mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity

Goal state before:
pp: case pos
X : Type u_1
R : Type u_2
A : Type u_3
p : A → Prop
inst✝¹² : CommSemiring R
inst✝¹¹ : StarRing R
inst✝¹⁰ : MetricSpace R
inst✝⁹ : Nontrivial R
inst✝⁸ : IsTopologicalSemiring R
inst✝⁷ : ContinuousStar R
inst✝⁶ : NonUnitalRing A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : Module R A
inst✝² : SMulCommClass R A A
inst✝¹ : IsScalarTower R A A
inst✝ : NonUnitalContinuousFunctionalCalculus R A p
l : Filter X
F : X → R → R
f : R → R
a : A
h_tendsto : TendstoUniformlyOn F f l (quasispectrum R a)
hF : ∀ᶠ (x : X) in l, ContinuousOn (F x) (quasispectrum R a)
hF0 : ∀ᶠ (x : X) in l, F x 0 = 0
hl : l.NeBot
hf : ContinuousOn f (quasispectrum R a)
hf0 : f 0 = 0
ha : p a
s : Set X := {x | ContinuousOn (F x) (quasispectrum R a) ∧ F x 0 = 0}
hs : s ∈ l
⊢ Tendsto ((fun x ↦ cfcₙ (F x) a) ∘ Subtype.val) (comap Subtype.val l) (𝓝 (cfcₙ f a))

Tactic:
]

Goal state after:
case pos
X : Type u_1
R : Type u_2
A : Type u_3
p : A → Prop
inst✝¹² : CommSemiring R
inst✝¹¹ : StarRing R
inst✝¹⁰ : MetricSpace R
inst✝⁹ : Nontrivial R
inst✝⁸ : IsTopologicalSemiring R
inst✝⁷ : ContinuousStar R
inst✝⁶ : NonUnitalRing A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : Module R A
inst✝² : SMulCommClass R A A
inst✝¹ : IsScalarTower R A A
inst✝ : NonUnitalContinuousFunctionalCalculus R A p
l : Filter X
F : X → R → R
f : R → R
a : A
h_tendsto : TendstoUniformlyOn F f l (quasispectrum R a)
hF : ∀ᶠ (x : X) in l, ContinuousOn (F x) (quasispectrum R a)
hF0 : ∀ᶠ (x : X) in l, F x 0 = 0
hl : l.NeBot
hf : ContinuousOn f (quasispectrum R a)
hf0 : f 0 = 0
ha : p a
s : Set X := {x | ContinuousOn (F x) (quasispectrum R a) ∧ F x 0 = 0}
hs : s ∈ l
⊢ Tendsto ((fun x ↦ cfcₙ (F x) a) ∘ Subtype.val) (comap Subtype.val l) (𝓝 (cfcₙ f a))
Module: Mathlib.Analysis.CStarAlgebra.Module.Constructions

Goal state before:
pp: A : Type u_1
inst✝³ : NonUnitalCStarAlgebra A
inst✝² : PartialOrder A
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℂ E
x : E
⊢ 0 ≤ RCLike.re ⟪x, x⟫_ℂ

Tactic:
]

Goal state after:
A : Type u_1
inst✝³ : NonUnitalCStarAlgebra A
inst✝² : PartialOrder A
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℂ E
x : E
⊢ 0 ≤ RCLike.re ⟪x, x⟫_ℂ
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity

Goal state before:
pp: case pos
X : Type u_1
R : Type u_2
A : Type u_3
p : A → Prop
inst✝¹² : CommSemiring R
inst✝¹¹ : StarRing R
inst✝¹⁰ : MetricSpace R
inst✝⁹ : Nontrivial R
inst✝⁸ : IsTopologicalSemiring R
inst✝⁷ : ContinuousStar R
inst✝⁶ : NonUnitalRing A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : Module R A
inst✝² : SMulCommClass R A A
inst✝¹ : IsScalarTower R A A
inst✝ : NonUnitalContinuousFunctionalCalculus R A p
l : Filter X
F : X → R → R
f : R → R
a : A
h_tendsto : TendstoUniformlyOn F f l (quasispectrum R a)
hF : ∀ᶠ (x : X) in l, ContinuousOn (F x) (quasispectrum R a)
hF0 : ∀ᶠ (x : X) in l, F x 0 = 0
hl : l.NeBot
hf : ContinuousOn f (quasispectrum R a)
hf0 : f 0 = 0
ha : p a
s : Set X := {x | ContinuousOn (F x) (quasispectrum R a) ∧ F x 0 = 0}
hs : s ∈ l
⊢ Tendsto
    (fun x ↦ (cfcₙHom ha) { toFun := (quasispectrum R a).domRestrict (F ↑x), continuous_toFun := ⋯, map_zero' := ⋯ })
    (comap Subtype.val l)
    (𝓝 ((cfcₙHom ha) { toFun := (quasispectrum R a).domRestrict f, continuous_toFun := ⋯, map_zero' := ⋯ }))

