-- Mathlib/Analysis/CStarAlgebra/Multiplier.lean
theorem norm_fst_eq_snd (a : 𝓜(𝕜, A)) : ‖a.fst‖ = ‖a.snd‖

Proof state:
𝕜 : Type u_1
A : Type u_2
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NonUnitalNormedRing A
inst✝⁴ : NormedSpace 𝕜 A
inst✝³ : SMulCommClass 𝕜 A A
inst✝² : IsScalarTower 𝕜 A A
inst✝¹ : StarRing A
inst✝ : CStarRing A
a : 𝓜(𝕜, A)
f : A →L[𝕜] A
C : ℝ≥0
h : ∀ (b : A), ‖f b‖₊ ^ 2 ≤ C * ‖f b‖₊ * ‖b‖₊
h1 : ∀ (b : A), C * ‖f b‖₊ * ‖b‖₊ ≤ C * ‖f‖₊ * ‖b‖₊ ^ 2
this : ‖f‖₊ / sqrt ‖f‖₊ ≤ sqrt C
⊢ ‖f‖₊ ≤ C

Tactic:
convert NNReal.rpow_le_rpow this two_pos.le

Result:
case h.e'_3
𝕜 : Type u_1
A : Type u_2
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NonUnitalNormedRing A
inst✝⁴ : NormedSpace 𝕜 A
inst✝³ : SMulCommClass 𝕜 A A
inst✝² : IsScalarTower 𝕜 A A
inst✝¹ : StarRing A
inst✝ : CStarRing A
a : 𝓜(𝕜, A)
f : A →L[𝕜] A
C : ℝ≥0
h : ∀ (b : A), ‖f b‖₊ ^ 2 ≤ C * ‖f b‖₊ * ‖b‖₊
h1 : ∀ (b : A), C * ‖f b‖₊ * ‖b‖₊ ≤ C * ‖f‖₊ * ‖b‖₊ ^ 2
this : ‖f‖₊ / sqrt ‖f‖₊ ≤ sqrt C
⊢ ‖f‖₊ = (‖f‖₊ / sqrt ‖f‖₊) ^ 2

case h.e'_4
𝕜 : Type u_1
A : Type u_2
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NonUnitalNormedRing A
inst✝⁴ : NormedSpace 𝕜 A
inst✝³ : SMulCommClass 𝕜 A A
inst✝² : IsScalarTower 𝕜 A A
inst✝¹ : StarRing A
inst✝ : CStarRing A
a : 𝓜(𝕜, A)
f : A →L[𝕜] A
C : ℝ≥0
h : ∀ (b : A), ‖f b‖₊ ^ 2 ≤ C * ‖f b‖₊ * ‖b‖₊
h1 : ∀ (b : A), C * ‖f b‖₊ * ‖b‖₊ ≤ C * ‖f‖₊ * ‖b‖₊ ^ 2
this : ‖f‖₊ / sqrt ‖f‖₊ ≤ sqrt C
⊢ C = sqrt C ^ 2

Proof state:
𝕜 : Type u_1
A : Type u_2
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NonUnitalNormedRing A
inst✝⁴ : NormedSpace 𝕜 A
inst✝³ : SMulCommClass 𝕜 A A
inst✝² : IsScalarTower 𝕜 A A
inst✝¹ : StarRing A
inst✝ : CStarRing A
a : 𝓜(𝕜, A)
h0 : ∀ (f : A →L[𝕜] A) (C : ℝ≥0), (∀ (b : A), ‖f b‖₊ ^ 2 ≤ C * ‖f b‖₊ * ‖b‖₊) → ‖f‖₊ ≤ C
b : A
⊢ ‖a.toProd.1 b‖₊ ^ 2 = ‖star (a.toProd.1 b) * a.toProd.1 b‖₊

Tactic:
simpa only [← sq] using CStarRing.nnnorm_star_mul_self.symm

Result:
no goals

Proof state:
𝕜 : Type u_1
A : Type u_2
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NonUnitalNormedRing A
inst✝⁴ : NormedSpace 𝕜 A
inst✝³ : SMulCommClass 𝕜 A A
inst✝² : IsScalarTower 𝕜 A A
inst✝¹ : StarRing A
inst✝ : CStarRing A
a : 𝓜(𝕜, A)
f : A →L[𝕜] A
C : ℝ≥0
h : ∀ (b : A), ‖f b‖₊ ^ 2 ≤ C * ‖f b‖₊ * ‖b‖₊
⊢ ‖f‖₊ ≤ C

Tactic:
have h1 : ∀ b, C * ‖f b‖₊ * ‖b‖₊ ≤ C * ‖f‖₊ * ‖b‖₊ ^ 2 := by
  intro b
  convert mul_le_mul_right' (mul_le_mul_left' (f.le_opNNNorm b) C) ‖b‖₊ using 1
  ring

Result:
𝕜 : Type u_1
A : Type u_2
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NonUnitalNormedRing A
inst✝⁴ : NormedSpace 𝕜 A
inst✝³ : SMulCommClass 𝕜 A A
inst✝² : IsScalarTower 𝕜 A A
inst✝¹ : StarRing A
inst✝ : CStarRing A
a : 𝓜(𝕜, A)
f : A →L[𝕜] A
C : ℝ≥0
h : ∀ (b : A), ‖f b‖₊ ^ 2 ≤ C * ‖f b‖₊ * ‖b‖₊
h1 : ∀ (b : A), C * ‖f b‖₊ * ‖b‖₊ ