awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric
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 : Field S
inst_6 : StarRing S
inst_7 : MetricSpace S
inst_8 : IsTopologicalRing S
inst_9 : ContinuousStar S
inst_10 : NonUnitalRing A
inst_11 : StarRing A
inst_12 : Module S A
inst_13 : IsScalarTower S A A
inst_14 : SMulCommClass S A A
inst_15 : Algebra R S
inst_16 : Module R A
inst_17 : IsScalarTower R S A
inst_18 : StarModule R S
inst_19 : ContinuousSMul R S
inst_20 : IsScalarTower R A A
inst_21 : SMulCommClass R A A
inst_22 : MetricSpace A
inst_23 : NonUnitalIsometricContinuousFunctionalCalculus S A q
inst_24 : CompleteSpace R
inst_25 : ContinuousMapZero.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 ∧ QuasispectrumRestricts.{u_1, u_2, u_3} (R := R) (S := S) a (⇑f : (a : S) → R)
a : A
ha : p a
ha' : q a
haf : QuasispectrumRestricts.{u_1, u_2, u_3} (R := R) (S := S) a (⇑f : (a : S) → R)
this : NonUnitalContinuousFunctionalCalculus R A p
g₁ g₂ : ContinuousMapZero (↑(σₙ R a) : Type u_1) R
x : (↑(σₙ S a) : Type u_2)
⊢ dist (α := S)
      (((↑(ContinuousMapZero.comp
                  { toFun := (⇑(StarAlgHom.ofId R S) : (a : R) → S), continuous_toFun := ⋯, map_zero' := ⋯ :
                    ContinuousMapZero R S }
                  (g₁.comp
                    {
                      toFun :=
                        Subtype.map (p := fun (x : S) ↦ x ∈ σₙ S a) (q := fun (x : R) ↦ x ∈ σₙ R a) (⇑f : (a : S) → R)
                          ⋯,
                      continuous_toFun := ⋯, map_zero' := ⋯ :
                      ContinuousMapZero.{u_2, u_1} (↑(σₙ S a) : Type u_2) (↑(σₙ R a) : Type u_1) })) :
              C((↑(σₙ S a) : Type u_2), S)) :
            (a : (↑(σₙ S a) : Type u_2)) → S)
          x :
        S)
      (((↑(ContinuousMapZero.comp
                  { toFun := (⇑(StarAlgHom.ofId R S) : (a : R) → S), continuous_toFun := ⋯, map_zero' := ⋯ :
                    ContinuousMapZero R S }
                  (g₂.comp
                    {
                      toFun :=
                        Subtype.map (p := fun (x : S) ↦ x ∈ σₙ S a) (q := fun (x : R) ↦ x ∈ σₙ R a) (⇑f : (a : S) → R)
                          ⋯,
                      continuous_toFun := ⋯, map_zero' := ⋯ :
                      ContinuousMapZero.{u_2, u_1} (↑(σₙ S a) : Type u_2) (↑(σₙ R a) : Type u_1) })) :
              C((↑(σₙ S a) : Type u_2), S)) :
            (a : (↑(σₙ S a) : Type u_2)) → S)
          x :
        S) ≤
    dist g₁ g₂

