-- Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/NonUnital.lean
lemma cfcₙ_neg : cfcₙ (fun x ↦ - (f x)) a = - (cfcₙ f a)

Proof state:
case neg.inr.inr
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹² : CommRing R
inst✝¹¹ : Nontrivial R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : IsTopologicalRing R
inst✝⁷ : ContinuousStar R
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module R A
inst✝² : IsScalarTower R A A
inst✝¹ : SMulCommClass R A A
inst✝ : NonUnitalContinuousFunctionalCalculus R p
f : R → R
a : A
h0 : ¬f 0 = 0
⊢ cfcₙ (fun x => -f x) a = -cfcₙ f a

Tactic:
rw [cfcₙ_apply_of_not_map_zero a h0, cfcₙ_apply_of_not_map_zero, neg_zero]

Result:
case neg.inr.inr.hf
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹² : CommRing R
inst✝¹¹ : Nontrivial R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : IsTopologicalRing R
inst✝⁷ : ContinuousStar R
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module R A
inst✝² : IsScalarTower R A A
inst✝¹ : SMulCommClass R A A
inst✝ : NonUnitalContinuousFunctionalCalculus R p
f : R → R
a : A
h0 : ¬f 0 = 0
⊢ ¬-f 0 = 0

Proof state:
case neg
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹² : CommRing R
inst✝¹¹ : Nontrivial R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : IsTopologicalRing R
inst✝⁷ : ContinuousStar R
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module R A
inst✝² : IsScalarTower R A A
inst✝¹ : SMulCommClass R A A
inst✝ : NonUnitalContinuousFunctionalCalculus R p
f : R → R
a : A
h : ¬(p a ∧ ContinuousOn f (σₙ R a) ∧ f 0 = 0)
⊢ cfcₙ (fun x => -f x) a = -cfcₙ f a

Tactic:
simp only [not_and_or] at h

Result:
case neg
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹² : CommRing R
inst✝¹¹ : Nontrivial R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : IsTopologicalRing R
inst✝⁷ : ContinuousStar R
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module R A
inst✝² : IsScalarTower R A A
inst✝¹ : SMulCommClass R A A
inst✝ : NonUnitalContinuousFunctionalCalculus R p
f : R → R
a : A
h : ¬p a ∨ ¬ContinuousOn f (σₙ R a) ∨ ¬f 0 = 0
⊢ cfcₙ (fun x => -f x) a = -cfcₙ f a

Proof state:
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹² : CommRing R
inst✝¹¹ : Nontrivial R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : IsTopologicalRing R
inst✝⁷ : ContinuousStar R
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module R A
inst✝² : IsScalarTower R A A
inst✝¹ : SMulCommClass R A A
inst✝ : NonUnitalContinuousFunctionalCalculus R p
f : R → R
a : A
⊢ cfcₙ (fun x => -f x) a = -cfcₙ f a

Tactic:
by_cases h : p a ∧ ContinuousOn f (σₙ R a) ∧ f 0 = 0

Result:
case pos
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹² : CommRing R
inst✝¹¹ : Nontrivial R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : IsTopologicalRing R
inst✝⁷ : ContinuousStar R
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module R A
inst✝² : IsScalarTower R A A
inst✝¹ : SMulCommClass R A A
inst✝ : NonUnitalContinuousFunctionalCalculus R p
f : R → R
a : A
h : p a ∧ ContinuousOn f (σₙ R a) ∧ f 0 = 0
⊢ cfcₙ (fun x => -f x) a = -cfcₙ f a

case neg
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹² : CommRing R
inst✝¹¹ : Nontrivial R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : IsTopologicalRing R
inst✝⁷ : ContinuousStar R
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module R A
inst✝² : IsScalarTower R A A
inst✝¹ : SMulCommClass R A A
inst✝ : NonUnitalContinuousFunctionalCalculus R p
f : R → R
a : A
h : ¬(p a ∧ ContinuousOn f (σₙ R a) ∧ f 0 = 0)
⊢ cfcₙ (fun x => -f x) a = -cfcₙ f a

Proof state:
case pos.intro.intro
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹² : CommRing R
inst✝¹¹ : Nontrivial R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : IsTopologicalRing R
inst✝⁷ : ContinuousStar R
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module R A
inst✝² : IsScalarTower R A A
inst✝¹ : SMulCommClass R A A
inst✝ : NonUnitalContinuousFunctionalCalculus R p
f : R → R
a : A
ha : p a
hf : ContinuousOn f (σₙ R a)
h0 : f 0 = 0
⊢ (cfcₙHom ha) { toFun := (σₙ R a).restrict fun x => -f x, continuous_toFun := ⋯, map_zero' := ⋯ } =
    (cfcₙHom ha) (-{ toFun := (σₙ R a).restrict f, continuous_toFun := ⋯, map_zero' := ⋯ })

Tactic:
congr

Result:
no goals

Proof state:
case pos
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹² : CommRing R
inst✝¹¹ : Nontrivial R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : IsTopologicalRing R
inst✝⁷ : ContinuousStar R
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module R A
inst✝² : IsScalarTower R A A
inst✝¹ : SMulCommClass R A A
inst✝ : NonUnitalContinuousFunctionalCalculus R p
f : R → R
a : A
h : p a ∧ ContinuousOn f (σₙ R a) ∧ f 0 = 0
⊢ cfcₙ (fun x => -f x) a = -cfcₙ f a

Tactic:
obtain ⟨ha, hf, h0⟩ := h

Result:
case pos.intro.intro
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹² : CommRing R
inst✝¹¹ : Nontrivial R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : IsTopologicalRing R
inst✝⁷ : ContinuousStar R
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module R A
inst✝² : IsScalarTower R A A
inst✝¹ : SMulCommClass R A A
inst✝ : NonUnitalContinuousFunctionalCalculus R p
f : R → R
a : A
ha : p a
hf : ContinuousOn f (σₙ R a)
h0 : f 0 = 0
⊢ cfcₙ (fun x => -f x) a = -cfcₙ f a

Proof state:
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹² : CommRing R
inst✝¹¹ : Nontrivial R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : IsTopologicalRing R
inst✝⁷ : ContinuousStar R
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module R A
inst✝² : IsScalarTower R A A
inst✝¹ : SMulCommClass R A A
inst✝ : NonUnitalContinuousFunctionalCalculus R p
f : R → R
a : A
hf : ¬ContinuousOn f (σₙ R a)
hf_neg : ContinuousOn (fun x => -f x) (σₙ R a)
⊢ ContinuousOn f (σₙ R a)

Tactic:
simpa using hf_neg.neg

Result:
no goals

Proof state:
case neg
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹² : CommRing R
inst✝¹¹ : Nontrivial R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : IsTopologicalRing R
inst✝⁷ : ContinuousStar R
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module R A
inst✝² : IsScalarTower R A A
inst✝¹ : SMulCommClass R A A
inst✝ : NonUnitalContinuousFunctionalCalculus R p
f : R → R
a : A
h : ¬p a ∨ ¬ContinuousOn f (σₙ R a) ∨ ¬f 0 = 0
⊢ cfcₙ (fun x => -f x) a = -cfcₙ f a

Tactic:
obtain (ha | hf | h0) := h

Result:
case neg.inl
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹² : CommRing R
inst✝¹¹ : Nontrivial R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : IsTopologicalRing R
inst✝⁷ : ContinuousStar R
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module R A
inst✝² : IsScalarTower R A A
inst✝¹ : SMulCommClass R A A
inst✝ : NonUnitalContinuousFunctionalCalculus R p
f : R → R
a : A
ha : ¬p a
⊢ cfcₙ (fun x => -f x) a = -cfcₙ f a

case neg.inr.inl
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹² : CommRing R
inst✝¹¹ : Nontrivial R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : IsTopologicalRing R
inst✝⁷ : ContinuousStar R
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module R A
inst✝² : IsScalarTower R A A
inst✝¹ : SMulCommClass R A A
inst✝ : NonUnitalContinuousFunctionalCalculus R p
f : R → R
a : A
hf : ¬ContinuousOn f (σₙ R a)
⊢ cfcₙ (fun x => -f x) a = -cfcₙ f a

case neg.inr.inr
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹² : CommRing R
inst✝¹¹ : Nontrivial R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : IsTopologicalRing R
inst✝⁷ : ContinuousStar R
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module R A
inst✝² : IsScalarTower R A A
inst✝¹ : SMulCommClass R A A
inst✝ : NonUnitalContinuousFunctionalCalculus R p
f : R → R
a : A
h0 : ¬f 0 = 0
⊢ cfcₙ (fun x => -f x) a = -cfcₙ f a