Theorem: spectrum_star_mul_self_nonneg
File: Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Instances.lean

Proof steps:
state_before: A : Type u_1
inst✝⁵ : NormedRing A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : CompleteSpace A
inst✝¹ : NormedAlgebra ℂ A
inst✝ : StarModule ℂ A
b : A
⊢ ∀ x ∈ σ ℝ (star b * b), 0 ≤ x
state_after: A : Type u_1
inst✝⁵ : NormedRing A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : CompleteSpace A
inst✝¹ : NormedAlgebra ℂ A
inst✝ : StarModule ℂ A
b : A
a : A := star b * b
⊢ ∀ x ∈ σ ℝ a, 0 ≤ x
tactic: set a := star b * b
state_before: A : Type u_1
inst✝⁵ : NormedRing A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : CompleteSpace A
inst✝¹ : NormedAlgebra ℂ A
inst✝ : StarModule ℂ A
b : A
a : A := star b * b
⊢ ∀ x ∈ σ ℝ a, 0 ≤ x
state_after: A : Type u_1
inst✝⁵ : NormedRing A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : CompleteSpace A
inst✝¹ : NormedAlgebra ℂ A
inst✝ : StarModule ℂ A
b : A
a : A := star b * b
a_def : a = star b * b
⊢ ∀ x ∈ σ ℝ a, 0 ≤ x
tactic: have a_def : a = star b * b := rfl
state_before: A : Type u_1
inst✝⁵ : NormedRing A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : CompleteSpace A
inst✝¹ : NormedAlgebra ℂ A
inst✝ : StarModule ℂ A
b : A
a : A := star b * b
a_def : a = star b * b
⊢ ∀ x ∈ σ ℝ a, 0 ≤ x
state_after: A : Type u_1
inst✝⁵ : NormedRing A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : CompleteSpace A
inst✝¹ : NormedAlgebra ℂ A
inst✝ : StarModule ℂ A
b : A
a : A := star b * b
a_def : a = star b * b
a_neg : A := cfc (fun x => (-ContinuousMap.id ℝ ⊔ 0) x) a
⊢ ∀ x ∈ σ ℝ a, 0 ≤ x
tactic: let a_neg : A := cfc (fun x ↦ (- ContinuousMap.id ℝ ⊔ 0) x) a
state_before: A : Type u_1
inst✝⁵ : NormedRing A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : CompleteSpace A
inst✝¹ : NormedAlgebra ℂ A
inst✝ : StarModule ℂ A
b : A
a : A := star b * b
a_def : a = star b * b
a_neg : A := cfc (fun x => (-ContinuousMap.id ℝ ⊔ 0) x) a
⊢ ∀ x ∈ σ ℝ a, 0 ≤ x
state_after: A : Type u_1
inst✝⁵ : NormedRing A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : CompleteSpace A
inst✝¹ : NormedAlgebra ℂ A
inst✝ : StarModule ℂ A
b : A
a : A := star b * b
a_def : a = star b * b
a_neg : A := cfc (fun x => (-ContinuousMap.id ℝ ⊔ 0) x) a
c : A := b * a_neg
⊢ ∀ x ∈ σ ℝ a, 0 ≤ x
tactic: set c := b * a_neg
state_before: A : Type u_1
inst✝⁵ : NormedRing A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : CompleteSpace A
inst✝¹ : NormedAlgebra ℂ A
inst✝ : StarModule ℂ A
b : A
a : A := star b * b
a_def : a = star b * b
a_neg : A := cfc (fun x => (-ContinuousMap.id ℝ ⊔ 0) x) a
c : A := b * a_neg
