Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital

Goal state before:
pp: case f'
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹² : CommSemiring R
inst✝¹¹ : Nontrivial R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : IsTopologicalSemiring R
inst✝⁷ : ContinuousStar R
inst✝⁶ : NonUnitalRing A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : Module R A
inst✝² : IsScalarTower R A A
inst✝¹ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
inst✝ : UniqueHom R A
g f : R → R
a : A
hg : ContinuousOn g (f '' σₙ R a)
hg0 : g 0 = 0
hf : ContinuousOn f (σₙ R a)
hf0 : f 0 = 0
ha : p a
this : ContinuousOn (g ∘ f) (σₙ R a)
sp_eq : σₙ R ((cfcₙHom ⋯) { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯, map_zero' := hf0 }) = f '' σₙ R a
⊢ C(↑(σₙ R a), ↑(σₙ R ((cfcₙHom ⋯) { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯, map_zero' := ⋯ })))₀

pp: R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹² : CommSemiring R
inst✝¹¹ : Nontrivial R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : IsTopologicalSemiring R
inst✝⁷ : ContinuousStar R
inst✝⁶ : NonUnitalRing A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : Module R A
inst✝² : IsScalarTower R A A
inst✝¹ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
inst✝ : UniqueHom R A
g f : R → R
a : A
hg : ContinuousOn g (f '' σₙ R a)
hg0 : g 0 = 0
hf : ContinuousOn f (σₙ R a)
hf0 : f 0 = 0
ha : p a
this : ContinuousOn (g ∘ f) (σₙ R a)
sp_eq : σₙ R ((cfcₙHom ⋯) { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯, map_zero' := hf0 }) = f '' σₙ R a
⊢ (cfcₙHom ha) { toFun := (σₙ R a).domRestrict (g ∘ f), continuous_toFun := ⋯, map_zero' := ⋯ } =
    (cfcₙHom ⋯)
      ({
            toFun :=
              (σₙ R
                    ((cfcₙHom ⋯)
                      { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯, map_zero' := ⋯ })).domRestrict
                g,
            continuous_toFun := ⋯, map_zero' := ⋯ }.comp
        ?f')

pp: case hff'
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹² : CommSemiring R
inst✝¹¹ : Nontrivial R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : IsTopologicalSemiring R
inst✝⁷ : ContinuousStar R
inst✝⁶ : NonUnitalRing A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : Module R A
inst✝² : IsScalarTower R A A
inst✝¹ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
inst✝ : UniqueHom R A
g f : R → R
a : A
hg : ContinuousOn g (f '' σₙ R a)
hg0 : g 0 = 0
hf : ContinuousOn f (σₙ R a)
hf0 : f 0 = 0
ha : p a
this : ContinuousOn (g ∘ f) (σₙ R a)
sp_eq : σₙ R ((cfcₙHom ⋯) { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯, map_zero' := hf0 }) = f '' σₙ R a
⊢ ∀ (x : ↑(σₙ R a)), { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯, map_zero' := ⋯ } x = ↑(?f' x)

Tactic:
· exact ⟨.mk _ <| hf.domRestrict.codRestrict fun x ↦ by rw [sp_eq]; use x.1; simp,
      Subtype.ext hf0⟩

Goal state after:
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹² : CommSemiring R
inst✝¹¹ : Nontrivial R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : IsTopologicalSemiring R
inst✝⁷ : ContinuousStar R
inst✝⁶ : NonUnitalRing A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : Module R A
inst✝² : IsScalarTower R A A
inst✝¹ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
inst✝ : UniqueHom R A
g f : R → R
a : A
hg : ContinuousOn g (f '' σₙ R a)
hg0 : g 0 = 0
hf : ContinuousOn f (σₙ R a)
hf0 : f 0 = 0
ha : p a
this : ContinuousOn (g ∘ f) (σₙ R a)
sp_eq : σₙ R ((cfcₙHom ⋯) { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯, map_zero' := hf0 }) = f '' σₙ R a
⊢ (cfcₙHom ha) { toFun := (σₙ R a).domRestrict (g ∘ f), continuous_toFun := ⋯, map_zero' := ⋯ } =
    (cfcₙHom ⋯)
      ({
            toFun :=
              (σₙ R
                    ((cfcₙHom ⋯)
                      { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯, map_zero' := ⋯ })).domRestrict
                g,
            continuous_toFun := ⋯, map_zero' := ⋯ }.comp
        {
          toFun :=
            Set.codRestrict ((σₙ R a).domRestrict f)
              (σₙ R ((cfcₙHom ⋯) { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯, map_zero' := ⋯ })) ⋯,
          continuous_toFun := ⋯, map_zero' := ⋯ })

case hff'
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹² : CommSemiring R
inst✝¹¹ : Nontrivial R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : IsTopologicalSemiring R
inst✝⁷ : ContinuousStar R
inst✝⁶ : NonUnitalRing A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : Module R A
inst✝² : IsScalarTower R A A
inst✝¹ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
inst✝ : UniqueHom R A
g f : R → R
a : A
hg : ContinuousOn g (f '' σₙ R a)
hg0 : g 0 = 0
hf : ContinuousOn f (σₙ R a)
hf0 : f 0 = 0
ha : p a
this : ContinuousOn (g ∘ f) (σₙ R a)
sp_eq : σₙ R ((cfcₙHom ⋯) { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯, map_zero' := hf0 }) = f '' σₙ R a
⊢ ∀ (x : ↑(σₙ R a)),
    { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯, map_zero' := ⋯ } x =
      ↑({
            toFun :=
              Set.codRestrict ((σₙ R a).domRestrict f)
                (σₙ R ((cfcₙHom ⋯) { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯, map_zero' := ⋯ })) ⋯,
            continuous_toFun := ⋯, map_zero' := ⋯ }
          x)