awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric (goal)
Goal:
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst : Semifield R
inst_1 : StarRing R
inst_2 : MetricSpace R
inst_3 : IsTopologicalSemiring R
inst_4 : ContinuousStar R
inst_5 : Semifield S
inst_6 : StarRing S
inst_7 : MetricSpace S
inst_8 : IsTopologicalSemiring S
inst_9 : ContinuousStar S
inst_10 : Ring A
inst_11 : StarRing A
inst_12 : Algebra S A
inst_13 : Algebra R S
inst_14 : Algebra R A
inst_15 : IsScalarTower R S A
inst_16 : StarModule R S
inst_17 : ContinuousSMul R S
inst_18 : MetricSpace A
inst_19 : IsometricContinuousFunctionalCalculus S A q
inst_20 : CompleteSpace R
inst_21 : ContinuousMap.UniqueHom R A
f : C(S, R)
halg : Isometry.{u_1, u_2} (α := R) (β := S) (⇑(algebraMap R S) : (a : R) → S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ SpectrumRestricts.{u_1, u_2, u_3} (R := R) (S := S) a (⇑f : (a : S) → R)
a : A
ha : p a
ha' : q a
haf : SpectrumRestricts.{u_1, u_2, u_3} (R := R) (S := S) a (⇑f : (a : S) → R)
this : ContinuousFunctionalCalculus R A p
g₁ g₂ : C((↑(σ R a) : Type u_1), R)
⊢ dist.{u_2} (α := C((↑(σ S a) : Type u_2), S))
      (ContinuousMap.comp { toFun := (⇑(StarAlgHom.ofId R S) : (a : R) → S), continuous_toFun := ⋯ : C(R, S) }
        (g₁.comp
          { toFun := Subtype.map (p := fun (x : S) ↦ x ∈ σ S a) (q := fun (x : R) ↦ x ∈ σ R a) (⇑f : (a : S) → R) ⋯,
            continuous_toFun := ⋯ : ContinuousMap.{u_2, u_1} (↑(σ S a) : Type u_2) (↑(σ R a) : Type u_1) }))
      (ContinuousMap.comp { toFun := (⇑(StarAlgHom.ofId R S) : (a : R) → S), continuous_toFun := ⋯ : C(R, S) }
        (g₂.comp
          { toFun := Subtype.map (p := fun (x : S) ↦ x ∈ σ S a) (q := fun (x : R) ↦ x ∈ σ R a) (⇑f : (a : S) → R) ⋯,
            continuous_toFun := ⋯ : ContinuousMap.{u_2, u_1} (↑(σ S a) : Type u_2) (↑(σ R a) : Type u_1) })) ≤
    dist g₁ g₂
have refine_1 : ∀ (x : (↑(σ S a) : Type u_2)),
  dist (α := S)
      ((ContinuousMap.comp { toFun := (⇑(StarAlgHom.ofId R S) : (a : R) → S), continuous_toFun := sorry : C(R, S) }
              (g₁.comp
                {
                  toFun :=
                    Subtype.map (p := fun (x : S) ↦ x ∈ σ S a) (q := fun (x : R) ↦ x ∈ σ R a) (⇑f : (a : S) → R) sorry,
                  continuous_toFun := sorry : ContinuousMap.{u_2, u_1} (↑(σ S a) : Type u_2) (↑(σ R a) : Type u_1) }) :
            (a : (↑(σ S a) : Type u_2)) → S)
          x :
        S)
      ((ContinuousMap.comp { toFun := (⇑(StarAlgHom.ofId R S) : (a : R) → S), continuous_toFun := sorry : C(R, S) }
              (g₂.comp
                {
                  toFun :=
                    Subtype.map (p := fun (x : S) ↦ x ∈ σ S a) (q := fun (x : R) ↦ x ∈ σ R a) (⇑f : (a : S) → R) sorry,
                  continuous_toFun := sorry : ContinuousMap.{u_2, u_1} (↑(σ S a) : Type u_2) (↑(σ R a) : Type u_1) }) :
