Theorem: cfc_star
File: Mathlib/Topology/ContinuousFunction/FunctionalCalculus.lean

tactic: by_cases h : p a ∧ ContinuousOn f (spectrum R a)
state_before: R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁹ : CommSemiring R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : TopologicalSemiring R
inst✝⁵ : ContinuousStar R
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra R A
inst✝ : ContinuousFunctionalCalculus R p
f✝ g : R → R
a✝ : A
ha : autoParam (p a✝) _auto✝
hf : autoParam (ContinuousOn f✝ (spectrum R a✝)) _auto✝
hg : autoParam (ContinuousOn g (spectrum R a✝)) _auto✝
f : R → R
a : A
⊢ cfc (fun x => star (f x)) a = star (cfc f a)
state_after: case pos
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁹ : CommSemiring R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : TopologicalSemiring R
inst✝⁵ : ContinuousStar R
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra R A
inst✝ : ContinuousFunctionalCalculus R p
f✝ g : R → R
a✝ : A
ha : autoParam (p a✝) _auto✝
hf : autoParam (ContinuousOn f✝ (spectrum R a✝)) _auto✝
hg : autoParam (ContinuousOn g (spectrum R a✝)) _auto✝
f : R → R
a : A
h : p a ∧ ContinuousOn f (spectrum R a)
⊢ cfc (fun x => star (f x)) a = star (cfc f a)

case neg
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁹ : CommSemiring R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : TopologicalSemiring R
inst✝⁵ : ContinuousStar R
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra R A
inst✝ : ContinuousFunctionalCalculus R p
f✝ g : R → R
a✝ : A
ha : autoParam (p a✝) _auto✝
hf : autoParam (ContinuousOn f✝ (spectrum R a✝)) _auto✝
hg : autoParam (ContinuousOn g (spectrum R a✝)) _auto✝
f : R → R
a : A
h : ¬(p a ∧ ContinuousOn f (spectrum R a))
⊢ cfc (fun x => star (f x)) a = star (cfc f a)

tactic: obtain ⟨ha, hf⟩ := h
state_before: case pos
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁹ : CommSemiring R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : TopologicalSemiring R
inst✝⁵ : ContinuousStar R
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra R A
inst✝ : ContinuousFunctionalCalculus R p
f✝ g : R → R
a✝ : A
ha : autoParam (p a✝) _auto✝
hf : autoParam (ContinuousOn f✝ (spectrum R a✝)) _auto✝
hg : autoParam (ContinuousOn g (spectrum R a✝)) _auto✝
f : R → R
a : A
h : p a ∧ ContinuousOn f (spectrum R a)
⊢ cfc (fun x => star (f x)) a = star (cfc f a)
state_after: case pos.intro
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁹ : CommSemiring R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : TopologicalSemiring R
inst✝⁵ : ContinuousStar R
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra R A
inst✝ : ContinuousFunctionalCalculus R p
f✝ g : R → R
a✝ : A
ha✝ : autoParam (p a✝) _auto✝
hf✝ : autoParam (ContinuousOn f✝ (spectrum R a✝)) _auto✝
hg : autoParam (ContinuousOn g (spectrum R a✝)) _auto✝
f : R → R
a : A
ha : p a
hf : ContinuousOn f (spectrum R a)
⊢ cfc (fun x => star (f x)) a = star (cfc f a)