Tactic:
]

Goal state after:
case pos
X : Type u_1
R : Type u_2
A : Type u_3
p : A → Prop
inst✝¹² : CommSemiring R
inst✝¹¹ : StarRing R
inst✝¹⁰ : MetricSpace R
inst✝⁹ : Nontrivial R
inst✝⁸ : IsTopologicalSemiring R
inst✝⁷ : ContinuousStar R
inst✝⁶ : NonUnitalRing A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : Module R A
inst✝² : SMulCommClass R A A
inst✝¹ : IsScalarTower R A A
inst✝ : NonUnitalContinuousFunctionalCalculus R A p
l : Filter X
F : X → R → R
f : R → R
a : A
h_tendsto : TendstoUniformlyOn F f l (quasispectrum R a)
hF : ∀ᶠ (x : X) in l, ContinuousOn (F x) (quasispectrum R a)
hF0 : ∀ᶠ (x : X) in l, F x 0 = 0
hl : l.NeBot
hf : ContinuousOn f (quasispectrum R a)
hf0 : f 0 = 0
ha : p a
s : Set X := {x | ContinuousOn (F x) (quasispectrum R a) ∧ F x 0 = 0}
hs : s ∈ l
⊢ Tendsto
    (fun x ↦ (cfcₙHom ha) { toFun := (quasispectrum R a).domRestrict (F ↑x), continuous_toFun := ⋯, map_zero' := ⋯ })
    (comap Subtype.val l)
    (𝓝 ((cfcₙHom ha) { toFun := (quasispectrum R a).domRestrict f, continuous_toFun := ⋯, map_zero' := ⋯ }))
Module: Mathlib.Analysis.CStarAlgebra.ApproximateUnit

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
f : ℝ≥0 → ℝ≥0 := fun x ↦ 1 - (1 + x)⁻¹
g : ℝ≥0 → ℝ≥0 := fun x ↦ x * (1 - x)⁻¹
a b : A
ha₁ : 0 ≤ a
ha₂ : ‖a‖ < 1
⊢ cfcₙ f (cfcₙ g a) = cfcₙ f (cfcₙ g a)

pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
f : ℝ≥0 → ℝ≥0 := fun x ↦ 1 - (1 + x)⁻¹
g : ℝ≥0 → ℝ≥0 := fun x ↦ x * (1 - x)⁻¹
a b : A
ha₁ : 0 ≤ a
ha₂ : ‖a‖ < 1
⊢ ContinuousOn f (g '' σₙ ℝ≥0 a)

pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
f : ℝ≥0 → ℝ≥0 := fun x ↦ 1 - (1 + x)⁻¹
g : ℝ≥0 → ℝ≥0 := fun x ↦ x * (1 - x)⁻¹
a b : A
ha₁ : 0 ≤ a
ha₂ : ‖a‖ < 1
⊢ ContinuousOn g (σₙ ℝ≥0 a)

Tactic:
]

Goal state after:
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
f : ℝ≥0 → ℝ≥0 := fun x ↦ 1 - (1 + x)⁻¹
g : ℝ≥0 → ℝ≥0 := fun x ↦ x * (1 - x)⁻¹
a b : A
ha₁ : 0 ≤ a
ha₂ : ‖a‖ < 1
⊢ ContinuousOn f (g '' σₙ ℝ≥0 a)

A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
f : ℝ≥0 → ℝ≥0 := fun x ↦ 1 - (1 + x)⁻¹
g : ℝ≥0 → ℝ≥0 := fun x ↦ x * (1 - x)⁻¹
a b : A
ha₁ : 0 ≤ a
ha₂ : ‖a‖ < 1
⊢ ContinuousOn g (σₙ ℝ≥0 a)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity

Goal state before:
pp: case pos
X : Type u_1
R : Type u_2
A : Type u_3
p : A → Prop
inst✝¹² : CommSemiring R
inst✝¹¹ : StarRing R
inst✝¹⁰ : MetricSpace R
inst✝⁹ : Nontrivial R
inst✝⁸ : IsTopologicalSemiring R
inst✝⁷ : ContinuousStar R
inst✝⁶ : NonUnitalRing A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : Module R A
inst✝² : SMulCommClass R A A
inst✝¹ : IsScalarTower R A A
inst✝ : NonUnitalContinuousFunctionalCalculus R A p
l : Filter X
F : X → R → R
f : R → R
a : A
h_tendsto : TendstoUniformlyOn F f l (quasispectrum R a)
hF : ∀ᶠ (x : X) in l, ContinuousOn (F x) (quasispectrum R a)
hF0 : ∀ᶠ (x : X) in l, F x 0 = 0
hl : l.NeBot
hf : ContinuousOn f (quasispectrum R a)
hf0 : f 0 = 0
ha : p a
s : Set X := {x | ContinuousOn (F x) (quasispectrum R a) ∧ F x 0 = 0}
hs : s ∈ l
⊢ Tendsto
    (toContinuousMap ∘ fun x ↦
      { toFun := (quasispectrum R a).domRestrict (F ↑x), continuous_toFun := ⋯, map_zero' := ⋯ })
    (comap Subtype.val l) (𝓝 ↑{ toFun := (quasispectrum R a).domRestrict f, continuous_toFun := ⋯, map_zero' := ⋯ })

Tactic:
]

Goal state after:
case pos
X : Type u_1
R : Type u_2
A : Type u_3
p : A → Prop
inst✝¹² : CommSemiring R
inst✝¹¹ : StarRing R
inst✝¹⁰ : MetricSpace R
inst✝⁹ : Nontrivial R
inst✝⁸ : IsTopologicalSemiring R
inst✝⁷ : ContinuousStar R
inst✝⁶ : NonUnitalRing A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : Module R A
inst✝² : SMulCommClass R A A
inst✝¹ : IsScalarTower R A A
inst✝ : NonUnitalContinuousFunctionalCalculus R A p
l : Filter X
F : X → R → R
f : R → R
a : A
h_tendsto : TendstoUniformlyOn F f l (quasispectrum R a)
hF : ∀ᶠ (x : X) in l, ContinuousOn (F x) (quasispectrum R a)
hF0 : ∀ᶠ (x : X) in l, F x 0 = 0
hl : l.NeBot
hf : ContinuousOn f (quasispectrum R a)
hf0 : f 0 = 0
ha : p a
s : Set X := {x | ContinuousOn (F x) (quasispectrum R a) ∧ F x 0 = 0}
hs : s ∈ l
⊢ Tendsto
    (toContinuousMap ∘ fun x ↦
      { toFun := (quasispectrum R a).domRestrict (F ↑x), continuous_toFun := ⋯, map_zero' := ⋯ })
    (comap Subtype.val l) (𝓝 ↑{ toFun := (quasispectrum R a).domRestrict f, continuous_toFun := ⋯, map_zero' := ⋯ })
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity

Goal state before:
pp: case pos
X : Type u_1
R : Type u_2
A : Type u_3
p : A → Prop
inst✝¹² : CommSemiring R
inst✝¹¹ : StarRing R
inst✝¹⁰ : MetricSpace R
inst✝⁹ : Nontrivial R
inst✝⁸ : IsTopologicalSemiring R
inst✝⁷ : ContinuousStar R
inst✝⁶ : NonUnitalRing A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : Module R A
inst✝² : SMulCommClass R A A
inst✝¹ : IsScalarTower R A A
inst✝ : NonUnitalContinuousFunctionalCalculus R A p
l : Filter X
F : X → R → R
f : R → R
a : A
h_tendsto : TendstoUniformlyOn F f l (quasispectrum R a)
hF : ∀ᶠ (x : X) in l, ContinuousOn (F x) (quasispectrum R a)
hF0 : ∀ᶠ (x : X) in l, F x 0 = 0
hl : l.NeBot
hf : ContinuousOn f (quasispectrum R a)
hf0 : f 0 = 0
ha : p a
s : Set X := {x | ContinuousOn (F x) (quasispectrum R a) ∧ F x 0 = 0}
hs : s ∈ l
⊢ TendstoUniformlyOn (fun x ↦ F ↑x) f (comap Subtype.val l) (quasispectrum R a)

Tactic:
]

