Theorem: elementalStarAlgebra.isUnit_of_isUnit_of_isStarNormal
File: Mathlib/Analysis/NormedSpace/Star/ContinuousFunctionalCalculus.lean

tactic: nontriviality A
state_before: A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
h : IsUnit a
⊢ IsUnit ⟨a, ⋯⟩
state_after: A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
h : IsUnit a
a✝ : Nontrivial A
⊢ IsUnit ⟨a, ⋯⟩

tactic: set a' : elementalStarAlgebra ℂ a := ⟨a, self_mem ℂ a⟩
state_before: A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
h : IsUnit a
a✝ : Nontrivial A
⊢ IsUnit ⟨a, ⋯⟩
state_after: A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
h : IsUnit a
a✝ : Nontrivial A
a' : ↥(elementalStarAlgebra ℂ a) := ⟨a, ⋯⟩
⊢ IsUnit a'

tactic: suffices IsUnit (star a' * a') from (IsUnit.mul_iff.1 this).2
state_before: A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
h : IsUnit a
a✝ : Nontrivial A
a' : ↥(elementalStarAlgebra ℂ a) := ⟨a, ⋯⟩
⊢ IsUnit a'
state_after: A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
h : IsUnit a
a✝ : Nontrivial A
a' : ↥(elementalStarAlgebra ℂ a) := ⟨a, ⋯⟩
⊢ IsUnit (star a' * a')

tactic: replace h := (show Commute (star a) a from star_comm_self' a).isUnit_mul_iff.2 ⟨h.star, h⟩
state_before: A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
h : IsUnit a
a✝ : Nontrivial A
a' : ↥(elementalStarAlgebra ℂ a) := ⟨a, ⋯⟩
⊢ IsUnit (star a' * a')
state_after: A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
a✝ : Nontrivial A
a' : ↥(elementalStarAlgebra ℂ a) := ⟨a, ⋯⟩
h : IsUnit (star a * a)
⊢ IsUnit (star a' * a')

tactic: have h₁ : (‖star a * a‖ : ℂ) ≠ 0 := Complex.ofReal_ne_zero.mpr (norm_ne_zero_iff.mpr h.ne_zero)
state_before: A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
a✝ : Nontrivial A
a' : ↥(elementalStarAlgebra ℂ a) := ⟨a, ⋯⟩
h : IsUnit (star a * a)
⊢ IsUnit (star a' * a')
state_after: A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
a✝ : Nontrivial A
a' : ↥(elementalStarAlgebra ℂ a) := ⟨a, ⋯⟩
h : IsUnit (star a * a)
h₁ : ↑‖star a * a‖ ≠ 0
⊢ IsUnit (star a' * a')

tactic: set u : Units (elementalStarAlgebra ℂ a) :=
  Units.map (algebraMap ℂ (elementalStarAlgebra ℂ a)).toMonoidHom (Units.mk0 _ h₁)
state_before: A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
a✝ : Nontrivial A
a' : ↥(elementalStarAlgebra ℂ a) := ⟨a, ⋯⟩
h : IsUnit (star a * a)
h₁ : ↑‖star a * a‖ ≠ 0
⊢ IsUnit (star a' * a')
state_after: A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
a✝ : Nontrivial A
a' : ↥(elementalStarAlgebra ℂ a) := ⟨a, ⋯⟩
h : IsUnit (star a * a)
h₁ : ↑‖star a * a‖ ≠ 0
u : (↥(elementalStarAlgebra ℂ a))ˣ :=
  (Units.map ↑(algebraMap ℂ ↥(elementalStarAlgebra ℂ a))) (Units.mk0 (↑‖star a * a‖) h₁)
⊢ IsUnit (star a' * a')

tactic: refine ⟨u.ofNearby _ ?_, rfl⟩
state_before: A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
a✝ : Nontrivial A
a' : ↥(elementalStarAlgebra ℂ a) := ⟨a, ⋯⟩
h : IsUnit (star a * a)
h₁ : ↑‖star a * a‖ ≠ 0
u : (↥(elementalStarAlgebra ℂ a))ˣ :=
  (Units.map ↑(algebraMap ℂ ↥(elementalStarAlgebra ℂ a))) (Units.mk0 (↑‖star a * a‖) h₁)
⊢ IsUnit (star a' * a')
state_after: A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
a✝ : Nontrivial A
a' : ↥(elementalStarAlgebra ℂ a) := ⟨a, ⋯⟩
h : IsUnit (star a * a)
h₁ : ↑‖star a * a‖ ≠ 0
u : (↥(elementalStarAlgebra ℂ a))ˣ :=
  (Units.map ↑(algebraMap ℂ ↥(elementalStarAlgebra ℂ a))) (Units.mk0 (↑‖star a * a‖) h₁)
⊢ ‖star a' * a' - ↑u‖ < ‖↑u⁻¹‖⁻¹