            (a : (↑(σ S a) : Type u_2)) → S)
          x :
        S) ≤
    dist g₁ g₂
Tactic: refine ContinuousMap.dist_le dist_nonneg |>.mpr fun x ↦ ?_
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric (goal)
Goal:
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst : Semifield R
inst_1 : StarRing R
inst_2 : MetricSpace R
inst_3 : IsTopologicalSemiring R
inst_4 : ContinuousStar R
inst_5 : Semifield S
inst_6 : StarRing S
inst_7 : MetricSpace S
inst_8 : IsTopologicalSemiring S
inst_9 : ContinuousStar S
inst_10 : Ring A
inst_11 : StarRing A
inst_12 : Algebra S A
inst_13 : Algebra R S
inst_14 : Algebra R A
inst_15 : IsScalarTower R S A
inst_16 : StarModule R S
inst_17 : ContinuousSMul R S
inst_18 : MetricSpace A
inst_19 : IsometricContinuousFunctionalCalculus S A q
inst_20 : CompleteSpace R
inst_21 : ContinuousMap.UniqueHom R A
f : C(S, R)
halg : Isometry.{u_1, u_2} (α := R) (β := S) (⇑(algebraMap R S) : (a : R) → S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ SpectrumRestricts.{u_1, u_2, u_3} (R := R) (S := S) a (⇑f : (a : S) → R)
a : A
ha : p a
ha' : q a
haf : SpectrumRestricts.{u_1, u_2, u_3} (R := R) (S := S) a (⇑f : (a : S) → R)
this : ContinuousFunctionalCalculus R A p
g₁ g₂ : C((↑(σ R a) : Type u_1), R)
⊢ dist g₁ g₂ ≤
    dist.{u_2} (α := C((↑(σ S a) : Type u_2), S))
      (ContinuousMap.comp { toFun := (⇑(StarAlgHom.ofId R S) : (a : R) → S), continuous_toFun := ⋯ }
        (g₁.comp
          { toFun := Subtype.map (p := fun (x : S) ↦ x ∈ σ S a) (q := fun (x : R) ↦ x ∈ σ R a) (⇑f : (a : S) → R) ⋯,
            continuous_toFun := ⋯ : ContinuousMap.{u_2, u_1} (↑(σ S a) : Type u_2) (↑(σ R a) : Type u_1) }))
      (ContinuousMap.comp { toFun := (⇑(StarAlgHom.ofId R S) : (a : R) → S), continuous_toFun := ⋯ }
        (g₂.comp
          { toFun := Subtype.map (p := fun (x : S) ↦ x ∈ σ S a) (q := fun (x : R) ↦ x ∈ σ R a) (⇑f : (a : S) → R) ⋯,
            continuous_toFun := ⋯ : ContinuousMap.{u_2, u_1} (↑(σ S a) : Type u_2) (↑(σ R a) : Type u_1) }))
have refine_2 : ∀ (x : (↑(σ R a) : Type u_1)),
  dist (α := R) ((g₁ : (a : (↑(σ R a) : Type u_1)) → R) x : R) ((g₂ : (a : (↑(σ R a) : Type u_1)) → R) x : R) ≤
    dist.{u_2} (α := C((↑(σ S a) : Type u_2), S))
      (ContinuousMap.comp { toFun := (⇑(StarAlgHom.ofId R S) : (a : R) → S), continuous_toFun := sorry }
        (g₁.comp
          { toFun := Subtype.map (p := fun (x : S) ↦ x ∈ σ S a) (q := fun (x : R) ↦ x ∈ σ R a) (⇑f : (a : S) → R) sorry,
            continuous_toFun := sorry : ContinuousMap.{u_2, u_1} (↑(σ S a) : Type u_2) (↑(σ R a) : Type u_1) }))
      (ContinuousMap.comp { toFun := (⇑(StarAlgHom.ofId R S) : (a : R) → S), continuous_toFun := sorry }
        (g₂.comp
          { toFun := Subtype.map (p := fun (x : S) ↦ x ∈ σ S a) (q := fun (x : R) ↦ x ∈ σ R a) (⇑f : (a : S) → R) sorry,
            continuous_toFun := sorry : ContinuousMap.{u_2, u_1} (↑(σ S a) : Type u_2) (↑(σ R a) : Type u_1) }))
