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

Goal state before:
pp: case pos
ι : Type u_1
R : Type u_2
S : Type u_3
A : ι → Type u_4
inst✝¹⁸ : CommSemiring R
inst✝¹⁷ : Nontrivial R
inst✝¹⁶ : StarRing R
inst✝¹⁵ : MetricSpace R
inst✝¹⁴ : IsTopologicalSemiring R
inst✝¹³ : ContinuousStar R
inst✝¹² : CommRing S
inst✝¹¹ : Algebra R S
inst✝¹⁰ : (i : ι) → NonUnitalRing (A i)
inst✝⁹ : (i : ι) → Module S (A i)
inst✝⁸ : (i : ι) → Module R (A i)
inst✝⁷ : ∀ (i : ι), IsScalarTower R S (A i)
inst✝⁶ : ∀ (i : ι), SMulCommClass R (A i) (A i)
inst✝⁵ : ∀ (i : ι), IsScalarTower R (A i) (A i)
inst✝⁴ : (i : ι) → StarRing (A i)
inst✝³ : (i : ι) → TopologicalSpace (A i)
p : ((i : ι) → A i) → Prop
q : (i : ι) → A i → Prop
inst✝² : NonUnitalContinuousFunctionalCalculus R ((i : ι) → A i) p
inst✝¹ : ∀ (i : ι), NonUnitalContinuousFunctionalCalculus R (A i) (q i)
inst✝ : ∀ (i : ι), ContinuousMapZero.UniqueHom R (A i)
f : R → R
a : (i : ι) → A i
hf : ContinuousOn f (⋃ i, quasispectrum R (a i))
ha : p a
ha' : ∀ (i : ι), q i (a i)
hf₀ : f 0 = 0
⊢ cfcₙ f a = fun i ↦ cfcₙ f (a i)

pp: case neg
ι : Type u_1
R : Type u_2
S : Type u_3
A : ι → Type u_4
inst✝¹⁸ : CommSemiring R
inst✝¹⁷ : Nontrivial R
inst✝¹⁶ : StarRing R
inst✝¹⁵ : MetricSpace R
inst✝¹⁴ : IsTopologicalSemiring R
inst✝¹³ : ContinuousStar R
inst✝¹² : CommRing S
inst✝¹¹ : Algebra R S
inst✝¹⁰ : (i : ι) → NonUnitalRing (A i)
inst✝⁹ : (i : ι) → Module S (A i)
inst✝⁸ : (i : ι) → Module R (A i)
inst✝⁷ : ∀ (i : ι), IsScalarTower R S (A i)
inst✝⁶ : ∀ (i : ι), SMulCommClass R (A i) (A i)
inst✝⁵ : ∀ (i : ι), IsScalarTower R (A i) (A i)
inst✝⁴ : (i : ι) → StarRing (A i)
inst✝³ : (i : ι) → TopologicalSpace (A i)
p : ((i : ι) → A i) → Prop
q : (i : ι) → A i → Prop
inst✝² : NonUnitalContinuousFunctionalCalculus R ((i : ι) → A i) p
inst✝¹ : ∀ (i : ι), NonUnitalContinuousFunctionalCalculus R (A i) (q i)
inst✝ : ∀ (i : ι), ContinuousMapZero.UniqueHom R (A i)
f : R → R
a : (i : ι) → A i
hf : ContinuousOn f (⋃ i, quasispectrum R (a i))
ha : p a
ha' : ∀ (i : ι), q i (a i)
hf₀ : ¬f 0 = 0
⊢ cfcₙ f a = fun i ↦ cfcₙ f (a i)

