∀ x ∈ t, spectrum 𝕜 (a x) ⊆ s
ha' : ∀ x ∈ t, p (a x)
hf : ContinuousOn f s
x : X
h_1 : x ∈ t
h : ∀ (x : X) (hx : x ∈ t), spectrum 𝕜 (a x) ⊆ s
⊢ ∀ᶠ (x : X) in 𝓝[t] x, spectrum 𝕜 (a x) ⊆ s

Tactic:
filter_upwards [self_mem_nhdsWithin] with x hx

Premises used: self_mem_nhdsWithin
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity
Goal:
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst : RCLike 𝕜
inst_1 : NormedRing A
inst_2 : StarRing A
inst_3 : NormedAlgebra 𝕜 A
inst_4 : IsometricContinuousFunctionalCalculus 𝕜 A p
inst_5 : ContinuousStar A
inst_6 : TopologicalSpace X
s : Set 𝕜
hs : IsCompact s
f : 𝕜 → 𝕜
a : X → A
t : Set X
ha_cont : ContinuousOn a t
ha : ∀ x ∈ t, spectrum 𝕜 (a x) ⊆ s
ha' : ∀ x ∈ t, p (a x)
hf : ContinuousOn f s
x_1 : X
h : x_1 ∈ t
x : X
hx : x ∈ t
⊢ spectrum 𝕜 (a x) ⊆ s

Tactic:
ha x hx