Tactic: refine ContinuousMap.dist_le dist_nonneg |>.mpr fun x ↦ ?_
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric (goal)
Goal:
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst : Semifield R
inst_1 : StarRing R
inst_2 : MetricSpace R
inst_3 : IsTopologicalSemiring R
inst_4 : ContinuousStar R
inst_5 : Semifield S
inst_6 : StarRing S
inst_7 : MetricSpace S
inst_8 : IsTopologicalSemiring S
inst_9 : ContinuousStar S
inst_10 : Ring A
inst_11 : StarRing A
inst_12 : Algebra S A
inst_13 : Algebra R S
inst_14 : Algebra R A
inst_15 : IsScalarTower R S A
inst_16 : StarModule R S
inst_17 : ContinuousSMul R S
inst_18 : MetricSpace A
inst_19 : IsometricContinuousFunctionalCalculus S A q
inst_20 : CompleteSpace R
inst_21 : ContinuousMap.UniqueHom R A
f : C(S, R)
halg : Isometry.{u_1, u_2} (α := R) (β := S) (⇑(algebraMap R S) : (a : R) → S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ SpectrumRestricts.{u_1, u_2, u_3} (R := R) (S := S) a (⇑f : (a : S) → R)
a : A
ha : p a
ha' : q a
haf : SpectrumRestricts.{u_1, u_2, u_3} (R := R) (S := S) a (⇑f : (a : S) → R)
this : ContinuousFunctionalCalculus R A p
g₁ g₂ : C((↑(σ R a) : Type u_1), R)
x : (↑(σ R a) : Type u_1)
x' : (↑(σ S a) : Type u_2) :=
  Subtype.map (p := fun (x : R) ↦ x ∈ σ R a) (q := fun (x : S) ↦ x ∈ σ S a) (⇑(algebraMap R S) : (a : R) → S) ⋯ x
⊢ dist (α := R) ((g₁ : (a : (↑(σ R a) : Type u_1)) → R) x : R) ((g₂ : (a : (↑(σ R a) : Type u_1)) → R) x : R) =
    dist (α := R)
      ((g₁ : (a : (↑(σ R a) : Type u_1)) → R)
          (Subtype.map (p := fun (x : S) ↦ x ∈ σ S a) (q := fun (x : R) ↦ x ∈ σ R a) (⇑f : (a : S) → R) ⋯
            (Subtype.map (p := fun (x : R) ↦ x ∈ σ R a) (q := fun (x : S) ↦ x ∈ σ S a) (⇑(algebraMap R S) : (a : R) → S)
              ⋯ x)) :
        R)
      ((g₂ : (a : (↑(σ R a) : Type u_1)) → R)
          (Subtype.map (p := fun (x : S) ↦ x ∈ σ S a) (q := fun (x : R) ↦ x ∈ σ R a) (⇑f : (a : S) → R) ⋯
            (Subtype.map (p := fun (x : R) ↦ x ∈ σ R a) (q := fun (x : S) ↦ x ∈ σ S a) (⇑(algebraMap R S) : (a : R) → S)
              ⋯ x)) :
        R)
have h₁ : x =
  Subtype.map (p := fun (x : S) ↦ x ∈ σ S a) (q := fun (x : R) ↦ x ∈ σ R a) (⇑f : (a : S) → R) sorry
    (Subtype.map (p := fun (x : R) ↦ x ∈ σ R a) (q := fun (x : S) ↦ x ∈ σ S a) (⇑(algebraMap R S) : (a : R) → S) sorry x)
Tactic: congr!
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric (goal)
Goal:
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst : Semifield R
inst_1 : StarRing R
inst_2 : MetricSpace R
inst_3 : IsTopologicalSemiring R
inst_4 : ContinuousStar R
inst_5 : Semifield S
inst_6 : StarRing S
inst_7 : MetricSpace S
inst_8 : IsTopologicalSemiring S
inst_9 : ContinuousStar S
inst_10 : Ring A
inst_11 : StarRing A
inst_12 : Algebra S A
inst_13 : Algebra R S
inst_14 : Algebra R A