Goal state after:
case pos
X : Type u_1
R : Type u_2
A : Type u_3
p : A → Prop
inst✝¹² : CommSemiring R
inst✝¹¹ : StarRing R
inst✝¹⁰ : MetricSpace R
inst✝⁹ : Nontrivial R
inst✝⁸ : IsTopologicalSemiring R
inst✝⁷ : ContinuousStar R
inst✝⁶ : NonUnitalRing A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : Module R A
inst✝² : SMulCommClass R A A
inst✝¹ : IsScalarTower R A A
inst✝ : NonUnitalContinuousFunctionalCalculus R A p
l : Filter X
F : X → R → R
f : R → R
a : A
h_tendsto : TendstoUniformlyOn F f l (quasispectrum R a)
hF : ∀ᶠ (x : X) in l, ContinuousOn (F x) (quasispectrum R a)
hF0 : ∀ᶠ (x : X) in l, F x 0 = 0
hl : l.NeBot
hf : ContinuousOn f (quasispectrum R a)
hf0 : f 0 = 0
ha : p a
s : Set X := {x | ContinuousOn (F x) (quasispectrum R a) ∧ F x 0 = 0}
hs : s ∈ l
⊢ TendstoUniformlyOn (fun x ↦ F ↑x) f (comap Subtype.val l) (quasispectrum R a)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity

Goal state before:
pp: X : Type u_1
R : Type u_2
A : Type u_3
p : A → Prop
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : MetricSpace R
inst✝¹⁰ : Nontrivial R
inst✝⁹ : IsTopologicalSemiring R
inst✝⁸ : ContinuousStar R
inst✝⁷ : NonUnitalRing A
inst✝⁶ : StarRing A
inst✝⁵ : TopologicalSpace A
inst✝⁴ : Module R A
inst✝³ : SMulCommClass R A A
inst✝² : IsScalarTower R A A
inst✝¹ : NonUnitalContinuousFunctionalCalculus R A p
inst✝ : TopologicalSpace X
f : X → R → R
a : A
s : Set X
h_cont : ∀ x ∈ s, ContinuousWithinAt (fun x ↦ (ofFun {quasispectrum R a}) (f x)) s x
hf : ∀ x ∈ s, ContinuousOn (f x) (quasispectrum R a)
hf0 : ∀ x ∈ s, f x 0 = 0
⊢ ∀ x ∈ s, ContinuousWithinAt (fun x ↦ cfcₙ (f x) a) s x

Tactic:
]

Goal state after:
X : Type u_1
R : Type u_2
A : Type u_3
p : A → Prop
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : MetricSpace R
inst✝¹⁰ : Nontrivial R
inst✝⁹ : IsTopologicalSemiring R
inst✝⁸ : ContinuousStar R
inst✝⁷ : NonUnitalRing A
inst✝⁶ : StarRing A
inst✝⁵ : TopologicalSpace A
inst✝⁴ : Module R A
inst✝³ : SMulCommClass R A A
inst✝² : IsScalarTower R A A
inst✝¹ : NonUnitalContinuousFunctionalCalculus R A p
inst✝ : TopologicalSpace X
f : X → R → R
a : A
s : Set X
h_cont : ∀ x ∈ s, ContinuousWithinAt (fun x ↦ (ofFun {quasispectrum R a}) (f x)) s x
hf : ∀ x ∈ s, ContinuousOn (f x) (quasispectrum R a)
hf0 : ∀ x ∈ s, f x 0 = 0
⊢ ∀ x ∈ s, ContinuousWithinAt (fun x ↦ cfcₙ (f x) a) s x
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity

Goal state before:
pp: X : Type u_1
R : Type u_2
A : Type u_3
p : A → Prop
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : MetricSpace R
inst✝¹⁰ : Nontrivial R
inst✝⁹ : IsTopologicalSemiring R
inst✝⁸ : ContinuousStar R
inst✝⁷ : NonUnitalRing A
inst✝⁶ : StarRing A
inst✝⁵ : TopologicalSpace A
inst✝⁴ : Module R A
inst✝³ : SMulCommClass R A A
inst✝² : IsScalarTower R A A
inst✝¹ : NonUnitalContinuousFunctionalCalculus R A p
inst✝ : TopologicalSpace X
f : X → R → R
a : A
h_cont : ContinuousOn (fun x ↦ (ofFun {quasispectrum R a}) (f x)) Set.univ
hf : ∀ (x : X), ContinuousOn (f x) (quasispectrum R a)
hf0 : ∀ (x : X), f x 0 = 0
⊢ ContinuousOn (fun x ↦ cfcₙ (f x) a) Set.univ

Tactic:
]