Tactic:
simpa [halg.dist_eq] using ContinuousMap.dist_apply_le_dist _

Premises used: ContinuousMap.dist_apply_le_dist, Isometry.dist_eq
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric
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 : Field S
inst_6 : StarRing S
inst_7 : MetricSpace S
inst_8 : IsTopologicalRing S
inst_9 : ContinuousStar S
inst_10 : NonUnitalRing A
inst_11 : StarRing A
inst_12 : Module S A
inst_13 : IsScalarTower S A A
inst_14 : SMulCommClass S A A
inst_15 : Algebra R S
inst_16 : Module R A
inst_17 : IsScalarTower R S A
inst_18 : StarModule R S
inst_19 : ContinuousSMul R S
inst_20 : IsScalarTower R A A
inst_21 : SMulCommClass R A A
inst_22 : MetricSpace A
inst_23 : NonUnitalIsometricContinuousFunctionalCalculus S A q
inst_24 : CompleteSpace R
inst_25 : ContinuousMapZero.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 ∧ QuasispectrumRestricts.{u_1, u_2, u_3} (R := R) (S := S) a (⇑f : (a : S) → R)
a : A
ha : p a
ha' : q a
haf : QuasispectrumRestricts.{u_1, u_2, u_3} (R := R) (S := S) a (⇑f : (a : S) → R)
this : NonUnitalContinuousFunctionalCalculus R A p
g₁ g₂ : ContinuousMapZero (↑(σₙ R a) : Type u_1) R
x : (↑(σₙ R a) : Type u_1)
x' : (↑(σₙ S a) : Type u_2) :=
  Subtype.map (p := fun (x : R) ↦ x ∈ σₙ R a) (q := fun (x : S) ↦ x ∈ σₙ S a) (⇑(algebraMap R S) : (a : R) → S) ⋯ x
h₁ h₂ :
  x =
    Subtype.map (p := fun (x : S) ↦ x ∈ σₙ S a) (q := fun (x : R) ↦ x ∈ σₙ R a) (⇑f : (a : S) → R) ⋯
      (Subtype.map (p := fun (x : R) ↦ x ∈ σₙ R a) (q := fun (x : S) ↦ x ∈ σₙ S a) (⇑(algebraMap R S) : (a : R) → S) ⋯
        x)
⊢ dist (α := R) ((g₁ : (a : (↑(σₙ R a) : Type u_1)) → R) x : R) ((g₂ : (a : (↑(σₙ R a) : Type u_1)) → R) x : R) =
    dist (α := R)
      ((g₁ : (a : (↑(σₙ R a) : Type u_1)) → R)
          (Subtype.map (p := fun (x : S) ↦ x ∈ σₙ S a) (q := fun (x : R) ↦ x ∈ σₙ R a) (⇑f : (a : S) → R) ⋯
            (Subtype.map (p := fun (x : R) ↦ x ∈ σₙ R a) (q := fun (x : S) ↦ x ∈ σₙ S a)
              (⇑(algebraMap R S) : (a : R) → S) ⋯ x)) :
        R)
      ((g₂ : (a : (↑(σₙ R a) : Type u_1)) → R)
          (Subtype.map (p := fun (x : S) ↦ x ∈ σₙ S a) (q := fun (x : R) ↦ x ∈ σₙ R a) (⇑f : (a : S) → R) ⋯
            (Subtype.map (p := fun (x : R) ↦ x ∈ σₙ R a) (q := fun (x : S) ↦ x ∈ σₙ S a)
              (⇑(algebraMap R S) : (a : R) → S) ⋯ x)) :
        R)

Tactic:
congr! 2
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric
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 : Field S
inst_6 : StarRing S
inst_7 : MetricSpace S
inst_8 : IsTopologicalRing S
inst_9 : ContinuousStar S
inst_10 : NonUnitalRing A
inst_11 : StarRing A
inst_12 : Module S A
inst_13 : IsScalarTower S A A
inst_14 : SMulCommClass S A A
inst_15 : Algebra R S
inst_16 : Module R A
inst_17 : IsScalarTower R S A
inst_18 : StarModule R S
inst_19 : ContinuousSMul R S
inst_20 : IsScalarTower R A A
inst_21 : SMulCommClass R A A
inst_22 : MetricSpace A
inst_23 : NonUnitalIsometricContinuousFunctionalCalculus S A q
inst_24 : CompleteSpace R
inst_25 : ContinuousMapZero.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 ∧ QuasispectrumRestricts.{u_1, u_2, u_3} (R := R) (S := S) a (⇑f : (a : S) → R)
a : A
ha : p a
ha' : q a
haf : QuasispectrumRestricts.{u_1, u_2, u_3} (R := R) (S := S) a (⇑f : (a : S) → R)
this : NonUnitalContinuousFunctionalCalculus R A p
g₁ g₂ : ContinuousMapZero (↑(σₙ R a) : Type u_1) R
x : (↑(σₙ R a) : Type u_1)
x' : (↑(σₙ S a) : Type u_2) :=
  Subtype.map (p := fun (x : R) ↦ x ∈ σₙ R a) (q := fun (x : S) ↦ x ∈ σₙ S a) (⇑(algebraMap R S) : (a : R) → S) ⋯ x
h₁ :
  (↑x : R) =
    (↑(Subtype.map (p := fun (x : S) ↦ x ∈ σₙ S a) (q := fun (x : R) ↦ x ∈ σₙ R a) (⇑f : (a : S) → R) ⋯
          (Subtype.map (p := fun (x : R) ↦ x ∈ σₙ R a) (q := fun (x : S) ↦ x ∈ σₙ S a) (⇑(algebraMap R S) : (a : R) → S)
            ⋯ x)) :
      R)
⊢ x =
    Subtype.map (p := fun (x : S) ↦ x ∈ σₙ S a) (q := fun (x : R) ↦ x ∈ σₙ R a) (⇑f : (a : S) → R) ⋯
      (Subtype.map (p := fun (x : R) ↦ x ∈ σₙ R a) (q := fun (x : S) ↦ x ∈ σₙ S a) (⇑(algebraMap R S) : (a : R) → S) ⋯
        x)