tactic: simp only [u, Units.coe_map, Units.val_inv_eq_inv_val, RingHom.toMonoidHom_eq_coe, Units.val_mk0,
  Units.coe_map_inv, MonoidHom.coe_coe, norm_algebraMap', norm_inv, Complex.norm_eq_abs,
  Complex.abs_ofReal, abs_norm, inv_inv]
state_before: A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
a✝ : Nontrivial A
a' : ↥(elementalStarAlgebra ℂ a) := ⟨a, ⋯⟩
h : IsUnit (star a * a)
h₁ : ↑‖star a * a‖ ≠ 0
u : (↥(elementalStarAlgebra ℂ a))ˣ :=
  (Units.map ↑(algebraMap ℂ ↥(elementalStarAlgebra ℂ a))) (Units.mk0 (↑‖star a * a‖) h₁)
⊢ ‖star a' * a' - ↑u‖ < ‖↑u⁻¹‖⁻¹
state_after: A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
a✝ : Nontrivial A
a' : ↥(elementalStarAlgebra ℂ a) := ⟨a, ⋯⟩
h : IsUnit (star a * a)
h₁ : ↑‖star a * a‖ ≠ 0
u : (↥(elementalStarAlgebra ℂ a))ˣ :=
  (Units.map ↑(algebraMap ℂ ↥(elementalStarAlgebra ℂ a))) (Units.mk0 (↑‖star a * a‖) h₁)
⊢ ‖star a' * a' - (algebraMap ℂ ↥(elementalStarAlgebra ℂ a)) ↑‖star a * a‖‖ < ‖star a * a‖

tactic: exact ENNReal.coe_lt_coe.1
  (calc
    (‖star a' * a' - algebraMap ℂ _ ‖star a * a‖‖₊ : ℝ≥0∞) =
        ‖algebraMap ℂ A ‖star a * a‖ - star a * a‖₊ := by
      rw [← nnnorm_neg, neg_sub]; rfl
    _ = spectralRadius ℂ (algebraMap ℂ A ‖star a * a‖ - star a * a) := by
      refine (IsSelfAdjoint.spectralRadius_eq_nnnorm ?_).symm
      rw [IsSelfAdjoint, star_sub, star_mul, star_star, ← algebraMap_star_comm]
      congr!
      exact RCLike.conj_ofReal _
    _ < ‖star a * a‖₊ := spectrum.spectralRadius_lt_of_forall_lt _ h₂)
state_before: A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
a✝ : Nontrivial A
a' : ↥(elementalStarAlgebra ℂ a) := ⟨a, ⋯⟩
h : IsUnit (star a * a)
h₁ : ↑‖star a * a‖ ≠ 0
u : (↥(elementalStarAlgebra ℂ a))ˣ :=
  (Units.map ↑(algebraMap ℂ ↥(elementalStarAlgebra ℂ a))) (Units.mk0 (↑‖star a * a‖) h₁)
h₂ : ∀ z ∈ spectrum ℂ ((algebraMap ℂ A) ↑‖star a * a‖ - star a * a), ‖z‖₊ < ‖star a * a‖₊
⊢ ‖star a' * a' - (algebraMap ℂ ↥(elementalStarAlgebra ℂ a)) ↑‖star a * a‖‖ < ‖star a * a‖
state_after: no goals

tactic: intro z hz
state_before: A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
a✝ : Nontrivial A
a' : ↥(elementalStarAlgebra ℂ a) := ⟨a, ⋯⟩
h : IsUnit (star a * a)
h₁ : ↑‖star a * a‖ ≠ 0
u : (↥(elementalStarAlgebra ℂ a))ˣ :=
  (Units.map ↑(algebraMap ℂ ↥(elementalStarAlgebra ℂ a))) (Units.mk0 (↑‖star a * a‖) h₁)
⊢ ∀ z ∈ spectrum ℂ ((algebraMap ℂ A) ↑‖star a * a‖ - star a * a), ‖z‖₊ < ‖star a * a‖₊
state_after: A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
a✝ : Nontrivial A
a' : ↥(elementalStarAlgebra ℂ a) := ⟨a, ⋯⟩
h : IsUnit (star a * a)
h₁ : ↑‖star a * a‖ ≠ 0
u : (↥(elementalStarAlgebra ℂ a))ˣ :=
  (Units.map ↑(algebraMap ℂ ↥(elementalStarAlgebra ℂ a))) (Units.mk0 (↑‖star a * a‖) h₁)
z : ℂ
hz : z ∈ spectrum ℂ ((algebraMap ℂ A) ↑‖star a * a‖ - star a * a)
⊢ ‖z‖₊ < ‖star a * a‖₊

