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

Goal state before:
pp: case e'_3
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²¹ : Semifield R
inst✝²⁰ : StarRing R
inst✝¹⁹ : MetricSpace R
inst✝¹⁸ : IsTopologicalSemiring R
inst✝¹⁷ : ContinuousStar R
inst✝¹⁶ : Semifield S
inst✝¹⁵ : StarRing S
inst✝¹⁴ : MetricSpace S
inst✝¹³ : IsTopologicalSemiring S
inst✝¹² : ContinuousStar S
inst✝¹¹ : Ring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Algebra S A
inst✝⁸ : Algebra R S
inst✝⁷ : Algebra R A
inst✝⁶ : IsScalarTower R S A
inst✝⁵ : StarModule R S
inst✝⁴ : ContinuousSMul R S
inst✝³ : MetricSpace A
inst✝² : IsometricContinuousFunctionalCalculus S A q
inst✝¹ : CompleteSpace R
inst✝ : ContinuousMap.UniqueHom R A
f : C(S, R)
halg : Isometry ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ SpectrumRestricts a ⇑f
a : A
ha : p a
ha' : q a
haf : SpectrumRestricts a ⇑f
this : ContinuousFunctionalCalculus R A p
g₁ g₂ : C(↑(σ R a), R)
x : ↑(σ R a)
x' : ↑(σ S a) := Subtype.map ⇑(algebraMap R S) ⋯ x
⊢ x = Subtype.map ⇑f ⋯ (Subtype.map ⇑(algebraMap R S) ⋯ x)

Tactic:
ext

Goal state after:
case e'_3
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²¹ : Semifield R
inst✝²⁰ : StarRing R
inst✝¹⁹ : MetricSpace R
inst✝¹⁸ : IsTopologicalSemiring R
inst✝¹⁷ : ContinuousStar R
inst✝¹⁶ : Semifield S
inst✝¹⁵ : StarRing S
inst✝¹⁴ : MetricSpace S
inst✝¹³ : IsTopologicalSemiring S
inst✝¹² : ContinuousStar S
inst✝¹¹ : Ring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Algebra S A
inst✝⁸ : Algebra R S
inst✝⁷ : Algebra R A
inst✝⁶ : IsScalarTower R S A
inst✝⁵ : StarModule R S
inst✝⁴ : ContinuousSMul R S
inst✝³ : MetricSpace A
inst✝² : IsometricContinuousFunctionalCalculus S A q
inst✝¹ : CompleteSpace R
inst✝ : ContinuousMap.UniqueHom R A
f : C(S, R)
halg : Isometry ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ SpectrumRestricts a ⇑f
a : A
ha : p a
ha' : q a
haf : SpectrumRestricts a ⇑f
this : ContinuousFunctionalCalculus R A p
g₁ g₂ : C(↑(σ R a), R)
x : ↑(σ R a)
x' : ↑(σ S a) := ⋯
⊢ ↑x = ↑(Subtype.map ⇑f ⋯ (Subtype.map ⇑(algebraMap R S) ⋯ x))
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric

Goal state before:
pp: case e'_4
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²¹ : Semifield R
inst✝²⁰ : StarRing R
inst✝¹⁹ : MetricSpace R
inst✝¹⁸ : IsTopologicalSemiring R
inst✝¹⁷ : ContinuousStar R
inst✝¹⁶ : Semifield S
inst✝¹⁵ : StarRing S
inst✝¹⁴ : MetricSpace S
inst✝¹³ : IsTopologicalSemiring S
inst✝¹² : ContinuousStar S
inst✝¹¹ : Ring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Algebra S A
inst✝⁸ : Algebra R S
inst✝⁷ : Algebra R A
inst✝⁶ : IsScalarTower R S A
inst✝⁵ : StarModule R S
inst✝⁴ : ContinuousSMul R S
inst✝³ : MetricSpace A
inst✝² : IsometricContinuousFunctionalCalculus S A q
inst✝¹ : CompleteSpace R
inst✝ : ContinuousMap.UniqueHom R A
f : C(S, R)
halg : Isometry ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ SpectrumRestricts a ⇑f
a : A
ha : p a
ha' : q a
haf : SpectrumRestricts a ⇑f
this : ContinuousFunctionalCalculus R A p
g₁ g₂ : C(↑(σ R a), R)
x : ↑(σ R a)
x' : ↑(σ S a) := Subtype.map ⇑(algebraMap R S) ⋯ x
⊢ x = Subtype.map ⇑f ⋯ (Subtype.map ⇑(algebraMap R S) ⋯ x)