Tactic:
ext
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric
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 : Field S
inst_6 : StarRing S
inst_7 : MetricSpace S
inst_8 : IsTopologicalRing S
inst_9 : ContinuousStar S
inst_10 : NonUnitalRing A
inst_11 : StarRing A
inst_12 : Module S A
inst_13 : IsScalarTower S A A
inst_14 : SMulCommClass S A A
inst_15 : Algebra R S
inst_16 : Module R A
inst_17 : IsScalarTower R S A
inst_18 : StarModule R S
inst_19 : ContinuousSMul R S
inst_20 : IsScalarTower R A A
inst_21 : SMulCommClass R A A
inst_22 : MetricSpace A
inst_23 : NonUnitalIsometricContinuousFunctionalCalculus S A q
inst_24 : CompleteSpace R
inst_25 : ContinuousMapZero.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 ∧ QuasispectrumRestricts.{u_1, u_2, u_3} (R := R) (S := S) a (⇑f : (a : S) → R)
a : A
ha : p a
ha' : q a
haf : QuasispectrumRestricts.{u_1, u_2, u_3} (R := R) (S := S) a (⇑f : (a : S) → R)
this : NonUnitalContinuousFunctionalCalculus R A p
g₁ g₂ : ContinuousMapZero (↑(σₙ R a) : Type u_1) R
x : (↑(σₙ R a) : Type u_1)
x' : (↑(σₙ S a) : Type u_2) :=
  Subtype.map (p := fun (x : R) ↦ x ∈ σₙ R a) (q := fun (x : S) ↦ x ∈ σₙ S a) (⇑(algebraMap R S) : (a : R) → S) ⋯ x
⊢ (↑x : R) =
    (↑(Subtype.map (p := fun (x : S) ↦ x ∈ σₙ S a) (q := fun (x : R) ↦ x ∈ σₙ R a) (⇑f : (a : S) → R) ⋯
          (Subtype.map (p := fun (x : R) ↦ x ∈ σₙ R a) (q := fun (x : S) ↦ x ∈ σₙ S a) (⇑(algebraMap R S) : (a : R) → S)
            ⋯ x)) :
      R)

