Theorem: cfcₙ_nonpos
File: Mathlib/Topology/ContinuousFunction/NonUnitalFunctionalCalculus.lean

tactic: by_cases hf : ContinuousOn f (σₙ R a) ∧ f 0 = 0
state_before: R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁸ : OrderedCommSemiring R
inst✝¹⁷ : Nontrivial R
inst✝¹⁶ : StarRing R
inst✝¹⁵ : StarOrderedRing R
inst✝¹⁴ : MetricSpace R
inst✝¹³ : TopologicalSemiring R
inst✝¹² : ContinuousStar R
inst✝¹¹ : ∀ (α : Type ?u.1171919) [inst : Zero α] [inst_1 : TopologicalSpace α], StarOrderedRing C(α, R)₀
inst✝¹⁰ : TopologicalSpace A
inst✝⁹ : NonUnitalRing A
inst✝⁸ : StarRing A
inst✝⁷ : PartialOrder A
inst✝⁶ : StarOrderedRing A
inst✝⁵ : Module R A
inst✝⁴ : IsScalarTower R A A
inst✝³ : SMulCommClass R A A
inst✝² : StarModule R A
inst✝¹ : NonUnitalContinuousFunctionalCalculus R p
inst✝ : NonnegSpectrumClass R A
f : R → R
a : A
h : ∀ x ∈ σₙ R a, f x ≤ 0
⊢ cfcₙ f a ≤ 0
state_after: case pos
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁸ : OrderedCommSemiring R
inst✝¹⁷ : Nontrivial R
inst✝¹⁶ : StarRing R
inst✝¹⁵ : StarOrderedRing R
inst✝¹⁴ : MetricSpace R
inst✝¹³ : TopologicalSemiring R
inst✝¹² : ContinuousStar R
inst✝¹¹ : ∀ (α : Type ?u.1171919) [inst : Zero α] [inst_1 : TopologicalSpace α], StarOrderedRing C(α, R)₀
inst✝¹⁰ : TopologicalSpace A
inst✝⁹ : NonUnitalRing A
inst✝⁸ : StarRing A
inst✝⁷ : PartialOrder A
inst✝⁶ : StarOrderedRing A
inst✝⁵ : Module R A
inst✝⁴ : IsScalarTower R A A
inst✝³ : SMulCommClass R A A
inst✝² : StarModule R A
inst✝¹ : NonUnitalContinuousFunctionalCalculus R p
inst✝ : NonnegSpectrumClass R A
f : R → R
a : A
h : ∀ x ∈ σₙ R a, f x ≤ 0
hf : ContinuousOn f (σₙ R a) ∧ f 0 = 0
⊢ cfcₙ f a ≤ 0

case neg
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁸ : OrderedCommSemiring R
inst✝¹⁷ : Nontrivial R
inst✝¹⁶ : StarRing R
inst✝¹⁵ : StarOrderedRing R
inst✝¹⁴ : MetricSpace R
inst✝¹³ : TopologicalSemiring R
inst✝¹² : ContinuousStar R
inst✝¹¹ : ∀ (α : Type ?u.1171919) [inst : Zero α] [inst_1 : TopologicalSpace α], StarOrderedRing C(α, R)₀
inst✝¹⁰ : TopologicalSpace A
inst✝⁹ : NonUnitalRing A
inst✝⁸ : StarRing A
inst✝⁷ : PartialOrder A
inst✝⁶ : StarOrderedRing A
inst✝⁵ : Module R A
inst✝⁴ : IsScalarTower R A A
inst✝³ : SMulCommClass R A A
inst✝² : StarModule R A
inst✝¹ : NonUnitalContinuousFunctionalCalculus R p
inst✝ : NonnegSpectrumClass R A
f : R → R
a : A
h : ∀ x ∈ σₙ R a, f x ≤ 0
hf : ¬(ContinuousOn f (σₙ R a) ∧ f 0 = 0)
⊢ cfcₙ f a ≤ 0

