Theorem: IsSelfAdjoint.le_algebraMap_norm_self
File: Mathlib/Analysis/NormedSpace/Star/ContinuousFunctionalCalculus/Order.lean

tactic: by_cases nontriv : Nontrivial A
state_before: A : Type u_1
inst✝⁷ : NormedRing A
inst✝⁶ : StarRing A
inst✝⁵ : CstarRing A
inst✝⁴ : CompleteSpace A
inst✝³ : NormedAlgebra ℂ A
inst✝² : StarModule ℂ A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
ha : autoParam (IsSelfAdjoint a) _auto✝
⊢ a ≤ (algebraMap ℝ A) ‖a‖
state_after: case pos
A : Type u_1
inst✝⁷ : NormedRing A
inst✝⁶ : StarRing A
inst✝⁵ : CstarRing A
inst✝⁴ : CompleteSpace A
inst✝³ : NormedAlgebra ℂ A
inst✝² : StarModule ℂ A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
ha : autoParam (IsSelfAdjoint a) _auto✝
nontriv : Nontrivial A
⊢ a ≤ (algebraMap ℝ A) ‖a‖

case neg
A : Type u_1
inst✝⁷ : NormedRing A
inst✝⁶ : StarRing A
inst✝⁵ : CstarRing A
inst✝⁴ : CompleteSpace A
inst✝³ : NormedAlgebra ℂ A
inst✝² : StarModule ℂ A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
ha : autoParam (IsSelfAdjoint a) _auto✝
nontriv : ¬Nontrivial A
⊢ a ≤ (algebraMap ℝ A) ‖a‖

tactic: refine le_algebraMap_of_spectrum_le fun r hr => ?_
state_before: case pos
A : Type u_1
inst✝⁷ : NormedRing A
inst✝⁶ : StarRing A
inst✝⁵ : CstarRing A
inst✝⁴ : CompleteSpace A
inst✝³ : NormedAlgebra ℂ A
inst✝² : StarModule ℂ A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
ha : autoParam (IsSelfAdjoint a) _auto✝
nontriv : Nontrivial A
⊢ a ≤ (algebraMap ℝ A) ‖a‖
state_after: case pos
A : Type u_1
inst✝⁷ : NormedRing A
inst✝⁶ : StarRing A
inst✝⁵ : CstarRing A
inst✝⁴ : CompleteSpace A
inst✝³ : NormedAlgebra ℂ A
inst✝² : StarModule ℂ A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
ha : autoParam (IsSelfAdjoint a) _auto✝
nontriv : Nontrivial A
r : ℝ
hr : r ∈ spectrum ℝ a
⊢ r ≤ ‖a‖

tactic: calc r ≤ ‖r‖ := Real.le_norm_self r
  _ ≤ ‖a‖ := spectrum.norm_le_norm_of_mem hr
state_before: case pos
A : Type u_1
inst✝⁷ : NormedRing A
inst✝⁶ : StarRing A
inst✝⁵ : CstarRing A
inst✝⁴ : CompleteSpace A
inst✝³ : NormedAlgebra ℂ A
inst✝² : StarModule ℂ A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
ha : autoParam (IsSelfAdjoint a) _auto✝
nontriv : Nontrivial A
r : ℝ
hr : r ∈ spectrum ℝ a
⊢ r ≤ ‖a‖
state_after: no goals

tactic: rw [not_nontrivial_iff_subsingleton] at nontriv
state_before: case neg
A : Type u_1
inst✝⁷ : NormedRing A
inst✝⁶ : StarRing A
inst✝⁵ : CstarRing A
inst✝⁴ : CompleteSpace A
inst✝³ : NormedAlgebra ℂ A
inst✝² : StarModule ℂ A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
ha : autoParam (IsSelfAdjoint a) _auto✝
nontriv : ¬Nontrivial A
⊢ a ≤ (algebraMap ℝ A) ‖a‖
state_after: case neg
A : Type u_1
inst✝⁷ : NormedRing A
inst✝⁶ : StarRing A
inst✝⁵ : CstarRing A
inst✝⁴ : CompleteSpace A
inst✝³ : NormedAlgebra ℂ A
inst✝² : StarModule ℂ A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
ha : autoParam (IsSelfAdjoint a) _auto✝
nontriv : Subsingleton A
⊢ a ≤ (algebraMap ℝ A) ‖a‖

tactic: simp
state_before: case neg
A : Type u_1
inst✝⁷ : NormedRing A
inst✝⁶ : StarRing A
inst✝⁵ : CstarRing A
inst✝⁴ : CompleteSpace A
inst✝³ : NormedAlgebra ℂ A
inst✝² : StarModule ℂ A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
ha : autoParam (IsSelfAdjoint a) _auto✝
nontriv : Subsingleton A
⊢ a ≤ (algebraMap ℝ A) ‖a‖
state_after: no goals