Tactic:
haf.left_inv _ |>.symm

Premises used: Eq.symm, QuasispectrumRestricts.left_inv
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric
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 : Field S
inst_6 : StarRing S
inst_7 : MetricSpace S
inst_8 : IsTopologicalRing S
inst_9 : ContinuousStar S
inst_10 : NonUnitalRing A
inst_11 : StarRing A
inst_12 : Module S A
inst_13 : IsScalarTower S A A
inst_14 : SMulCommClass S A A
inst_15 : Algebra R S
inst_16 : Module R A
inst_17 : IsScalarTower R S A
inst_18 : StarModule R S
inst_19 : ContinuousSMul R S
inst_20 : IsScalarTower R A A
inst_21 : SMulCommClass R A A
inst_22 : MetricSpace A
inst_23 : NonUnitalIsometricContinuousFunctionalCalculus S A q
inst_24 : CompleteSpace R
inst_25 : ContinuousMapZero.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 ∧ QuasispectrumRestricts.{u_1, u_2, u_3} (R := R) (S := S) a (⇑f : (a : S) → R)
a : A
ha : p a
ha' : q a
haf : QuasispectrumRestricts.{u_1, u_2, u_3} (R := R) (S := S) a (⇑f : (a : S) → R)
this : NonUnitalContinuousFunctionalCalculus R A p
g₁ g₂ : ContinuousMapZero (↑(σₙ R a) : Type u_1) R
x : (↑(σₙ R a) : Type u_1)
x' : (↑(σₙ S a) : Type u_2) :=
  Subtype.map (p := fun (x : R) ↦ x ∈ σₙ R a) (q := fun (x : S) ↦ x ∈ σₙ S a) (⇑(algebraMap R S) : (a : R) → S) ⋯ x
⊢ (↑x : R) =
    (↑(Subtype.map (p := fun (x : S) ↦ x ∈ σₙ S a) (q := fun (x : R) ↦ x ∈ σₙ R a) (⇑f : (a : S) → R) ⋯
          (Subtype.map (p := fun (x : R) ↦ x ∈ σₙ R a) (q := fun (x : S) ↦ x ∈ σₙ S a) (⇑(algebraMap R S) : (a : R) → S)
            ⋯ x)) :
      R)

Tactic:
exact haf.left_inv _ |>.symm

Premises used: Eq.symm, QuasispectrumRestricts.left_inv
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric
Goal:
A : Type u_1
inst : NormedRing A
inst_1 : PartialOrder A
inst_2 : StarRing A
inst_3 : StarOrderedRing A
inst_4 : NormedAlgebra ℝ A
inst_5 : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst_6 : NonnegSpectrumClass ℝ A
⊢ IsometricContinuousFunctionalCalculus ℝ≥0 A fun (x : A) ↦ 0 ≤ x

Tactic:
SpectrumRestricts.isometric_cfc (q := IsSelfAdjoint) ContinuousMap.realToNNReal
    NNReal.isometry_coe le_rfl (fun _ ↦ nonneg_iff_isSelfAdjoint_and_quasispectrumRestricts)

Premises used: le_rfl, SpectrumRestricts.isometric_cfc, ContinuousMap.realToNNReal, nonneg_iff_isSelfAdjoint_and_quasispectrumRestricts, NNReal.isometry_coe
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric
Goal:
A : Type u_1
inst : NonUnitalNormedRing A
inst_1 : PartialOrder A
inst_2 : StarRing A
inst_3 : StarOrderedRing A
inst_4 : NormedSpace ℝ A
inst_5 : IsScalarTower ℝ A A
inst_6 : SMulCommClass ℝ A A
inst_7 : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst_8 : NonnegSpectrumClass ℝ A
⊢ NonUnitalIsometricContinuousFunctionalCalculus ℝ≥0 A fun (x : A) ↦ 0 ≤ x

Tactic:
QuasispectrumRestricts.isometric_cfc (q := IsSelfAdjoint) ContinuousMap.realToNNReal
    NNReal.isometry_coe le_rfl (fun _ ↦ nonneg_iff_isSelfAdjoint_and_quasispectrumRestricts)

Premises used: le_rfl, ContinuousMap.realToNNReal, nonneg_iff_isSelfAdjoint_and_quasispectrumRestricts, NNReal.isometry_coe, QuasispectrumRestricts.isometric_cfc
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric
Goal:
A : Type u_1
inst : NormedRing A
inst_1 : StarRing A
inst_2 : NormedAlgebra ℝ A
inst_3 : PartialOrder A
inst_4 : StarOrderedRing A
inst_5 : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst_6 : NonnegSpectrumClass ℝ A
inst_7 : Nontrivial A
f : ℝ≥0 → ℝ≥0
a : A
hf : ContinuousOn f (σ ℝ≥0 a)
ha : 0 ≤ a
rw : IsGreatest (f '' σ ℝ≥0 a) ‖cfc (p := IsSelfAdjoint) (fun (x : ℝ) ↦ (↑(f x.toNNReal) : ℝ)) a‖₊
⊢ IsGreatest (f '' σ ℝ≥0 a) ‖cfc (p := fun (x : A) ↦ 0 ≤ x) f a‖₊

Tactic:
rw [cfc_nnreal_eq_real ..]

Premises used: cfc_nnreal_eq_real
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric
Goal:
A : Type u_1
inst : NormedRing A
inst_1 : StarRing A
inst_2 : NormedAlgebra ℝ A
inst_3 : PartialOrder A
inst_4 : StarOrderedRing A
inst_5 : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst_6 : NonnegSpectrumClass ℝ A
inst_7 : Nontrivial A
f : ℝ≥0 → ℝ≥0
a : A
hf : ContinuousOn f (σ ℝ≥0 a)
ha : 0 ≤ a
⊢ QuasispectrumRestricts (R := ℝ≥0) (S := ℝ) a (⇑ContinuousMap.realToNNReal : (a : ℝ) → ℝ≥0)

Tactic:
obtain ⟨-, ha'⟩ := nonneg_iff_isSelfAdjoint_and_quasispectrumRestricts.mp ha

Premises used: Iff.mp, nonneg_iff_isSelfAdjoint_and_quasispectrumRestricts
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric
Goal:
A : Type u_1
inst : NormedRing A
inst_1 : StarRing A
inst_2 : NormedAlgebra ℝ A
inst_3 : PartialOrder A
inst_4 : StarOrderedRing A
inst_5 : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst_6 : NonnegSpectrumClass ℝ A
inst_7 : Nontrivial A
f : ℝ≥0 → ℝ≥0
a : A
hf : ContinuousOn f (σ ℝ≥0 a)
ha : 0 ≤ a
ha' : QuasispectrumRestricts (R := ℝ≥0) (S := ℝ) a (⇑ContinuousMap.realToNNReal : (a : ℝ) → ℝ≥0)
⊢ SpectrumRestricts (R := ℝ≥0) (S := ℝ) a (⇑ContinuousMap.realToNNReal : (a : ℝ) → ℝ≥0)

Tactic:
rw [← SpectrumRestricts] at ha'

Premises used: SpectrumRestricts
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric
Goal:
A : Type u_1
inst : NormedRing A
inst_1 : StarRing A
inst_2 : NormedAlgebra ℝ A
inst_3 : PartialOrder A
inst_4 : StarOrderedRing A
inst_5 : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst_6 : NonnegSpectrumClass ℝ A
inst_7 : Nontrivial A
f : ℝ≥0 → ℝ≥0
a : A
hf : ContinuousOn f (σ ℝ≥0 a)
ha : 0 ≤ a
ha' : SpectrumRestricts (R := ℝ≥0) (S := ℝ) a (⇑ContinuousMap.realToNNReal : (a : ℝ) → ℝ≥0)
⊢ IsGreatest ((fun (x : ℝ) ↦ ‖(↑(f x.toNNReal) : ℝ)‖₊) '' σ ℝ a)
    ‖cfc (p := IsSelfAdjoint) (fun (x : ℝ) ↦ (↑(f x.toNNReal) : ℝ)) a‖₊

Tactic:
IsGreatest.nnnorm_cfc (fun x : ℝ ↦ (f x.toNNReal : ℝ)) a ?hf_cont

Premises used: IsGreatest.nnnorm_cfc, NNReal.toReal, Real.toNNReal
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric
Goal:
A : Type u_1
inst : NormedRing A
inst_1 : StarRing A
inst_2 : NormedAlgebra ℝ A
inst_3 : PartialOrder A
inst_4 : StarOrderedRing A
inst_5 : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst_6 : NonnegSpectrumClass ℝ A
inst_7 : Nontrivial A
f : ℝ≥0 → ℝ≥0
a : A
hf : ContinuousOn f (σ ℝ≥0 a)
ha : 0 ≤ a
ha' : SpectrumRestricts (R := ℝ≥0) (S := ℝ) a (⇑ContinuousMap.realToNNReal : (a : ℝ) → ℝ≥0)
h : f '' σ ℝ≥0 a = (fun (x : ℝ) ↦ ‖(↑(f x.toNNReal) : ℝ)‖₊) '' σ ℝ a
convert! : ContinuousOn (Y := ℝ) (fun (x : ℝ) ↦ (↑(f x.toNNReal) : ℝ)) (σ ℝ a)
⊢ IsGreatest (f '' σ ℝ≥0 a) ‖cfc (p := IsSelfAdjoint) (fun (x : ℝ) ↦ (↑(f x.toNNReal) : ℝ)) a‖₊

Tactic:
convert! IsGreatest.nnnorm_cfc (fun x : ℝ ↦ (f x.toNNReal : ℝ)) a ?hf_cont

Premises used: IsGreatest.nnnorm_cfc, NNReal.toReal, Real.toNNReal
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric
Goal:
A : Type u_1
inst : NormedRing A
inst_1 : StarRing A
inst_2 : NormedAlgebra ℝ A
inst_3 : PartialOrder A
inst_4 : StarOrderedRing A
inst_5 : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst_6 : NonnegSpectrumClass ℝ A
inst_7 : Nontrivial A
f : ℝ≥0 → ℝ≥0
a : A
hf : ContinuousOn f (σ ℝ≥0 a)
ha : 0 ≤ a
ha' : SpectrumRestricts (R := ℝ≥0) (S := ℝ) a (⇑ContinuousMap.realToNNReal : (a : ℝ) → ℝ≥0)
⊢ ContinuousOn (Y := ℝ) (fun (x : ℝ) ↦ (↑(f x.toNNReal) : ℝ)) (σ ℝ a)

Tactic:
continuous_subtype_val.comp_continuousOn <|
    ContinuousOn.comp ‹_› continuous_real_toNNReal.continuousOn <| ha'.image ▸ Set.mapsTo_image ..

Premises used: Continuous.continuousOn, continuous_subtype_val, Continuous.comp_continuousOn, ContinuousOn.comp, continuous_real_toNNReal, NNReal.toReal
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric
Goal:
A : Type u_1
inst : NormedRing A
inst_1 : StarRing A
inst_2 : NormedAlgebra ℝ A
inst_3 : PartialOrder A
inst_4 : StarOrderedRing A
inst_5 : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst_6 : NonnegSpectrumClass ℝ A
inst_7 : Nontrivial A
f : ℝ≥0 → ℝ≥0
a : A
hf : ContinuousOn f (σ ℝ≥0 a)
ha : 0 ≤ a
ha' : SpectrumRestricts (R := ℝ≥0) (S := ℝ) a (⇑ContinuousMap.realToNNReal : (a : ℝ) → ℝ≥0)
⊢ ContinuousOn (Y := ℝ) (fun (x : ℝ) ↦ (↑(f x.toNNReal) : ℝ)) (σ ℝ a)

Tactic:
exact continuous_subtype_val.comp_continuousOn <|
    ContinuousOn.comp ‹_› continuous_real_toNNReal.continuousOn <| ha'.image ▸ Set.mapsTo_image ..

Premises used: Continuous.continuousOn, continuous_subtype_val, Continuous.comp_continuousOn, ContinuousOn.comp, continuous_real_toNNReal
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric
Goal:
A : Type u_1
inst : NormedRing A
inst_1 : StarRing A
inst_2 : NormedAlgebra ℝ A
inst_3 : PartialOrder A
inst_4 : StarOrderedRing A
inst_5 : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst_6 : NonnegSpectrumClass ℝ A
inst_7 : Nontrivial A
f : ℝ≥0 → ℝ≥0
a : A
hf : ContinuousOn f (σ ℝ≥0 a)
ha : 0 ≤ a
ha' : SpectrumRestricts (R := ℝ≥0) (S := ℝ) a (⇑ContinuousMap.realToNNReal : (a : ℝ) → ℝ≥0)
⊢ ContinuousOn (Y := ℝ) (fun (x : ℝ) ↦ (↑(f x.toNNReal) : ℝ)) (σ ℝ a)

Tactic:
case hf_cont => exact continuous_subtype_val.comp_continuousOn <|
    ContinuousOn.comp ‹_› continuous_real_toNNReal.continuousOn <| ha'.image ▸ Set.mapsTo_image ..