h_eq_a_neg : -(star c * c) = a_neg ^ 3
h_c_spec₀ : SpectrumRestricts (-(star c * c)) fun x => ContinuousMap.realToNNReal x
⊢ ∀ x ∈ σ ℝ a, 0 ≤ x
state_after: A : Type u_1
inst✝⁵ : NormedRing A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : CompleteSpace A
inst✝¹ : NormedAlgebra ℂ A
inst✝ : StarModule ℂ A
b : A
a : A := star b * b
a_def : a = star b * b
a_neg : A := cfc (fun x => (-ContinuousMap.id ℝ ⊔ 0) x) a
c : A := b * a_neg
h_eq_a_neg : -(star c * c) = a_neg ^ 3
h_c_spec₀ : SpectrumRestricts (-(star c * c)) fun x => ContinuousMap.realToNNReal x
c_eq : star c * c + c * star c = 2 • (↑(ℜ c) * ↑(ℜ c) + ↑(ℑ c) * ↑(ℑ c))
⊢ ∀ x ∈ σ ℝ a, 0 ≤ x
tactic: have c_eq := star_mul_self_add_self_mul_star c
state_before: A : Type u_1
inst✝⁵ : NormedRing A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : CompleteSpace A
inst✝¹ : NormedAlgebra ℂ A
inst✝ : StarModule ℂ A
b : A
a : A := star b * b
a_def : a = star b * b
a_neg : A := cfc (fun x => (-ContinuousMap.id ℝ ⊔ 0) x) a
c : A := b * a_neg
h_eq_a_neg : -(star c * c) = a_neg ^ 3
h_c_spec₀ : SpectrumRestricts (-(star c * c)) fun x => ContinuousMap.realToNNReal x
c_eq : star c * c + c * star c = 2 • (↑(ℜ c) * ↑(ℜ c) + ↑(ℑ c) * ↑(ℑ c))
⊢ ∀ x ∈ σ ℝ a, 0 ≤ x
state_after: A : Type u_1
inst✝⁵ : NormedRing A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : CompleteSpace A
inst✝¹ : NormedAlgebra ℂ A
inst✝ : StarModule ℂ A
b : A
a : A := star b * b
a_def : a = star b * b
a_neg : A := cfc (fun x => (-ContinuousMap.id ℝ ⊔ 0) x) a
c : A := b * a_neg
h_eq_a_neg : -(star c * c) = a_neg ^ 3
h_c_spec₀ : SpectrumRestricts (-(star c * c)) fun x => ContinuousMap.realToNNReal x
c_eq : c * star c = 2 • (↑(ℜ c) ^ 2 + ↑(ℑ c) ^ 2) + -(star c * c)
⊢ ∀ x ∈ σ ℝ a, 0 ≤ x
tactic: rw [← eq_sub_iff_add_eq', sub_eq_add_neg, ← sq, ← sq] at c_eq
state_before: A : Type u_1
inst✝⁵ : NormedRing A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : CompleteSpace A
inst✝¹ : NormedAlgebra ℂ A
inst✝ : StarModule ℂ A
b : A
a : A := star b * b
a_def : a = star b * b
a_neg : A := cfc (fun x => (-ContinuousMap.id ℝ ⊔ 0) x) a
c : A := b * a_neg
h_eq_a_neg : -(star c * c) = a_neg ^ 3
h_c_spec₀ : SpectrumRestricts (-(star c * c)) fun x => ContinuousMap.realToNNReal x
c_eq : c * star c = 2 • (↑(ℜ c) ^ 2 + ↑(ℑ c) ^ 2) + -(star c * c)
h_c_spec₁ : SpectrumRestricts (c * star c) ⇑ContinuousMap.realToNNReal
⊢ ∀ x ∈ σ ℝ a, 0 ≤ x
state_after: A : Type u_1
inst✝⁵ : NormedRing A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : CompleteSpace A
inst✝¹ : NormedAlgebra ℂ A
inst✝ : StarModule ℂ A
b : A
a : A := star b * b
a_def : a = star b * b