tactic: rw [← spectrum.singleton_sub_eq, Set.singleton_sub] at hz
state_before: A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
a✝ : Nontrivial A
a' : ↥(elementalStarAlgebra ℂ a) := ⟨a, ⋯⟩
h : IsUnit (star a * a)
h₁ : ↑‖star a * a‖ ≠ 0
u : (↥(elementalStarAlgebra ℂ a))ˣ :=
  (Units.map ↑(algebraMap ℂ ↥(elementalStarAlgebra ℂ a))) (Units.mk0 (↑‖star a * a‖) h₁)
z : ℂ
hz : z ∈ spectrum ℂ ((algebraMap ℂ A) ↑‖star a * a‖ - star a * a)
⊢ ‖z‖₊ < ‖star a * a‖₊
state_after: A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
a✝ : Nontrivial A
a' : ↥(elementalStarAlgebra ℂ a) := ⟨a, ⋯⟩
h : IsUnit (star a * a)
h₁ : ↑‖star a * a‖ ≠ 0
u : (↥(elementalStarAlgebra ℂ a))ˣ :=
  (Units.map ↑(algebraMap ℂ ↥(elementalStarAlgebra ℂ a))) (Units.mk0 (↑‖star a * a‖) h₁)
z : ℂ
hz : z ∈ (fun x x_1 => x - x_1) ↑‖star a * a‖ '' spectrum ℂ (star a * a)
⊢ ‖z‖₊ < ‖star a * a‖₊

tactic: have h₃ : z ∈ Set.Icc (0 : ℂ) ‖star a * a‖ := by
  replace hz := Set.image_subset _ (spectrum_star_mul_self_of_isStarNormal a) hz
  rwa [Set.image_const_sub_Icc, sub_self, sub_zero] at hz
state_before: A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
a✝ : Nontrivial A
a' : ↥(elementalStarAlgebra ℂ a) := ⟨a, ⋯⟩
h : IsUnit (star a * a)
h₁ : ↑‖star a * a‖ ≠ 0
u : (↥(elementalStarAlgebra ℂ a))ˣ :=
  (Units.map ↑(algebraMap ℂ ↥(elementalStarAlgebra ℂ a))) (Units.mk0 (↑‖star a * a‖) h₁)
z : ℂ
hz : z ∈ (fun x x_1 => x - x_1) ↑‖star a * a‖ '' spectrum ℂ (star a * a)
⊢ ‖z‖₊ < ‖star a * a‖₊
state_after: A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
a✝ : Nontrivial A
a' : ↥(elementalStarAlgebra ℂ a) := ⟨a, ⋯⟩
h : IsUnit (star a * a)
h₁ : ↑‖star a * a‖ ≠ 0
u : (↥(elementalStarAlgebra ℂ a))ˣ :=
  (Units.map ↑(algebraMap ℂ ↥(elementalStarAlgebra ℂ a))) (Units.mk0 (↑‖star a * a‖) h₁)
z : ℂ
hz : z ∈ (fun x x_1 => x - x_1) ↑‖star a * a‖ '' spectrum ℂ (star a * a)
h₃ : z ∈ Set.Icc 0 ↑‖star a * a‖
⊢ ‖z‖₊ < ‖star a * a‖₊

tactic: refine lt_of_le_of_ne (Complex.real_le_real.1 <| Complex.eq_coe_norm_of_nonneg h₃.1 ▸ h₃.2) ?_
state_before: A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
a✝ : Nontrivial A
a' : ↥(elementalStarAlgebra ℂ a) := ⟨a, ⋯⟩
h : IsUnit (star a * a)
h₁ : ↑‖star a * a‖ ≠ 0
u : (↥(elementalStarAlgebra ℂ a))ˣ :=
  (Units.map ↑(algebraMap ℂ ↥(elementalStarAlgebra ℂ a))) (Units.mk0 (↑‖star a * a‖) h₁)
z : ℂ
hz : z ∈ (fun x x_1 => x - x_1) ↑‖star a * a‖ '' spectrum ℂ (star a * a)
h₃ : z ∈ Set.Icc 0 ↑‖star a * a‖
⊢ ‖z‖₊ < ‖star a * a‖₊
state_after: A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
a✝ : Nontrivial A
a' : ↥(elementalStarAlgebra ℂ a) := ⟨a, ⋯⟩
h : IsUnit (star a * a)
h₁ : ↑‖star a * a‖ ≠ 0
u : (↥(elementalStarAlgebra ℂ a))ˣ :=
  (Units.map ↑(algebraMap ℂ ↥(elementalStarAlgebra ℂ a))) (Units.mk0 (↑‖star a * a‖) h₁)
z : ℂ
hz : z ∈ (fun x x_1 => x - x_1) ↑‖star a * a‖ '' spectrum ℂ (star a * a)
h₃ : z ∈ Set.Icc 0 ↑‖star a * a‖
⊢ ‖z‖₊ ≠ ‖star a * a‖₊