Premises used: Continuous.continuousOn, continuous_subtype_val, Continuous.comp_continuousOn, ContinuousOn.comp, continuous_real_toNNReal
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric
Goal:
A : Type u_1
inst : NormedRing A
inst_1 : StarRing A
inst_2 : NormedAlgebra ℝ A
inst_3 : PartialOrder A
inst_4 : StarOrderedRing A
inst_5 : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst_6 : NonnegSpectrumClass ℝ A
inst_7 : Nontrivial A
f : ℝ≥0 → ℝ≥0
a : A
hf : ContinuousOn f (σ ℝ≥0 a)
ha : 0 ≤ a
ha' : SpectrumRestricts (R := ℝ≥0) (S := ℝ) a (⇑ContinuousMap.realToNNReal : (a : ℝ) → ℝ≥0)
⊢ f '' σ ℝ≥0 a = (fun (x : ℝ) ↦ ‖(↑(f x.toNNReal) : ℝ)‖₊) '' σ ℝ a

Tactic:
simp [Set.image_image, ← ha'.image]

Premises used: Set.image_image, SpectrumRestricts.image
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric
Goal:
A : Type u_1
inst : NormedRing A
inst_1 : StarRing A
inst_2 : NormedAlgebra ℝ A
inst_3 : PartialOrder A
inst_4 : StarOrderedRing A
inst_5 : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst_6 : NonnegSpectrumClass ℝ A
f : ℝ≥0 → ℝ≥0
a : A
x : ℝ≥0
hx : x ∈ σ ℝ≥0 a
hf : ContinuousOn f (σ ℝ≥0 a)
ha : 0 ≤ a
revert : ∀ (hx : x ∈ σ ℝ≥0 a), f x ≤ ‖cfc (p := fun (x : A) ↦ 0 ≤ x) f a‖₊
⊢ f x ≤ ‖cfc (p := fun (x : A) ↦ 0 ≤ x) f a‖₊

Tactic:
revert hx
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric
Goal:
A : Type u_1
inst : NormedRing A
inst_1 : StarRing A
inst_2 : NormedAlgebra ℝ A
inst_3 : PartialOrder A
inst_4 : StarOrderedRing A
inst_5 : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst_6 : NonnegSpectrumClass ℝ A
f : ℝ≥0 → ℝ≥0
a : A
x : ℝ≥0
hf : ContinuousOn f (σ ℝ≥0 a)
ha : 0 ≤ a
⊢ Nontrivial A

Tactic:
nontriviality A
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric
Goal:
A : Type u_1
inst : NormedRing A
inst_1 : StarRing A
inst_2 : NormedAlgebra ℝ A
inst_3 : PartialOrder A
inst_4 : StarOrderedRing A
inst_5 : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst_6 : NonnegSpectrumClass ℝ A
f : ℝ≥0 → ℝ≥0
a : A
x : ℝ≥0
hf : ContinuousOn f (σ ℝ≥0 a)
ha : 0 ≤ a
h : Nontrivial A
⊢ ∀ (hx : x ∈ σ ℝ≥0 a), f x ≤ ‖cfc (p := fun (x : A) ↦ 0 ≤ x) f a‖₊

Tactic:
(IsGreatest.nnnorm_cfc_nnreal f a hf ha |>.2 ⟨x, ·, rfl⟩)

Premises used: IsGreatest.nnnorm_cfc_nnreal, And.right, And.intro, Exists.intro, rfl
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric
Goal:
A : Type u_1
inst : NormedRing A
inst_1 : StarRing A
inst_2 : NormedAlgebra ℝ A
inst_3 : PartialOrder A
inst_4 : StarOrderedRing A
inst_5 : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst_6 : NonnegSpectrumClass ℝ A
f : ℝ≥0 → ℝ≥0
a : A
x : ℝ≥0
hf : ContinuousOn f (σ ℝ≥0 a)
ha : 0 ≤ a
h : Nontrivial A
⊢ ∀ (hx : x ∈ σ ℝ≥0 a), f x ≤ ‖cfc (p := fun (x : A) ↦ 0 ≤ x) f a‖₊

Tactic:
exact (IsGreatest.nnnorm_cfc_nnreal f a hf ha |>.2 ⟨x, ·, rfl⟩)