inst_15 : IsScalarTower R S A
inst_16 : StarModule R S
inst_17 : ContinuousSMul R S
inst_18 : MetricSpace A
inst_19 : IsometricContinuousFunctionalCalculus S A q
inst_20 : CompleteSpace R
inst_21 : ContinuousMap.UniqueHom R A
f : C(S, R)
halg : Isometry.{u_1, u_2} (α := R) (β := S) (⇑(algebraMap R S) : (a : R) → S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ SpectrumRestricts.{u_1, u_2, u_3} (R := R) (S := S) a (⇑f : (a : S) → R)
a : A
ha : p a
ha' : q a
haf : SpectrumRestricts.{u_1, u_2, u_3} (R := R) (S := S) a (⇑f : (a : S) → R)
this : ContinuousFunctionalCalculus R A p
g₁ g₂ : C((↑(σ R a) : Type u_1), R)
x : (↑(σ R a) : Type u_1)
x' : (↑(σ S a) : Type u_2) :=
  Subtype.map (p := fun (x : R) ↦ x ∈ σ R a) (q := fun (x : S) ↦ x ∈ σ S a) (⇑(algebraMap R S) : (a : R) → S) ⋯ x
⊢ x =
    Subtype.map (p := fun (x : S) ↦ x ∈ σ S a) (q := fun (x : R) ↦ x ∈ σ R a) (⇑f : (a : S) → R) ⋯
      (Subtype.map (p := fun (x : R) ↦ x ∈ σ R a) (q := fun (x : S) ↦ x ∈ σ S a) (⇑(algebraMap R S) : (a : R) → S) ⋯ x)
have h₁ : (↑x : R) =
  (↑(Subtype.map (p := fun (x : S) ↦ x ∈ σ S a) (q := fun (x : R) ↦ x ∈ σ R a) (⇑f : (a : S) → R) sorry
        (Subtype.map (p := fun (x : R) ↦ x ∈ σ R a) (q := fun (x : S) ↦ x ∈ σ S a) (⇑(algebraMap R S) : (a : R) → S) sorry
          x)) :
    R)
Tactic: ext
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric (goal)
Goal:
𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst : RCLike 𝕜
inst_1 : NonUnitalNormedRing A
inst_2 : StarRing A
inst_3 : NormedSpace 𝕜 A
inst_4 : IsScalarTower 𝕜 A A
inst_5 : SMulCommClass 𝕜 A A
inst_6 : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p
f : 𝕜 → 𝕜
a : A
hf : ContinuousOn f (σₙ 𝕜 a)
hf₀ : f 0 = 0
ha : p a
⊢ IsGreatest ((fun (x : 𝕜) ↦ ‖f x‖) '' σₙ 𝕜 a) ‖cfcₙ (p := p) f a‖
have obtain : ∀ (x : ℝ) (hx : IsGreatest ((fun (x : 𝕜) ↦ ‖f x‖) '' σₙ 𝕜 a) x),
  IsGreatest ((fun (x : 𝕜) ↦ ‖f x‖) '' σₙ 𝕜 a) ‖cfcₙ (p := p) f a‖
Tactic: obtain ⟨x, hx⟩ := NonUnitalContinuousFunctionalCalculus.isCompact_quasispectrum a
      |>.image_of_continuousOn hf.norm |>.exists_isGreatest <|
      (quasispectrum.nonempty 𝕜 a).image _
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric (goal)
Goal:
𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst : RCLike 𝕜
inst_1 : NonUnitalNormedRing A
inst_2 : StarRing A
inst_3 : NormedSpace 𝕜 A
inst_4 : IsScalarTower 𝕜 A A
inst_5 : SMulCommClass 𝕜 A A
inst_6 : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p
f : 𝕜 → 𝕜
a : A
hf : ContinuousOn f (σₙ 𝕜 a)
hf₀ : f 0 = 0
ha : p a
x : ℝ
hx : IsGreatest ((fun (x : 𝕜) ↦ ‖f x‖) '' σₙ 𝕜 a) x
⊢ IsGreatest ((fun (x : 𝕜) ↦ ‖f x‖) '' σₙ 𝕜 a) ‖cfcₙ (p := p) f a‖
have obtain : ∀ (x : 𝕜) (hx' : x ∈ σₙ 𝕜 a) (hx : IsGreatest (α := ℝ) ((fun (x : 𝕜) ↦ ‖f x‖) '' σₙ 𝕜 a) ((fun (x : 𝕜) ↦ ‖f x‖) x)),
  IsGreatest ((fun (x : 𝕜) ↦ ‖f x‖) '' σₙ 𝕜 a) ‖cfcₙ (p := p) f a‖
Tactic: obtain ⟨x, hx', rfl⟩ := hx.1