tactic: replace hz := Set.image_subset _ (spectrum_star_mul_self_of_isStarNormal a) hz
state_before: A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
a✝ : Nontrivial A
a' : ↥(elementalStarAlgebra ℂ a) := ⟨a, ⋯⟩
h : IsUnit (star a * a)
h₁ : ↑‖star a * a‖ ≠ 0
u : (↥(elementalStarAlgebra ℂ a))ˣ :=
  (Units.map ↑(algebraMap ℂ ↥(elementalStarAlgebra ℂ a))) (Units.mk0 (↑‖star a * a‖) h₁)
z : ℂ
hz : z ∈ (fun x x_1 => x - x_1) ↑‖star a * a‖ '' spectrum ℂ (star a * a)
⊢ z ∈ Set.Icc 0 ↑‖star a * a‖
state_after: A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
a✝ : Nontrivial A
a' : ↥(elementalStarAlgebra ℂ a) := ⟨a, ⋯⟩
h : IsUnit (star a * a)
h₁ : ↑‖star a * a‖ ≠ 0
u : (↥(elementalStarAlgebra ℂ a))ˣ :=
  (Units.map ↑(algebraMap ℂ ↥(elementalStarAlgebra ℂ a))) (Units.mk0 (↑‖star a * a‖) h₁)
z : ℂ
hz : z ∈ (fun x x_1 => x - x_1) ↑‖star a * a‖ '' Set.Icc 0 ↑‖star a * a‖
⊢ z ∈ Set.Icc 0 ↑‖star a * a‖

tactic: rwa [Set.image_const_sub_Icc, sub_self, sub_zero] at hz
state_before: A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
a✝ : Nontrivial A
a' : ↥(elementalStarAlgebra ℂ a) := ⟨a, ⋯⟩
h : IsUnit (star a * a)
h₁ : ↑‖star a * a‖ ≠ 0
u : (↥(elementalStarAlgebra ℂ a))ˣ :=
  (Units.map ↑(algebraMap ℂ ↥(elementalStarAlgebra ℂ a))) (Units.mk0 (↑‖star a * a‖) h₁)
z : ℂ
hz : z ∈ (fun x x_1 => x - x_1) ↑‖star a * a‖ '' Set.Icc 0 ↑‖star a * a‖
⊢ z ∈ Set.Icc 0 ↑‖star a * a‖
state_after: no goals

tactic: intro hz'
state_before: A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
a✝ : Nontrivial A
a' : ↥(elementalStarAlgebra ℂ a) := ⟨a, ⋯⟩
h : IsUnit (star a * a)
h₁ : ↑‖star a * a‖ ≠ 0
u : (↥(elementalStarAlgebra ℂ a))ˣ :=
  (Units.map ↑(algebraMap ℂ ↥(elementalStarAlgebra ℂ a))) (Units.mk0 (↑‖star a * a‖) h₁)
z : ℂ
hz : z ∈ (fun x x_1 => x - x_1) ↑‖star a * a‖ '' spectrum ℂ (star a * a)
h₃ : z ∈ Set.Icc 0 ↑‖star a * a‖
⊢ ‖z‖₊ ≠ ‖star a * a‖₊
state_after: A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
a✝ : Nontrivial A
a' : ↥(elementalStarAlgebra ℂ a) := ⟨a, ⋯⟩
h : IsUnit (star a * a)
h₁ : ↑‖star a * a‖ ≠ 0
u : (↥(elementalStarAlgebra ℂ a))ˣ :=
  (Units.map ↑(algebraMap ℂ ↥(elementalStarAlgebra ℂ a))) (Units.mk0 (↑‖star a * a‖) h₁)
z : ℂ
hz : z ∈ (fun x x_1 => x - x_1) ↑‖star a * a‖ '' spectrum ℂ (star a * a)
h₃ : z ∈ Set.Icc 0 ↑‖star a * a‖
hz' : ‖z‖₊ = ‖star a * a‖₊
⊢ False

tactic: replace hz' := congr_arg (fun x : ℝ≥0 => ((x : ℝ) : ℂ)) hz'
state_before: A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
a✝ : Nontrivial A
a' : ↥(elementalStarAlgebra ℂ a) := ⟨a, ⋯⟩
h : IsUnit (star a * a)
h₁ : ↑‖star a * a‖ ≠ 0
u : (↥(elementalStarAlgebra ℂ a))ˣ :=
  (Units.map ↑(algebraMap ℂ ↥(elementalStarAlgebra ℂ a))) (Units.mk0 (↑‖star a * a‖) h₁)
z : ℂ
hz : z ∈ (fun x x_1 => x - x_1) ↑‖star a * a‖ '' spectrum ℂ (star a * a)
h₃ : z ∈ Set.Icc 0 ↑‖star a * a‖
hz' : ‖z‖₊ = ‖star a * a‖₊
⊢ False
state_after: A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
a✝ : Nontrivial A
a' : ↥(elementalStarAlgebra ℂ a) := ⟨a, ⋯⟩
h : IsUnit (star a * a)
h₁ : ↑‖star a * a‖ ≠ 0
u : (↥(elementalStarAlgebra ℂ a))ˣ :=
  (Units.map ↑(algebraMap ℂ ↥(elementalStarAlgebra ℂ a))) (Units.mk0 (↑‖star a * a‖) h₁)
z : ℂ
hz : z ∈ (fun x x_1 => x - x_1) ↑‖star a * a‖ '' spectrum ℂ (star a * a)
h₃ : z ∈ Set.Icc 0 ↑‖star a * a‖
hz' : (fun x => ↑↑x) ‖z‖₊ = (fun x => ↑↑x) ‖star a * a‖₊
⊢ False