a_neg : A := cfc (fun x => (-ContinuousMap.id ℝ ⊔ 0) x) a
c : A := b * a_neg
h_eq_a_neg : -(star c * c) = a_neg ^ 3
h_c_spec₀ : SpectrumRestricts (-(star c * c)) fun x => ContinuousMap.realToNNReal x
c_eq : c * star c = 2 • (↑(ℜ c) ^ 2 + ↑(ℑ c) ^ 2) + -(star c * c)
h_c_spec₁ : SpectrumRestricts (c * star c) ⇑ContinuousMap.realToNNReal
h_c_spec₂ : SpectrumRestricts (star c * c) ⇑ContinuousMap.realToNNReal
⊢ ∀ x ∈ σ ℝ a, 0 ≤ x
tactic: have h_c_spec₂ : SpectrumRestricts (star c * c) ContinuousMap.realToNNReal := by
    rw [SpectrumRestricts.nnreal_iff] at h_c_spec₁ ⊢
    intro x hx
    replace hx := Set.subset_diff_union _ {(0 : ℝ)} hx
    rw [spectrum.nonzero_mul_eq_swap_mul, Set.diff_union_self, Set.union_singleton,
      Set.mem_insert_iff] at hx
    obtain (rfl | hx) := hx
    exacts [le_rfl, h_c_spec₁ x hx]
state_before: A : Type u_1
inst✝⁵ : NormedRing A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : CompleteSpace A
inst✝¹ : NormedAlgebra ℂ A
inst✝ : StarModule ℂ A
b : A
a : A := star b * b
a_def : a = star b * b
a_neg : A := cfc (fun x => (-ContinuousMap.id ℝ ⊔ 0) x) a
c : A := b * a_neg
h_eq_a_neg : -(star c * c) = a_neg ^ 3
h_c_spec₀ : SpectrumRestricts (-(star c * c)) fun x => ContinuousMap.realToNNReal x
c_eq : c * star c = 2 • (↑(ℜ c) ^ 2 + ↑(ℑ c) ^ 2) + -(star c * c)
h_c_spec₁ : SpectrumRestricts (c * star c) ⇑ContinuousMap.realToNNReal
h_c_spec₂ : SpectrumRestricts (star c * c) ⇑ContinuousMap.realToNNReal
⊢ ∀ x ∈ σ ℝ a, 0 ≤ x
state_after: A : Type u_1
inst✝⁵ : NormedRing A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : CompleteSpace A
inst✝¹ : NormedAlgebra ℂ A
inst✝ : StarModule ℂ A
b : A
a : A := star b * b
a_def : a = star b * b
a_neg : A := cfc (fun x => (-ContinuousMap.id ℝ ⊔ 0) x) a
c : A := b * a_neg
h_eq_a_neg : 0 = a_neg ^ 3
h_c_spec₀ : SpectrumRestricts (-(star c * c)) fun x => ContinuousMap.realToNNReal x
c_eq : c * star c = 2 • (↑(ℜ c) ^ 2 + ↑(ℑ c) ^ 2) + -(star c * c)
h_c_spec₁ : SpectrumRestricts (c * star c) ⇑ContinuousMap.realToNNReal
h_c_spec₂ : SpectrumRestricts (star c * c) ⇑ContinuousMap.realToNNReal
⊢ ∀ x ∈ σ ℝ a, 0 ≤ x
tactic: rw [h_c_spec₂.eq_zero_of_neg (.star_mul_self c) h_c_spec₀, neg_zero] at h_eq_a_neg
state_before: A : Type u_1
inst✝⁵ : NormedRing A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : CompleteSpace A
inst✝¹ : NormedAlgebra ℂ A
inst✝ : StarModule ℂ A
b : A
a : A := star b * b
a_def : a = star b * b
a_neg : A := cfc (fun x => (-ContinuousMap.id ℝ ⊔ 0) x) a
c : A := b * a_neg
h_eq_a_neg : 0 = a_neg ^ 3
h_c_spec₀ : SpectrumRestricts (-(star c * c)) fun x => ContinuousMap.realToNNReal x
c_eq : c * star c = 2 • (↑(ℜ c) ^ 2 + ↑(ℑ c) ^ 2) + -(star c * c)
