Theorem: CFC.exp_eq_normedSpace_exp
File: Mathlib/Analysis/SpecialFunctions/ContinuousFunctionalCalculus/ExpLog.lean

Proof steps:
state_before: 𝕜 : Type u_1
A : Type u_2
inst✝⁹ : RCLike 𝕜
p : A → Prop
inst✝⁸ : PartialOrder A
inst✝⁷ : NormedRing A
inst✝⁶ : StarRing A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : TopologicalRing A
inst✝³ : NormedAlgebra 𝕜 A
inst✝² : CompleteSpace A
inst✝¹ : ContinuousFunctionalCalculus 𝕜 p
inst✝ : UniqueContinuousFunctionalCalculus 𝕜 A
a : A
ha : autoParam (p a) _auto✝
⊢ cfc (exp 𝕜) a = exp 𝕜 a
state_after: 𝕜 : Type u_1
A : Type u_2
inst✝⁹ : RCLike 𝕜
p : A → Prop
inst✝⁸ : PartialOrder A
inst✝⁷ : NormedRing A
inst✝⁶ : StarRing A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : TopologicalRing A
inst✝³ : NormedAlgebra 𝕜 A
inst✝² : CompleteSpace A
inst✝¹ : ContinuousFunctionalCalculus 𝕜 p
inst✝ : UniqueContinuousFunctionalCalculus 𝕜 A
a : A
ha : autoParam (p a) _auto✝
⊢ cfc (exp 𝕜) a = exp 𝕜 ((cfcHom ha) { toFun := (spectrum 𝕜 a).restrict id, continuous_toFun := ⋯ })
tactic: conv_rhs => rw [← cfc_id 𝕜 a ha, cfc_apply id a ha]
state_before: 𝕜 : Type u_1
A : Type u_2
inst✝⁹ : RCLike 𝕜
p : A → Prop
inst✝⁸ : PartialOrder A
inst✝⁷ : NormedRing A
inst✝⁶ : StarRing A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : TopologicalRing A
inst✝³ : NormedAlgebra 𝕜 A
inst✝² : CompleteSpace A
inst✝¹ : ContinuousFunctionalCalculus 𝕜 p
inst✝ : UniqueContinuousFunctionalCalculus 𝕜 A
a : A
ha : autoParam (p a) _auto✝
⊢ cfc (exp 𝕜) a = exp 𝕜 ((cfcHom ha) { toFun := (spectrum 𝕜 a).restrict id, continuous_toFun := ⋯ })
state_after: 𝕜 : Type u_1
A : Type u_2
inst✝⁹ : RCLike 𝕜
p : A → Prop
inst✝⁸ : PartialOrder A
inst✝⁷ : NormedRing A
inst✝⁶ : StarRing A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : TopologicalRing A
inst✝³ : NormedAlgebra 𝕜 A
inst✝² : CompleteSpace A
inst✝¹ : ContinuousFunctionalCalculus 𝕜 p
inst✝ : UniqueContinuousFunctionalCalculus 𝕜 A
a : A
ha : autoParam (p a) _auto✝
h : Continuous ⇑(cfcHom ⋯)
⊢ cfc (exp 𝕜) a = exp 𝕜 ((cfcHom ha) { toFun := (spectrum 𝕜 a).restrict id, continuous_toFun := ⋯ })
tactic: have h := (cfcHom_closedEmbedding (R := 𝕜) (show p a from ha)).continuous
state_before: 𝕜 : Type u_1
A : Type u_2
inst✝⁹ : RCLike 𝕜
p : A → Prop
inst✝⁸ : PartialOrder A
inst✝⁷ : NormedRing A
inst✝⁶ : StarRing A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : TopologicalRing A
inst✝³ : NormedAlgebra 𝕜 A
inst✝² : CompleteSpace A
inst✝¹ : ContinuousFunctionalCalculus 𝕜 p
inst✝ : UniqueContinuousFunctionalCalculus 𝕜 A
a : A
ha : autoParam (p a) _auto✝
h : Continuous ⇑(cfcHom ⋯)
