mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic

Goal state before:
pp: 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
⊢ cfcₙ (fun x ↦ x⁻) a * cfcₙ (fun x ↦ x⁺) a = 0

Tactic:
]

Goal state after:
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
⊢ 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⁻ * x⁺) a = cfcₙ 0 a

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⁻ * x⁺) a = cfcₙ 0 a
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic

Goal state before:
pp: 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 = a

Tactic:
]

Goal state after:
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 = a
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic

Goal state before:
pp: 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⁺ - x⁻) a = a

Tactic:
]

Goal state after:
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⁺ - x⁻) a = a
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Pi

Goal state before:
pp: A : Type u_1
B : Type u_2
R : Type u_3
S : Type u_4
inst✝²⁸ : CommSemiring R
inst✝²⁷ : CommRing S
inst✝²⁶ : Nontrivial R
inst✝²⁵ : StarRing R
inst✝²⁴ : MetricSpace R
inst✝²³ : IsTopologicalSemiring R
inst✝²² : ContinuousStar R
inst✝²¹ : Algebra R S
inst✝²⁰ : NonUnitalRing A
inst✝¹⁹ : NonUnitalRing B
inst✝¹⁸ : Module S A
inst✝¹⁷ : Module R A
inst✝¹⁶ : Module R B
inst✝¹⁵ : Module S B
inst✝¹⁴ : SMulCommClass R A A
inst✝¹³ : SMulCommClass R B B
inst✝¹² : IsScalarTower R A A
inst✝¹¹ : IsScalarTower R B B
inst✝¹⁰ : StarRing A
inst✝⁹ : StarRing B
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : IsScalarTower R S A
inst✝⁵ : IsScalarTower R S B
pab : A × B → Prop
pa : A → Prop
pb : B → Prop
inst✝⁴ : NonUnitalContinuousFunctionalCalculus R (A × B) pab
inst✝³ : NonUnitalContinuousFunctionalCalculus R A pa
inst✝² : NonUnitalContinuousFunctionalCalculus R B pb
inst✝¹ : ContinuousMapZero.UniqueHom R A
inst✝ : ContinuousMapZero.UniqueHom R B
f : R → R
a : A
b : B
hf : ContinuousOn f (quasispectrum R a ∪ quasispectrum R b)
hab : pab (a, b)
ha : pa a
hb : pb b
hf₀ : f 0 = 0
φ : A × B →⋆ₙₐ[S] A := NonUnitalStarAlgHom.fst S A B
⊢ ContinuousOn f (quasispectrum R a ∪ quasispectrum R b)

Tactic:
]

Goal state after:
A : Type u_1
B : Type u_2
R : Type u_3
S : Type u_4
inst✝²⁸ : CommSemiring R
inst✝²⁷ : CommRing S
inst✝²⁶ : Nontrivial R
inst✝²⁵ : StarRing R
inst✝²⁴ : MetricSpace R
inst✝²³ : IsTopologicalSemiring R
inst✝²² : ContinuousStar R
inst✝²¹ : Algebra R S
inst✝²⁰ : NonUnitalRing A
inst✝¹⁹ : NonUnitalRing B
inst✝¹⁸ : Module S A
inst✝¹⁷ : Module R A
inst✝¹⁶ : Module R B
inst✝¹⁵ : Module S B
inst✝¹⁴ : SMulCommClass R A A
inst✝¹³ : SMulCommClass R B B
inst✝¹² : IsScalarTower R A A
inst✝¹¹ : IsScalarTower R B B
inst✝¹⁰ : StarRing A
inst✝⁹ : StarRing B
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : IsScalarTower R S A
inst✝⁵ : IsScalarTower R S B
pab : A × B → Prop
pa : A → Prop
pb : B → Prop
inst✝⁴ : NonUnitalContinuousFunctionalCalculus R (A × B) pab
inst✝³ : NonUnitalContinuousFunctionalCalculus R A pa
inst✝² : NonUnitalContinuousFunctionalCalculus R B pb
inst✝¹ : ContinuousMapZero.UniqueHom R A
inst✝ : ContinuousMapZero.UniqueHom R B
f : R → R
a : A
b : B
hf : ContinuousOn f (quasispectrum R a ∪ quasispectrum R b)
hab : pab (a, b)
ha : pa a
hb : pb b
hf₀ : f 0 = 0
φ : A × B →⋆ₙₐ[S] A := NonUnitalStarAlgHom.fst S A B
⊢ ContinuousOn f (quasispectrum R a ∪ quasispectrum R b)
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:
]