h_c_spec₁ : SpectrumRestricts (c * star c) ⇑ContinuousMap.realToNNReal
h_c_spec₂ : SpectrumRestricts (star c * c) ⇑ContinuousMap.realToNNReal
⊢ ∀ x ∈ σ ℝ a, 0 ≤ x
state_after: A : Type u_1
inst✝⁵ : NormedRing A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : CompleteSpace A
inst✝¹ : NormedAlgebra ℂ A
inst✝ : StarModule ℂ A
b : A
a : A := star b * b
a_def : a = star b * b
a_neg : A := cfc (fun x => (-ContinuousMap.id ℝ ⊔ 0) x) a
c : A := b * a_neg
h_eq_a_neg : 0 = cfc (fun x => (-ContinuousMap.id ℝ ⊔ 0) x) a ^ 3
h_c_spec₀ : SpectrumRestricts (-(star c * c)) fun x => ContinuousMap.realToNNReal x
c_eq : c * star c = 2 • (↑(ℜ c) ^ 2 + ↑(ℑ c) ^ 2) + -(star c * c)
h_c_spec₁ : SpectrumRestricts (c * star c) ⇑ContinuousMap.realToNNReal
h_c_spec₂ : SpectrumRestricts (star c * c) ⇑ContinuousMap.realToNNReal
⊢ ∀ x ∈ σ ℝ a, 0 ≤ x
tactic: simp only [a_neg] at h_eq_a_neg
state_before: A : Type u_1
inst✝⁵ : NormedRing A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : CompleteSpace A
inst✝¹ : NormedAlgebra ℂ A
inst✝ : StarModule ℂ A
b : A
a : A := star b * b
a_def : a = star b * b
a_neg : A := cfc (fun x => (-ContinuousMap.id ℝ ⊔ 0) x) a
c : A := b * a_neg
h_eq_a_neg : 0 = cfc (fun x => (-ContinuousMap.id ℝ ⊔ 0) x) a ^ 3
h_c_spec₀ : SpectrumRestricts (-(star c * c)) fun x => ContinuousMap.realToNNReal x
c_eq : c * star c = 2 • (↑(ℜ c) ^ 2 + ↑(ℑ c) ^ 2) + -(star c * c)
h_c_spec₁ : SpectrumRestricts (c * star c) ⇑ContinuousMap.realToNNReal
h_c_spec₂ : SpectrumRestricts (star c * c) ⇑ContinuousMap.realToNNReal
⊢ ∀ x ∈ σ ℝ a, 0 ≤ x
state_after: A : Type u_1
inst✝⁵ : NormedRing A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : CompleteSpace A
inst✝¹ : NormedAlgebra ℂ A
inst✝ : StarModule ℂ A
b : A
a : A := star b * b
a_def : a = star b * b
a_neg : A := cfc (fun x => (-ContinuousMap.id ℝ ⊔ 0) x) a
c : A := b * a_neg
h_eq_a_neg : cfc 0 a = cfc (fun x => (-ContinuousMap.id ℝ ⊔ 0) x ^ 3) a
h_c_spec₀ : SpectrumRestricts (-(star c * c)) fun x => ContinuousMap.realToNNReal x
c_eq : c * star c = 2 • (↑(ℜ c) ^ 2 + ↑(ℑ c) ^ 2) + -(star c * c)
h_c_spec₁ : SpectrumRestricts (c * star c) ⇑ContinuousMap.realToNNReal
h_c_spec₂ : SpectrumRestricts (star c * c) ⇑ContinuousMap.realToNNReal
⊢ ∀ x ∈ σ ℝ a, 0 ≤ x
tactic: rw [← cfc_pow _ _ (ha := .star_mul_self b), ← cfc_zero a (R := ℝ)] at h_eq_a_neg
state_before: A : Type u_1
inst✝⁵ : NormedRing A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : CompleteSpace A
inst✝¹ : NormedAlgebra ℂ A
inst✝ : StarModule ℂ A
b : A
a : A := star b * b
a_def : a = star b * b
a_neg : A := cfc (fun x => (-ContinuousMap.id ℝ ⊔ 0) x) a