⊢ cfc (exp 𝕜) a = exp 𝕜 ((cfcHom ha) { toFun := (spectrum 𝕜 a).restrict id, continuous_toFun := ⋯ })
state_after: 𝕜 : Type u_1
A : Type u_2
inst✝⁹ : RCLike 𝕜
p : A → Prop
inst✝⁸ : PartialOrder A
inst✝⁷ : NormedRing A
inst✝⁶ : StarRing A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : TopologicalRing A
inst✝³ : NormedAlgebra 𝕜 A
inst✝² : CompleteSpace A
inst✝¹ : ContinuousFunctionalCalculus 𝕜 p
inst✝ : UniqueContinuousFunctionalCalculus 𝕜 A
a : A
ha : autoParam (p a) _auto✝
h : Continuous ⇑(cfcHom ⋯)
x✝ : ContinuousOn (exp 𝕜) (spectrum 𝕜 a)
⊢ cfc (exp 𝕜) a = exp 𝕜 ((cfcHom ha) { toFun := (spectrum 𝕜 a).restrict id, continuous_toFun := ⋯ })
tactic: have _ : ContinuousOn (exp 𝕜) (spectrum 𝕜 a) := exp_continuous.continuousOn
state_before: 𝕜 : Type u_1
A : Type u_2
inst✝⁹ : RCLike 𝕜
p : A → Prop
inst✝⁸ : PartialOrder A
inst✝⁷ : NormedRing A
inst✝⁶ : StarRing A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : TopologicalRing A
inst✝³ : NormedAlgebra 𝕜 A
inst✝² : CompleteSpace A
inst✝¹ : ContinuousFunctionalCalculus 𝕜 p
inst✝ : UniqueContinuousFunctionalCalculus 𝕜 A
a : A
ha : autoParam (p a) _auto✝
h : Continuous ⇑(cfcHom ⋯)
x✝ : ContinuousOn (exp 𝕜) (spectrum 𝕜 a)
⊢ cfc (exp 𝕜) a = exp 𝕜 ((cfcHom ha) { toFun := (spectrum 𝕜 a).restrict id, continuous_toFun := ⋯ })
state_after: 𝕜 : Type u_1
A : Type u_2
inst✝⁹ : RCLike 𝕜
p : A → Prop
inst✝⁸ : PartialOrder A
inst✝⁷ : NormedRing A
inst✝⁶ : StarRing A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : TopologicalRing A
inst✝³ : NormedAlgebra 𝕜 A
inst✝² : CompleteSpace A
inst✝¹ : ContinuousFunctionalCalculus 𝕜 p
inst✝ : UniqueContinuousFunctionalCalculus 𝕜 A
a : A
ha : autoParam (p a) _auto✝
h : Continuous ⇑(cfcHom ⋯)
x✝ : ContinuousOn (exp 𝕜) (spectrum 𝕜 a)
⊢ (cfcHom ha) { toFun := (spectrum 𝕜 a).restrict (exp 𝕜), continuous_toFun := ⋯ } =
    (cfcHom ⋯) (exp 𝕜 { toFun := (spectrum 𝕜 a).restrict id, continuous_toFun := ⋯ })
tactic: simp_rw [← map_exp 𝕜 _ h, cfc_apply (exp 𝕜) a ha]
state_before: 𝕜 : Type u_1
A : Type u_2
inst✝⁹ : RCLike 𝕜
p : A → Prop
inst✝⁸ : PartialOrder A
inst✝⁷ : NormedRing A
inst✝⁶ : StarRing A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : TopologicalRing A
inst✝³ : NormedAlgebra 𝕜 A
inst✝² : CompleteSpace A
inst✝¹ : ContinuousFunctionalCalculus 𝕜 p
inst✝ : UniqueContinuousFunctionalCalculus 𝕜 A
a : A
ha : autoParam (p a) _auto✝
h : Continuous ⇑(cfcHom ⋯)
x✝ : ContinuousOn (exp 𝕜) (spectrum 𝕜 a)
⊢ (cfcHom ha) { toFun := (spectrum 𝕜 a).restrict (exp 𝕜), continuous_toFun := ⋯ } =
    (cfcHom ⋯) (exp 𝕜 { toFun := (spectrum 𝕜 a).restrict id, continuous_toFun := ⋯ })