tactic: rw [cfc_apply f a, ← map_star, cfc_apply _ a]
state_before: case pos.intro
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁹ : CommSemiring R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : TopologicalSemiring R
inst✝⁵ : ContinuousStar R
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra R A
inst✝ : ContinuousFunctionalCalculus R p
f✝ g : R → R
a✝ : A
ha✝ : autoParam (p a✝) _auto✝
hf✝ : autoParam (ContinuousOn f✝ (spectrum R a✝)) _auto✝
hg : autoParam (ContinuousOn g (spectrum R a✝)) _auto✝
f : R → R
a : A
ha : p a
hf : ContinuousOn f (spectrum R a)
⊢ cfc (fun x => star (f x)) a = star (cfc f a)
state_after: case pos.intro
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁹ : CommSemiring R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : TopologicalSemiring R
inst✝⁵ : ContinuousStar R
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra R A
inst✝ : ContinuousFunctionalCalculus R p
f✝ g : R → R
a✝ : A
ha✝ : autoParam (p a✝) _auto✝
hf✝ : autoParam (ContinuousOn f✝ (spectrum R a✝)) _auto✝
hg : autoParam (ContinuousOn g (spectrum R a✝)) _auto✝
f : R → R
a : A
ha : p a
hf : ContinuousOn f (spectrum R a)
⊢ (cfcHom ha) { toFun := (spectrum R a).restrict fun x => star (f x), continuous_toFun := ⋯ } =
    (cfcHom ha) (star { toFun := (spectrum R a).restrict f, continuous_toFun := ⋯ })

tactic: congr
state_before: case pos.intro
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁹ : CommSemiring R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : TopologicalSemiring R
inst✝⁵ : ContinuousStar R
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra R A
inst✝ : ContinuousFunctionalCalculus R p
f✝ g : R → R
a✝ : A
ha✝ : autoParam (p a✝) _auto✝
hf✝ : autoParam (ContinuousOn f✝ (spectrum R a✝)) _auto✝
hg : autoParam (ContinuousOn g (spectrum R a✝)) _auto✝
f : R → R
a : A
ha : p a
hf : ContinuousOn f (spectrum R a)
⊢ (cfcHom ha) { toFun := (spectrum R a).restrict fun x => star (f x), continuous_toFun := ⋯ } =
    (cfcHom ha) (star { toFun := (spectrum R a).restrict f, continuous_toFun := ⋯ })
state_after: no goals

tactic: obtain (ha | hf) := not_and_or.mp h
state_before: case neg
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁹ : CommSemiring R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : TopologicalSemiring R
inst✝⁵ : ContinuousStar R
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra R A
inst✝ : ContinuousFunctionalCalculus R p
f✝ g : R → R
a✝ : A
ha : autoParam (p a✝) _auto✝
hf : autoParam (ContinuousOn f✝ (spectrum R a✝)) _auto✝
hg : autoParam (ContinuousOn g (spectrum R a✝)) _auto✝
f : R → R
a : A
h : ¬(p a ∧ ContinuousOn f (spectrum R a))
⊢ cfc (fun x => star (f x)) a = star (cfc f a)
state_after: case neg.inl
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁹ : CommSemiring R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : TopologicalSemiring R
inst✝⁵ : ContinuousStar R
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra R A
inst✝ : ContinuousFunctionalCalculus R p
f✝ g : R → R
a✝ : A
ha✝ : autoParam (p a✝) _auto✝
hf : autoParam (ContinuousOn f✝ (spectrum R a✝)) _auto✝
hg : autoParam (ContinuousOn g (spectrum R a✝)) _auto✝
f : R → R
a : A
h : ¬(p a ∧ ContinuousOn f (spectrum R a))
ha : ¬p a
⊢ cfc (fun x => star (f x)) a = star (cfc f a)

case neg.inr
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁹ : CommSemiring R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : TopologicalSemiring R
inst✝⁵ : ContinuousStar R
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra R A
inst✝ : ContinuousFunctionalCalculus R p
f✝ g : R → R
a✝ : A
ha : autoParam (p a✝) _auto✝
hf✝ : autoParam (ContinuousOn f✝ (spectrum R a✝)) _auto✝
hg : autoParam (ContinuousOn g (spectrum R a✝)) _auto✝
f : R → R
a : A
h : ¬(p a ∧ ContinuousOn f (spectrum R a))
hf : ¬ContinuousOn f (spectrum R a)
⊢ cfc (fun x => star (f x)) a = star (cfc f a)