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.CStarAlgebra.ContinuousFunctionalCalculus.Pi

Goal state before:
pp: A : Type u_1
B : Type u_2
R : Type u_3
S : Type u_4
inst✝²⁸ : CommSemiring R
inst✝²⁷ : CommRing S
inst✝²⁶ : Nontrivial R
inst✝²⁵ : StarRing R
inst✝²⁴ : MetricSpace R
inst✝²³ : IsTopologicalSemiring R
inst✝²² : ContinuousStar R
inst✝²¹ : Algebra R S
inst✝²⁰ : NonUnitalRing A
inst✝¹⁹ : NonUnitalRing B
inst✝¹⁸ : Module S A
inst✝¹⁷ : Module R A
inst✝¹⁶ : Module R B
inst✝¹⁵ : Module S B
inst✝¹⁴ : SMulCommClass R A A
inst✝¹³ : SMulCommClass R B B
inst✝¹² : IsScalarTower R A A
inst✝¹¹ : IsScalarTower R B B
inst✝¹⁰ : StarRing A
inst✝⁹ : StarRing B
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : IsScalarTower R S A
inst✝⁵ : IsScalarTower R S B
pab : A × B → Prop
pa : A → Prop
pb : B → Prop
inst✝⁴ : NonUnitalContinuousFunctionalCalculus R (A × B) pab
inst✝³ : NonUnitalContinuousFunctionalCalculus R A pa
inst✝² : NonUnitalContinuousFunctionalCalculus R B pb
inst✝¹ : ContinuousMapZero.UniqueHom R A
inst✝ : ContinuousMapZero.UniqueHom R B
f : R → R
a : A
b : B
hf : ContinuousOn f (quasispectrum R a ∪ quasispectrum R b)
hab : pab (a, b)
ha : pa a
hb : pb b
hf₀ : f 0 = 0
φ : A × B →⋆ₙₐ[S] B := NonUnitalStarAlgHom.snd S A B
⊢ ContinuousOn f (quasispectrum R a ∪ quasispectrum R b)

Tactic:
]

Goal state after:
A : Type u_1
B : Type u_2
R : Type u_3
S : Type u_4
inst✝²⁸ : CommSemiring R
inst✝²⁷ : CommRing S
inst✝²⁶ : Nontrivial R
inst✝²⁵ : StarRing R
inst✝²⁴ : MetricSpace R
inst✝²³ : IsTopologicalSemiring R
inst✝²² : ContinuousStar R
inst✝²¹ : Algebra R S
inst✝²⁰ : NonUnitalRing A
inst✝¹⁹ : NonUnitalRing B
inst✝¹⁸ : Module S A
inst✝¹⁷ : Module R A
inst✝¹⁶ : Module R B
inst✝¹⁵ : Module S B
inst✝¹⁴ : SMulCommClass R A A
inst✝¹³ : SMulCommClass R B B
inst✝¹² : IsScalarTower R A A
inst✝¹¹ : IsScalarTower R B B
inst✝¹⁰ : StarRing A
inst✝⁹ : StarRing B
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : IsScalarTower R S A
inst✝⁵ : IsScalarTower R S B
pab : A × B → Prop
pa : A → Prop
pb : B → Prop
inst✝⁴ : NonUnitalContinuousFunctionalCalculus R (A × B) pab
inst✝³ : NonUnitalContinuousFunctionalCalculus R A pa
inst✝² : NonUnitalContinuousFunctionalCalculus R B pb
inst✝¹ : ContinuousMapZero.UniqueHom R A
inst✝ : ContinuousMapZero.UniqueHom R B
f : R → R
a : A
b : B
hf : ContinuousOn f (quasispectrum R a ∪ quasispectrum R b)
hab : pab (a, b)
ha : pa a
hb : pb b
hf₀ : f 0 = 0
φ : A × B →⋆ₙₐ[S] B := NonUnitalStarAlgHom.snd S A B
⊢ ContinuousOn f (quasispectrum R a ∪ quasispectrum R b)
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
ha' : ¬IsSelfAdjoint (-a)
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic

Goal state before:
pp: 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
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: ι : Type u_1
R : Type u_2
S : Type u_3
A : ι → Type u_4
inst✝¹⁵ : CommSemiring R
inst✝¹⁴ : StarRing R
inst✝¹³ : MetricSpace R
inst✝¹² : IsTopologicalSemiring R
inst✝¹¹ : ContinuousStar R
inst✝¹⁰ : CommRing S
inst✝⁹ : Algebra R S
inst✝⁸ : (i : ι) → Ring (A i)
inst✝⁷ : (i : ι) → Algebra S (A i)
inst✝⁶ : (i : ι) → Algebra R (A i)
inst✝⁵ : ∀ (i : ι), IsScalarTower R S (A i)
hinst : IsScalarTower R S ((i : ι) → 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 : ContinuousFunctionalCalculus R ((i : ι) → A i) p
inst✝¹ : ∀ (i : ι), ContinuousFunctionalCalculus R (A i) (q i)
inst✝ : ∀ (i : ι), ContinuousMap.UniqueHom R (A i)
f : R → R
a : (i : ι) → A i
hf : ContinuousOn f (⋃ i, spectrum R (a i))
ha : p a
ha' : ∀ (i : ι), q i (a i)
i : ι
φ : ((i : ι) → A i) →⋆ₐ[S] A i := Pi.evalStarAlgHom S A i
⊢ ContinuousOn f (⋃ i, spectrum R (a i))

Tactic:
]

Goal state after:
ι : Type u_1
R : Type u_2
S : Type u_3
A : ι → Type u_4
inst✝¹⁵ : CommSemiring R
inst✝¹⁴ : StarRing R
inst✝¹³ : MetricSpace R
inst✝¹² : IsTopologicalSemiring R
inst✝¹¹ : ContinuousStar R
inst✝¹⁰ : CommRing S
inst✝⁹ : Algebra R S
inst✝⁸ : (i : ι) → Ring (A i)
inst✝⁷ : (i : ι) → Algebra S (A i)
inst✝⁶ : (i : ι) → Algebra R (A i)
inst✝⁵ : ∀ (i : ι), IsScalarTower R S (A i)
hinst : IsScalarTower R S ((i : ι) → 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 : ContinuousFunctionalCalculus R ((i : ι) → A i) p
inst✝¹ : ∀ (i : ι), ContinuousFunctionalCalculus R (A i) (q i)
inst✝ : ∀ (i : ι), ContinuousMap.UniqueHom R (A i)
f : R → R
a : (i : ι) → A i
hf : ContinuousOn f (⋃ i, spectrum R (a i))
ha : p a
ha' : ∀ (i : ι), q i (a i)
i : ι
φ : ((i : ι) → A i) →⋆ₐ[S] A i := Pi.evalStarAlgHom S A i
⊢ ContinuousOn f (⋃ i, spectrum R (a i))
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
⊢ cfcₙ (fun x ↦ (↑r • x)⁺) a = cfcₙ (fun x ↦ ↑r • x⁺) 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
⊢ cfcₙ (fun x ↦ (↑r • x)⁺) a = cfcₙ (fun x ↦ ↑r • x⁺) a
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Pi