state_after: case h.e_6.h
𝕜 : Type u_1
A : Type u_2
inst✝⁹ : RCLike 𝕜
p : A → Prop
inst✝⁸ : PartialOrder A
inst✝⁷ : NormedRing A
inst✝⁶ : StarRing A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : TopologicalRing A
inst✝³ : NormedAlgebra 𝕜 A
inst✝² : CompleteSpace A
inst✝¹ : ContinuousFunctionalCalculus 𝕜 p
inst✝ : UniqueContinuousFunctionalCalculus 𝕜 A
a : A
ha : autoParam (p a) _auto✝
h : Continuous ⇑(cfcHom ⋯)
x✝ : ContinuousOn (exp 𝕜) (spectrum 𝕜 a)
⊢ { toFun := (spectrum 𝕜 a).restrict (exp 𝕜), continuous_toFun := ⋯ } =
    exp 𝕜 { toFun := (spectrum 𝕜 a).restrict id, continuous_toFun := ⋯ }
tactic: congr 1
state_before: case h.e_6.h
𝕜 : Type u_1
A : Type u_2
inst✝⁹ : RCLike 𝕜
p : A → Prop
inst✝⁸ : PartialOrder A
inst✝⁷ : NormedRing A
inst✝⁶ : StarRing A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : TopologicalRing A
inst✝³ : NormedAlgebra 𝕜 A
inst✝² : CompleteSpace A
inst✝¹ : ContinuousFunctionalCalculus 𝕜 p
inst✝ : UniqueContinuousFunctionalCalculus 𝕜 A
a : A
ha : autoParam (p a) _auto✝
h : Continuous ⇑(cfcHom ⋯)
x✝ : ContinuousOn (exp 𝕜) (spectrum 𝕜 a)
⊢ { toFun := (spectrum 𝕜 a).restrict (exp 𝕜), continuous_toFun := ⋯ } =
    exp 𝕜 { toFun := (spectrum 𝕜 a).restrict id, continuous_toFun := ⋯ }
state_after: case h.e_6.h.h
𝕜 : Type u_1
A : Type u_2
inst✝⁹ : RCLike 𝕜
p : A → Prop
inst✝⁸ : PartialOrder A
inst✝⁷ : NormedRing A
inst✝⁶ : StarRing A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : TopologicalRing A
inst✝³ : NormedAlgebra 𝕜 A
inst✝² : CompleteSpace A
inst✝¹ : ContinuousFunctionalCalculus 𝕜 p
inst✝ : UniqueContinuousFunctionalCalculus 𝕜 A
a : A
ha : autoParam (p a) _auto✝
h : Continuous ⇑(cfcHom ⋯)
x✝ : ContinuousOn (exp 𝕜) (spectrum 𝕜 a)
a✝ : ↑(spectrum 𝕜 a)
⊢ { toFun := (spectrum 𝕜 a).restrict (exp 𝕜), continuous_toFun := ⋯ } a✝ =
    (exp 𝕜 { toFun := (spectrum 𝕜 a).restrict id, continuous_toFun := ⋯ }) a✝
tactic: ext
state_before: case h.e_6.h.h
𝕜 : Type u_1
A : Type u_2
inst✝⁹ : RCLike 𝕜
p : A → Prop
inst✝⁸ : PartialOrder A
inst✝⁷ : NormedRing A
inst✝⁶ : StarRing A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : TopologicalRing A
inst✝³ : NormedAlgebra 𝕜 A
inst✝² : CompleteSpace A
inst✝¹ : ContinuousFunctionalCalculus 𝕜 p
inst✝ : UniqueContinuousFunctionalCalculus 𝕜 A
a : A
ha : autoParam (p a) _auto✝
h : Continuous ⇑(cfcHom ⋯)
x✝ : ContinuousOn (exp 𝕜) (spectrum 𝕜 a)
a✝ : ↑(spectrum 𝕜 a)
⊢ { toFun := (spectrum 𝕜 a).restrict (exp 𝕜), continuous_toFun := ⋯ } a✝ =
    (exp 𝕜 { toFun := (spectrum 𝕜 a).restrict id, continuous_toFun := ⋯ }) a✝
state_after: no goals
tactic: simp [exp_continuousMap_eq]