Tactic:
·

Goal state after:
case pos
ι : Type u_1
R : Type u_2
S : Type u_3
A : ι → Type u_4
inst✝¹⁸ : CommSemiring R
inst✝¹⁷ : Nontrivial R
inst✝¹⁶ : StarRing R
inst✝¹⁵ : MetricSpace R
inst✝¹⁴ : IsTopologicalSemiring R
inst✝¹³ : ContinuousStar R
inst✝¹² : CommRing S
inst✝¹¹ : Algebra R S
inst✝¹⁰ : (i : ι) → NonUnitalRing (A i)
inst✝⁹ : (i : ι) → Module S (A i)
inst✝⁸ : (i : ι) → Module R (A i)
inst✝⁷ : ∀ (i : ι), IsScalarTower R S (A i)
inst✝⁶ : ∀ (i : ι), SMulCommClass R (A i) (A i)
inst✝⁵ : ∀ (i : ι), IsScalarTower R (A i) (A i)
inst✝⁴ : (i : ι) → StarRing (A i)
inst✝³ : (i : ι) → TopologicalSpace (A i)
p : ((i : ι) → A i) → Prop
q : (i : ι) → A i → Prop
inst✝² : NonUnitalContinuousFunctionalCalculus R ((i : ι) → A i) p
inst✝¹ : ∀ (i : ι), NonUnitalContinuousFunctionalCalculus R (A i) (q i)
inst✝ : ∀ (i : ι), ContinuousMapZero.UniqueHom R (A i)
f : R → R
a : (i : ι) → A i
hf : ContinuousOn f (⋃ i, quasispectrum R (a i))
ha : p a
ha' : ∀ (i : ι), q i (a i)
hf₀ : f 0 = 0
⊢ cfcₙ f a = fun i ↦ cfcₙ f (a i)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Pi

Goal state before:
pp: case neg
ι : Type u_1
R : Type u_2
S : Type u_3
A : ι → Type u_4
inst✝¹⁸ : CommSemiring R
inst✝¹⁷ : Nontrivial R
inst✝¹⁶ : StarRing R
inst✝¹⁵ : MetricSpace R
inst✝¹⁴ : IsTopologicalSemiring R
inst✝¹³ : ContinuousStar R
inst✝¹² : CommRing S
inst✝¹¹ : Algebra R S
inst✝¹⁰ : (i : ι) → NonUnitalRing (A i)
inst✝⁹ : (i : ι) → Module S (A i)
inst✝⁸ : (i : ι) → Module R (A i)
inst✝⁷ : ∀ (i : ι), IsScalarTower R S (A i)
inst✝⁶ : ∀ (i : ι), SMulCommClass R (A i) (A i)
inst✝⁵ : ∀ (i : ι), IsScalarTower R (A i) (A i)
inst✝⁴ : (i : ι) → StarRing (A i)
inst✝³ : (i : ι) → TopologicalSpace (A i)
p : ((i : ι) → A i) → Prop
q : (i : ι) → A i → Prop
inst✝² : NonUnitalContinuousFunctionalCalculus R ((i : ι) → A i) p
inst✝¹ : ∀ (i : ι), NonUnitalContinuousFunctionalCalculus R (A i) (q i)
inst✝ : ∀ (i : ι), ContinuousMapZero.UniqueHom R (A i)
f : R → R
a : (i : ι) → A i
hf : ContinuousOn f (⋃ i, quasispectrum R (a i))
ha : p a
ha' : ∀ (i : ι), q i (a i)
hf₀ : ¬f 0 = 0
⊢ cfcₙ f a = fun i ↦ cfcₙ f (a i)

Tactic:
·

Goal state after:
case neg
ι : Type u_1
R : Type u_2
S : Type u_3
A : ι → Type u_4
inst✝¹⁸ : CommSemiring R
inst✝¹⁷ : Nontrivial R
inst✝¹⁶ : StarRing R
inst✝¹⁵ : MetricSpace R
inst✝¹⁴ : IsTopologicalSemiring R
inst✝¹³ : ContinuousStar R
inst✝¹² : CommRing S
inst✝¹¹ : Algebra R S
inst✝¹⁰ : (i : ι) → NonUnitalRing (A i)
inst✝⁹ : (i : ι) → Module S (A i)
inst✝⁸ : (i : ι) → Module R (A i)
inst✝⁷ : ∀ (i : ι), IsScalarTower R S (A i)
inst✝⁶ : ∀ (i : ι), SMulCommClass R (A i) (A i)
inst✝⁵ : ∀ (i : ι), IsScalarTower R (A i) (A i)
inst✝⁴ : (i : ι) → StarRing (A i)
inst✝³ : (i : ι) → TopologicalSpace (A i)
p : ((i : ι) → A i) → Prop
q : (i : ι) → A i → Prop
inst✝² : NonUnitalContinuousFunctionalCalculus R ((i : ι) → A i) p
inst✝¹ : ∀ (i : ι), NonUnitalContinuousFunctionalCalculus R (A i) (q i)
inst✝ : ∀ (i : ι), ContinuousMapZero.UniqueHom R (A i)
f : R → R
a : (i : ι) → A i
hf : ContinuousOn f (⋃ i, quasispectrum R (a i))
ha : p a
ha' : ∀ (i : ι), q i (a i)
hf₀ : ¬f 0 = 0
⊢ cfcₙ f a = fun i ↦ cfcₙ f (a i)
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic

Goal state before:
pp: case pos
A : Type u_1
inst✝⁶ : NonUnitalRing A
inst✝⁵ : Module ℝ A
inst✝⁴ : SMulCommClass ℝ A A
inst✝³ : IsScalarTower ℝ A A
inst✝² : StarRing A
inst✝¹ : TopologicalSpace A
inst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha : IsSelfAdjoint a
⊢ cfcₙ (fun x ↦ x⁺) a * cfcₙ (fun x ↦ x⁻) a = 0

pp: case neg
A : Type u_1
inst✝⁶ : NonUnitalRing A
inst✝⁵ : Module ℝ A
inst✝⁴ : SMulCommClass ℝ A A
inst✝³ : IsScalarTower ℝ A A
inst✝² : StarRing A
inst✝¹ : TopologicalSpace A
inst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha : ¬IsSelfAdjoint a
⊢ cfcₙ (fun x ↦ x⁺) a * cfcₙ (fun x ↦ x⁻) a = 0

Tactic:
·

Goal state after:
case pos
A : Type u_1
inst✝⁶ : NonUnitalRing A
inst✝⁵ : Module ℝ A
inst✝⁴ : SMulCommClass ℝ A A
inst✝³ : IsScalarTower ℝ A A
inst✝² : StarRing A
inst✝¹ : TopologicalSpace A
inst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha : IsSelfAdjoint a
⊢ cfcₙ (fun x ↦ x⁺) a * cfcₙ (fun x ↦ x⁻) a = 0
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic

Goal state before:
pp: case neg
A : Type u_1
inst✝⁶ : NonUnitalRing A
inst✝⁵ : Module ℝ A
inst✝⁴ : SMulCommClass ℝ A A
inst✝³ : IsScalarTower ℝ A A
inst✝² : StarRing A
inst✝¹ : TopologicalSpace A
inst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha : ¬IsSelfAdjoint a
⊢ cfcₙ (fun x ↦ x⁺) a * cfcₙ (fun x ↦ x⁻) a = 0

Tactic:
·

Goal state after:
case neg
A : Type u_1
inst✝⁶ : NonUnitalRing A
inst✝⁵ : Module ℝ A
inst✝⁴ : SMulCommClass ℝ A A
inst✝³ : IsScalarTower ℝ A A
inst✝² : StarRing A
inst✝¹ : TopologicalSpace A
inst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha : ¬IsSelfAdjoint a
⊢ cfcₙ (fun x ↦ x⁺) a * cfcₙ (fun x ↦ x⁻) a = 0
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic

Goal state before:
pp: case pos
A : Type u_1
inst✝⁶ : NonUnitalRing A
inst✝⁵ : Module ℝ A
inst✝⁴ : SMulCommClass ℝ A A
inst✝³ : IsScalarTower ℝ A A
inst✝² : StarRing A
inst✝¹ : TopologicalSpace A
inst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha : IsSelfAdjoint a
⊢ cfcₙ (fun x ↦ x⁻) a * cfcₙ (fun x ↦ x⁺) a = 0

pp: case neg
A : Type u_1
inst✝⁶ : NonUnitalRing A
inst✝⁵ : Module ℝ A
inst✝⁴ : SMulCommClass ℝ A A
inst✝³ : IsScalarTower ℝ A A
inst✝² : StarRing A
inst✝¹ : TopologicalSpace A
inst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha : ¬IsSelfAdjoint a
⊢ cfcₙ (fun x ↦ x⁻) a * cfcₙ (fun x ↦ x⁺) a = 0

Tactic:
·

Goal state after:
case pos
A : Type u_1
inst✝⁶ : NonUnitalRing A
inst✝⁵ : Module ℝ A
inst✝⁴ : SMulCommClass ℝ A A
inst✝³ : IsScalarTower ℝ A A
inst✝² : StarRing A
inst✝¹ : TopologicalSpace A
inst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha : IsSelfAdjoint a
⊢ cfcₙ (fun x ↦ x⁻) a * cfcₙ (fun x ↦ x⁺) a = 0
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic

Goal state before:
pp: case neg
A : Type u_1
inst✝⁶ : NonUnitalRing A
inst✝⁵ : Module ℝ A
inst✝⁴ : SMulCommClass ℝ A A
inst✝³ : IsScalarTower ℝ A A
inst✝² : StarRing A
inst✝¹ : TopologicalSpace A
inst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha : ¬IsSelfAdjoint a
⊢ cfcₙ (fun x ↦ x⁻) a * cfcₙ (fun x ↦ x⁺) a = 0