tactic: obtain ⟨h₁, h₂⟩ := hf
state_before: case pos
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁸ : OrderedCommSemiring R
inst✝¹⁷ : Nontrivial R
inst✝¹⁶ : StarRing R
inst✝¹⁵ : StarOrderedRing R
inst✝¹⁴ : MetricSpace R
inst✝¹³ : TopologicalSemiring R
inst✝¹² : ContinuousStar R
inst✝¹¹ : ∀ (α : Type ?u.1171919) [inst : Zero α] [inst_1 : TopologicalSpace α], StarOrderedRing C(α, R)₀
inst✝¹⁰ : TopologicalSpace A
inst✝⁹ : NonUnitalRing A
inst✝⁸ : StarRing A
inst✝⁷ : PartialOrder A
inst✝⁶ : StarOrderedRing A
inst✝⁵ : Module R A
inst✝⁴ : IsScalarTower R A A
inst✝³ : SMulCommClass R A A
inst✝² : StarModule R A
inst✝¹ : NonUnitalContinuousFunctionalCalculus R p
inst✝ : NonnegSpectrumClass R A
f : R → R
a : A
h : ∀ x ∈ σₙ R a, f x ≤ 0
hf : ContinuousOn f (σₙ R a) ∧ f 0 = 0
⊢ cfcₙ f a ≤ 0
state_after: case pos.intro
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁸ : OrderedCommSemiring R
inst✝¹⁷ : Nontrivial R
inst✝¹⁶ : StarRing R
inst✝¹⁵ : StarOrderedRing R
inst✝¹⁴ : MetricSpace R
inst✝¹³ : TopologicalSemiring R
inst✝¹² : ContinuousStar R
inst✝¹¹ : ∀ (α : Type ?u.1171919) [inst : Zero α] [inst_1 : TopologicalSpace α], StarOrderedRing C(α, R)₀
inst✝¹⁰ : TopologicalSpace A
inst✝⁹ : NonUnitalRing A
inst✝⁸ : StarRing A
inst✝⁷ : PartialOrder A
inst✝⁶ : StarOrderedRing A
inst✝⁵ : Module R A
inst✝⁴ : IsScalarTower R A A
inst✝³ : SMulCommClass R A A
inst✝² : StarModule R A
inst✝¹ : NonUnitalContinuousFunctionalCalculus R p
inst✝ : NonnegSpectrumClass R A
f : R → R
a : A
h : ∀ x ∈ σₙ R a, f x ≤ 0
h₁ : ContinuousOn f (σₙ R a)
h₂ : f 0 = 0
⊢ cfcₙ f a ≤ 0

tactic: simpa using cfcₙ_mono h
state_before: case pos.intro
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁸ : OrderedCommSemiring R
inst✝¹⁷ : Nontrivial R
inst✝¹⁶ : StarRing R
inst✝¹⁵ : StarOrderedRing R
inst✝¹⁴ : MetricSpace R
inst✝¹³ : TopologicalSemiring R
inst✝¹² : ContinuousStar R
inst✝¹¹ : ∀ (α : Type ?u.1171919) [inst : Zero α] [inst_1 : TopologicalSpace α], StarOrderedRing C(α, R)₀
inst✝¹⁰ : TopologicalSpace A
inst✝⁹ : NonUnitalRing A
inst✝⁸ : StarRing A
inst✝⁷ : PartialOrder A
inst✝⁶ : StarOrderedRing A
inst✝⁵ : Module R A
inst✝⁴ : IsScalarTower R A A
inst✝³ : SMulCommClass R A A
inst✝² : StarModule R A
inst✝¹ : NonUnitalContinuousFunctionalCalculus R p
inst✝ : NonnegSpectrumClass R A
f : R → R
a : A
h : ∀ x ∈ σₙ R a, f x ≤ 0
h₁ : ContinuousOn f (σₙ R a)
h₂ : f 0 = 0
⊢ cfcₙ f a ≤ 0
state_after: no goals