Cleared hypotheses: x, hx
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric (goal)
Goal:
𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst : RCLike 𝕜
inst_1 : NonUnitalNormedRing A
inst_2 : StarRing A
inst_3 : NormedSpace 𝕜 A
inst_4 : IsScalarTower 𝕜 A A
inst_5 : SMulCommClass 𝕜 A A
inst_6 : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p
f : 𝕜 → 𝕜
a : A
hf : ContinuousOn f (σₙ 𝕜 a)
hf₀ : f 0 = 0
ha : p a
x : 𝕜
hx' : x ∈ σₙ 𝕜 a
hx : IsGreatest (α := ℝ) ((fun (x : 𝕜) ↦ ‖f x‖) '' σₙ 𝕜 a) ((fun (x : 𝕜) ↦ ‖f x‖) x)
⊢ IsGreatest ((fun (x : 𝕜) ↦ ‖f x‖) '' σₙ 𝕜 a) ‖cfcₙ (p := p) f a‖
have h : ‖cfcₙ (p := p) f a‖ = (fun (x : 𝕜) ↦ ‖f x‖) x
Tactic: convert! hx
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric (goal)
Goal:
𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst : RCLike 𝕜
inst_1 : NonUnitalNormedRing A
inst_2 : StarRing A
inst_3 : NormedSpace 𝕜 A
inst_4 : IsScalarTower 𝕜 A A
inst_5 : SMulCommClass 𝕜 A A
inst_6 : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p
f : 𝕜 → 𝕜
a : A
hf : ContinuousOn f (σₙ 𝕜 a)
hf₀ : f 0 = 0
ha : p a
x : 𝕜
hx' : x ∈ σₙ 𝕜 a
hx : IsGreatest (α := ℝ) ((fun (x : 𝕜) ↦ ‖f x‖) '' σₙ 𝕜 a) ((fun (x : 𝕜) ↦ ‖f x‖) x)
⊢ ‖cfcₙ (p := p) f a‖ = (fun (x : 𝕜) ↦ ‖f x‖) x
have h : norm (E := C((↑(σₙ 𝕜 a) : Type u_1), 𝕜)₀)
    { toFun := (σₙ 𝕜 a).restrict (π := fun (a : 𝕜) ↦ 𝕜) f, continuous_toFun := sorry, map_zero' := sorry :
      C((↑(σₙ 𝕜 a) : Type u_1), 𝕜)₀ } =
  (fun (x : 𝕜) ↦ ‖f x‖) x
Tactic: rw [cfcₙ_apply f a, norm_cfcₙHom a _]
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric (goal)
Goal:
𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst : RCLike 𝕜
inst_1 : NonUnitalNormedRing A
inst_2 : StarRing A
inst_3 : NormedSpace 𝕜 A
inst_4 : IsScalarTower 𝕜 A A
inst_5 : SMulCommClass 𝕜 A A
inst_6 : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p
f : 𝕜 → 𝕜
a : A
hf : ContinuousOn f (σₙ 𝕜 a)
hf₀ : f 0 = 0
ha : p a
x : 𝕜
hx' : x ∈ σₙ 𝕜 a
hx : IsGreatest (α := ℝ) ((fun (x : 𝕜) ↦ ‖f x‖) '' σₙ 𝕜 a) ((fun (x : 𝕜) ↦ ‖f x‖) x)
⊢ norm (E := C((↑(σₙ 𝕜 a) : Type u_1), 𝕜)₀)
      { toFun := (σₙ 𝕜 a).restrict (π := fun (a : 𝕜) ↦ 𝕜) f, continuous_toFun := ⋯, map_zero' := ⋯ :
        C((↑(σₙ 𝕜 a) : Type u_1), 𝕜)₀ } =
    (fun (x : 𝕜) ↦ ‖f x‖) x
have h : norm (E := C((↑(σₙ 𝕜 a) : Type u_1), 𝕜)₀)
    { toFun := (σₙ 𝕜 a).restrict (π := fun (a : 𝕜) ↦ 𝕜) f, continuous_toFun := sorry, map_zero' := sorry :
      C((↑(σₙ 𝕜 a) : Type u_1), 𝕜)₀ } ≤
  (fun (x : 𝕜) ↦ ‖f x‖) x
Tactic: apply le_antisymm
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric (goal)
Goal:
𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst : RCLike 𝕜
inst_1 : NonUnitalNormedRing A
inst_2 : StarRing A
inst_3 : NormedSpace 𝕜 A
inst_4 : IsScalarTower 𝕜 A A
inst_5 : SMulCommClass 𝕜 A A
inst_6 : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p
f : 𝕜 → 𝕜
a : A
hf : ContinuousOn f (σₙ 𝕜 a)
