Theorem: eqOn_of_cfc_eq_cfc
File: Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Unital.lean

Proof steps:
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
h : cfc f a = cfc g a
hf : autoParam (ContinuousOn f (spectrum R a)) _auto✝
hg : autoParam (ContinuousOn g (spectrum R a)) _auto✝
ha : autoParam (p a) _auto✝
⊢ Set.EqOn f g (spectrum R a)
state_after: 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
hf : autoParam (ContinuousOn f (spectrum R a)) _auto✝
hg : autoParam (ContinuousOn g (spectrum R a)) _auto✝
ha : autoParam (p a) _auto✝
h :
  (cfcHom ha) { toFun := (spectrum R a).restrict f, continuous_toFun := ⋯ } =
    (cfcHom ha) { toFun := (spectrum R a).restrict g, continuous_toFun := ⋯ }
⊢ Set.EqOn f g (spectrum R a)
tactic: rw [cfc_apply f a, cfc_apply g a] at h
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
hf : autoParam (ContinuousOn f (spectrum R a)) _auto✝
hg : autoParam (ContinuousOn g (spectrum R a)) _auto✝
ha : autoParam (p a) _auto✝
h :
  (cfcHom ha) { toFun := (spectrum R a).restrict f, continuous_toFun := ⋯ } =
    (cfcHom ha) { toFun := (spectrum R a).restrict g, continuous_toFun := ⋯ }
⊢ Set.EqOn f g (spectrum R a)
state_after: 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
hf : autoParam (ContinuousOn f (spectrum R a)) _auto✝
hg : autoParam (ContinuousOn g (spectrum R a)) _auto✝
ha : autoParam (p a) _auto✝
h :
  (cfcHom ha) { toFun := (spectrum R a).restrict f, continuous_toFun := ⋯ } =
    (cfcHom ha) { toFun := (spectrum R a).restrict g, continuous_toFun := ⋯ }
this :
  { toFun := (spectrum R a).restrict f, continuous_toFun := ⋯ } =
    { toFun := (spectrum R a).restrict g, continuous_toFun := ⋯ }
⊢ Set.EqOn f g (spectrum R a)
tactic: have := (cfcHom_closedEmbedding (show p a from ha) (R := R)).inj h
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
hf : autoParam (ContinuousOn f (spectrum R a)) _auto✝
hg : autoParam (ContinuousOn g (spectrum R a)) _auto✝
ha : autoParam (p a) _auto✝
h :
  (cfcHom ha) { toFun := (spectrum R a).restrict f, continuous_toFun := ⋯ } =
    (cfcHom ha) { toFun := (spectrum R a).restrict g, continuous_toFun := ⋯ }
this :
  { toFun := (spectrum R a).restrict f, continuous_toFun := ⋯ } =
    { toFun := (spectrum R a).restrict g, continuous_toFun := ⋯ }
⊢ Set.EqOn f g (spectrum R a)
state_after: 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
hf : autoParam (ContinuousOn f (spectrum R a)) _auto✝
hg : autoParam (ContinuousOn g (spectrum R a)) _auto✝
ha : autoParam (p a) _auto✝
h :
  (cfcHom ha) { toFun := (spectrum R a).restrict f, continuous_toFun := ⋯ } =
    (cfcHom ha) { toFun := (spectrum R a).restrict g, continuous_toFun := ⋯ }
this :
  { toFun := (spectrum R a).restrict f, continuous_toFun := ⋯ } =
    { toFun := (spectrum R a).restrict g, continuous_toFun := ⋯ }
x : R
hx : x ∈ spectrum R a
⊢ f x = g x
tactic: intro x hx
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
hf : autoParam (ContinuousOn f (spectrum R a)) _auto✝
hg : autoParam (ContinuousOn g (spectrum R a)) _auto✝
ha : autoParam (p a) _auto✝
h :
  (cfcHom ha) { toFun := (spectrum R a).restrict f, continuous_toFun := ⋯ } =
    (cfcHom ha) { toFun := (spectrum R a).restrict g, continuous_toFun := ⋯ }
this :
  { toFun := (spectrum R a).restrict f, continuous_toFun := ⋯ } =
    { toFun := (spectrum R a).restrict g, continuous_toFun := ⋯ }
x : R
hx : x ∈ spectrum R a
⊢ f x = g x
state_after: no goals
tactic: congrm($(this) ⟨x, hx⟩)