Theorem: StarAlgHom.nnnorm_apply_le
File: Mathlib/Analysis/NormedSpace/Star/Spectrum.lean

tactic: suffices ∀ s : A, IsSelfAdjoint s → ‖φ s‖₊ ≤ ‖s‖₊ by
  exact nonneg_le_nonneg_of_sq_le_sq zero_le' <| by
    simpa only [nnnorm_star_mul_self, map_star, map_mul]
    using this _ (IsSelfAdjoint.star_mul_self a)
state_before: F : Type u_1
A : Type u_2
B : Type u_3
inst✝¹² : NormedRing A
inst✝¹¹ : NormedAlgebra ℂ A
inst✝¹⁰ : CompleteSpace A
inst✝⁹ : StarRing A
inst✝⁸ : CstarRing A
inst✝⁷ : NormedRing B
inst✝⁶ : NormedAlgebra ℂ B
inst✝⁵ : CompleteSpace B
inst✝⁴ : StarRing B
inst✝³ : CstarRing B
inst✝² : FunLike F A B
inst✝¹ : AlgHomClass F ℂ A B
inst✝ : StarAlgHomClass F ℂ A B
φ : F
a : A
⊢ ‖φ a‖₊ ≤ ‖a‖₊
state_after: F : Type u_1
A : Type u_2
B : Type u_3
inst✝¹² : NormedRing A
inst✝¹¹ : NormedAlgebra ℂ A
inst✝¹⁰ : CompleteSpace A
inst✝⁹ : StarRing A
inst✝⁸ : CstarRing A
inst✝⁷ : NormedRing B
inst✝⁶ : NormedAlgebra ℂ B
inst✝⁵ : CompleteSpace B
inst✝⁴ : StarRing B
inst✝³ : CstarRing B
inst✝² : FunLike F A B
inst✝¹ : AlgHomClass F ℂ A B
inst✝ : StarAlgHomClass F ℂ A B
φ : F
a : A
⊢ ∀ (s : A), IsSelfAdjoint s → ‖φ s‖₊ ≤ ‖s‖₊

tactic: intro s hs
state_before: F : Type u_1
A : Type u_2
B : Type u_3
inst✝¹² : NormedRing A
inst✝¹¹ : NormedAlgebra ℂ A
inst✝¹⁰ : CompleteSpace A
inst✝⁹ : StarRing A
inst✝⁸ : CstarRing A
inst✝⁷ : NormedRing B
inst✝⁶ : NormedAlgebra ℂ B
inst✝⁵ : CompleteSpace B
inst✝⁴ : StarRing B
inst✝³ : CstarRing B
inst✝² : FunLike F A B
inst✝¹ : AlgHomClass F ℂ A B
inst✝ : StarAlgHomClass F ℂ A B
φ : F
a : A
⊢ ∀ (s : A), IsSelfAdjoint s → ‖φ s‖₊ ≤ ‖s‖₊
state_after: F : Type u_1
A : Type u_2
B : Type u_3
inst✝¹² : NormedRing A
inst✝¹¹ : NormedAlgebra ℂ A
inst✝¹⁰ : CompleteSpace A
inst✝⁹ : StarRing A
inst✝⁸ : CstarRing A
inst✝⁷ : NormedRing B
inst✝⁶ : NormedAlgebra ℂ B
inst✝⁵ : CompleteSpace B
inst✝⁴ : StarRing B
inst✝³ : CstarRing B
inst✝² : FunLike F A B
inst✝¹ : AlgHomClass F ℂ A B
inst✝ : StarAlgHomClass F ℂ A B
φ : F
a s : A
hs : IsSelfAdjoint s
⊢ ‖φ s‖₊ ≤ ‖s‖₊

tactic: simpa only [hs.spectralRadius_eq_nnnorm, (hs.starHom_apply φ).spectralRadius_eq_nnnorm,
  coe_le_coe] using
  show spectralRadius ℂ (φ s) ≤ spectralRadius ℂ s from
    iSup_le_iSup_of_subset (AlgHom.spectrum_apply_subset φ s)
state_before: F : Type u_1
A : Type u_2
B : Type u_3
inst✝¹² : NormedRing A
inst✝¹¹ : NormedAlgebra ℂ A
inst✝¹⁰ : CompleteSpace A
inst✝⁹ : StarRing A
inst✝⁸ : CstarRing A
inst✝⁷ : NormedRing B
inst✝⁶ : NormedAlgebra ℂ B
inst✝⁵ : CompleteSpace B
inst✝⁴ : StarRing B
inst✝³ : CstarRing B
inst✝² : FunLike F A B
inst✝¹ : AlgHomClass F ℂ A B
inst✝ : StarAlgHomClass F ℂ A B
φ : F
a s : A
hs : IsSelfAdjoint s
⊢ ‖φ s‖₊ ≤ ‖s‖₊
state_after: no goals

tactic: exact nonneg_le_nonneg_of_sq_le_sq zero_le' <| by
  simpa only [nnnorm_star_mul_self, map_star, map_mul]
  using this _ (IsSelfAdjoint.star_mul_self a)
state_before: F : Type u_1
A : Type u_2
B : Type u_3
inst✝¹² : NormedRing A
inst✝¹¹ : NormedAlgebra ℂ A
inst✝¹⁰ : CompleteSpace A
inst✝⁹ : StarRing A
inst✝⁸ : CstarRing A
inst✝⁷ : NormedRing B
inst✝⁶ : NormedAlgebra ℂ B
inst✝⁵ : CompleteSpace B
inst✝⁴ : StarRing B
inst✝³ : CstarRing B
inst✝² : FunLike F A B
inst✝¹ : AlgHomClass F ℂ A B
inst✝ : StarAlgHomClass F ℂ A B
φ : F
a : A
this : ∀ (s : A), IsSelfAdjoint s → ‖φ s‖₊ ≤ ‖s‖₊
⊢ ‖φ a‖₊ ≤ ‖a‖₊
state_after: no goals

tactic: simpa only [nnnorm_star_mul_self, map_star, map_mul]
using this _ (IsSelfAdjoint.star_mul_self a)
state_before: F : Type u_1
A : Type u_2
B : Type u_3
inst✝¹² : NormedRing A
inst✝¹¹ : NormedAlgebra ℂ A
inst✝¹⁰ : CompleteSpace A
inst✝⁹ : StarRing A
inst✝⁸ : CstarRing A
inst✝⁷ : NormedRing B
inst✝⁶ : NormedAlgebra ℂ B
inst✝⁵ : CompleteSpace B
inst✝⁴ : StarRing B
inst✝³ : CstarRing B
inst✝² : FunLike F A B
inst✝¹ : AlgHomClass F ℂ A B
inst✝ : StarAlgHomClass F ℂ A B
φ : F
a : A
this : ∀ (s : A), IsSelfAdjoint s → ‖φ s‖₊ ≤ ‖s‖₊
⊢ ‖φ a‖₊ * ‖φ a‖₊ ≤ ‖a‖₊ * ‖a‖₊
state_after: no goals