Tactic:
ext

Goal state after:
case e'_4
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²¹ : Semifield R
inst✝²⁰ : StarRing R
inst✝¹⁹ : MetricSpace R
inst✝¹⁸ : IsTopologicalSemiring R
inst✝¹⁷ : ContinuousStar R
inst✝¹⁶ : Semifield S
inst✝¹⁵ : StarRing S
inst✝¹⁴ : MetricSpace S
inst✝¹³ : IsTopologicalSemiring S
inst✝¹² : ContinuousStar S
inst✝¹¹ : Ring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Algebra S A
inst✝⁸ : Algebra R S
inst✝⁷ : Algebra R A
inst✝⁶ : IsScalarTower R S A
inst✝⁵ : StarModule R S
inst✝⁴ : ContinuousSMul R S
inst✝³ : MetricSpace A
inst✝² : IsometricContinuousFunctionalCalculus S A q
inst✝¹ : CompleteSpace R
inst✝ : ContinuousMap.UniqueHom R A
f : C(S, R)
halg : Isometry ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ SpectrumRestricts a ⇑f
a : A
ha : p a
ha' : q a
haf : SpectrumRestricts a ⇑f
this : ContinuousFunctionalCalculus R A p
g₁ g₂ : C(↑(σ R a), R)
x : ↑(σ R a)
x' : ↑(σ S a) := ⋯
⊢ ↑x = ↑(Subtype.map ⇑f ⋯ (Subtype.map ⇑(algebraMap R S) ⋯ x))
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances

Goal state before:
pp: case refine_2
A : Type u_1
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module ℂ A
inst✝² : IsScalarTower ℂ A A
inst✝¹ : SMulCommClass ℂ A A
inst✝ : NonUnitalContinuousFunctionalCalculus ℂ A IsStarNormal
a : A
ha₁ : IsStarNormal a
ha₂ : QuasispectrumRestricts a ⇑Complex.reCLM
⊢ IsSelfAdjoint a

Tactic:
isSelfAdjoint_iff

Goal state after:
case refine_2
A : Type u_1
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module ℂ A
inst✝² : IsScalarTower ℂ A A
inst✝¹ : SMulCommClass ℂ A A
inst✝ : NonUnitalContinuousFunctionalCalculus ℂ A IsStarNormal
a : A
ha₁ : IsStarNormal a
ha₂ : QuasispectrumRestricts a ⇑Complex.reCLM
⊢ star a = a
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances

Goal state before:
pp: case right
A : Type u_1
inst✝⁶ : NonUnitalRing A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : Module ℝ A
inst✝² : IsScalarTower ℝ A A
inst✝¹ : SMulCommClass ℝ A A
inst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha₁ : IsSelfAdjoint a
ha₂ : QuasispectrumRestricts a ⇑ContinuousMap.realToNNReal
⊢ QuasispectrumRestricts (cfcₙ (fun x ↦ √x) a) ⇑ContinuousMap.realToNNReal ∧
    cfcₙ (fun x ↦ √x) a * cfcₙ (fun x ↦ √x) a = a

Tactic:
constructor

Goal state after:
case right.left
A : Type u_1
inst✝⁶ : NonUnitalRing A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : Module ℝ A
inst✝² : IsScalarTower ℝ A A
inst✝¹ : SMulCommClass ℝ A A
inst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha₁ : IsSelfAdjoint a
ha₂ : QuasispectrumRestricts a ⇑ContinuousMap.realToNNReal
⊢ QuasispectrumRestricts (cfcₙ (fun x ↦ √x) a) ⇑ContinuousMap.realToNNReal

case right.right
A : Type u_1
inst✝⁶ : NonUnitalRing A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : Module ℝ A
inst✝² : IsScalarTower ℝ A A
inst✝¹ : SMulCommClass ℝ A A
inst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha₁ : IsSelfAdjoint a
ha₂ : QuasispectrumRestricts a ⇑ContinuousMap.realToNNReal
⊢ cfcₙ (fun x ↦ √x) a * cfcₙ (fun x ↦ √x) a = a
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric

Goal state before:
pp: case e'_3
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²⁵ : Semifield R
inst✝²⁴ : StarRing R
inst✝²³ : MetricSpace R
inst✝²² : IsTopologicalSemiring R
inst✝²¹ : ContinuousStar R
inst✝²⁰ : Field S
inst✝¹⁹ : StarRing S
inst✝¹⁸ : MetricSpace S
inst✝¹⁷ : IsTopologicalRing S
inst✝¹⁶ : ContinuousStar S
inst✝¹⁵ : NonUnitalRing A
inst✝¹⁴ : StarRing A
inst✝¹³ : Module S A
inst✝¹² : IsScalarTower S A A
inst✝¹¹ : SMulCommClass S A A
inst✝¹⁰ : Algebra R S
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R S A
inst✝⁷ : StarModule R S
inst✝⁶ : ContinuousSMul R S
inst✝⁵ : IsScalarTower R A A
inst✝⁴ : SMulCommClass R A A
inst✝³ : MetricSpace A
inst✝² : NonUnitalIsometricContinuousFunctionalCalculus S A q
inst✝¹ : CompleteSpace R
inst✝ : ContinuousMapZero.UniqueHom R A
f : C(S, R)
halg : Isometry ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ QuasispectrumRestricts a ⇑f
a : A
ha : p a
ha' : q a
haf : QuasispectrumRestricts a ⇑f
this : NonUnitalContinuousFunctionalCalculus R A p
g₁ g₂ : ContinuousMapZero (↑(σₙ R a)) R
x : ↑(σₙ R a)
x' : ↑(σₙ S a) := Subtype.map ⇑(algebraMap R S) ⋯ x
⊢ x = Subtype.map ⇑f ⋯ (Subtype.map ⇑(algebraMap R S) ⋯ x)

Tactic:
ext

Goal state after:
case e'_3
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²⁵ : Semifield R
inst✝²⁴ : StarRing R
inst✝²³ : MetricSpace R
inst✝²² : IsTopologicalSemiring R
inst✝²¹ : ContinuousStar R
inst✝²⁰ : Field S
inst✝¹⁹ : StarRing S
inst✝¹⁸ : MetricSpace S
inst✝¹⁷ : IsTopologicalRing S
inst✝¹⁶ : ContinuousStar S
inst✝¹⁵ : NonUnitalRing A
inst✝¹⁴ : StarRing A
inst✝¹³ : Module S A
inst✝¹² : IsScalarTower S A A
inst✝¹¹ : SMulCommClass S A A
inst✝¹⁰ : Algebra R S
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R S A
inst✝⁷ : StarModule R S
inst✝⁶ : ContinuousSMul R S
inst✝⁵ : IsScalarTower R A A
inst✝⁴ : SMulCommClass R A A
inst✝³ : MetricSpace A
inst✝² : NonUnitalIsometricContinuousFunctionalCalculus S A q
inst✝¹ : CompleteSpace R
inst✝ : ContinuousMapZero.UniqueHom R A
f : C(S, R)
halg : Isometry ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ QuasispectrumRestricts a ⇑f
a : A
ha : p a
ha' : q a
haf : QuasispectrumRestricts a ⇑f
this : NonUnitalContinuousFunctionalCalculus R A p
g₁ g₂ : ContinuousMapZero (↑(σₙ R a)) R
x : ↑(σₙ R a)
x' : ↑(σₙ S a) := ⋯
⊢ ↑x = ↑(Subtype.map ⇑f ⋯ (Subtype.map ⇑(algebraMap R S) ⋯ x))
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric

Goal state before:
pp: case e'_4
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²⁵ : Semifield R
inst✝²⁴ : StarRing R
inst✝²³ : MetricSpace R
inst✝²² : IsTopologicalSemiring R
inst✝²¹ : ContinuousStar R
inst✝²⁰ : Field S
inst✝¹⁹ : StarRing S
inst✝¹⁸ : MetricSpace S
inst✝¹⁷ : IsTopologicalRing S
inst✝¹⁶ : ContinuousStar S
inst✝¹⁵ : NonUnitalRing A
inst✝¹⁴ : StarRing A
inst✝¹³ : Module S A
inst✝¹² : IsScalarTower S A A
inst✝¹¹ : SMulCommClass S A A
inst✝¹⁰ : Algebra R S
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R S A
inst✝⁷ : StarModule R S
inst✝⁶ : ContinuousSMul R S
inst✝⁵ : IsScalarTower R A A
inst✝⁴ : SMulCommClass R A A
inst✝³ : MetricSpace A
inst✝² : NonUnitalIsometricContinuousFunctionalCalculus S A q
inst✝¹ : CompleteSpace R
inst✝ : ContinuousMapZero.UniqueHom R A
f : C(S, R)
halg : Isometry ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ QuasispectrumRestricts a ⇑f
a : A
ha : p a
ha' : q a
haf : QuasispectrumRestricts a ⇑f
this : NonUnitalContinuousFunctionalCalculus R A p
g₁ g₂ : ContinuousMapZero (↑(σₙ R a)) R
x : ↑(σₙ R a)
x' : ↑(σₙ S a) := Subtype.map ⇑(algebraMap R S) ⋯ x
⊢ x = Subtype.map ⇑f ⋯ (Subtype.map ⇑(algebraMap R S) ⋯ x)

Tactic:
ext

Goal state after:
case e'_4
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²⁵ : Semifield R
inst✝²⁴ : StarRing R
inst✝²³ : MetricSpace R
inst✝²² : IsTopologicalSemiring R
inst✝²¹ : ContinuousStar R
inst✝²⁰ : Field S
inst✝¹⁹ : StarRing S
inst✝¹⁸ : MetricSpace S
inst✝¹⁷ : IsTopologicalRing S
inst✝¹⁶ : ContinuousStar S
inst✝¹⁵ : NonUnitalRing A
inst✝¹⁴ : StarRing A
inst✝¹³ : Module S A
inst✝¹² : IsScalarTower S A A
inst✝¹¹ : SMulCommClass S A A
inst✝¹⁰ : Algebra R S
inst✝⁹ : Module R A
inst✝⁸ : IsScalarTower R S A
inst✝⁷ : StarModule R S
inst✝⁶ : ContinuousSMul R S
inst✝⁵ : IsScalarTower R A A
inst✝⁴ : SMulCommClass R A A
inst✝³ : MetricSpace A
inst✝² : NonUnitalIsometricContinuousFunctionalCalculus S A q
inst✝¹ : CompleteSpace R
inst✝ : ContinuousMapZero.UniqueHom R A
f : C(S, R)
halg : Isometry ⇑(algebraMap R S)
h0 : p 0
h : ∀ (a : A), p a ↔ q a ∧ QuasispectrumRestricts a ⇑f
a : A
ha : p a
ha' : q a
haf : QuasispectrumRestricts a ⇑f
this : NonUnitalContinuousFunctionalCalculus R A p
g₁ g₂ : ContinuousMapZero (↑(σₙ R a)) R
x : ↑(σₙ R a)
x' : ↑(σₙ S a) := ⋯
⊢ ↑x = ↑(Subtype.map ⇑f ⋯ (Subtype.map ⇑(algebraMap R S) ⋯ x))
Module: Mathlib.Analysis.CStarAlgebra.Spectrum

Goal state before:
pp: A : Type u_1
inst✝ : NonUnitalCStarAlgebra A
a : A
x : ℝ≥0
hx : x ∈ quasispectrum ℝ≥0 a
⊢ x ≤ ‖a‖₊

Tactic:
by
  rw [Unitization.quasispectrum_eq_spectrum_inr' ℝ≥0 ℂ] at hx
  simpa [Unitization.nnnorm_inr] using spectrum.le_nnnorm_of_mem hx