tactic: simp [cfc_apply_of_not_predicate a ha]
state_before: case neg.inl
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁹ : CommSemiring R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : TopologicalSemiring R
inst✝⁵ : ContinuousStar R
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra R A
inst✝ : ContinuousFunctionalCalculus R p
f✝ g : R → R
a✝ : A
ha✝ : autoParam (p a✝) _auto✝
hf : autoParam (ContinuousOn f✝ (spectrum R a✝)) _auto✝
hg : autoParam (ContinuousOn g (spectrum R a✝)) _auto✝
f : R → R
a : A
h : ¬(p a ∧ ContinuousOn f (spectrum R a))
ha : ¬p a
⊢ cfc (fun x => star (f x)) a = star (cfc f a)
state_after: no goals

tactic: rw [cfc_apply_of_not_continuousOn a hf, cfc_apply_of_not_continuousOn, star_zero]
state_before: case neg.inr
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁹ : CommSemiring R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : TopologicalSemiring R
inst✝⁵ : ContinuousStar R
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra R A
inst✝ : ContinuousFunctionalCalculus R p
f✝ g : R → R
a✝ : A
ha : autoParam (p a✝) _auto✝
hf✝ : autoParam (ContinuousOn f✝ (spectrum R a✝)) _auto✝
hg : autoParam (ContinuousOn g (spectrum R a✝)) _auto✝
f : R → R
a : A
h : ¬(p a ∧ ContinuousOn f (spectrum R a))
hf : ¬ContinuousOn f (spectrum R a)
⊢ cfc (fun x => star (f x)) a = star (cfc f a)
state_after: case neg.inr.hf
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁹ : CommSemiring R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : TopologicalSemiring R
inst✝⁵ : ContinuousStar R
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra R A
inst✝ : ContinuousFunctionalCalculus R p
f✝ g : R → R
a✝ : A
ha : autoParam (p a✝) _auto✝
hf✝ : autoParam (ContinuousOn f✝ (spectrum R a✝)) _auto✝
hg : autoParam (ContinuousOn g (spectrum R a✝)) _auto✝
f : R → R
a : A
h : ¬(p a ∧ ContinuousOn f (spectrum R a))
hf : ¬ContinuousOn f (spectrum R a)
⊢ ¬ContinuousOn (fun x => star (f x)) (spectrum R a)

tactic: exact fun hf_star ↦ hf <| by simpa using hf_star.star
state_before: case neg.inr.hf
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁹ : CommSemiring R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : TopologicalSemiring R
inst✝⁵ : ContinuousStar R
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra R A
inst✝ : ContinuousFunctionalCalculus R p
f✝ g : R → R
a✝ : A
ha : autoParam (p a✝) _auto✝
hf✝ : autoParam (ContinuousOn f✝ (spectrum R a✝)) _auto✝
hg : autoParam (ContinuousOn g (spectrum R a✝)) _auto✝
f : R → R
a : A
h : ¬(p a ∧ ContinuousOn f (spectrum R a))
hf : ¬ContinuousOn f (spectrum R a)
⊢ ¬ContinuousOn (fun x => star (f x)) (spectrum R a)
state_after: no goals

tactic: simpa using hf_star.star
state_before: R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁹ : CommSemiring R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : TopologicalSemiring R
inst✝⁵ : ContinuousStar R
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra R A
inst✝ : ContinuousFunctionalCalculus R p
f✝ g : R → R
a✝ : A
ha : autoParam (p a✝) _auto✝
hf✝ : autoParam (ContinuousOn f✝ (spectrum R a✝)) _auto✝
hg : autoParam (ContinuousOn g (spectrum R a✝)) _auto✝
f : R → R
a : A
h : ¬(p a ∧ ContinuousOn f (spectrum R a))
hf : ¬ContinuousOn f (spectrum R a)
hf_star : ContinuousOn (fun x => star (f x)) (spectrum R a)
⊢ ContinuousOn f (spectrum R a)
state_after: no goals