Theorem: spectrum.nonempty
File: Mathlib/Analysis/NormedSpace/Spectrum.lean

tactic: by_contra! h
state_before: 𝕜 : Type u_1
A : Type u_2
inst✝³ : NormedRing A
inst✝² : NormedAlgebra ℂ A
inst✝¹ : CompleteSpace A
inst✝ : Nontrivial A
a : A
⊢ (spectrum ℂ a).Nonempty
state_after: 𝕜 : Type u_1
A : Type u_2
inst✝³ : NormedRing A
inst✝² : NormedAlgebra ℂ A
inst✝¹ : CompleteSpace A
inst✝ : Nontrivial A
a : A
h : spectrum ℂ a = ∅
⊢ False

tactic: have H₀ : resolventSet ℂ a = Set.univ := by rwa [spectrum, Set.compl_empty_iff] at h
state_before: 𝕜 : Type u_1
A : Type u_2
inst✝³ : NormedRing A
inst✝² : NormedAlgebra ℂ A
inst✝¹ : CompleteSpace A
inst✝ : Nontrivial A
a : A
h : spectrum ℂ a = ∅
⊢ False
state_after: 𝕜 : Type u_1
A : Type u_2
inst✝³ : NormedRing A
inst✝² : NormedAlgebra ℂ A
inst✝¹ : CompleteSpace A
inst✝ : Nontrivial A
a : A
h : spectrum ℂ a = ∅
H₀ : resolventSet ℂ a = Set.univ
⊢ False

tactic: have H₁ : Differentiable ℂ fun z : ℂ => resolvent a z := fun z =>
  (hasDerivAt_resolvent (H₀.symm ▸ Set.mem_univ z : z ∈ resolventSet ℂ a)).differentiableAt
state_before: 𝕜 : Type u_1
A : Type u_2
inst✝³ : NormedRing A
inst✝² : NormedAlgebra ℂ A
inst✝¹ : CompleteSpace A
inst✝ : Nontrivial A
a : A
h : spectrum ℂ a = ∅
H₀ : resolventSet ℂ a = Set.univ
⊢ False
state_after: 𝕜 : Type u_1
A : Type u_2
inst✝³ : NormedRing A
inst✝² : NormedAlgebra ℂ A
inst✝¹ : CompleteSpace A
inst✝ : Nontrivial A
a : A
h : spectrum ℂ a = ∅
H₀ : resolventSet ℂ a = Set.univ
H₁ : Differentiable ℂ fun z => resolvent a z
⊢ False

tactic: have H₃ := H₁.apply_eq_of_tendsto_cocompact 0 <| by
  simpa [Metric.cobounded_eq_cocompact] using resolvent_tendsto_cobounded a (𝕜 := ℂ)
state_before: 𝕜 : Type u_1
A : Type u_2
inst✝³ : NormedRing A
inst✝² : NormedAlgebra ℂ A
inst✝¹ : CompleteSpace A
inst✝ : Nontrivial A
a : A
h : spectrum ℂ a = ∅
H₀ : resolventSet ℂ a = Set.univ
H₁ : Differentiable ℂ fun z => resolvent a z
⊢ False
state_after: 𝕜 : Type u_1
A : Type u_2
inst✝³ : NormedRing A
inst✝² : NormedAlgebra ℂ A
inst✝¹ : CompleteSpace A
inst✝ : Nontrivial A
a : A
h : spectrum ℂ a = ∅
H₀ : resolventSet ℂ a = Set.univ
H₁ : Differentiable ℂ fun z => resolvent a z
H₃ : resolvent a 0 = 0
⊢ False

tactic: exact not_isUnit_zero <| H₃ ▸ (isUnit_resolvent.mp <| H₀.symm ▸ Set.mem_univ 0)
state_before: 𝕜 : Type u_1
A : Type u_2
inst✝³ : NormedRing A
inst✝² : NormedAlgebra ℂ A
inst✝¹ : CompleteSpace A
inst✝ : Nontrivial A
a : A
h : spectrum ℂ a = ∅
H₀ : resolventSet ℂ a = Set.univ
H₁ : Differentiable ℂ fun z => resolvent a z
H₃ : resolvent a 0 = 0
⊢ False
state_after: no goals

tactic: rwa [spectrum, Set.compl_empty_iff] at h
state_before: 𝕜 : Type u_1
A : Type u_2
inst✝³ : NormedRing A
inst✝² : NormedAlgebra ℂ A
inst✝¹ : CompleteSpace A
inst✝ : Nontrivial A
a : A
h : spectrum ℂ a = ∅
⊢ resolventSet ℂ a = Set.univ
state_after: no goals

tactic: simpa [Metric.cobounded_eq_cocompact] using resolvent_tendsto_cobounded a (𝕜 := ℂ)
state_before: 𝕜 : Type u_1
A : Type u_2
inst✝³ : NormedRing A
inst✝² : NormedAlgebra ℂ A
inst✝¹ : CompleteSpace A
inst✝ : Nontrivial A
a : A
h : spectrum ℂ a = ∅
H₀ : resolventSet ℂ a = Set.univ
H₁ : Differentiable ℂ fun z => resolvent a z
⊢ Filter.Tendsto (fun z => resolvent a z) (Filter.cocompact ℂ) (nhds ?m.242830)
state_after: no goals