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric

Goal state before:
pp: A : Type u_1
inst✝⁷ : NormedRing A
inst✝⁶ : StarRing A
inst✝⁵ : NormedAlgebra ℝ A
inst✝⁴ : PartialOrder A
inst✝³ : StarOrderedRing A
inst✝² : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝¹ : NonnegSpectrumClass ℝ A
inst✝ : Nontrivial A
f : ℝ≥0 → ℝ≥0
a : A
hf : ContinuousOn f (σ ℝ≥0 a)
ha : 0 ≤ a
ha' : SpectrumRestricts a ⇑ContinuousMap.realToNNReal
⊢ f '' σ ℝ≥0 a = (fun x ↦ ‖↑(f x.toNNReal)‖₊) '' σ ℝ a

pp: case hf_cont
A : Type u_1
inst✝⁷ : NormedRing A
inst✝⁶ : StarRing A
inst✝⁵ : NormedAlgebra ℝ A
inst✝⁴ : PartialOrder A
inst✝³ : StarOrderedRing A
inst✝² : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝¹ : NonnegSpectrumClass ℝ A
inst✝ : Nontrivial A
f : ℝ≥0 → ℝ≥0
a : A
hf : ContinuousOn f (σ ℝ≥0 a)
ha : 0 ≤ a
ha' : SpectrumRestricts a ⇑ContinuousMap.realToNNReal
⊢ ContinuousOn (fun x ↦ ↑(f x.toNNReal)) (σ ℝ a)

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

Goal state after:
A : Type u_1
inst✝⁷ : NormedRing A
inst✝⁶ : StarRing A
inst✝⁵ : NormedAlgebra ℝ A
inst✝⁴ : PartialOrder A
inst✝³ : StarOrderedRing A
inst✝² : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝¹ : NonnegSpectrumClass ℝ A
inst✝ : Nontrivial A
f : ℝ≥0 → ℝ≥0
a : A
hf : ContinuousOn f (σ ℝ≥0 a)
ha : 0 ≤ a
ha' : SpectrumRestricts a ⇑ContinuousMap.realToNNReal
⊢ f '' σ ℝ≥0 a = (fun x ↦ ‖↑(f x.toNNReal)‖₊) '' σ ℝ a
Module: Mathlib.Analysis.Normed.Algebra.GelfandFormula

Goal state before:
pp: A : Type u_2
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra ℂ A
inst✝ : CompleteSpace A
a : A
r : ℝ≥0
r_pos : 0 < r
r_lt : ↑r < (spectralRadius ℂ a)⁻¹
p : FormalMultilinearSeries ℂ ℂ A := fun n ↦ ContinuousMultilinearMap.mkPiRing ℂ (Fin n) (a ^ n)
h : ↑r ≤ p.radius
⊢ ↑r ≤ liminf (fun i ↦ 1 / ↑‖a ^ i‖₊ ^ (1 / ↑i)) atTop

Tactic:
convert! h
    simp only [p, p.radius_eq_liminf, ← norm_toNNReal, norm_mkPiRing]
    congr
    ext n
    rw [norm_toNNReal, ENNReal.coe_rpow_def ‖a ^ n‖₊ (1 / n : ℝ), if_neg]
    exact fun ha => (lt_self_iff_false _).mp
      (ha.2.trans_le (one_div_nonneg.mpr n.cast_nonneg : 0 ≤ (1 / n : ℝ)))

Goal state after:
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
f : α → β
⊢ ContinuousSup (Compacts α)

Tactic:
constructor

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
f : α → β
⊢ Continuous fun p ↦ p.1 ⊔ p.2
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric

Goal state before:
pp: A : Type u_1
inst✝⁸ : NonUnitalNormedRing A
inst✝⁷ : StarRing A
inst✝⁶ : NormedSpace ℝ A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : SMulCommClass ℝ A A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
inst✝¹ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
f : ℝ≥0 → ℝ≥0
a : A
hf : ContinuousOn f (σₙ ℝ≥0 a)
hf0 : f 0 = 0
ha : 0 ≤ a
ha' : QuasispectrumRestricts a ⇑ContinuousMap.realToNNReal
⊢ f '' σₙ ℝ≥0 a = (fun x ↦ ‖↑(f x.toNNReal)‖₊) '' σₙ ℝ a

pp: case hf_cont
A : Type u_1
inst✝⁸ : NonUnitalNormedRing A
inst✝⁷ : StarRing A
inst✝⁶ : NormedSpace ℝ A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : SMulCommClass ℝ A A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
inst✝¹ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
f : ℝ≥0 → ℝ≥0
a : A
hf : ContinuousOn f (σₙ ℝ≥0 a)
hf0 : f 0 = 0
ha : 0 ≤ a
ha' : QuasispectrumRestricts a ⇑ContinuousMap.realToNNReal
⊢ ContinuousOn (fun x ↦ ↑(f x.toNNReal)) (σₙ ℝ a)

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

Goal state after:
A : Type u_1
inst✝⁸ : NonUnitalNormedRing A
inst✝⁷ : StarRing A
inst✝⁶ : NormedSpace ℝ A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : SMulCommClass ℝ A A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
inst✝¹ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
f : ℝ≥0 → ℝ≥0
a : A
hf : ContinuousOn f (σₙ ℝ≥0 a)
hf0 : f 0 = 0
ha : 0 ≤ a
ha' : QuasispectrumRestricts a ⇑ContinuousMap.realToNNReal
⊢ f '' σₙ ℝ≥0 a = (fun x ↦ ‖↑(f x.toNNReal)‖₊) '' σₙ ℝ a
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
f : α → Compacts β
g : α → β → γ
hf : Continuous[inst✝², _] f
hg : Continuous[instTopologicalSpaceProd, inst✝] (Function.uncurry g)
x : α
⊢ Compacts.map (g x) ⋯ (f x) = Compacts.map (Function.uncurry g) hg ({x} ×ˢ f x)

Tactic:
ext

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
f : α → Compacts β
g : α → β → γ
hf : Continuous[inst✝², _] f
hg : Continuous[instTopologicalSpaceProd, inst✝] (Function.uncurry g)
x : α
x✝ : γ
⊢ x✝ ∈ ↑(Compacts.map (g x) ⋯ (f x)) ↔ x✝ ∈ ↑(Compacts.map (Function.uncurry g) hg ({x} ×ˢ f x))
Module: Mathlib.Analysis.CStarAlgebra.Spectrum

Goal state before:
pp: F : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : NonUnitalCStarAlgebra A
inst✝³ : NonUnitalCStarAlgebra B
inst✝² : FunLike F A B
inst✝¹ : NonUnitalAlgHomClass F ℂ A B
inst✝ : StarHomClass F A B
φ : F
⊢ ∀ (x : A), ‖φ x‖ ≤ 1 * ‖x‖

Tactic:
by simpa only [one_mul] using! nnnorm_apply_le φ

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

Goal state before:
pp: A : Type u_1
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra ℂ A
inst✝³ : StarRing A
inst✝² : ContinuousStar A
inst✝¹ : CompleteSpace A
inst✝ : StarModule ℂ A
⊢ expUnitary 0 = 1

Tactic:
ext

Goal state after:
A : Type u_1
inst✝⁵ : NormedRing A
inst✝⁴ : NormedAlgebra ℂ A
inst✝³ : StarRing A
inst✝² : ContinuousStar A
inst✝¹ : CompleteSpace A
inst✝ : StarModule ℂ A
⊢ ↑(expUnitary 0) = ↑1
Module: Mathlib.Topology.ContinuousMap.ZeroAtInfty

Goal state before:
pp: F : Type u_1
α : Type u
β : Type v
γ : Type w
inst✝⁷ : TopologicalSpace α
inst✝⁶ : TopologicalSpace β
x : α
R : Type u_2
inst✝⁵ : Semiring R
inst✝⁴ : NonUnitalNonAssocSemiring β
inst✝³ : IsTopologicalSemiring β
inst✝² : Module R β
inst✝¹ : ContinuousConstSMul R β
inst✝ : IsScalarTower R β β
r : R
f g : α →C₀ β
⊢ (r • f) • g = r • f • g