hf₀ : f 0 = 0
ha : p a
x : 𝕜
hx' : x ∈ σₙ 𝕜 a
hx : IsGreatest (α := ℝ) ((fun (x : 𝕜) ↦ ‖f x‖) '' σₙ 𝕜 a) ((fun (x : 𝕜) ↦ ‖f x‖) x)
h :
  norm (E := C((↑(σₙ 𝕜 a) : Type u_1), 𝕜)₀)
      { toFun := (σₙ 𝕜 a).restrict (π := fun (a : 𝕜) ↦ 𝕜) f, continuous_toFun := ⋯, map_zero' := ⋯ :
        C((↑(σₙ 𝕜 a) : Type u_1), 𝕜)₀ } ≤
    (fun (x : 𝕜) ↦ ‖f x‖) x
⊢ norm (E := C((↑(σₙ 𝕜 a) : Type u_1), 𝕜)₀)
      { toFun := (σₙ 𝕜 a).restrict (π := fun (a : 𝕜) ↦ 𝕜) f, continuous_toFun := ⋯, map_zero' := ⋯ :
        C((↑(σₙ 𝕜 a) : Type u_1), 𝕜)₀ } =
    (fun (x : 𝕜) ↦ ‖f x‖) x
have h₁ : (fun (x : 𝕜) ↦ ‖f x‖) x ≤
  norm (E := C((↑(σₙ 𝕜 a) : Type u_1), 𝕜)₀)
    { toFun := (σₙ 𝕜 a).restrict (π := fun (a : 𝕜) ↦ 𝕜) f, continuous_toFun := sorry, map_zero' := sorry :
      C((↑(σₙ 𝕜 a) : Type u_1), 𝕜)₀ }
Tactic: apply le_antisymm
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric (goal)
Goal:
𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst : RCLike 𝕜
inst_1 : NonUnitalNormedRing A
inst_2 : StarRing A
inst_3 : NormedSpace 𝕜 A
inst_4 : IsScalarTower 𝕜 A A
inst_5 : SMulCommClass 𝕜 A A
inst_6 : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p
f : 𝕜 → 𝕜
a : A
hf : ContinuousOn f (σₙ 𝕜 a)
hf₀ : f 0 = 0
ha : p a
x : 𝕜
hx' : x ∈ σₙ 𝕜 a
hx : IsGreatest (α := ℝ) ((fun (x : 𝕜) ↦ ‖f x‖) '' σₙ 𝕜 a) ((fun (x : 𝕜) ↦ ‖f x‖) x)
⊢ norm (E := C((↑(σₙ 𝕜 a) : Type u_1), 𝕜)₀)
      { toFun := (σₙ 𝕜 a).restrict (π := fun (a : 𝕜) ↦ 𝕜) f, continuous_toFun := ⋯, map_zero' := ⋯ :
        C((↑(σₙ 𝕜 a) : Type u_1), 𝕜)₀ } ≤
    (fun (x : 𝕜) ↦ ‖f x‖) x
have h : ∀ (x_1 : (↑(σₙ 𝕜 a) : Type u_1)),
  norm (E := 𝕜)
      (((↑{ toFun := (σₙ 𝕜 a).restrict (π := fun (a : 𝕜) ↦ 𝕜) f, continuous_toFun := sorry, map_zero' := sorry :
                  C((↑(σₙ 𝕜 a) : Type u_1), 𝕜)₀ } :
              C((↑(σₙ 𝕜 a) : Type u_1), 𝕜)) :
            (a : (↑(σₙ 𝕜 a) : Type u_1)) → 𝕜)
          x_1 :
        𝕜) ≤
    ‖f x‖
Tactic: apply ContinuousMap.norm_le _ (norm_nonneg _) |>.mpr
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric (goal)
Goal:
𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst : RCLike 𝕜
inst_1 : NonUnitalNormedRing A
inst_2 : StarRing A
inst_3 : NormedSpace 𝕜 A
inst_4 : IsScalarTower 𝕜 A A
inst_5 : SMulCommClass 𝕜 A A
inst_6 : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p
f : 𝕜 → 𝕜
a : A
hf : ContinuousOn f (σₙ 𝕜 a)
hf₀ : f 0 = 0
ha : p a
⊢ IsGreatest ((fun (x : 𝕜) ↦ ‖f x‖₊) '' σₙ 𝕜 a) ‖cfcₙ (p := p) f a‖₊
have h : (fun (x : 𝕜) ↦ ‖f x‖₊) '' σₙ 𝕜 a = Real.toNNReal '' (fun (x : 𝕜) ↦ ‖f x‖) '' σₙ 𝕜 a
Tactic: convert! Real.toNNReal_monotone.map_isGreatest (.norm_cfcₙ f a)
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric (goal)
Goal:
𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst : RCLike 𝕜
inst_1 : NonUnitalNormedRing A
inst_2 : StarRing A
inst_3 : NormedSpace 𝕜 A
inst_4 : IsScalarTower 𝕜 A A