c : A := b * a_neg
h_eq_a_neg : cfc 0 a = cfc (fun x => (-ContinuousMap.id ℝ ⊔ 0) x ^ 3) a
h_c_spec₀ : SpectrumRestricts (-(star c * c)) fun x => ContinuousMap.realToNNReal x
c_eq : c * star c = 2 • (↑(ℜ c) ^ 2 + ↑(ℑ c) ^ 2) + -(star c * c)
h_c_spec₁ : SpectrumRestricts (c * star c) ⇑ContinuousMap.realToNNReal
h_c_spec₂ : SpectrumRestricts (star c * c) ⇑ContinuousMap.realToNNReal
⊢ ∀ x ∈ σ ℝ a, 0 ≤ x
state_after: A : Type u_1
inst✝⁵ : NormedRing A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : CompleteSpace A
inst✝¹ : NormedAlgebra ℂ A
inst✝ : StarModule ℂ A
b : A
a : A := star b * b
a_def : a = star b * b
a_neg : A := cfc (fun x => (-ContinuousMap.id ℝ ⊔ 0) x) a
c : A := b * a_neg
h_eq_a_neg : cfc 0 a = cfc (fun x => (-ContinuousMap.id ℝ ⊔ 0) x ^ 3) a
h_c_spec₀ : SpectrumRestricts (-(star c * c)) fun x => ContinuousMap.realToNNReal x
c_eq : c * star c = 2 • (↑(ℜ c) ^ 2 + ↑(ℑ c) ^ 2) + -(star c * c)
h_c_spec₁ : SpectrumRestricts (c * star c) ⇑ContinuousMap.realToNNReal
h_c_spec₂ : SpectrumRestricts (star c * c) ⇑ContinuousMap.realToNNReal
x : ℝ
hx : x ∈ σ ℝ a
⊢ 0 ≤ x
tactic: intro x hx
state_before: A : Type u_1
inst✝⁵ : NormedRing A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : CompleteSpace A
inst✝¹ : NormedAlgebra ℂ A
inst✝ : StarModule ℂ A
b : A
a : A := star b * b
a_def : a = star b * b
a_neg : A := cfc (fun x => (-ContinuousMap.id ℝ ⊔ 0) x) a
c : A := b * a_neg
h_eq_a_neg : cfc 0 a = cfc (fun x => (-ContinuousMap.id ℝ ⊔ 0) x ^ 3) a
h_c_spec₀ : SpectrumRestricts (-(star c * c)) fun x => ContinuousMap.realToNNReal x
c_eq : c * star c = 2 • (↑(ℜ c) ^ 2 + ↑(ℑ c) ^ 2) + -(star c * c)
h_c_spec₁ : SpectrumRestricts (c * star c) ⇑ContinuousMap.realToNNReal
h_c_spec₂ : SpectrumRestricts (star c * c) ⇑ContinuousMap.realToNNReal
x : ℝ
hx : x ∈ σ ℝ a
⊢ 0 ≤ x
state_after: A : Type u_1
inst✝⁵ : NormedRing A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : CompleteSpace A
inst✝¹ : NormedAlgebra ℂ A
inst✝ : StarModule ℂ A
b : A
a : A := star b * b
a_def : a = star b * b
a_neg : A := cfc (fun x => (-ContinuousMap.id ℝ ⊔ 0) x) a
c : A := b * a_neg
h_eq_a_neg : cfc 0 a = cfc (fun x => (-ContinuousMap.id ℝ ⊔ 0) x ^ 3) a
h_c_spec₀ : SpectrumRestricts (-(star c * c)) fun x => ContinuousMap.realToNNReal x
c_eq : c * star c = 2 • (↑(ℜ c) ^ 2 + ↑(ℑ c) ^ 2) + -(star c * c)
h_c_spec₁ : SpectrumRestricts (c * star c) ⇑ContinuousMap.realToNNReal
h_c_spec₂ : SpectrumRestricts (star c * c) ⇑ContinuousMap.realToNNReal
x : ℝ
hx : x ∈ σ ℝ a
hx' : x < 0
⊢ False
tactic: by_contra! hx'
state_before: A : Type u_1
inst✝⁵ : NormedRing A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : CompleteSpace A