Tactic:
ext

Goal state after:
F : Type u_1
α : Type u
β : Type v
γ : Type w
inst✝⁷ : TopologicalSpace α
inst✝⁶ : TopologicalSpace β
x : α
R : Type u_2
inst✝⁵ : Semiring R
inst✝⁴ : NonUnitalNonAssocSemiring β
inst✝³ : IsTopologicalSemiring β
inst✝² : Module R β
inst✝¹ : ContinuousConstSMul R β
inst✝ : IsScalarTower R β β
r : R
f g : α →C₀ β
x✝ : α
⊢ ((r • f) • g) x✝ = (r • f • g) x✝
Module: Mathlib.Topology.ContinuousMap.ZeroAtInfty

Goal state before:
pp: F : Type u_1
α : Type u
β : Type v
γ : Type w
inst✝⁷ : TopologicalSpace α
inst✝⁶ : TopologicalSpace β
x : α
R : Type u_2
inst✝⁵ : Semiring R
inst✝⁴ : NonUnitalNonAssocSemiring β
inst✝³ : IsTopologicalSemiring β
inst✝² : Module R β
inst✝¹ : ContinuousConstSMul R β
inst✝ : SMulCommClass R β β
r : R
f g : α →C₀ β
⊢ r • f • g = f • r • g

Tactic:
ext

Goal state after:
F : Type u_1
α : Type u
β : Type v
γ : Type w
inst✝⁷ : TopologicalSpace α
inst✝⁶ : TopologicalSpace β
x : α
R : Type u_2
inst✝⁵ : Semiring R
inst✝⁴ : NonUnitalNonAssocSemiring β
inst✝³ : IsTopologicalSemiring β
inst✝² : Module R β
inst✝¹ : ContinuousConstSMul R β
inst✝ : SMulCommClass R β β
r : R
f g : α →C₀ β
x✝ : α
⊢ (r • f • g) x✝ = (f • r • g) x✝
Module: Mathlib.Analysis.LocallyConvex.Polar

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NormedCommRing 𝕜
inst✝³ : AddCommMonoid E
inst✝² : AddCommMonoid F
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜
s : Set E
⊢ B.polar s = ⋂ x ∈ s, {y | ‖(B x) y‖ ≤ 1}

Tactic:
ext

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NormedCommRing 𝕜
inst✝³ : AddCommMonoid E
inst✝² : AddCommMonoid F
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜
s : Set E
x✝ : F
⊢ x✝ ∈ B.polar s ↔ x✝ ∈ ⋂ x ∈ s, {y | ‖(B x) y‖ ≤ 1}
Module: Mathlib.Analysis.CStarAlgebra.Fuglede

Goal state before:
pp: case refine_1
A : Type u_1
inst✝ : CStarAlgebra A
a : A
x✝ : NormedAlgebra ℚ A := NormedAlgebra.restrictScalars ℚ ℂ A
this : IsAddTorsionFree A
ha : IsStarNormal a
x : ℝ
⊢ NormedSpace.exp (x • a) * NormedSpace.exp (-x • star a) ∈ unitary A

Tactic:
convert! (selfAdjoint.expUnitary (x • (2 : ℝ) • ℑ a)).2

Goal state after:
A : Type u_1
inst✝ : CStarAlgebra A
a : A
x✝ : NormedAlgebra ℚ A := NormedAlgebra.restrictScalars ℚ ℂ A
this : IsAddTorsionFree A
ha : IsStarNormal a
x : ℝ
⊢ NormedSpace.exp (x • a) * NormedSpace.exp (-x • star a) = ↑(expUnitary (x • 2 • ℑ a))
Module: Mathlib.Analysis.LocallyConvex.Polar

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : AddCommMonoid E
inst✝⁴ : AddCommMonoid F
inst✝³ : Module 𝕜 E
inst✝² : Module 𝕜 F
B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜
S : Type u_4
inst✝¹ : SetLike S E
inst✝ : SMulMemClass S 𝕜 E
m : S
y : F
⊢ y ∈ B.polar ↑m ↔ y ∈ {y | ∀ x ∈ m, (B x) y = 0}