Premises used: IsGreatest.nnnorm_cfc_nnreal, And.right, And.intro, Exists.intro, rfl
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric
Goal:
A : Type u_1
inst : NormedRing A
inst_1 : StarRing A
inst_2 : NormedAlgebra ℝ A
inst_3 : PartialOrder A
inst_4 : StarOrderedRing A
inst_5 : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst_6 : NonnegSpectrumClass ℝ A
f : ℝ≥0 → ℝ≥0
a : A
c : ℝ≥0
h : ∀ x ∈ σ ℝ≥0 a, f x ≤ c
inl : Subsingleton A → ‖cfc (p := fun (x : A) ↦ 0 ≤ x) f a‖₊ ≤ c
inr : Nontrivial A → ‖cfc (p := fun (x : A) ↦ 0 ≤ x) f a‖₊ ≤ c
⊢ ‖cfc (p := fun (x : A) ↦ 0 ≤ x) f a‖₊ ≤ c

Tactic:
obtain (_ | _) := subsingleton_or_nontrivial A

Premises used: subsingleton_or_nontrivial
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric
Goal:
A : Type u_1
inst : NormedRing A
inst_1 : StarRing A
inst_2 : NormedAlgebra ℝ A
inst_3 : PartialOrder A
inst_4 : StarOrderedRing A
inst_5 : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst_6 : NonnegSpectrumClass ℝ A
f : ℝ≥0 → ℝ≥0
a : A
c : ℝ≥0
h : ∀ x ∈ σ ℝ≥0 a, f x ≤ c
h_1 : Subsingleton A
inl : nnnorm.{u_1} (E := A) 0 ≤ c
⊢ ‖cfc (p := fun (x : A) ↦ 0 ≤ x) f a‖₊ ≤ c

Tactic:
rw [Subsingleton.elim (cfc f a) 0]

Premises used: cfc, Subsingleton.elim, OfNat.ofNat
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric
Goal:
A : Type u_1
inst : NormedRing A
inst_1 : StarRing A
inst_2 : NormedAlgebra ℝ A
inst_3 : PartialOrder A
inst_4 : StarOrderedRing A
inst_5 : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst_6 : NonnegSpectrumClass ℝ A
f : ℝ≥0 → ℝ≥0
a : A
c : ℝ≥0
h : ∀ x ∈ σ ℝ≥0 a, f x ≤ c
h_1 : Subsingleton A
⊢ nnnorm.{u_1} (E := A) 0 ≤ c

Tactic:
simp
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric
Goal:
A : Type u_1
inst : NormedRing A
inst_1 : StarRing A
inst_2 : NormedAlgebra ℝ A
inst_3 : PartialOrder A
inst_4 : StarOrderedRing A
inst_5 : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst_6 : NonnegSpectrumClass ℝ A
f : ℝ≥0 → ℝ≥0
a : A
c : ℝ≥0
h : ∀ x ∈ σ ℝ≥0 a, f x ≤ c
h_1 : Nontrivial A
⊢ (fun (x : A) ↦ ‖x‖₊ ≤ c) 0

Tactic:
simp
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric
Goal:
A : Type u_1
inst : NormedRing A
inst_1 : StarRing A
inst_2 : NormedAlgebra ℝ A
inst_3 : PartialOrder A
inst_4 : StarOrderedRing A
inst_5 : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst_6 : NonnegSpectrumClass ℝ A
f : ℝ≥0 → ℝ≥0
a : A
c : ℝ≥0
h : ∀ x ∈ σ ℝ≥0 a, f x ≤ c
h_1 : Nontrivial A
inr :
  ∀ (hf : ContinuousOn f (σ ℝ≥0 a)) (ha : 0 ≤ a),
    (fun (x : A) ↦ ‖x‖₊ ≤ c)
      ((cfcHom (p := fun (x : A) ↦ 0 ≤ x) ha : (a : C((↑(σ ℝ≥0 a) : Type), ℝ≥0)) → A)
        { toFun := (σ ℝ≥0 a).restrict (π := fun (a : ℝ≥0) ↦ ℝ≥0) f, continuous_toFun := ⋯ :
          C((↑(σ ℝ≥0 a) : Type), ℝ≥0) })
⊢ ‖cfc (p := fun (x : A) ↦ 0 ≤ x) f a‖₊ ≤ c

Tactic:
refine cfc_cases (‖·‖₊ ≤ c) a f (by simp) fun hf ha ↦ ?_

Premises used: NNNorm.nnnorm, LE.le, cfc_cases