Goal state after:
X : Type u_1
R : Type u_2
A : Type u_3
p : A → Prop
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : MetricSpace R
inst✝¹⁰ : Nontrivial R
inst✝⁹ : IsTopologicalSemiring R
inst✝⁸ : ContinuousStar R
inst✝⁷ : NonUnitalRing A
inst✝⁶ : StarRing A
inst✝⁵ : TopologicalSpace A
inst✝⁴ : Module R A
inst✝³ : SMulCommClass R A A
inst✝² : IsScalarTower R A A
inst✝¹ : NonUnitalContinuousFunctionalCalculus R A p
inst✝ : TopologicalSpace X
f : X → R → R
a : A
h_cont : ContinuousOn (fun x ↦ (ofFun {quasispectrum R a}) (f x)) Set.univ
hf : ∀ (x : X), ContinuousOn (f x) (quasispectrum R a)
hf0 : ∀ (x : X), f x 0 = 0
⊢ ContinuousOn (fun x ↦ cfcₙ (f x) a) Set.univ
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity

Goal state before:
pp: case pos
R : Type u_2
A : Type u_3
p : A → Prop
inst✝¹² : CommSemiring R
inst✝¹¹ : StarRing R
inst✝¹⁰ : MetricSpace R
inst✝⁹ : Nontrivial R
inst✝⁸ : IsTopologicalSemiring R
inst✝⁷ : ContinuousStar R
inst✝⁶ : NonUnitalRing A
inst✝⁵ : StarRing A
inst✝⁴ : MetricSpace A
inst✝³ : Module R A
inst✝² : SMulCommClass R A A
inst✝¹ : IsScalarTower R A A
inst✝ : NonUnitalIsometricContinuousFunctionalCalculus R A p
a : A
ha : p a
f : R →ᵤ[{quasispectrum R a}] R
hf : ContinuousOn ((toFun {quasispectrum R a}) f) (quasispectrum R a)
hf0 : f 0 = 0
g : R →ᵤ[{quasispectrum R a}] R
hg : ContinuousOn ((toFun {quasispectrum R a}) g) (quasispectrum R a)
hg0 : g 0 = 0
⊢ edist
      { toFun := (quasispectrum R a).domRestrict ((toFun {quasispectrum R a}) f), continuous_toFun := ⋯,
        map_zero' := hf0 }
      { toFun := (quasispectrum R a).domRestrict ((toFun {quasispectrum R a}) g), continuous_toFun := ⋯,
        map_zero' := hg0 } ≤
    ↑1 * edist f g

Tactic:
]

Goal state after:
case pos
R : Type u_2
A : Type u_3
p : A → Prop
inst✝¹² : CommSemiring R
inst✝¹¹ : StarRing R
inst✝¹⁰ : MetricSpace R
inst✝⁹ : Nontrivial R
inst✝⁸ : IsTopologicalSemiring R
inst✝⁷ : ContinuousStar R
inst✝⁶ : NonUnitalRing A
inst✝⁵ : StarRing A
inst✝⁴ : MetricSpace A
inst✝³ : Module R A
inst✝² : SMulCommClass R A A
inst✝¹ : IsScalarTower R A A
inst✝ : NonUnitalIsometricContinuousFunctionalCalculus R A p
a : A
ha : p a
f : R →ᵤ[{quasispectrum R a}] R
hf : ContinuousOn ((toFun {quasispectrum R a}) f) (quasispectrum R a)
hf0 : f 0 = 0
g : R →ᵤ[{quasispectrum R a}] R
hg : ContinuousOn ((toFun {quasispectrum R a}) g) (quasispectrum R a)
hg0 : g 0 = 0
⊢ edist
      { toFun := (quasispectrum R a).domRestrict ((toFun {quasispectrum R a}) f), continuous_toFun := ⋯,
        map_zero' := hf0 }
      { toFun := (quasispectrum R a).domRestrict ((toFun {quasispectrum R a}) g), continuous_toFun := ⋯,
        map_zero' := hg0 } ≤
    ↑1 * edist f g
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity

Goal state before:
pp: case pos
R : Type u_2
A : Type u_3
p : A → Prop
inst✝¹² : CommSemiring R
inst✝¹¹ : StarRing R
inst✝¹⁰ : MetricSpace R
inst✝⁹ : Nontrivial R
inst✝⁸ : IsTopologicalSemiring R
inst✝⁷ : ContinuousStar R
inst✝⁶ : NonUnitalRing A
inst✝⁵ : StarRing A
inst✝⁴ : MetricSpace A
inst✝³ : Module R A
inst✝² : SMulCommClass R A A
inst✝¹ : IsScalarTower R A A
inst✝ : NonUnitalIsometricContinuousFunctionalCalculus R A p
a : A
ha : p a
f : R →ᵤ[{quasispectrum R a}] R
hf : ContinuousOn ((toFun {quasispectrum R a}) f) (quasispectrum R a)
hf0 : f 0 = 0
g : R →ᵤ[{quasispectrum R a}] R
hg : ContinuousOn ((toFun {quasispectrum R a}) g) (quasispectrum R a)
hg0 : g 0 = 0
⊢ edist f g ≤ edist f g