tactic: simp only [coe_nnnorm] at hz'
state_before: A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
a✝ : Nontrivial A
a' : ↥(elementalStarAlgebra ℂ a) := ⟨a, ⋯⟩
h : IsUnit (star a * a)
h₁ : ↑‖star a * a‖ ≠ 0
u : (↥(elementalStarAlgebra ℂ a))ˣ :=
  (Units.map ↑(algebraMap ℂ ↥(elementalStarAlgebra ℂ a))) (Units.mk0 (↑‖star a * a‖) h₁)
z : ℂ
hz : z ∈ (fun x x_1 => x - x_1) ↑‖star a * a‖ '' spectrum ℂ (star a * a)
h₃ : z ∈ Set.Icc 0 ↑‖star a * a‖
hz' : (fun x => ↑↑x) ‖z‖₊ = (fun x => ↑↑x) ‖star a * a‖₊
⊢ False
state_after: A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
a✝ : Nontrivial A
a' : ↥(elementalStarAlgebra ℂ a) := ⟨a, ⋯⟩
h : IsUnit (star a * a)
h₁ : ↑‖star a * a‖ ≠ 0
u : (↥(elementalStarAlgebra ℂ a))ˣ :=
  (Units.map ↑(algebraMap ℂ ↥(elementalStarAlgebra ℂ a))) (Units.mk0 (↑‖star a * a‖) h₁)
z : ℂ
hz : z ∈ (fun x x_1 => x - x_1) ↑‖star a * a‖ '' spectrum ℂ (star a * a)
h₃ : z ∈ Set.Icc 0 ↑‖star a * a‖
hz' : ↑‖z‖ = ↑‖star a * a‖
⊢ False

tactic: rw [← Complex.eq_coe_norm_of_nonneg h₃.1] at hz'
state_before: A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
a✝ : Nontrivial A
a' : ↥(elementalStarAlgebra ℂ a) := ⟨a, ⋯⟩
h : IsUnit (star a * a)
h₁ : ↑‖star a * a‖ ≠ 0
u : (↥(elementalStarAlgebra ℂ a))ˣ :=
  (Units.map ↑(algebraMap ℂ ↥(elementalStarAlgebra ℂ a))) (Units.mk0 (↑‖star a * a‖) h₁)
z : ℂ
hz : z ∈ (fun x x_1 => x - x_1) ↑‖star a * a‖ '' spectrum ℂ (star a * a)
h₃ : z ∈ Set.Icc 0 ↑‖star a * a‖
hz' : ↑‖z‖ = ↑‖star a * a‖
⊢ False
state_after: A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
a✝ : Nontrivial A
a' : ↥(elementalStarAlgebra ℂ a) := ⟨a, ⋯⟩
h : IsUnit (star a * a)
h₁ : ↑‖star a * a‖ ≠ 0
u : (↥(elementalStarAlgebra ℂ a))ˣ :=
  (Units.map ↑(algebraMap ℂ ↥(elementalStarAlgebra ℂ a))) (Units.mk0 (↑‖star a * a‖) h₁)
z : ℂ
hz : z ∈ (fun x x_1 => x - x_1) ↑‖star a * a‖ '' spectrum ℂ (star a * a)
h₃ : z ∈ Set.Icc 0 ↑‖star a * a‖
hz' : z = ↑‖star a * a‖
⊢ False

tactic: obtain ⟨w, hw₁, hw₂⟩ := hz
state_before: A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
a✝ : Nontrivial A
a' : ↥(elementalStarAlgebra ℂ a) := ⟨a, ⋯⟩
h : IsUnit (star a * a)
h₁ : ↑‖star a * a‖ ≠ 0
u : (↥(elementalStarAlgebra ℂ a))ˣ :=
  (Units.map ↑(algebraMap ℂ ↥(elementalStarAlgebra ℂ a))) (Units.mk0 (↑‖star a * a‖) h₁)
z : ℂ
hz : z ∈ (fun x x_1 => x - x_1) ↑‖star a * a‖ '' spectrum ℂ (star a * a)
h₃ : z ∈ Set.Icc 0 ↑‖star a * a‖
hz' : z = ↑‖star a * a‖
⊢ False
state_after: case intro.intro
A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
a✝ : Nontrivial A
a' : ↥(elementalStarAlgebra ℂ a) := ⟨a, ⋯⟩
h : IsUnit (star a * a)
h₁ : ↑‖star a * a‖ ≠ 0
u : (↥(elementalStarAlgebra ℂ a))ˣ :=
  (Units.map ↑(algebraMap ℂ ↥(elementalStarAlgebra ℂ a))) (Units.mk0 (↑‖star a * a‖) h₁)
z : ℂ
h₃ : z ∈ Set.Icc 0 ↑‖star a * a‖
hz' : z = ↑‖star a * a‖
w : ℂ
hw₁ : w ∈ spectrum ℂ (star a * a)
hw₂ : (fun x x_1 => x - x_1) (↑‖star a * a‖) w = z
⊢ False

tactic: refine (spectrum.zero_not_mem_iff ℂ).mpr h ?_
state_before: case intro.intro
A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
a✝ : Nontrivial A
a' : ↥(elementalStarAlgebra ℂ a) := ⟨a, ⋯⟩
h : IsUnit (star a * a)
h₁ : ↑‖star a * a‖ ≠ 0
u : (↥(elementalStarAlgebra ℂ a))ˣ :=
  (Units.map ↑(algebraMap ℂ ↥(elementalStarAlgebra ℂ a))) (Units.mk0 (↑‖star a * a‖) h₁)
z : ℂ
h₃ : z ∈ Set.Icc 0 ↑‖star a * a‖
hz' : z = ↑‖star a * a‖
w : ℂ
hw₁ : w ∈ spectrum ℂ (star a * a)
hw₂ : (fun x x_1 => x - x_1) (↑‖star a * a‖) w = z
⊢ False
state_after: case intro.intro
A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
a✝ : Nontrivial A
a' : ↥(elementalStarAlgebra ℂ a) := ⟨a, ⋯⟩
h : IsUnit (star a * a)
h₁ : ↑‖star a * a‖ ≠ 0
u : (↥(elementalStarAlgebra ℂ a))ˣ :=
  (Units.map ↑(algebraMap ℂ ↥(elementalStarAlgebra ℂ a))) (Units.mk0 (↑‖star a * a‖) h₁)
z : ℂ
h₃ : z ∈ Set.Icc 0 ↑‖star a * a‖
hz' : z = ↑‖star a * a‖
w : ℂ
hw₁ : w ∈ spectrum ℂ (star a * a)
hw₂ : (fun x x_1 => x - x_1) (↑‖star a * a‖) w = z
⊢ 0 ∈ spectrum ℂ (star a * a)

tactic: rw [hz', sub_eq_self] at hw₂
state_before: case intro.intro
A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
a✝ : Nontrivial A
a' : ↥(elementalStarAlgebra ℂ a) := ⟨a, ⋯⟩
h : IsUnit (star a * a)
h₁ : ↑‖star a * a‖ ≠ 0
u : (↥(elementalStarAlgebra ℂ a))ˣ :=
  (Units.map ↑(algebraMap ℂ ↥(elementalStarAlgebra ℂ a))) (Units.mk0 (↑‖star a * a‖) h₁)
z : ℂ
h₃ : z ∈ Set.Icc 0 ↑‖star a * a‖
hz' : z = ↑‖star a * a‖
w : ℂ
hw₁ : w ∈ spectrum ℂ (star a * a)
hw₂ : (fun x x_1 => x - x_1) (↑‖star a * a‖) w = z
⊢ 0 ∈ spectrum ℂ (star a * a)
state_after: case intro.intro
A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
a✝ : Nontrivial A
a' : ↥(elementalStarAlgebra ℂ a) := ⟨a, ⋯⟩
h : IsUnit (star a * a)
h₁ : ↑‖star a * a‖ ≠ 0
u : (↥(elementalStarAlgebra ℂ a))ˣ :=
  (Units.map ↑(algebraMap ℂ ↥(elementalStarAlgebra ℂ a))) (Units.mk0 (↑‖star a * a‖) h₁)
z : ℂ
h₃ : z ∈ Set.Icc 0 ↑‖star a * a‖
hz' : z = ↑‖star a * a‖
w : ℂ
hw₁ : w ∈ spectrum ℂ (star a * a)
hw₂ : w = 0
⊢ 0 ∈ spectrum ℂ (star a * a)