inst✝¹ : NormedAlgebra ℂ A
inst✝ : StarModule ℂ A
b : A
a : A := star b * b
a_def : a = star b * b
a_neg : A := cfc (fun x => (-ContinuousMap.id ℝ ⊔ 0) x) a
c : A := b * a_neg
h_eq_a_neg : cfc 0 a = cfc (fun x => (-ContinuousMap.id ℝ ⊔ 0) x ^ 3) a
h_c_spec₀ : SpectrumRestricts (-(star c * c)) fun x => ContinuousMap.realToNNReal x
c_eq : c * star c = 2 • (↑(ℜ c) ^ 2 + ↑(ℑ c) ^ 2) + -(star c * c)
h_c_spec₁ : SpectrumRestricts (c * star c) ⇑ContinuousMap.realToNNReal
h_c_spec₂ : SpectrumRestricts (star c * c) ⇑ContinuousMap.realToNNReal
x : ℝ
hx : x ∈ σ ℝ a
hx' : x < 0
⊢ False
state_after: A : Type u_1
inst✝⁵ : NormedRing A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : CompleteSpace A
inst✝¹ : NormedAlgebra ℂ A
inst✝ : StarModule ℂ A
b : A
a : A := star b * b
a_def : a = star b * b
a_neg : A := cfc (fun x => (-ContinuousMap.id ℝ ⊔ 0) x) a
c : A := b * a_neg
h_eq_a_neg : cfc 0 a = cfc (fun x => (-ContinuousMap.id ℝ ⊔ 0) x ^ 3) a
h_c_spec₀ : SpectrumRestricts (-(star c * c)) fun x => ContinuousMap.realToNNReal x
c_eq : c * star c = 2 • (↑(ℜ c) ^ 2 + ↑(ℑ c) ^ 2) + -(star c * c)
h_c_spec₁ : SpectrumRestricts (c * star c) ⇑ContinuousMap.realToNNReal
h_c_spec₂ : SpectrumRestricts (star c * c) ⇑ContinuousMap.realToNNReal
x : ℝ
hx : x ∈ σ ℝ a
hx' : 0 < -x
⊢ False
tactic: rw [← neg_pos] at hx'
state_before: A : Type u_1
inst✝⁵ : NormedRing A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : CompleteSpace A
inst✝¹ : NormedAlgebra ℂ A
inst✝ : StarModule ℂ A
b : A
a : A := star b * b
a_def : a = star b * b
a_neg : A := cfc (fun x => (-ContinuousMap.id ℝ ⊔ 0) x) a
c : A := b * a_neg
h_eq_a_neg : cfc 0 a = cfc (fun x => (-ContinuousMap.id ℝ ⊔ 0) x ^ 3) a
h_c_spec₀ : SpectrumRestricts (-(star c * c)) fun x => ContinuousMap.realToNNReal x
c_eq : c * star c = 2 • (↑(ℜ c) ^ 2 + ↑(ℑ c) ^ 2) + -(star c * c)
h_c_spec₁ : SpectrumRestricts (c * star c) ⇑ContinuousMap.realToNNReal
h_c_spec₂ : SpectrumRestricts (star c * c) ⇑ContinuousMap.realToNNReal
x : ℝ
hx : x ∈ σ ℝ a
hx' : 0 < -x
⊢ False
state_after: A : Type u_1
inst✝⁵ : NormedRing A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : CompleteSpace A
inst✝¹ : NormedAlgebra ℂ A
inst✝ : StarModule ℂ A
b : A
a : A := star b * b
a_def : a = star b * b
a_neg : A := cfc (fun x => (-ContinuousMap.id ℝ ⊔ 0) x) a
c : A := b * a_neg
h_eq_a_neg : cfc 0 a = cfc (fun x => (-ContinuousMap.id ℝ ⊔ 0) x ^ 3) a
h_c_spec₀ : SpectrumRestricts (-(star c * c)) fun x => ContinuousMap.realToNNReal x
c_eq : c * star c = 2 • (↑(ℜ c) ^ 2 + ↑(ℑ c) ^ 2) + -(star c * c)
h_c_spec₁ : SpectrumRestricts (c * star c) ⇑ContinuousMap.realToNNReal
h_c_spec₂ : SpectrumRestricts (star c * c) ⇑ContinuousMap.realToNNReal
x : ℝ
hx : x ∈ σ ℝ a
hx' : 0 < -x
⊢ 0 = (-x) ^ 3
tactic: apply (pow_pos hx' 3).ne
state_before: A : Type u_1
inst✝⁵ : NormedRing A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : CompleteSpace A
inst✝¹ : NormedAlgebra ℂ A
inst✝ : StarModule ℂ A
b : A
a : A := star b * b
a_def : a = star b * b
a_neg : A := cfc (fun x => (-ContinuousMap.id ℝ ⊔ 0) x) a
c : A := b * a_neg
h_eq_a_neg : cfc 0 a = cfc (fun x => (-ContinuousMap.id ℝ ⊔ 0) x ^ 3) a
h_c_spec₀ : SpectrumRestricts (-(star c * c)) fun x => ContinuousMap.realToNNReal x
c_eq : c * star c = 2 • (↑(ℜ c) ^ 2 + ↑(ℑ c) ^ 2) + -(star c * c)
h_c_spec₁ : SpectrumRestricts (c * star c) ⇑ContinuousMap.realToNNReal
h_c_spec₂ : SpectrumRestricts (star c * c) ⇑ContinuousMap.realToNNReal
x : ℝ
hx : x ∈ σ ℝ a
hx' : 0 < -x
⊢ 0 = (-x) ^ 3
state_after: A : Type u_1
inst✝⁵ : NormedRing A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : CompleteSpace A
inst✝¹ : NormedAlgebra ℂ A
inst✝ : StarModule ℂ A
b : A
a : A := star b * b
a_def : a = star b * b
a_neg : A := cfc (fun x => (-ContinuousMap.id ℝ ⊔ 0) x) a
c : A := b * a_neg
h_eq_a_neg : cfc 0 a = cfc (fun x => (-ContinuousMap.id ℝ ⊔ 0) x ^ 3) a
h_c_spec₀ : SpectrumRestricts (-(star c * c)) fun x => ContinuousMap.realToNNReal x
c_eq : c * star c = 2 • (↑(ℜ c) ^ 2 + ↑(ℑ c) ^ 2) + -(star c * c)
h_c_spec₁ : SpectrumRestricts (c * star c) ⇑ContinuousMap.realToNNReal
h_c_spec₂ : SpectrumRestricts (star c * c) ⇑ContinuousMap.realToNNReal
x : ℝ
hx : x ∈ σ ℝ a
hx' : 0 < -x
h_eqOn : Set.EqOn 0 (fun x => (-ContinuousMap.id ℝ ⊔ 0) x ^ 3) (σ ℝ a)
⊢ 0 = (-x) ^ 3
tactic: have h_eqOn := eqOn_of_cfc_eq_cfc (ha := IsSelfAdjoint.star_mul_self b) h_eq_a_neg
state_before: A : Type u_1
inst✝⁵ : NormedRing A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : CompleteSpace A
inst✝¹ : NormedAlgebra ℂ A
inst✝ : StarModule ℂ A
b : A
a : A := star b * b
a_def : a = star b * b
a_neg : A := cfc (fun x => (-ContinuousMap.id ℝ ⊔ 0) x) a
c : A := b * a_neg
h_eq_a_neg : cfc 0 a = cfc (fun x => (-ContinuousMap.id ℝ ⊔ 0) x ^ 3) a
h_c_spec₀ : SpectrumRestricts (-(star c * c)) fun x => ContinuousMap.realToNNReal x
c_eq : c * star c = 2 • (↑(ℜ c) ^ 2 + ↑(ℑ c) ^ 2) + -(star c * c)
h_c_spec₁ : SpectrumRestricts (c * star c) ⇑ContinuousMap.realToNNReal
h_c_spec₂ : SpectrumRestricts (star c * c) ⇑ContinuousMap.realToNNReal
x : ℝ
hx : x ∈ σ ℝ a
hx' : 0 < -x
h_eqOn : Set.EqOn 0 (fun x => (-ContinuousMap.id ℝ ⊔ 0) x ^ 3) (σ ℝ a)
⊢ 0 = (-x) ^ 3
state_after: no goals
tactic: simpa [sup_eq_left.mpr hx'.le] using h_eqOn hx