Tactic:
]

Goal state after:
Module: Mathlib.Topology.UniformSpace.Matrix

Goal state before:
pp: m : Type u_1
n : Type u_2
𝕜 : Type u_3
inst✝ : UniformSpace 𝕜
⊢ ⨅ i, Filter.comap (fun a ↦ (a.1 i, a.2 i)) (𝓤 (n → 𝕜)) = ⨅ i, ⨅ j, Filter.comap (fun a ↦ (a.1 i j, a.2 i j)) (𝓤 𝕜)

Tactic:
]

Goal state after:
m : Type u_1
n : Type u_2
𝕜 : Type u_3
inst✝ : UniformSpace 𝕜
⊢ ⨅ i, Filter.comap (fun a ↦ (a.1 i, a.2 i)) (𝓤 (n → 𝕜)) = ⨅ i, ⨅ j, Filter.comap (fun a ↦ (a.1 i j, a.2 i j)) (𝓤 𝕜)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity

Goal state before:
pp: X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst✝⁸ : RCLike 𝕜
inst✝⁷ : NonUnitalNormedRing A
inst✝⁶ : StarRing A
inst✝⁵ : NormedSpace 𝕜 A
inst✝⁴ : IsScalarTower 𝕜 A A
inst✝³ : SMulCommClass 𝕜 A A
inst✝² : ContinuousStar A
inst✝¹ : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p
inst✝ : TopologicalSpace X
s : Set 𝕜
hs : CompactSpace ↑s
hs0 : Fact (0 ∈ s)
f : C(↑s, 𝕜)₀
a : X → A
ha_cont : Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] a
ha : ∀ (x : X), quasispectrum 𝕜 (a x) ⊆ s
ha' : ∀ (x : X), p (a x)
⊢ CompactSpace ↑s

Tactic:
]

Goal state after:
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst✝⁸ : RCLike 𝕜
inst✝⁷ : NonUnitalNormedRing A
inst✝⁶ : StarRing A
inst✝⁵ : NormedSpace 𝕜 A
inst✝⁴ : IsScalarTower 𝕜 A A
inst✝³ : SMulCommClass 𝕜 A A
inst✝² : ContinuousStar A
inst✝¹ : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p
inst✝ : TopologicalSpace X
s : Set 𝕜
hs : CompactSpace ↑s
hs0 : Fact (0 ∈ s)
f : C(↑s, 𝕜)₀
a : X → A
ha_cont : Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] a
ha : ∀ (x : X), quasispectrum 𝕜 (a x) ⊆ s
ha' : ∀ (x : X), p (a x)
⊢ CompactSpace ↑s
Module: Mathlib.Analysis.CStarAlgebra.ApproximateUnit

Goal state before:
pp: A : Type u_2
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
x y z : A
hx₀ : 0 ≤ x
hy : y ∈ Set.Icc x 1
c : ℝ≥0
h : ‖star z * (1 - x) * z‖₊ ≤ c ^ 2
⊢ ‖1 * z - y * z‖₊ ≤ c

Tactic:
]

Goal state after:
A : Type u_2
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
x y z : A
hx₀ : 0 ≤ x
hy : y ∈ Set.Icc x 1
c : ℝ≥0
h : ‖star z * (1 - x) * z‖₊ ≤ c ^ 2
⊢ ‖1 * z - y * z‖₊ ≤ c
Module: Mathlib.Analysis.CStarAlgebra.ApproximateUnit

Goal state before:
pp: A : Type u_2
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
x y z : A
hx₀ : 0 ≤ x
hy : y ∈ Set.Icc x 1
c : ℝ≥0
h : ‖star z * (1 - x) * z‖₊ ≤ c ^ 2
⊢ ‖star ((1 - y) * z) * ((1 - y) * z)‖₊ ≤ c ^ 2

Tactic:
]

Goal state after:
A : Type u_2
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
x y z : A
hx₀ : 0 ≤ x
hy : y ∈ Set.Icc x 1
c : ℝ≥0
h : ‖star z * (1 - x) * z‖₊ ≤ c ^ 2
⊢ ‖star ((1 - y) * z) * ((1 - y) * z)‖₊ ≤ c ^ 2
Module: Mathlib.Analysis.CStarAlgebra.ApproximateUnit

