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 : 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

Tactic:
le_antisymm ?_ ?_

Premises used: le_antisymm
-- 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 : 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

Tactic:
ContinuousMap.dist_le dist_nonneg |>.mpr fun x ↦ ?_

Premises used: Iff.mpr, ContinuousMap.dist_le, setOf, dist_nonneg
-- 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 : 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)
refine_1 :
  ∀ (x : (↑(σ S a) : Type u_2)),
    dist (α := 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) }) :
              (a : (↑(σ S a) : Type u_2)) → S)
            x :
          S)
        ((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) }) :
              (a : (↑(σ S a) : Type u_2)) → S)
            x :
          S) ≤
      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 := ⋯ : 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

Tactic:
refine ContinuousMap.dist_le dist_nonneg |>.mpr fun x ↦ ?_

Premises used: Iff.mpr, ContinuousMap.dist_le, dist_nonneg
-- 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 : 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) }))

Tactic:
ContinuousMap.dist_le dist_nonneg |>.mpr fun x ↦ ?_

Premises used: Iff.mpr, ContinuousMap.dist_le, dist_nonneg
-- 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 : 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)
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 := ⋯ }
          (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) }))
⊢ 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) }))

Tactic:
refine ContinuousMap.dist_le dist_nonneg |>.mpr fun x ↦ ?_

Premises used: Iff.mpr, ContinuousMap.dist_le, dist_nonneg
-- 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 : 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 S a) : Type u_2)
⊢ dist (α := 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) }) :
            (a : (↑(σ S a) : Type u_2)) → S)
          x :
        S)
      ((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) }) :
            (a : (↑(σ 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 : 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
h₁ h₂ :
  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)
⊢ 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)

Tactic:
congr!
-- 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 : 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
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) ⋯
          (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)
⊢ 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)

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 : 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 : 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)

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 : 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 : 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)

Tactic:
exact haf.left_inv _ |>.symm

Premises used: Eq.symm, QuasispectrumRestricts.left_inv
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric
Goal:
R : Type u_1
A : Type u_2
p : A → Prop
inst : CommSemiring R
inst_1 : Nontrivial R
inst_2 : StarRing R
inst_3 : MetricSpace R
inst_4 : IsTopologicalSemiring R
inst_5 : ContinuousStar R
inst_6 : NonUnitalRing A
inst_7 : StarRing A
inst_8 : MetricSpace A
inst_9 : Module R A
inst_10 : IsScalarTower R A A
inst_11 : SMulCommClass R A A
inst_12 : NonUnitalIsometricContinuousFunctionalCalculus R A p
a : A
ha : p a
⊢ Isometry.{u_1, u_2} (α := ContinuousMapZero (↑(σₙ R a) : Type u_1) R) (β := A)
    (⇑(cfcₙHom (p := p) ⋯) : (a : ContinuousMapZero (↑(σₙ R a) : Type u_1) R) → A)

Tactic:
NonUnitalIsometricContinuousFunctionalCalculus.isometric a ha

Premises used: NonUnitalIsometricContinuousFunctionalCalculus.isometric
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric
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
a : A
f : C((↑(σₙ 𝕜 a) : Type u_1), 𝕜)₀
ha : p a
⊢ norm.{u_2} (E := A) ((cfcₙHom (p := p) ⋯ : (a : C((↑(σₙ 𝕜 a) : Type u_1), 𝕜)₀) → A) f : A) = ‖f‖

Tactic:
isometry_cfcₙHom a |>.norm_map_of_map_zero (map_zero _) f

Premises used: map_zero, Isometry.norm_map_of_map_zero
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric
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
a : A
f : C((↑(σₙ 𝕜 a) : Type u_1), 𝕜)₀
ha : p a
⊢ norm.{u_2} (E := A) ((cfcₙHom (p := p) ⋯ : (a : C((↑(σₙ 𝕜 a) : Type u_1), 𝕜)₀) → A) f : A) = ‖f‖

Tactic:
refine isometry_cfcₙHom a |>.norm_map_of_map_zero (map_zero _) f

Premises used: map_zero, Isometry.norm_map_of_map_zero
-- Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric
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
a : A
f : C((↑(σₙ 𝕜 a) : Type u_1), 𝕜)₀
ha : p a
⊢ nnnorm.{u_2} (E := A) ((cfcₙHom (p := p) ⋯ : (a : C((↑(σₙ 𝕜 a) : Type u_1), 𝕜)₀) → A) f : A) = ‖f‖₊

Tactic:
Subtype.ext <| norm_cfcₙHom a f ha

Premises used: norm_cfcₙHom, NNReal, Subtype.ext