tactic: simp only [not_and_or] at hf
state_before: case neg
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁸ : OrderedCommSemiring R
inst✝¹⁷ : Nontrivial R
inst✝¹⁶ : StarRing R
inst✝¹⁵ : StarOrderedRing R
inst✝¹⁴ : MetricSpace R
inst✝¹³ : TopologicalSemiring R
inst✝¹² : ContinuousStar R
inst✝¹¹ : ∀ (α : Type u_1) [inst : Zero α] [inst_1 : TopologicalSpace α], StarOrderedRing C(α, R)₀
inst✝¹⁰ : TopologicalSpace A
inst✝⁹ : NonUnitalRing A
inst✝⁸ : StarRing A
inst✝⁷ : PartialOrder A
inst✝⁶ : StarOrderedRing A
inst✝⁵ : Module R A
inst✝⁴ : IsScalarTower R A A
inst✝³ : SMulCommClass R A A
inst✝² : StarModule R A
inst✝¹ : NonUnitalContinuousFunctionalCalculus R p
inst✝ : NonnegSpectrumClass R A
f : R → R
a : A
h : ∀ x ∈ σₙ R a, f x ≤ 0
hf : ¬(ContinuousOn f (σₙ R a) ∧ f 0 = 0)
⊢ cfcₙ f a ≤ 0
state_after: case neg
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁸ : OrderedCommSemiring R
inst✝¹⁷ : Nontrivial R
inst✝¹⁶ : StarRing R
inst✝¹⁵ : StarOrderedRing R
inst✝¹⁴ : MetricSpace R
inst✝¹³ : TopologicalSemiring R
inst✝¹² : ContinuousStar R
inst✝¹¹ : ∀ (α : Type u_1) [inst : Zero α] [inst_1 : TopologicalSpace α], StarOrderedRing C(α, R)₀
inst✝¹⁰ : TopologicalSpace A
inst✝⁹ : NonUnitalRing A
inst✝⁸ : StarRing A
inst✝⁷ : PartialOrder A
inst✝⁶ : StarOrderedRing A
inst✝⁵ : Module R A
inst✝⁴ : IsScalarTower R A A
inst✝³ : SMulCommClass R A A
inst✝² : StarModule R A
inst✝¹ : NonUnitalContinuousFunctionalCalculus R p
inst✝ : NonnegSpectrumClass R A
f : R → R
a : A
h : ∀ x ∈ σₙ R a, f x ≤ 0
hf : ¬ContinuousOn f (σₙ R a) ∨ ¬f 0 = 0
⊢ cfcₙ f a ≤ 0

tactic: obtain (hf | hf) := hf
state_before: case neg
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁸ : OrderedCommSemiring R
inst✝¹⁷ : Nontrivial R
inst✝¹⁶ : StarRing R
inst✝¹⁵ : StarOrderedRing R
inst✝¹⁴ : MetricSpace R
inst✝¹³ : TopologicalSemiring R
inst✝¹² : ContinuousStar R
inst✝¹¹ : ∀ (α : Type u_1) [inst : Zero α] [inst_1 : TopologicalSpace α], StarOrderedRing C(α, R)₀
inst✝¹⁰ : TopologicalSpace A
inst✝⁹ : NonUnitalRing A
inst✝⁸ : StarRing A
inst✝⁷ : PartialOrder A
inst✝⁶ : StarOrderedRing A
inst✝⁵ : Module R A
inst✝⁴ : IsScalarTower R A A
inst✝³ : SMulCommClass R A A
inst✝² : StarModule R A
inst✝¹ : NonUnitalContinuousFunctionalCalculus R p
inst✝ : NonnegSpectrumClass R A
f : R → R
a : A
h : ∀ x ∈ σₙ R a, f x ≤ 0
hf : ¬ContinuousOn f (σₙ R a) ∨ ¬f 0 = 0
⊢ cfcₙ f a ≤ 0
state_after: case neg.inl
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁸ : OrderedCommSemiring R
inst✝¹⁷ : Nontrivial R
inst✝¹⁶ : StarRing R
inst✝¹⁵ : StarOrderedRing R
inst✝¹⁴ : MetricSpace R
inst✝¹³ : TopologicalSemiring R
inst✝¹² : ContinuousStar R
inst✝¹¹ : ∀ (α : Type u_1) [inst : Zero α] [inst_1 : TopologicalSpace α], StarOrderedRing C(α, R)₀
inst✝¹⁰ : TopologicalSpace A
inst✝⁹ : NonUnitalRing A
inst✝⁸ : StarRing A
inst✝⁷ : PartialOrder A
inst✝⁶ : StarOrderedRing A
inst✝⁵ : Module R A
inst✝⁴ : IsScalarTower R A A
inst✝³ : SMulCommClass R A A
inst✝² : StarModule R A
inst✝¹ : NonUnitalContinuousFunctionalCalculus R p
inst✝ : NonnegSpectrumClass R A
f : R → R
a : A
h : ∀ x ∈ σₙ R a, f x ≤ 0
hf : ¬ContinuousOn f (σₙ R a)
⊢ cfcₙ f a ≤ 0