Tactic:
·

Goal state after:
case neg
A : Type u_1
inst✝⁶ : NonUnitalRing A
inst✝⁵ : Module ℝ A
inst✝⁴ : SMulCommClass ℝ A A
inst✝³ : IsScalarTower ℝ A A
inst✝² : StarRing A
inst✝¹ : TopologicalSpace A
inst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
a : A
ha : ¬IsSelfAdjoint a
⊢ cfcₙ (fun x ↦ x⁻) a * cfcₙ (fun x ↦ x⁺) a = 0
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic

Goal state before:
pp: case pos
A : Type u_1
inst✝⁷ : NonUnitalRing A
inst✝⁶ : Module ℝ A
inst✝⁵ : SMulCommClass ℝ A A
inst✝⁴ : IsScalarTower ℝ A A
inst✝³ : StarRing A
inst✝² : TopologicalSpace A
inst✝¹ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : T2Space A
a : A
ha : IsSelfAdjoint a
⊢ (-a)⁺ = a⁻

pp: case neg
A : Type u_1
inst✝⁷ : NonUnitalRing A
inst✝⁶ : Module ℝ A
inst✝⁵ : SMulCommClass ℝ A A
inst✝⁴ : IsScalarTower ℝ A A
inst✝³ : StarRing A
inst✝² : TopologicalSpace A
inst✝¹ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : T2Space A
a : A
ha : ¬IsSelfAdjoint a
⊢ (-a)⁺ = a⁻

Tactic:
·

Goal state after:
case pos
A : Type u_1
inst✝⁷ : NonUnitalRing A
inst✝⁶ : Module ℝ A
inst✝⁵ : SMulCommClass ℝ A A
inst✝⁴ : IsScalarTower ℝ A A
inst✝³ : StarRing A
inst✝² : TopologicalSpace A
inst✝¹ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : T2Space A
a : A
ha : IsSelfAdjoint a
⊢ (-a)⁺ = a⁻
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic

Goal state before:
pp: case pos
A : Type u_1
inst✝⁷ : NonUnitalRing A
inst✝⁶ : Module ℝ A
inst✝⁵ : SMulCommClass ℝ A A
inst✝⁴ : IsScalarTower ℝ A A
inst✝³ : StarRing A
inst✝² : TopologicalSpace A
inst✝¹ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : T2Space A
a : A
ha : IsSelfAdjoint a
⊢ cfcₙ (fun x ↦ x⁺) (-a) = cfcₙ (fun x ↦ x⁻) a

Tactic:
← cfcₙ_comp_neg _ _

Goal state after:
case pos
A : Type u_1
inst✝⁷ : NonUnitalRing A
inst✝⁶ : Module ℝ A
inst✝⁵ : SMulCommClass ℝ A A
inst✝⁴ : IsScalarTower ℝ A A
inst✝³ : StarRing A
inst✝² : TopologicalSpace A
inst✝¹ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : T2Space A
a : A
ha : IsSelfAdjoint a
⊢ cfcₙ (fun x ↦ (-x)⁺) a = cfcₙ (fun x ↦ x⁻) a
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic

Goal state before:
pp: case neg
A : Type u_1
inst✝⁷ : NonUnitalRing A
inst✝⁶ : Module ℝ A
inst✝⁵ : SMulCommClass ℝ A A
inst✝⁴ : IsScalarTower ℝ A A
inst✝³ : StarRing A
inst✝² : TopologicalSpace A
inst✝¹ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : T2Space A
a : A
ha : ¬IsSelfAdjoint a
⊢ (-a)⁺ = a⁻

Tactic:
·

Goal state after:
case neg
A : Type u_1
inst✝⁷ : NonUnitalRing A
inst✝⁶ : Module ℝ A
inst✝⁵ : SMulCommClass ℝ A A
inst✝⁴ : IsScalarTower ℝ A A
inst✝³ : StarRing A
inst✝² : TopologicalSpace A
inst✝¹ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : T2Space A
a : A
ha : ¬IsSelfAdjoint a
⊢ (-a)⁺ = a⁻
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic

Goal state before:
pp: case pos
A : Type u_1
inst✝⁸ : NonUnitalRing A
inst✝⁷ : Module ℝ A
inst✝⁶ : SMulCommClass ℝ A A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝¹ : T2Space A
inst✝ : StarModule ℝ A
r : ℝ≥0
a : A
ha : IsSelfAdjoint a
⊢ (r • a)⁺ = r • a⁺

pp: case neg
A : Type u_1
inst✝⁸ : NonUnitalRing A
inst✝⁷ : Module ℝ A
inst✝⁶ : SMulCommClass ℝ A A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝¹ : T2Space A
inst✝ : StarModule ℝ A
r : ℝ≥0
a : A
ha : ¬IsSelfAdjoint a
⊢ (r • a)⁺ = r • a⁺

Tactic:
·

Goal state after:
case pos
A : Type u_1
inst✝⁸ : NonUnitalRing A
inst✝⁷ : Module ℝ A
inst✝⁶ : SMulCommClass ℝ A A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝¹ : T2Space A
inst✝ : StarModule ℝ A
r : ℝ≥0
a : A
ha : IsSelfAdjoint a
⊢ (r • a)⁺ = r • a⁺
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic

Goal state before:
pp: case neg
A : Type u_1
inst✝⁸ : NonUnitalRing A
inst✝⁷ : Module ℝ A
inst✝⁶ : SMulCommClass ℝ A A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝¹ : T2Space A
inst✝ : StarModule ℝ A
r : ℝ≥0
a : A
ha : ¬IsSelfAdjoint a
⊢ (r • a)⁺ = r • a⁺

Tactic:
·

Goal state after:
case neg
A : Type u_1
inst✝⁸ : NonUnitalRing A
inst✝⁷ : Module ℝ A
inst✝⁶ : SMulCommClass ℝ A A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝¹ : T2Space A
inst✝ : StarModule ℝ A
r : ℝ≥0
a : A
ha : ¬IsSelfAdjoint a
⊢ (r • a)⁺ = r • a⁺
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic

Goal state before:
pp: case neg.inl
A : Type u_1
inst✝⁸ : NonUnitalRing A
inst✝⁷ : Module ℝ A
inst✝⁶ : SMulCommClass ℝ A A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝¹ : T2Space A
inst✝ : StarModule ℝ A
a : A
ha : ¬IsSelfAdjoint a
⊢ (0 • a)⁺ = 0 • a⁺

pp: case neg.inr
A : Type u_1
inst✝⁸ : NonUnitalRing A
inst✝⁷ : Module ℝ A
inst✝⁶ : SMulCommClass ℝ A A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝¹ : T2Space A
inst✝ : StarModule ℝ A
r : ℝ≥0
a : A
ha : ¬IsSelfAdjoint a
hr : r ≠ 0
⊢ (r • a)⁺ = r • a⁺

Tactic:
·

Goal state after:
case neg.inl
A : Type u_1
inst✝⁸ : NonUnitalRing A
inst✝⁷ : Module ℝ A
inst✝⁶ : SMulCommClass ℝ A A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝¹ : T2Space A
inst✝ : StarModule ℝ A
a : A
ha : ¬IsSelfAdjoint a
⊢ (0 • a)⁺ = 0 • a⁺
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic

Goal state before:
pp: case neg.inr
A : Type u_1
inst✝⁸ : NonUnitalRing A
inst✝⁷ : Module ℝ A
inst✝⁶ : SMulCommClass ℝ A A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝¹ : T2Space A
inst✝ : StarModule ℝ A
r : ℝ≥0
a : A
ha : ¬IsSelfAdjoint a
hr : r ≠ 0
⊢ (r • a)⁺ = r • a⁺

Tactic:
·

Goal state after:
case neg.inr
A : Type u_1
inst✝⁸ : NonUnitalRing A
inst✝⁷ : Module ℝ A
inst✝⁶ : SMulCommClass ℝ A A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝¹ : T2Space A
inst✝ : StarModule ℝ A
r : ℝ≥0
a : A
ha : ¬IsSelfAdjoint a
hr : r ≠ 0
⊢ (r • a)⁺ = r • a⁺
Module: Mathlib.Topology.UrysohnsLemma