tactic: rwa [hw₂] at hw₁
state_before: case intro.intro
A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
a✝ : Nontrivial A
a' : ↥(elementalStarAlgebra ℂ a) := ⟨a, ⋯⟩
h : IsUnit (star a * a)
h₁ : ↑‖star a * a‖ ≠ 0
u : (↥(elementalStarAlgebra ℂ a))ˣ :=
  (Units.map ↑(algebraMap ℂ ↥(elementalStarAlgebra ℂ a))) (Units.mk0 (↑‖star a * a‖) h₁)
z : ℂ
h₃ : z ∈ Set.Icc 0 ↑‖star a * a‖
hz' : z = ↑‖star a * a‖
w : ℂ
hw₁ : w ∈ spectrum ℂ (star a * a)
hw₂ : w = 0
⊢ 0 ∈ spectrum ℂ (star a * a)
state_after: no goals

tactic: rw [← nnnorm_neg, neg_sub]
state_before: A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
a✝ : Nontrivial A
a' : ↥(elementalStarAlgebra ℂ a) := ⟨a, ⋯⟩
h : IsUnit (star a * a)
h₁ : ↑‖star a * a‖ ≠ 0
u : (↥(elementalStarAlgebra ℂ a))ˣ :=
  (Units.map ↑(algebraMap ℂ ↥(elementalStarAlgebra ℂ a))) (Units.mk0 (↑‖star a * a‖) h₁)
h₂ : ∀ z ∈ spectrum ℂ ((algebraMap ℂ A) ↑‖star a * a‖ - star a * a), ‖z‖₊ < ‖star a * a‖₊
⊢ ↑‖star a' * a' - (algebraMap ℂ ↥(elementalStarAlgebra ℂ a)) ↑‖star a * a‖‖₊ =
    ↑‖(algebraMap ℂ A) ↑‖star a * a‖ - star a * a‖₊
state_after: A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
a✝ : Nontrivial A
a' : ↥(elementalStarAlgebra ℂ a) := ⟨a, ⋯⟩
h : IsUnit (star a * a)
h₁ : ↑‖star a * a‖ ≠ 0
u : (↥(elementalStarAlgebra ℂ a))ˣ :=
  (Units.map ↑(algebraMap ℂ ↥(elementalStarAlgebra ℂ a))) (Units.mk0 (↑‖star a * a‖) h₁)
h₂ : ∀ z ∈ spectrum ℂ ((algebraMap ℂ A) ↑‖star a * a‖ - star a * a), ‖z‖₊ < ‖star a * a‖₊
⊢ ↑‖(algebraMap ℂ ↥(elementalStarAlgebra ℂ a)) ↑‖star a * a‖ - star a' * a'‖₊ =
    ↑‖(algebraMap ℂ A) ↑‖star a * a‖ - star a * a‖₊

tactic: rfl
state_before: A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
a✝ : Nontrivial A
a' : ↥(elementalStarAlgebra ℂ a) := ⟨a, ⋯⟩
h : IsUnit (star a * a)
h₁ : ↑‖star a * a‖ ≠ 0
u : (↥(elementalStarAlgebra ℂ a))ˣ :=
  (Units.map ↑(algebraMap ℂ ↥(elementalStarAlgebra ℂ a))) (Units.mk0 (↑‖star a * a‖) h₁)
h₂ : ∀ z ∈ spectrum ℂ ((algebraMap ℂ A) ↑‖star a * a‖ - star a * a), ‖z‖₊ < ‖star a * a‖₊
⊢ ↑‖(algebraMap ℂ ↥(elementalStarAlgebra ℂ a)) ↑‖star a * a‖ - star a' * a'‖₊ =
    ↑‖(algebraMap ℂ A) ↑‖star a * a‖ - star a * a‖₊
state_after: no goals

tactic: refine (IsSelfAdjoint.spectralRadius_eq_nnnorm ?_).symm
state_before: A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
a✝ : Nontrivial A
a' : ↥(elementalStarAlgebra ℂ a) := ⟨a, ⋯⟩
h : IsUnit (star a * a)
h₁ : ↑‖star a * a‖ ≠ 0
u : (↥(elementalStarAlgebra ℂ a))ˣ :=
  (Units.map ↑(algebraMap ℂ ↥(elementalStarAlgebra ℂ a))) (Units.mk0 (↑‖star a * a‖) h₁)
h₂ : ∀ z ∈ spectrum ℂ ((algebraMap ℂ A) ↑‖star a * a‖ - star a * a), ‖z‖₊ < ‖star a * a‖₊
⊢ ↑‖(algebraMap ℂ A) ↑‖star a * a‖ - star a * a‖₊ = spectralRadius ℂ ((algebraMap ℂ A) ↑‖star a * a‖ - star a * a)
state_after: A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
a✝ : Nontrivial A
a' : ↥(elementalStarAlgebra ℂ a) := ⟨a, ⋯⟩
h : IsUnit (star a * a)
h₁ : ↑‖star a * a‖ ≠ 0
u : (↥(elementalStarAlgebra ℂ a))ˣ :=
  (Units.map ↑(algebraMap ℂ ↥(elementalStarAlgebra ℂ a))) (Units.mk0 (↑‖star a * a‖) h₁)
h₂ : ∀ z ∈ spectrum ℂ ((algebraMap ℂ A) ↑‖star a * a‖ - star a * a), ‖z‖₊ < ‖star a * a‖₊
⊢ IsSelfAdjoint ((algebraMap ℂ A) ↑‖star a * a‖ - star a * a)