Goal state before:
pp: A : Type u_2
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
x y z : A
hx₀ : 0 ≤ x
hy : y ∈ Set.Icc x 1
c : ℝ≥0
h : ‖star z * (1 - x) * z‖₊ ≤ c ^ 2
hy₀ : y ∈ Set.Icc 0 1
hy' : 1 - y ∈ Set.Icc 0 1
⊢ ‖star z * (1 - y) ^ 2 * z‖₊ ≤ c ^ 2

Tactic:
]

Goal state after:
A : Type u_2
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
x y z : A
hx₀ : 0 ≤ x
hy : y ∈ Set.Icc x 1
c : ℝ≥0
h : ‖star z * (1 - x) * z‖₊ ≤ c ^ 2
hy₀ : y ∈ Set.Icc 0 1
hy' : 1 - y ∈ Set.Icc 0 1
⊢ ‖star z * (1 - y) ^ 2 * z‖₊ ≤ c ^ 2
Module: Mathlib.Analysis.CStarAlgebra.ApproximateUnit

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
x y z : A
hx₀ : 0 ≤ x
hxy : x ≤ y
hy₁ : ‖y‖ ≤ 1
c : ℝ
hc : 0 ≤ c
h : ‖star ↑z * (1 - ↑x) * ↑z‖ ≤ c ^ 2
⊢ ‖↑z - ↑y * ↑z‖ ≤ c

Tactic:
]

Goal state after:
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
x y z : A
hx₀ : 0 ≤ x
hxy : x ≤ y
hy₁ : ‖y‖ ≤ 1
c : ℝ
hc : 0 ≤ c
h : ‖star ↑z * (1 - ↑x) * ↑z‖ ≤ c ^ 2
⊢ ‖↑z - ↑y * ↑z‖ ≤ c
Module: Mathlib.Analysis.Matrix.Normed

Goal state before:
pp: case e_f.refine_1.inl
n : Type u_4
α : Type u_5
inst✝² : Fintype n
inst✝¹ : SeminormedAddCommGroup α
inst✝ : DecidableEq n
v : n → α
i : n
hj : i ∈ Finset.univ
⊢ ‖v i‖₊ ≤ ‖v i‖₊

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.ApproximateUnit

Goal state before:
pp: case refine_1
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
x y z : A
hx₀ : 0 ≤ x
hxy : x ≤ y
hy₁ : ‖y‖ ≤ 1
c : ℝ
hc : 0 ≤ c
h : ‖star ↑z * (1 - ↑x) * ↑z‖ ≤ c ^ 2
⊢ 0 ≤ x

Tactic:
]

Goal state after:
case refine_1
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
x y z : A
hx₀ : 0 ≤ x
hxy : x ≤ y
hy₁ : ‖y‖ ≤ 1
c : ℝ
hc : 0 ≤ c
h : ‖star ↑z * (1 - ↑x) * ↑z‖ ≤ c ^ 2
⊢ 0 ≤ x
Module: Mathlib.Analysis.Matrix.Normed

Goal state before:
pp: case e_f.refine_2
n : Type u_4
α : Type u_5
inst✝² : Fintype n
inst✝¹ : SeminormedAddCommGroup α
inst✝ : DecidableEq n
v : n → α
i : n
⊢ ‖v i‖₊ = ‖v i‖₊

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.Matrix.Normed

Goal state before:
pp: m : Type u_3
α : Type u_5
ι : Type u_7
inst✝² : Fintype m
inst✝¹ : Unique ι
inst✝ : SeminormedAddCommGroup α
v : m → α
⊢ (Finset.univ.sup fun i ↦ ∑ j, ‖replicateCol ι v i j‖₊) = Finset.univ.sup fun b ↦ ‖v b‖₊

Tactic:
]

Goal state after:
m : Type u_3
α : Type u_5
ι : Type u_7
inst✝² : Fintype m
inst✝¹ : Unique ι
inst✝ : SeminormedAddCommGroup α
v : m → α
⊢ (Finset.univ.sup fun i ↦ ∑ j, ‖replicateCol ι v i j‖₊) = Finset.univ.sup fun b ↦ ‖v b‖₊
Module: Mathlib.Analysis.CStarAlgebra.ApproximateUnit

