Theorem: CstarRing.conjugate_le_norm_smul'
File: Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Order.lean

Proof steps:
state_before: A : Type u_1
inst✝⁹ : NonUnitalNormedRing A
inst✝⁸ : CompleteSpace A
inst✝⁷ : PartialOrder A
inst✝⁶ : StarRing A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : CstarRing A
inst✝³ : NormedSpace ℂ A
inst✝² : StarModule ℂ A
inst✝¹ : SMulCommClass ℂ A A
inst✝ : IsScalarTower ℂ A A
a b : A
hb : autoParam (IsSelfAdjoint b) _auto✝
⊢ a * b * star a ≤ ‖b‖ • (a * star a)
state_after: A : Type u_1
inst✝⁹ : NonUnitalNormedRing A
inst✝⁸ : CompleteSpace A
inst✝⁷ : PartialOrder A
inst✝⁶ : StarRing A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : CstarRing A
inst✝³ : NormedSpace ℂ A
inst✝² : StarModule ℂ A
inst✝¹ : SMulCommClass ℂ A A
inst✝ : IsScalarTower ℂ A A
a b : A
hb : autoParam (IsSelfAdjoint b) _auto✝
h₁ : a * b * star a = star (star a) * b * star a
⊢ a * b * star a ≤ ‖b‖ • (a * star a)
tactic: have h₁ : a * b * star a = star (star a) * b * star a := by simp
state_before: A : Type u_1
inst✝⁹ : NonUnitalNormedRing A
inst✝⁸ : CompleteSpace A
inst✝⁷ : PartialOrder A
inst✝⁶ : StarRing A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : CstarRing A
inst✝³ : NormedSpace ℂ A
inst✝² : StarModule ℂ A
inst✝¹ : SMulCommClass ℂ A A
inst✝ : IsScalarTower ℂ A A
a b : A
hb : autoParam (IsSelfAdjoint b) _auto✝
h₁ : a * b * star a = star (star a) * b * star a
⊢ a * b * star a ≤ ‖b‖ • (a * star a)
state_after: A : Type u_1
inst✝⁹ : NonUnitalNormedRing A
inst✝⁸ : CompleteSpace A
inst✝⁷ : PartialOrder A
inst✝⁶ : StarRing A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : CstarRing A
inst✝³ : NormedSpace ℂ A
inst✝² : StarModule ℂ A
inst✝¹ : SMulCommClass ℂ A A
inst✝ : IsScalarTower ℂ A A
a b : A
hb : autoParam (IsSelfAdjoint b) _auto✝
h₁ : a * b * star a = star (star a) * b * star a
h₂ : a * star a = star (star a) * star a
⊢ a * b * star a ≤ ‖b‖ • (a * star a)
tactic: have h₂ : a * star a = star (star a) * star a := by simp
state_before: A : Type u_1
inst✝⁹ : NonUnitalNormedRing A
inst✝⁸ : CompleteSpace A
inst✝⁷ : PartialOrder A
inst✝⁶ : StarRing A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : CstarRing A
inst✝³ : NormedSpace ℂ A
inst✝² : StarModule ℂ A
inst✝¹ : SMulCommClass ℂ A A
inst✝ : IsScalarTower ℂ A A
a b : A
hb : autoParam (IsSelfAdjoint b) _auto✝
h₁ : a * b * star a = star (star a) * b * star a
h₂ : a * star a = star (star a) * star a
⊢ a * b * star a ≤ ‖b‖ • (a * star a)
state_after: A : Type u_1
inst✝⁹ : NonUnitalNormedRing A
inst✝⁸ : CompleteSpace A
inst✝⁷ : PartialOrder A
inst✝⁶ : StarRing A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : CstarRing A
inst✝³ : NormedSpace ℂ A
inst✝² : StarModule ℂ A
inst✝¹ : SMulCommClass ℂ A A
inst✝ : IsScalarTower ℂ A A
a b : A
hb : autoParam (IsSelfAdjoint b) _auto✝
h₁ : a * b * star a = star (star a) * b * star a
h₂ : a * star a = star (star a) * star a
⊢ star (star a) * b * star a ≤ ‖b‖ • (star (star a) * star a)
tactic: simp only [h₁, h₂]
state_before: A : Type u_1
inst✝⁹ : NonUnitalNormedRing A
inst✝⁸ : CompleteSpace A
inst✝⁷ : PartialOrder A
inst✝⁶ : StarRing A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : CstarRing A
inst✝³ : NormedSpace ℂ A
inst✝² : StarModule ℂ A
inst✝¹ : SMulCommClass ℂ A A
inst✝ : IsScalarTower ℂ A A
a b : A
hb : autoParam (IsSelfAdjoint b) _auto✝
h₁ : a * b * star a = star (star a) * b * star a
h₂ : a * star a = star (star a) * star a
⊢ star (star a) * b * star a ≤ ‖b‖ • (star (star a) * star a)
state_after: no goals
tactic: exact conjugate_le_norm_smul