Goal state before:
pp: case h.refine_1
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : R1Space X
s t : Set X
hs : IsOpen[inst✝¹] s
hscp : IsCompact (closure[inst✝¹] sᶜᶜ)
ht : IsClosed[inst✝¹] t
hst : t ⊆ s
u v : Set X
huIsOpen : IsOpen[inst✝¹] u
hvIsOpen : IsOpen[inst✝¹] v
hscompl_subset_u : sᶜ ⊆ u
ht_subset_v : t ⊆ v
hDisjointuv : u ⊆ vᶜ
huvc : closure[inst✝¹] u ⊆ vᶜ
P : Set X → Set X → Prop := fun C x ↦ sᶜ ⊆ C
c : Urysohns.CU P := { C := closure[inst✝¹] u, U := tᶜ, P_C_U := ⋯, closed_C := ⋯, open_U := ⋯, subset := ⋯, hP := ⋯ }
⊢ tsupport c.lim ⊆ s

pp: case h.refine_2
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : R1Space X
s t : Set X
hs : IsOpen[inst✝¹] s
hscp : IsCompact (closure[inst✝¹] sᶜᶜ)
ht : IsClosed[inst✝¹] t
hst : t ⊆ s
u v : Set X
huIsOpen : IsOpen[inst✝¹] u
hvIsOpen : IsOpen[inst✝¹] v
hscompl_subset_u : sᶜ ⊆ u
ht_subset_v : t ⊆ v
hDisjointuv : u ⊆ vᶜ
huvc : closure[inst✝¹] u ⊆ vᶜ
P : Set X → Set X → Prop := fun C x ↦ sᶜ ⊆ C
c : Urysohns.CU P := { C := closure[inst✝¹] u, U := tᶜ, P_C_U := ⋯, closed_C := ⋯, open_U := ⋯, subset := ⋯, hP := ⋯ }
⊢ EqOn c.lim 1 t

Tactic:
·

Goal state after:
case h.refine_1
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : R1Space X
s t : Set X
hs : IsOpen[inst✝¹] s
hscp : IsCompact (closure[inst✝¹] sᶜᶜ)
ht : IsClosed[inst✝¹] t
hst : t ⊆ s
u v : Set X
huIsOpen : IsOpen[inst✝¹] u
hvIsOpen : IsOpen[inst✝¹] v
hscompl_subset_u : sᶜ ⊆ u
ht_subset_v : t ⊆ v
hDisjointuv : u ⊆ vᶜ
huvc : closure[inst✝¹] u ⊆ vᶜ
P : Set X → Set X → Prop := fun C x ↦ sᶜ ⊆ C
c : Urysohns.CU P := { C := closure[inst✝¹] u, U := tᶜ, P_C_U := ⋯, closed_C := ⋯, open_U := ⋯, subset := ⋯, hP := ⋯ }
⊢ tsupport c.lim ⊆ s
Module: Mathlib.Topology.UrysohnsLemma

Goal state before:
pp: case h.refine_2
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : R1Space X
s t : Set X
hs : IsOpen[inst✝¹] s
hscp : IsCompact (closure[inst✝¹] sᶜᶜ)
ht : IsClosed[inst✝¹] t
hst : t ⊆ s
u v : Set X
huIsOpen : IsOpen[inst✝¹] u
hvIsOpen : IsOpen[inst✝¹] v
hscompl_subset_u : sᶜ ⊆ u
ht_subset_v : t ⊆ v
hDisjointuv : u ⊆ vᶜ
huvc : closure[inst✝¹] u ⊆ vᶜ
P : Set X → Set X → Prop := fun C x ↦ sᶜ ⊆ C
c : Urysohns.CU P := { C := closure[inst✝¹] u, U := tᶜ, P_C_U := ⋯, closed_C := ⋯, open_U := ⋯, subset := ⋯, hP := ⋯ }
⊢ EqOn c.lim 1 t

Tactic:
·

Goal state after:
case h.refine_2
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : R1Space X
s t : Set X
hs : IsOpen[inst✝¹] s
hscp : IsCompact (closure[inst✝¹] sᶜᶜ)
ht : IsClosed[inst✝¹] t
hst : t ⊆ s
u v : Set X
huIsOpen : IsOpen[inst✝¹] u
hvIsOpen : IsOpen[inst✝¹] v
hscompl_subset_u : sᶜ ⊆ u
ht_subset_v : t ⊆ v
hDisjointuv : u ⊆ vᶜ
huvc : closure[inst✝¹] u ⊆ vᶜ
P : Set X → Set X → Prop := fun C x ↦ sᶜ ⊆ C
c : Urysohns.CU P := { C := closure[inst✝¹] u, U := tᶜ, P_C_U := ⋯, closed_C := ⋯, open_U := ⋯, subset := ⋯, hP := ⋯ }
⊢ EqOn c.lim 1 t
Module: Mathlib.Topology.MetricSpace.UniformConvergence