inst_5 : SMulCommClass 𝕜 A A
inst_6 : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p
f : 𝕜 → 𝕜
a : A
hf : ContinuousOn f (σₙ 𝕜 a)
hf₀ : f 0 = 0
ha : p a
h : (fun (x : 𝕜) ↦ ‖f x‖₊) '' σₙ 𝕜 a = Real.toNNReal '' (fun (x : 𝕜) ↦ ‖f x‖) '' σₙ 𝕜 a
⊢ IsGreatest ((fun (x : 𝕜) ↦ ‖f x‖₊) '' σₙ 𝕜 a) ‖cfcₙ (p := p) f a‖₊
have h₁ : ‖cfcₙ (p := p) f a‖₊ = ‖cfcₙ (p := p) f a‖.toNNReal
Tactic: convert! Real.toNNReal_monotone.map_isGreatest (.norm_cfcₙ f a)
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric (goal)
Goal:
𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst : RCLike 𝕜
inst_1 : NonUnitalNormedRing A
inst_2 : StarRing A
inst_3 : NormedSpace 𝕜 A
inst_4 : IsScalarTower 𝕜 A A
inst_5 : SMulCommClass 𝕜 A A
inst_6 : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p
f : 𝕜 → 𝕜
a : A
c : ℝ
h : ∀ x ∈ σₙ 𝕜 a, ‖f x‖ ≤ c
⊢ ‖cfcₙ (p := p) f a‖ ≤ c
have refine_1 : (fun (x : A) ↦ ‖x‖ ≤ c) 0
Tactic: refine cfcₙ_cases (‖·‖ ≤ c) a f ?_ fun hf hf0 ha ↦ ?_
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric (goal)
Goal:
𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst : RCLike 𝕜
inst_1 : NonUnitalNormedRing A
inst_2 : StarRing A
inst_3 : NormedSpace 𝕜 A
inst_4 : IsScalarTower 𝕜 A A
inst_5 : SMulCommClass 𝕜 A A
inst_6 : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p
f : 𝕜 → 𝕜
a : A
c : ℝ
h : ∀ x ∈ σₙ 𝕜 a, ‖f x‖ ≤ c
refine_1 : norm.{u_2} (E := A) 0 ≤ c
⊢ ‖cfcₙ (p := p) f a‖ ≤ c
have refine_2 : ∀ (hf : ContinuousOn f (σₙ 𝕜 a))
  (hf0 :
    Eq (α := 𝕜)
      (({ toFun := (σₙ 𝕜 a).restrict (π := fun (a : 𝕜) ↦ 𝕜) f, continuous_toFun := sorry : C((↑(σₙ 𝕜 a) : Type u_1), 𝕜) } :
            (a : (↑(σₙ 𝕜 a) : Type u_1)) → 𝕜)
          0 :
        𝕜)
      (0 : 𝕜))
  (ha : p a),
  (fun (x : A) ↦ ‖x‖ ≤ c)
    ((cfcₙHom (p := p) ha : (a : C((↑(σₙ 𝕜 a) : Type u_1), 𝕜)₀) → A)
      { toFun := (σₙ 𝕜 a).restrict (π := fun (a : 𝕜) ↦ 𝕜) f, continuous_toFun := sorry, map_zero' := hf0 :
        C((↑(σₙ 𝕜 a) : Type u_1), 𝕜)₀ })
Tactic: refine cfcₙ_cases (‖·‖ ≤ c) a f ?_ fun hf hf0 ha ↦ ?_
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric (goal)
Goal:
𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst : RCLike 𝕜
inst_1 : NonUnitalNormedRing A
inst_2 : StarRing A
inst_3 : NormedSpace 𝕜 A
inst_4 : IsScalarTower 𝕜 A A
inst_5 : SMulCommClass 𝕜 A A
inst_6 : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p
f : 𝕜 → 𝕜
a : A
c : ℝ
h : ∀ x ∈ σₙ 𝕜 a, ‖f x‖ ≤ c
hf : ContinuousOn f (σₙ 𝕜 a)
hf0 :
  Eq (α := 𝕜)
    (({ toFun := (σₙ 𝕜 a).restrict (π := fun (a : 𝕜) ↦ 𝕜) f, continuous_toFun := ⋯ : C((↑(σₙ 𝕜 a) : Type u_1), 𝕜) } :
          (a : (↑(σₙ 𝕜 a) : Type u_1)) → 𝕜)
        0 :
      𝕜)
    (0 : 𝕜)
ha : p a
⊢ (fun (x : A) ↦ ‖x‖ ≤ c)
    ((cfcₙHom (p := p) ha : (a : C((↑(σₙ 𝕜 a) : Type u_1), 𝕜)₀) → A)
      { toFun := (σₙ 𝕜 a).restrict (π := fun (a : 𝕜) ↦ 𝕜) f, continuous_toFun := ⋯, map_zero' := hf0 :