Goal state before:
pp: case refine_2
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
x y z : A
hx₀ : 0 ≤ x
hxy : x ≤ y
hy₁ : ‖y‖ ≤ 1
c : ℝ
hc : 0 ≤ c
h : ‖star ↑z * (1 - ↑x) * ↑z‖ ≤ c ^ 2
hy : 0 ≤ y
⊢ x ≤ y ∧ ‖y‖ ≤ 1

Tactic:
]

Goal state after:
case refine_2
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
x y z : A
hx₀ : 0 ≤ x
hxy : x ≤ y
hy₁ : ‖y‖ ≤ 1
c : ℝ
hc : 0 ≤ c
h : ‖star ↑z * (1 - ↑x) * ↑z‖ ≤ c ^ 2
hy : 0 ≤ y
⊢ x ≤ y ∧ ‖y‖ ≤ 1
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity

Goal state before:
pp: case frequently
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst✝⁸ : RCLike 𝕜
inst✝⁷ : NonUnitalNormedRing A
inst✝⁶ : StarRing A
inst✝⁵ : NormedSpace 𝕜 A
inst✝⁴ : IsScalarTower 𝕜 A A
inst✝³ : SMulCommClass 𝕜 A A
inst✝² : ContinuousStar A
inst✝¹ : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p
inst✝ : TopologicalSpace X
s : Set 𝕜
hs : IsCompact s
hs0 : Fact (0 ∈ s)
a : X → A
ha_cont : Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] a
ha : ∀ (x : X), quasispectrum 𝕜 (a x) ⊆ s
ha' : ∀ (x : X), p (a x)
this : CompactSpace ↑s
f : C(↑s, 𝕜)₀
hf :
  ∃ᶠ (g : C(↑s, 𝕜)₀) in 𝓝 f,
    Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun x ↦ (cfcₙHomSuperset ⋯ ⋯) g
⊢ ∀ (i : ℝ),
    0 < i →
      ∃ F,
        Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] F ∧
          ∀ (y : X), ((cfcₙHomSuperset ⋯ ⋯) f, F y) ∈ {p | dist p.1 p.2 ≤ i}

Tactic:
]

Goal state after:
case frequently
X : Type u_1
𝕜 : Type u_2
A : Type u_3
p : A → Prop
inst✝⁸ : RCLike 𝕜
inst✝⁷ : NonUnitalNormedRing A
inst✝⁶ : StarRing A
inst✝⁵ : NormedSpace 𝕜 A
inst✝⁴ : IsScalarTower 𝕜 A A
inst✝³ : SMulCommClass 𝕜 A A
inst✝² : ContinuousStar A
inst✝¹ : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p
inst✝ : TopologicalSpace X
s : Set 𝕜
hs : IsCompact s
hs0 : Fact (0 ∈ s)
a : X → A
ha_cont : Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] a
ha : ∀ (x : X), quasispectrum 𝕜 (a x) ⊆ s
ha' : ∀ (x : X), p (a x)
this : CompactSpace ↑s
f : C(↑s, 𝕜)₀
hf :
  ∃ᶠ (g : C(↑s, 𝕜)₀) in 𝓝 f,
    Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun x ↦ (cfcₙHomSuperset ⋯ ⋯) g
⊢ ∀ (i : ℝ),
    0 < i →
      ∃ F,
        Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] F ∧
          ∀ (y : X), ((cfcₙHomSuperset ⋯ ⋯) f, F y) ∈ {p | dist p.1 p.2 ≤ i}
Module: Mathlib.Analysis.Matrix.Normed

Goal state before:
pp: m : Type u_3
α : Type u_5
inst✝² : Fintype m
inst✝¹ : SeminormedAddCommGroup α
inst✝ : DecidableEq m
v : m → α
⊢ (Finset.univ.sup fun i ↦ ∑ j, ‖diagonal v i j‖₊) = Finset.univ.sup fun b ↦ ‖v b‖₊

Tactic:
]

Goal state after:
m : Type u_3
α : Type u_5
inst✝² : Fintype m
inst✝¹ : SeminormedAddCommGroup α
inst✝ : DecidableEq m
v : m → α
⊢ (Finset.univ.sup fun i ↦ ∑ j, ‖diagonal v i j‖₊) = Finset.univ.sup fun b ↦ ‖v b‖₊
Module: Mathlib.Analysis.Matrix.Normed

Goal state before:
pp: case e_f.refine_1
m : Type u_3
α : Type u_5
inst✝² : Fintype m
inst✝¹ : SeminormedAddCommGroup α
inst✝ : DecidableEq m
v : m → α
i j : m
_hj : j ∈ Finset.univ
hij : j ≠ i
⊢ 0 = 0

Tactic:
]

Goal state after: