Theorem: ContinuousLinearMap.opNorm_mul_flip_apply
File: Mathlib/Analysis/NormedSpace/Star/Unitization.lean

tactic: refine le_antisymm
  (opNorm_le_bound _ (norm_nonneg _) fun b => by simpa only [mul_comm] using norm_mul_le b a) ?_
state_before: 𝕜 : Type u_1
E : Type u_2
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NonUnitalNormedRing E
inst✝⁵ : StarRing E
inst✝⁴ : NormedStarGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : IsScalarTower 𝕜 E E
inst✝¹ : SMulCommClass 𝕜 E E
inst✝ : RegularNormedAlgebra 𝕜 E
a : E
⊢ ‖(mul 𝕜 E).flip a‖ = ‖a‖
state_after: 𝕜 : Type u_1
E : Type u_2
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NonUnitalNormedRing E
inst✝⁵ : StarRing E
inst✝⁴ : NormedStarGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : IsScalarTower 𝕜 E E
inst✝¹ : SMulCommClass 𝕜 E E
inst✝ : RegularNormedAlgebra 𝕜 E
a : E
⊢ ‖a‖ ≤ ‖(mul 𝕜 E).flip a‖

tactic: suffices ‖mul 𝕜 E (star a)‖ ≤ ‖(mul 𝕜 E).flip a‖ by simpa using this
state_before: 𝕜 : Type u_1
E : Type u_2
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NonUnitalNormedRing E
inst✝⁵ : StarRing E
inst✝⁴ : NormedStarGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : IsScalarTower 𝕜 E E
inst✝¹ : SMulCommClass 𝕜 E E
inst✝ : RegularNormedAlgebra 𝕜 E
a : E
⊢ ‖a‖ ≤ ‖(mul 𝕜 E).flip a‖
state_after: 𝕜 : Type u_1
E : Type u_2
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NonUnitalNormedRing E
inst✝⁵ : StarRing E
inst✝⁴ : NormedStarGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : IsScalarTower 𝕜 E E
inst✝¹ : SMulCommClass 𝕜 E E
inst✝ : RegularNormedAlgebra 𝕜 E
a : E
⊢ ‖(mul 𝕜 E) a⋆‖ ≤ ‖(mul 𝕜 E).flip a‖

tactic: refine opNorm_le_bound _ (norm_nonneg _) fun b => ?_
state_before: 𝕜 : Type u_1
E : Type u_2
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NonUnitalNormedRing E
inst✝⁵ : StarRing E
inst✝⁴ : NormedStarGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : IsScalarTower 𝕜 E E
inst✝¹ : SMulCommClass 𝕜 E E
inst✝ : RegularNormedAlgebra 𝕜 E
a : E
⊢ ‖(mul 𝕜 E) a⋆‖ ≤ ‖(mul 𝕜 E).flip a‖
state_after: 𝕜 : Type u_1
E : Type u_2
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NonUnitalNormedRing E
inst✝⁵ : StarRing E
inst✝⁴ : NormedStarGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : IsScalarTower 𝕜 E E
inst✝¹ : SMulCommClass 𝕜 E E
inst✝ : RegularNormedAlgebra 𝕜 E
a b : E
⊢ ‖((mul 𝕜 E) a⋆) b‖ ≤ ‖(mul 𝕜 E).flip a‖ * ‖b‖

tactic: calc ‖mul 𝕜 E (star a) b‖ = ‖(mul 𝕜 E).flip a (star b)‖ := by simpa using norm_star (star b * a)
  _ ≤ ‖(mul 𝕜 E).flip a‖ * ‖b‖ := by simpa using le_opNorm ((mul 𝕜 E).flip a) (star b)
state_before: 𝕜 : Type u_1
E : Type u_2
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NonUnitalNormedRing E
inst✝⁵ : StarRing E
inst✝⁴ : NormedStarGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : IsScalarTower 𝕜 E E
inst✝¹ : SMulCommClass 𝕜 E E
inst✝ : RegularNormedAlgebra 𝕜 E
a b : E
⊢ ‖((mul 𝕜 E) a⋆) b‖ ≤ ‖(mul 𝕜 E).flip a‖ * ‖b‖
state_after: no goals

tactic: simpa only [mul_comm] using norm_mul_le b a
state_before: 𝕜 : Type u_1
E : Type u_2
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NonUnitalNormedRing E
inst✝⁵ : StarRing E
inst✝⁴ : NormedStarGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : IsScalarTower 𝕜 E E
inst✝¹ : SMulCommClass 𝕜 E E
inst✝ : RegularNormedAlgebra 𝕜 E
a b : E
⊢ ‖((mul 𝕜 E).flip a) b‖ ≤ ‖a‖ * ‖b‖
state_after: no goals

tactic: simpa using this
state_before: 𝕜 : Type u_1
E : Type u_2
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NonUnitalNormedRing E
inst✝⁵ : StarRing E
inst✝⁴ : NormedStarGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : IsScalarTower 𝕜 E E
inst✝¹ : SMulCommClass 𝕜 E E
inst✝ : RegularNormedAlgebra 𝕜 E
a : E
this : ‖(mul 𝕜 E) a⋆‖ ≤ ‖(mul 𝕜 E).flip a‖
⊢ ‖a‖ ≤ ‖(mul 𝕜 E).flip a‖
state_after: no goals

tactic: simpa using norm_star (star b * a)
state_before: 𝕜 : Type u_1
E : Type u_2
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NonUnitalNormedRing E
inst✝⁵ : StarRing E
inst✝⁴ : NormedStarGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : IsScalarTower 𝕜 E E
inst✝¹ : SMulCommClass 𝕜 E E
inst✝ : RegularNormedAlgebra 𝕜 E
a b : E
⊢ ‖((mul 𝕜 E) a⋆) b‖ = ‖((mul 𝕜 E).flip a) b⋆‖
state_after: no goals

tactic: simpa using le_opNorm ((mul 𝕜 E).flip a) (star b)
state_before: 𝕜 : Type u_1
E : Type u_2
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NonUnitalNormedRing E
inst✝⁵ : StarRing E
inst✝⁴ : NormedStarGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : IsScalarTower 𝕜 E E
inst✝¹ : SMulCommClass 𝕜 E E
inst✝ : RegularNormedAlgebra 𝕜 E
a b : E
⊢ ‖((mul 𝕜 E).flip a) b⋆‖ ≤ ‖(mul 𝕜 E).flip a‖ * ‖b‖
state_after: no goals