case neg.inr
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁸ : OrderedCommSemiring R
inst✝¹⁷ : Nontrivial R
inst✝¹⁶ : StarRing R
inst✝¹⁵ : StarOrderedRing R
inst✝¹⁴ : MetricSpace R
inst✝¹³ : TopologicalSemiring R
inst✝¹² : ContinuousStar R
inst✝¹¹ : ∀ (α : Type u_1) [inst : Zero α] [inst_1 : TopologicalSpace α], StarOrderedRing C(α, R)₀
inst✝¹⁰ : TopologicalSpace A
inst✝⁹ : NonUnitalRing A
inst✝⁸ : StarRing A
inst✝⁷ : PartialOrder A
inst✝⁶ : StarOrderedRing A
inst✝⁵ : Module R A
inst✝⁴ : IsScalarTower R A A
inst✝³ : SMulCommClass R A A
inst✝² : StarModule R A
inst✝¹ : NonUnitalContinuousFunctionalCalculus R p
inst✝ : NonnegSpectrumClass R A
f : R → R
a : A
h : ∀ x ∈ σₙ R a, f x ≤ 0
hf : ¬f 0 = 0
⊢ cfcₙ f a ≤ 0

tactic: simp only [cfcₙ_apply_of_not_continuousOn _ hf, le_rfl]
state_before: case neg.inl
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁸ : OrderedCommSemiring R
inst✝¹⁷ : Nontrivial R
inst✝¹⁶ : StarRing R
inst✝¹⁵ : StarOrderedRing R
inst✝¹⁴ : MetricSpace R
inst✝¹³ : TopologicalSemiring R
inst✝¹² : ContinuousStar R
inst✝¹¹ : ∀ (α : Type u_1) [inst : Zero α] [inst_1 : TopologicalSpace α], StarOrderedRing C(α, R)₀
inst✝¹⁰ : TopologicalSpace A
inst✝⁹ : NonUnitalRing A
inst✝⁸ : StarRing A
inst✝⁷ : PartialOrder A
inst✝⁶ : StarOrderedRing A
inst✝⁵ : Module R A
inst✝⁴ : IsScalarTower R A A
inst✝³ : SMulCommClass R A A
inst✝² : StarModule R A
inst✝¹ : NonUnitalContinuousFunctionalCalculus R p
inst✝ : NonnegSpectrumClass R A
f : R → R
a : A
h : ∀ x ∈ σₙ R a, f x ≤ 0
hf : ¬ContinuousOn f (σₙ R a)
⊢ cfcₙ f a ≤ 0
state_after: no goals

tactic: simp only [cfcₙ_apply_of_not_map_zero _ hf, le_rfl]
state_before: case neg.inr
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁸ : OrderedCommSemiring R
inst✝¹⁷ : Nontrivial R
inst✝¹⁶ : StarRing R
inst✝¹⁵ : StarOrderedRing R
inst✝¹⁴ : MetricSpace R
inst✝¹³ : TopologicalSemiring R
inst✝¹² : ContinuousStar R
inst✝¹¹ : ∀ (α : Type u_1) [inst : Zero α] [inst_1 : TopologicalSpace α], StarOrderedRing C(α, R)₀
inst✝¹⁰ : TopologicalSpace A
inst✝⁹ : NonUnitalRing A
inst✝⁸ : StarRing A
inst✝⁷ : PartialOrder A
inst✝⁶ : StarOrderedRing A
inst✝⁵ : Module R A
inst✝⁴ : IsScalarTower R A A
inst✝³ : SMulCommClass R A A
inst✝² : StarModule R A
inst✝¹ : NonUnitalContinuousFunctionalCalculus R p
inst✝ : NonnegSpectrumClass R A
f : R → R
a : A
h : ∀ x ∈ σₙ R a, f x ≤ 0
hf : ¬f 0 = 0
⊢ cfcₙ f a ≤ 0
state_after: no goals