Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital

Goal state before:
pp: case neg.inl
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Nontrivial R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
P : A → Prop
a : A
f : R → R
h₀ : P 0
haf :
  ∀ (hf : ContinuousOn f (σₙ R a)) (h0 : { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯ } 0 = 0) (ha : p a),
    P ((cfcₙHom ha) { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯, map_zero' := h0 })
h : ¬ContinuousOn f (σₙ R a)
⊢ P (cfcₙ f a)

pp: case neg.inr.inl
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Nontrivial R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
P : A → Prop
a : A
f : R → R
h₀ : P 0
haf :
  ∀ (hf : ContinuousOn f (σₙ R a)) (h0 : { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯ } 0 = 0) (ha : p a),
    P ((cfcₙHom ha) { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯, map_zero' := h0 })
h : ¬f 0 = 0
⊢ P (cfcₙ f a)

pp: case neg.inr.inr
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Nontrivial R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
P : A → Prop
a : A
f : R → R
h₀ : P 0
haf :
  ∀ (hf : ContinuousOn f (σₙ R a)) (h0 : { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯ } 0 = 0) (ha : p a),
    P ((cfcₙHom ha) { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯, map_zero' := h0 })
h : ¬p a
⊢ P (cfcₙ f a)

Tactic:
· rwa [cfcₙ_apply_of_not_continuousOn _ h]

Goal state after:
case neg.inr.inl
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Nontrivial R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
P : A → Prop
a : A
f : R → R
h₀ : P 0
haf :
  ∀ (hf : ContinuousOn f (σₙ R a)) (h0 : { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯ } 0 = 0) (ha : p a),
    P ((cfcₙHom ha) { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯, map_zero' := h0 })
h : ¬f 0 = 0
⊢ P (cfcₙ f a)

case neg.inr.inr
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Nontrivial R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
inst✝ : SMulCommClass R A A
instCFCₙ : NonUnitalContinuousFunctionalCalculus R A p
P : A → Prop
a : A
f : R → R
h₀ : P 0
haf :
  ∀ (hf : ContinuousOn f (σₙ R a)) (h0 : { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯ } 0 = 0) (ha : p a),
    P ((cfcₙHom ha) { toFun := (σₙ R a).domRestrict f, continuous_toFun := ⋯, map_zero' := h0 })
h : ¬p a
⊢ P (cfcₙ f a)