File: Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Order.lean

full_name: Unitization.instPartialOrder
code: instance instPartialOrder : PartialOrder (Unitization ℂ A) 
goal: PartialOrder (Unitization ℂ A)

full_name: Unitization.instStarOrderedRing
code: instance instStarOrderedRing : StarOrderedRing (Unitization ℂ A) 
goal: StarOrderedRing (Unitization ℂ A)

full_name: Unitization.inr_le_iff
code: lemma inr_le_iff (a b : A) (ha : IsSelfAdjoint a := by cfc_tac)
    (hb : IsSelfAdjoint b := by cfc_tac) :
    (a : Unitization ℂ A) ≤ (b : Unitization ℂ A) ↔ a ≤ b 
goal: ↑a ≤ ↑b ↔ a ≤ b

full_name: Unitization.inr_nonneg_iff
code: @[simp, norm_cast]
lemma inr_nonneg_iff {a : A} : 0 ≤ (a : Unitization ℂ A) ↔ 0 ≤ a 
goal: 0 ≤ ↑a ↔ 0 ≤ a

full_name: IsSelfAdjoint.le_algebraMap_norm_self
code: lemma IsSelfAdjoint.le_algebraMap_norm_self {a : A} (ha : IsSelfAdjoint a := by cfc_tac) :
    a ≤ algebraMap ℝ A ‖a‖ 
goal: a ≤ (algebraMap ℝ A) ‖a‖

full_name: IsSelfAdjoint.neg_algebraMap_norm_le_self
code: lemma IsSelfAdjoint.neg_algebraMap_norm_le_self {a : A} (ha : IsSelfAdjoint a := by cfc_tac) :
    - (algebraMap ℝ A ‖a‖) ≤ a 
goal: -(algebraMap ℝ A) ‖a‖ ≤ a

full_name: CstarRing.mul_star_le_algebraMap_norm_sq
code: lemma CstarRing.mul_star_le_algebraMap_norm_sq {a : A} : a * star a ≤ algebraMap ℝ A (‖a‖ ^ 2) 
goal: a * star a ≤ (algebraMap ℝ A) (‖a‖ ^ 2)

full_name: CstarRing.star_mul_le_algebraMap_norm_sq
code: lemma CstarRing.star_mul_le_algebraMap_norm_sq {a : A} : star a * a ≤ algebraMap ℝ A (‖a‖ ^ 2) 
goal: star a * a ≤ (algebraMap ℝ A) (‖a‖ ^ 2)

full_name: IsSelfAdjoint.toReal_spectralRadius_eq_norm
code: lemma IsSelfAdjoint.toReal_spectralRadius_eq_norm {a : A} (ha : IsSelfAdjoint a) :
    (spectralRadius ℝ a).toReal = ‖a‖ 
goal: (spectralRadius ℝ a).toReal = ‖a‖

full_name: CstarRing.norm_or_neg_norm_mem_spectrum
code: lemma CstarRing.norm_or_neg_norm_mem_spectrum [Nontrivial A] {a : A}
    (ha : IsSelfAdjoint a := by cfc_tac) : ‖a‖ ∈ spectrum ℝ a ∨ -‖a‖ ∈ spectrum ℝ a 
goal: ‖a‖ ∈ spectrum ℝ a ∨ -‖a‖ ∈ spectrum ℝ a

full_name: CstarRing.nnnorm_mem_spectrum_of_nonneg
code: lemma CstarRing.nnnorm_mem_spectrum_of_nonneg [Nontrivial A] {a : A} (ha : 0 ≤ a := by cfc_tac) :
    ‖a‖₊ ∈ spectrum ℝ≥0 a 
goal: ‖a‖₊ ∈ spectrum ℝ≥0 a

full_name: CstarRing.norm_mem_spectrum_of_nonneg
code: lemma CstarRing.norm_mem_spectrum_of_nonneg [Nontrivial A] {a : A} (ha : 0 ≤ a := by cfc_tac) :
    ‖a‖ ∈ spectrum ℝ a 
goal: ‖a‖ ∈ spectrum ℝ a

full_name: CstarRing.instNonnegSpectrumClassComplexNonUnital
code: instance CstarRing.instNonnegSpectrumClassComplexNonUnital : NonnegSpectrumClass ℂ A 
goal: NonnegSpectrumClass ℂ A

full_name: CstarRing.norm_le_norm_of_nonneg_of_le
code: lemma CstarRing.norm_le_norm_of_nonneg_of_le {a b : A} (ha : 0 ≤ a := by cfc_tac) (hab : a ≤ b) :
    ‖a‖ ≤ ‖b‖ 
goal: ‖a‖ ≤ ‖b‖

full_name: CstarRing.conjugate_le_norm_smul
code: lemma CstarRing.conjugate_le_norm_smul {a b : A} (hb : IsSelfAdjoint b := by cfc_tac) :
    star a * b * a ≤ ‖b‖ • (star a * a) 
goal: star a * b * a ≤ ‖b‖ • (star a * a)

full_name: CstarRing.conjugate_le_norm_smul'
code: lemma CstarRing.conjugate_le_norm_smul' {a b : A} (hb : IsSelfAdjoint b := by cfc_tac) :
    a * b * star a ≤ ‖b‖ • (a * star a) 
goal: a * b * star a ≤ ‖b‖ • (a * star a)