Goal state before:
pp: A : Type u_1
B : Type u_2
R : Type u_3
S : Type u_4
inst✝²³ : CommSemiring R
inst✝²² : StarRing R
inst✝²¹ : MetricSpace R
inst✝²⁰ : IsTopologicalSemiring R
inst✝¹⁹ : ContinuousStar R
inst✝¹⁸ : CommRing S
inst✝¹⁷ : Algebra R S
inst✝¹⁶ : Ring A
inst✝¹⁵ : Ring B
inst✝¹⁴ : Algebra S A
inst✝¹³ : Algebra S B
inst✝¹² : Algebra R A
inst✝¹¹ : Algebra R B
inst✝¹⁰ : IsScalarTower R S A
inst✝⁹ : IsScalarTower R S B
inst✝⁸ : StarRing A
inst✝⁷ : StarRing B
inst✝⁶ : TopologicalSpace A
inst✝⁵ : TopologicalSpace B
pab : A × B → Prop
pa : A → Prop
pb : B → Prop
inst✝⁴ : ContinuousFunctionalCalculus R (A × B) pab
inst✝³ : ContinuousFunctionalCalculus R A pa
inst✝² : ContinuousFunctionalCalculus R B pb
inst✝¹ : ContinuousMap.UniqueHom R A
inst✝ : ContinuousMap.UniqueHom R B
f : R → R
a : A
b : B
hf : ContinuousOn f (spectrum R a ∪ spectrum R b)
hab : pab (a, b)
ha : pa a
hb : pb b
φ : A × B →⋆ₐ[S] A := StarAlgHom.fst S A B
⊢ ContinuousOn f (spectrum R a ∪ spectrum R b)

Tactic:
]

Goal state after:
A : Type u_1
B : Type u_2
R : Type u_3
S : Type u_4
inst✝²³ : CommSemiring R
inst✝²² : StarRing R
inst✝²¹ : MetricSpace R
inst✝²⁰ : IsTopologicalSemiring R
inst✝¹⁹ : ContinuousStar R
inst✝¹⁸ : CommRing S
inst✝¹⁷ : Algebra R S
inst✝¹⁶ : Ring A
inst✝¹⁵ : Ring B
inst✝¹⁴ : Algebra S A
inst✝¹³ : Algebra S B
inst✝¹² : Algebra R A
inst✝¹¹ : Algebra R B
inst✝¹⁰ : IsScalarTower R S A
inst✝⁹ : IsScalarTower R S B
inst✝⁸ : StarRing A
inst✝⁷ : StarRing B
inst✝⁶ : TopologicalSpace A
inst✝⁵ : TopologicalSpace B
pab : A × B → Prop
pa : A → Prop
pb : B → Prop
inst✝⁴ : ContinuousFunctionalCalculus R (A × B) pab
inst✝³ : ContinuousFunctionalCalculus R A pa
inst✝² : ContinuousFunctionalCalculus R B pb
inst✝¹ : ContinuousMap.UniqueHom R A
inst✝ : ContinuousMap.UniqueHom R B
f : R → R
a : A
b : B
hf : ContinuousOn f (spectrum R a ∪ spectrum R b)
hab : pab (a, b)
ha : pa a
hb : pb b
φ : A × B →⋆ₐ[S] A := StarAlgHom.fst S A B
⊢ ContinuousOn f (spectrum R a ∪ spectrum R b)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Pi