Goal state before:
pp: case inl
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : PseudoEMetricSpace γ
inst✝¹ : PseudoMetricSpace β
inst✝ : BoundedSpace β
f g : α →ᵤ β
h✝ : IsEmpty α
⊢ edist f g = ENNReal.ofReal (⨆ x, dist (toFun f x) (toFun g x))

pp: case inr
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : PseudoEMetricSpace γ
inst✝¹ : PseudoMetricSpace β
inst✝ : BoundedSpace β
f g : α →ᵤ β
h✝ : Nonempty α
⊢ edist f g = ENNReal.ofReal (⨆ x, dist (toFun f x) (toFun g x))

Tactic:
·

Goal state after:
case inl
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : PseudoEMetricSpace γ
inst✝¹ : PseudoMetricSpace β
inst✝ : BoundedSpace β
f g : α →ᵤ β
h✝ : IsEmpty α
⊢ edist f g = ENNReal.ofReal (⨆ x, dist (toFun f x) (toFun g x))
Module: Mathlib.Topology.MetricSpace.UniformConvergence

Goal state before:
pp: case inl
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : PseudoEMetricSpace γ
𝔖 𝔗 : Set (Set α)
inst✝² : Finite ↑𝔖
inst✝¹ : PseudoMetricSpace β
inst✝ : BoundedSpace β
f g : α →ᵤ[𝔖] β
h✝ : IsEmpty ↑(⋃₀ 𝔖)
⊢ edist f g = ENNReal.ofReal (⨆ x, dist ((toFun 𝔖) f ↑x) ((toFun 𝔖) g ↑x))

pp: case inr
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : PseudoEMetricSpace γ
𝔖 𝔗 : Set (Set α)
inst✝² : Finite ↑𝔖
inst✝¹ : PseudoMetricSpace β
inst✝ : BoundedSpace β
f g : α →ᵤ[𝔖] β
h✝ : Nonempty ↑(⋃₀ 𝔖)
⊢ edist f g = ENNReal.ofReal (⨆ x, dist ((toFun 𝔖) f ↑x) ((toFun 𝔖) g ↑x))

Tactic:
·

Goal state after:
case inl
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : PseudoEMetricSpace γ
𝔖 𝔗 : Set (Set α)
inst✝² : Finite ↑𝔖
inst✝¹ : PseudoMetricSpace β
inst✝ : BoundedSpace β
f g : α →ᵤ[𝔖] β
h✝ : IsEmpty ↑(⋃₀ 𝔖)
⊢ edist f g = ENNReal.ofReal (⨆ x, dist ((toFun 𝔖) f ↑x) ((toFun 𝔖) g ↑x))
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic

Goal state before:
pp: case refine_1
A : Type u_1
inst✝¹ : CStarAlgebra A
a : A
inst✝ : IsStarNormal a
φ ψ : ↑(characterSpace ℂ ↥(elemental ℂ a))
h : characterSpaceToSpectrum a φ = characterSpaceToSpectrum a ψ
⊢ Continuous ⇑φ

pp: case refine_2
A : Type u_1
inst✝¹ : CStarAlgebra A
a : A
inst✝ : IsStarNormal a
φ ψ : ↑(characterSpace ℂ ↥(elemental ℂ a))
h : characterSpaceToSpectrum a φ = characterSpaceToSpectrum a ψ
⊢ Continuous ⇑ψ

pp: case refine_3
A : Type u_1
inst✝¹ : CStarAlgebra A
a : A
inst✝ : IsStarNormal a
φ ψ : ↑(characterSpace ℂ ↥(elemental ℂ a))
h : characterSpaceToSpectrum a φ = characterSpaceToSpectrum a ψ
⊢ φ ⟨a, ⋯⟩ = ψ ⟨a, ⋯⟩

pp: case refine_4
A : Type u_1
inst✝¹ : CStarAlgebra A
a : A
inst✝ : IsStarNormal a
⊢ Function.Surjective (characterSpaceToSpectrum a)

Tactic:
·

Goal state after:
case refine_1
A : Type u_1
inst✝¹ : CStarAlgebra A
a : A
inst✝ : IsStarNormal a
φ ψ : ↑(characterSpace ℂ ↥(elemental ℂ a))
h : characterSpaceToSpectrum a φ = characterSpaceToSpectrum a ψ
⊢ Continuous ⇑φ
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic

Goal state before:
pp: case refine_2
A : Type u_1
inst✝¹ : CStarAlgebra A
a : A
inst✝ : IsStarNormal a
φ ψ : ↑(characterSpace ℂ ↥(elemental ℂ a))
h : characterSpaceToSpectrum a φ = characterSpaceToSpectrum a ψ
⊢ Continuous ⇑ψ

pp: case refine_3
A : Type u_1
inst✝¹ : CStarAlgebra A
a : A
inst✝ : IsStarNormal a
φ ψ : ↑(characterSpace ℂ ↥(elemental ℂ a))
h : characterSpaceToSpectrum a φ = characterSpaceToSpectrum a ψ
⊢ φ ⟨a, ⋯⟩ = ψ ⟨a, ⋯⟩

pp: case refine_4
A : Type u_1
inst✝¹ : CStarAlgebra A
a : A
inst✝ : IsStarNormal a
⊢ Function.Surjective (characterSpaceToSpectrum a)

Tactic:
·

Goal state after:
case refine_2
A : Type u_1
inst✝¹ : CStarAlgebra A
a : A
inst✝ : IsStarNormal a
φ ψ : ↑(characterSpace ℂ ↥(elemental ℂ a))
h : characterSpaceToSpectrum a φ = characterSpaceToSpectrum a ψ
⊢ Continuous ⇑ψ
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic

Goal state before:
pp: case refine_3
A : Type u_1
inst✝¹ : CStarAlgebra A
a : A
inst✝ : IsStarNormal a
φ ψ : ↑(characterSpace ℂ ↥(elemental ℂ a))
h : characterSpaceToSpectrum a φ = characterSpaceToSpectrum a ψ
⊢ φ ⟨a, ⋯⟩ = ψ ⟨a, ⋯⟩

pp: case refine_4
A : Type u_1
inst✝¹ : CStarAlgebra A
a : A
inst✝ : IsStarNormal a
⊢ Function.Surjective (characterSpaceToSpectrum a)

Tactic:
·

Goal state after:
case refine_3
A : Type u_1
inst✝¹ : CStarAlgebra A
a : A
inst✝ : IsStarNormal a
φ ψ : ↑(characterSpace ℂ ↥(elemental ℂ a))
h : characterSpaceToSpectrum a φ = characterSpaceToSpectrum a ψ
⊢ φ ⟨a, ⋯⟩ = ψ ⟨a, ⋯⟩
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic

Goal state before:
pp: case refine_4
A : Type u_1
inst✝¹ : CStarAlgebra A
a : A
inst✝ : IsStarNormal a
⊢ Function.Surjective (characterSpaceToSpectrum a)

Tactic:
·

Goal state after:
case refine_4
A : Type u_1
inst✝¹ : CStarAlgebra A
a : A
inst✝ : IsStarNormal a
⊢ Function.Surjective (characterSpaceToSpectrum a)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic

Goal state before:
pp: case refine_1
A : Type u_1
inst✝ : CStarAlgebra A
a : A
ha : IsStarNormal a
⊢ Continuous ⇑((elemental ℂ a).subtype.comp ↑(continuousFunctionalCalculus a))

pp: case refine_2
A : Type u_1
inst✝ : CStarAlgebra A
a : A
ha : IsStarNormal a
⊢ ((elemental ℂ a).subtype.comp ↑(continuousFunctionalCalculus a))
      (ContinuousMap.restrict (spectrum ℂ a) (ContinuousMap.id ℂ)) =
    a

Tactic:
·

Goal state after:
case refine_1
A : Type u_1
inst✝ : CStarAlgebra A
a : A
ha : IsStarNormal a
⊢ Continuous ⇑((elemental ℂ a).subtype.comp ↑(continuousFunctionalCalculus a))
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic

Goal state before:
pp: case refine_2
A : Type u_1
inst✝ : CStarAlgebra A
a : A
ha : IsStarNormal a
⊢ ((elemental ℂ a).subtype.comp ↑(continuousFunctionalCalculus a))
      (ContinuousMap.restrict (spectrum ℂ a) (ContinuousMap.id ℂ)) =
    a

Tactic:
·

Goal state after:
case refine_2
A : Type u_1
inst✝ : CStarAlgebra A
a : A
ha : IsStarNormal a
⊢ ((elemental ℂ a).subtype.comp ↑(continuousFunctionalCalculus a))
      (ContinuousMap.restrict (spectrum ℂ a) (ContinuousMap.id ℂ)) =
    a