        C((↑(σₙ 𝕜 a) : Type u_1), 𝕜)₀ })
have refine_2 : Membership.mem (γ := Set ℝ) (upperBounds ((fun (x : 𝕜) ↦ ‖f x‖) '' σₙ 𝕜 a)) c
Tactic: simp only [← cfcₙ_apply f a, isLUB_le_iff (IsGreatest.norm_cfcₙ f a hf hf0 ha |>.isLUB)]
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric (goal)
Goal:
𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst : RCLike 𝕜
inst_1 : NonUnitalNormedRing A
inst_2 : StarRing A
inst_3 : NormedSpace 𝕜 A
inst_4 : IsScalarTower 𝕜 A A
inst_5 : SMulCommClass 𝕜 A A
inst_6 : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p
f : 𝕜 → 𝕜
a : A
c : ℝ
h : ∀ x ∈ σₙ 𝕜 a, ‖f x‖ < c
⊢ ‖cfcₙ (p := p) f a‖ < c
have refine_1 : (fun (x : A) ↦ ‖x‖ < c) 0
Tactic: refine cfcₙ_cases (‖·‖ < c) a f ?_ fun hf hf0 ha ↦ ?_
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric (goal)
Goal:
𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst : RCLike 𝕜
inst_1 : NonUnitalNormedRing A
inst_2 : StarRing A
inst_3 : NormedSpace 𝕜 A
inst_4 : IsScalarTower 𝕜 A A
inst_5 : SMulCommClass 𝕜 A A
inst_6 : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p
f : 𝕜 → 𝕜
a : A
c : ℝ
h : ∀ x ∈ σₙ 𝕜 a, ‖f x‖ < c
refine_1 : norm.{u_2} (E := A) 0 < c
⊢ ‖cfcₙ (p := p) f a‖ < c
have refine_2 : ∀ (hf : ContinuousOn f (σₙ 𝕜 a))
  (hf0 :
    Eq (α := 𝕜)
      (({ toFun := (σₙ 𝕜 a).restrict (π := fun (a : 𝕜) ↦ 𝕜) f, continuous_toFun := sorry : C((↑(σₙ 𝕜 a) : Type u_1), 𝕜) } :
            (a : (↑(σₙ 𝕜 a) : Type u_1)) → 𝕜)
          0 :
        𝕜)
      (0 : 𝕜))
  (ha : p a),
  (fun (x : A) ↦ ‖x‖ < c)
    ((cfcₙHom (p := p) ha : (a : C((↑(σₙ 𝕜 a) : Type u_1), 𝕜)₀) → A)
      { toFun := (σₙ 𝕜 a).restrict (π := fun (a : 𝕜) ↦ 𝕜) f, continuous_toFun := sorry, map_zero' := hf0 :
        C((↑(σₙ 𝕜 a) : Type u_1), 𝕜)₀ })
Tactic: refine cfcₙ_cases (‖·‖ < c) a f ?_ fun hf hf0 ha ↦ ?_
-- Lean 4 proof step from Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric (goal)
Goal:
𝕜 : Type u_1
A : Type u_2
p : A → Prop
inst : RCLike 𝕜
inst_1 : NonUnitalNormedRing A
inst_2 : StarRing A
inst_3 : NormedSpace 𝕜 A
inst_4 : IsScalarTower 𝕜 A A
inst_5 : SMulCommClass 𝕜 A A
inst_6 : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p
f : 𝕜 → 𝕜
a : A
c : ℝ
h : ∀ x ∈ σₙ 𝕜 a, ‖f x‖ < c
hf : ContinuousOn f (σₙ 𝕜 a)
hf0 :
  Eq (α := 𝕜)
    (({ toFun := (σₙ 𝕜 a).restrict (π := fun (a : 𝕜) ↦ 𝕜) f, continuous_toFun := ⋯ : C((↑(σₙ 𝕜 a) : Type u_1), 𝕜) } :
          (a : (↑(σₙ 𝕜 a) : Type u_1)) → 𝕜)
        0 :
      𝕜)
    (0 : 𝕜)
ha : p a
⊢ (fun (x : A) ↦ ‖x‖ < c)
    ((cfcₙHom (p := p) ha : (a : C((↑(σₙ 𝕜 a) : Type u_1), 𝕜)₀) → A)
      { toFun := (σₙ 𝕜 a).restrict (π := fun (a : 𝕜) ↦ 𝕜) f, continuous_toFun := ⋯, map_zero' := hf0 :
        C((↑(σₙ 𝕜 a) : Type u_1), 𝕜)₀ })
have refine_2 : ∀ (x : ℝ), Membership.mem (γ := Set ℝ) ((fun (x : 𝕜) ↦ ‖f x‖) '' σₙ 𝕜 a) x → x < c
Tactic: simp only [← cfcₙ_apply f a, (IsGreatest.norm_cfcₙ f a hf hf0 ha |>.lt_iff)]