Goal state before:
pp: A : Type u_1
B : Type u_2
R : Type u_3
S : Type u_4
inst✝²³ : CommSemiring R
inst✝²² : StarRing R
inst✝²¹ : MetricSpace R
inst✝²⁰ : IsTopologicalSemiring R
inst✝¹⁹ : ContinuousStar R
inst✝¹⁸ : CommRing S
inst✝¹⁷ : Algebra R S
inst✝¹⁶ : Ring A
inst✝¹⁵ : Ring B
inst✝¹⁴ : Algebra S A
inst✝¹³ : Algebra S B
inst✝¹² : Algebra R A
inst✝¹¹ : Algebra R B
inst✝¹⁰ : IsScalarTower R S A
inst✝⁹ : IsScalarTower R S B
inst✝⁸ : StarRing A
inst✝⁷ : StarRing B
inst✝⁶ : TopologicalSpace A
inst✝⁵ : TopologicalSpace B
pab : A × B → Prop
pa : A → Prop
pb : B → Prop
inst✝⁴ : ContinuousFunctionalCalculus R (A × B) pab
inst✝³ : ContinuousFunctionalCalculus R A pa
inst✝² : ContinuousFunctionalCalculus R B pb
inst✝¹ : ContinuousMap.UniqueHom R A
inst✝ : ContinuousMap.UniqueHom R B
f : R → R
a : A
b : B
hf : ContinuousOn f (spectrum R a ∪ spectrum R b)
hab : pab (a, b)
ha : pa a
hb : pb b
φ : A × B →⋆ₐ[S] B := StarAlgHom.snd S A B
⊢ ContinuousOn f (spectrum R a ∪ spectrum R b)

Tactic:
]

Goal state after:
A : Type u_1
B : Type u_2
R : Type u_3
S : Type u_4
inst✝²³ : CommSemiring R
inst✝²² : StarRing R
inst✝²¹ : MetricSpace R
inst✝²⁰ : IsTopologicalSemiring R
inst✝¹⁹ : ContinuousStar R
inst✝¹⁸ : CommRing S
inst✝¹⁷ : Algebra R S
inst✝¹⁶ : Ring A
inst✝¹⁵ : Ring B
inst✝¹⁴ : Algebra S A
inst✝¹³ : Algebra S B
inst✝¹² : Algebra R A
inst✝¹¹ : Algebra R B
inst✝¹⁰ : IsScalarTower R S A
inst✝⁹ : IsScalarTower R S B
inst✝⁸ : StarRing A
inst✝⁷ : StarRing B
inst✝⁶ : TopologicalSpace A
inst✝⁵ : TopologicalSpace B
pab : A × B → Prop
pa : A → Prop
pb : B → Prop
inst✝⁴ : ContinuousFunctionalCalculus R (A × B) pab
inst✝³ : ContinuousFunctionalCalculus R A pa
inst✝² : ContinuousFunctionalCalculus R B pb
inst✝¹ : ContinuousMap.UniqueHom R A
inst✝ : ContinuousMap.UniqueHom R B
f : R → R
a : A
b : B
hf : ContinuousOn f (spectrum R a ∪ spectrum R b)
hab : pab (a, b)
ha : pa a
hb : pb b
φ : A × B →⋆ₐ[S] B := StarAlgHom.snd S A B
⊢ ContinuousOn f (spectrum R a ∪ spectrum R b)
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic

Goal state before:
pp: 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
hr : r ≤ 0
a : A
⊢ (- -r • a)⁺ = -r • a⁻

Tactic:
]

Goal state after:
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
hr : r ≤ 0
a : A
⊢ (- -r • a)⁺ = -r • a⁻
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic

Goal state before:
pp: 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
hr : r ≤ 0
a : A
⊢ -r • a⁻ = -r • a⁻

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic

Goal state before:
pp: 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
hr : r ≤ 0
a : A
⊢ (r • a)⁺ = -r • a⁻

Tactic:
nth_rw 1 [← neg_neg r]
  rw [neg_smul, ← smul_neg, posPart_smul_of_nonneg (neg_nonneg.mpr hr), posPart_neg]

Goal state after:
Module: Mathlib.Topology.UrysohnsLemma

Goal state before:
pp: 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 ⊆ closure[inst✝¹] uᶜ

Tactic:
]

Goal state after:
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 ⊆ closure[inst✝¹] uᶜ
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic

Goal state before:
pp: 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 : PartialOrder A
inst✝ : StarOrderedRing A
a b : A
hab : a⁺ - a⁻ = b - a⁻
hb : 0 ≤ b
ha : IsSelfAdjoint a
⊢ a⁺ = b

Tactic:
]

Goal state after:
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 : PartialOrder A
inst✝ : StarOrderedRing A
a b : A
hab : a⁺ - a⁻ = b - a⁻
hb : 0 ≤ b
ha : IsSelfAdjoint a
⊢ a⁺ = b
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic

Goal state before:
pp: 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 : PartialOrder A
inst✝ : StarOrderedRing A
a c : A
hac : a⁺ - a⁻ = a⁺ - c
hc : 0 ≤ c
ha : IsSelfAdjoint a
⊢ a⁻ = c

Tactic:
]

Goal state after:
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 : PartialOrder A
inst✝ : StarOrderedRing A
a c : A
hac : a⁺ - a⁻ = a⁺ - c
hc : 0 ≤ c
ha : IsSelfAdjoint a
⊢ a⁻ = c
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic

Goal state before:
pp: 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 : PartialOrder A
inst✝ : StarOrderedRing A
a : A
ha : IsSelfAdjoint a
⊢ a ≤ a⁺

Tactic:
simpa [posPart_sub_negPart a] using sub_le_self a⁺ (negPart_nonneg a)

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic

Goal state before:
pp: A : Type u_1
inst✝⁹ : NonUnitalRing A
inst✝⁸ : Module ℝ A
inst✝⁷ : SMulCommClass ℝ A A
inst✝⁶ : IsScalarTower ℝ A A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : NonnegSpectrumClass ℝ A
a : A
ha : 0 ≤ a
⊢ cfcₙ (fun x ↦ x⁺) a = cfcₙ id a

Tactic:
]

Goal state after:
A : Type u_1
inst✝⁹ : NonUnitalRing A
inst✝⁸ : Module ℝ A
inst✝⁷ : SMulCommClass ℝ A A
inst✝⁶ : IsScalarTower ℝ A A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : NonnegSpectrumClass ℝ A
a : A
ha : 0 ≤ a
⊢ cfcₙ (fun x ↦ x⁺) a = cfcₙ id a
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic

Goal state before:
pp: A : Type u_1
inst✝⁹ : NonUnitalRing A
inst✝⁸ : Module ℝ A
inst✝⁷ : SMulCommClass ℝ A A
inst✝⁶ : IsScalarTower ℝ A A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : NonnegSpectrumClass ℝ A
a : A
ha : IsSelfAdjoint a
⊢ a⁻ = 0 ↔ a = a⁺

Tactic:
]

Goal state after:
A : Type u_1
inst✝⁹ : NonUnitalRing A
inst✝⁸ : Module ℝ A
inst✝⁷ : SMulCommClass ℝ A A
inst✝⁶ : IsScalarTower ℝ A A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : NonnegSpectrumClass ℝ A
a : A
ha : IsSelfAdjoint a
⊢ a⁻ = 0 ↔ a = a⁺
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic

Goal state before:
pp: A : Type u_1
inst✝⁹ : NonUnitalRing A
inst✝⁸ : Module ℝ A
inst✝⁷ : SMulCommClass ℝ A A
inst✝⁶ : IsScalarTower ℝ A A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : NonnegSpectrumClass ℝ A
a : A
ha : IsSelfAdjoint a
⊢ a⁻ = 0 ↔ a⁺ - a⁻ = a⁺

Tactic:
]

Goal state after:
A : Type u_1
inst✝⁹ : NonUnitalRing A
inst✝⁸ : Module ℝ A
inst✝⁷ : SMulCommClass ℝ A A
inst✝⁶ : IsScalarTower ℝ A A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : NonnegSpectrumClass ℝ A
a : A
ha : IsSelfAdjoint a
⊢ a⁻ = 0 ↔ a⁺ - a⁻ = a⁺