tactic: rw [IsSelfAdjoint, star_sub, star_mul, star_star, ← algebraMap_star_comm]
state_before: A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
a✝ : Nontrivial A
a' : ↥(elementalStarAlgebra ℂ a) := ⟨a, ⋯⟩
h : IsUnit (star a * a)
h₁ : ↑‖star a * a‖ ≠ 0
u : (↥(elementalStarAlgebra ℂ a))ˣ :=
  (Units.map ↑(algebraMap ℂ ↥(elementalStarAlgebra ℂ a))) (Units.mk0 (↑‖star a * a‖) h₁)
h₂ : ∀ z ∈ spectrum ℂ ((algebraMap ℂ A) ↑‖star a * a‖ - star a * a), ‖z‖₊ < ‖star a * a‖₊
⊢ IsSelfAdjoint ((algebraMap ℂ A) ↑‖star a * a‖ - star a * a)
state_after: A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
a✝ : Nontrivial A
a' : ↥(elementalStarAlgebra ℂ a) := ⟨a, ⋯⟩
h : IsUnit (star a * a)
h₁ : ↑‖star a * a‖ ≠ 0
u : (↥(elementalStarAlgebra ℂ a))ˣ :=
  (Units.map ↑(algebraMap ℂ ↥(elementalStarAlgebra ℂ a))) (Units.mk0 (↑‖star a * a‖) h₁)
h₂ : ∀ z ∈ spectrum ℂ ((algebraMap ℂ A) ↑‖star a * a‖ - star a * a), ‖z‖₊ < ‖star a * a‖₊
⊢ (algebraMap ℂ A) (star ↑‖star a * a‖) - star a * a = (algebraMap ℂ A) ↑‖star a * a‖ - star a * a

tactic: congr!
state_before: A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
a✝ : Nontrivial A
a' : ↥(elementalStarAlgebra ℂ a) := ⟨a, ⋯⟩
h : IsUnit (star a * a)
h₁ : ↑‖star a * a‖ ≠ 0
u : (↥(elementalStarAlgebra ℂ a))ˣ :=
  (Units.map ↑(algebraMap ℂ ↥(elementalStarAlgebra ℂ a))) (Units.mk0 (↑‖star a * a‖) h₁)
h₂ : ∀ z ∈ spectrum ℂ ((algebraMap ℂ A) ↑‖star a * a‖ - star a * a), ‖z‖₊ < ‖star a * a‖₊
⊢ (algebraMap ℂ A) (star ↑‖star a * a‖) - star a * a = (algebraMap ℂ A) ↑‖star a * a‖ - star a * a
state_after: case h.e'_5.h.e'_6
A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
a✝ : Nontrivial A
a' : ↥(elementalStarAlgebra ℂ a) := ⟨a, ⋯⟩
h : IsUnit (star a * a)
h₁ : ↑‖star a * a‖ ≠ 0
u : (↥(elementalStarAlgebra ℂ a))ˣ :=
  (Units.map ↑(algebraMap ℂ ↥(elementalStarAlgebra ℂ a))) (Units.mk0 (↑‖star a * a‖) h₁)
h₂ : ∀ z ∈ spectrum ℂ ((algebraMap ℂ A) ↑‖star a * a‖ - star a * a), ‖z‖₊ < ‖star a * a‖₊
⊢ star ↑‖star a * a‖ = ↑‖star a * a‖

tactic: exact RCLike.conj_ofReal _
state_before: case h.e'_5.h.e'_6
A : Type u_1
inst✝⁶ : NormedRing A
inst✝⁵ : NormedAlgebra ℂ A
inst✝⁴ : StarRing A
inst✝³ : CstarRing A
inst✝² : StarModule ℂ A
inst✝¹ : CompleteSpace A
a : A
inst✝ : IsStarNormal a
S : StarSubalgebra ℂ A
a✝ : Nontrivial A
a' : ↥(elementalStarAlgebra ℂ a) := ⟨a, ⋯⟩
h : IsUnit (star a * a)
h₁ : ↑‖star a * a‖ ≠ 0
u : (↥(elementalStarAlgebra ℂ a))ˣ :=
  (Units.map ↑(algebraMap ℂ ↥(elementalStarAlgebra ℂ a))) (Units.mk0 (↑‖star a * a‖) h₁)
h₂ : ∀ z ∈ spectrum ℂ ((algebraMap ℂ A) ↑‖star a * a‖ - star a * a), ‖z‖₊ < ‖star a * a‖₊
⊢ star ↑‖star a * a‖ = ↑‖star a * a‖
state_after: no goals