Tactic:
constructor

Goal state after:
case mp
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : AddCommMonoid E
inst✝⁴ : AddCommMonoid F
inst✝³ : Module 𝕜 E
inst✝² : Module 𝕜 F
B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜
S : Type u_4
inst✝¹ : SetLike S E
inst✝ : SMulMemClass S 𝕜 E
m : S
y : F
⊢ y ∈ B.polar ↑m → y ∈ {y | ∀ x ∈ m, (B x) y = 0}

case mpr
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : AddCommMonoid E
inst✝⁴ : AddCommMonoid F
inst✝³ : Module 𝕜 E
inst✝² : Module 𝕜 F
B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜
S : Type u_4
inst✝¹ : SetLike S E
inst✝ : SMulMemClass S 𝕜 E
m : S
y : F
⊢ y ∈ {y | ∀ x ∈ m, (B x) y = 0} → y ∈ B.polar ↑m
Module: Mathlib.Analysis.LocallyConvex.AbsConvex

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : SeminormedRing 𝕜
inst✝² : SMul 𝕜 E
inst✝¹ : AddCommMonoid E
inst✝ : PartialOrder 𝕜
s : Set E
⊢ (absConvexHull 𝕜) s = ∅ ↔ s = ∅

Tactic:
constructor

Goal state after:
case mp
𝕜 : Type u_1
E : Type u_2
inst✝³ : SeminormedRing 𝕜
inst✝² : SMul 𝕜 E
inst✝¹ : AddCommMonoid E
inst✝ : PartialOrder 𝕜
s : Set E
⊢ (absConvexHull 𝕜) s = ∅ → s = ∅

case mpr
𝕜 : Type u_1
E : Type u_2
inst✝³ : SeminormedRing 𝕜
inst✝² : SMul 𝕜 E
inst✝¹ : AddCommMonoid E
inst✝ : PartialOrder 𝕜
s : Set E
⊢ s = ∅ → (absConvexHull 𝕜) s = ∅
Module: Mathlib.Analysis.LocallyConvex.Polar

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : AddCommMonoid E
inst✝⁴ : AddCommMonoid F
inst✝³ : Module 𝕜 E
inst✝² : Module 𝕜 F
B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜
S : Type u_4
inst✝¹ : SetLike S E
inst✝ : SMulMemClass S 𝕜 E
m : S
⊢ B.polar ↑m = ↑(⨅ x ∈ m, (B x).ker)

Tactic:
ext

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : AddCommMonoid E
inst✝⁴ : AddCommMonoid F
inst✝³ : Module 𝕜 E
inst✝² : Module 𝕜 F
B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜
S : Type u_4
inst✝¹ : SetLike S E
inst✝ : SMulMemClass S 𝕜 E
m : S
x✝ : F
⊢ x✝ ∈ B.polar ↑m ↔ x✝ ∈ ↑(⨅ x ∈ m, (B x).ker)
Module: Mathlib.Analysis.LocallyConvex.AbsConvexOpen

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁶ : RCLike 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : TopologicalSpace E
inst✝³ : Module 𝕜 E
inst✝² : Module ℝ E
inst✝¹ : IsScalarTower ℝ 𝕜 E
inst✝ : ContinuousSMul ℝ E
s : AbsConvexOpenSets 𝕜 E
⊢ (gaugeSeminorm ⋯ ⋯ ⋯).ball 0 1 = ↑s

Tactic:
Seminorm.ball_zero_eq

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝⁶ : RCLike 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : TopologicalSpace E
inst✝³ : Module 𝕜 E
inst✝² : Module ℝ E
inst✝¹ : IsScalarTower ℝ 𝕜 E
inst✝ : ContinuousSMul ℝ E
s : AbsConvexOpenSets 𝕜 E
⊢ {y | (gaugeSeminorm ⋯ ⋯ ⋯) y < 1} = ↑s
Module: Mathlib.Topology.Sets.VietorisTopology

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
f : α → β
⊢ ContinuousSup (NonemptyCompacts α)

Tactic:
constructor

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
f : α → β
⊢ Continuous fun p ↦ p.1 ⊔ p.2