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

tactic: by_cases h : p a ∧ ContinuousOn g (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 g : R → R
a : A
hfg : Set.EqOn f g (spectrum R a)
⊢ cfc f a = cfc g 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 g : R → R
a : A
hfg : Set.EqOn f g (spectrum R a)
h : p a ∧ ContinuousOn g (spectrum R a)
⊢ cfc f a = cfc g 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 g : R → R
a : A
hfg : Set.EqOn f g (spectrum R a)
h : ¬(p a ∧ ContinuousOn g (spectrum R a))
⊢ cfc f a = cfc g a

tactic: rw [cfc_apply (ha := h.1) (hf := h.2.congr hfg), cfc_apply (ha := h.1) (hf := h.2)]
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 g : R → R
a : A
hfg : Set.EqOn f g (spectrum R a)
h : p a ∧ ContinuousOn g (spectrum R a)
⊢ cfc f a = cfc g 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 g : R → R
a : A
hfg : Set.EqOn f g (spectrum R a)
h : p a ∧ ContinuousOn g (spectrum R a)
⊢ (cfcHom ⋯) { toFun := (spectrum R a).restrict f, continuous_toFun := ⋯ } =
    (cfcHom ⋯) { toFun := (spectrum R a).restrict g, continuous_toFun := ⋯ }

tactic: congr
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 g : R → R
a : A
hfg : Set.EqOn f g (spectrum R a)
h : p a ∧ ContinuousOn g (spectrum R a)
⊢ (cfcHom ⋯) { toFun := (spectrum R a).restrict f, continuous_toFun := ⋯ } =
    (cfcHom ⋯) { toFun := (spectrum R a).restrict g, continuous_toFun := ⋯ }
state_after: case pos.h.e_6.h.e_toFun
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 g : R → R
a : A
hfg : Set.EqOn f g (spectrum R a)
h : p a ∧ ContinuousOn g (spectrum R a)
⊢ (spectrum R a).restrict f = (spectrum R a).restrict g

tactic: exact Set.restrict_eq_iff.mpr hfg
state_before: case pos.h.e_6.h.e_toFun
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 g : R → R
a : A
hfg : Set.EqOn f g (spectrum R a)
h : p a ∧ ContinuousOn g (spectrum R a)
⊢ (spectrum R a).restrict f = (spectrum R a).restrict g
state_after: no goals

tactic: obtain (ha | hg) := 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 g : R → R
a : A
hfg : Set.EqOn f g (spectrum R a)
h : ¬(p a ∧ ContinuousOn g (spectrum R a))
⊢ cfc f a = cfc g 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 g : R → R
a : A
hfg : Set.EqOn f g (spectrum R a)
h : ¬(p a ∧ ContinuousOn g (spectrum R a))
ha : ¬p a
⊢ cfc f a = cfc g 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 g : R → R
a : A
hfg : Set.EqOn f g (spectrum R a)
h : ¬(p a ∧ ContinuousOn g (spectrum R a))
hg : ¬ContinuousOn g (spectrum R a)
⊢ cfc f a = cfc g 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 g : R → R
a : A
hfg : Set.EqOn f g (spectrum R a)
h : ¬(p a ∧ ContinuousOn g (spectrum R a))
ha : ¬p a
⊢ cfc f a = cfc g a
state_after: no goals

tactic: rw [cfc_apply_of_not_continuousOn a hg, cfc_apply_of_not_continuousOn]
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 g : R → R
a : A
hfg : Set.EqOn f g (spectrum R a)
h : ¬(p a ∧ ContinuousOn g (spectrum R a))
hg : ¬ContinuousOn g (spectrum R a)
⊢ cfc f a = cfc g 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 g : R → R
a : A
hfg : Set.EqOn f g (spectrum R a)
h : ¬(p a ∧ ContinuousOn g (spectrum R a))
hg : ¬ContinuousOn g (spectrum R a)
⊢ ¬ContinuousOn f (spectrum R a)

tactic: exact fun hf ↦ hg (hf.congr hfg.symm)
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 g : R → R
a : A
hfg : Set.EqOn f g (spectrum R a)
h : ¬(p a ∧ ContinuousOn g (spectrum R a))
hg : ¬ContinuousOn g (spectrum R a)
⊢ ¬ContinuousOn f (spectrum R a)
state_after: no goals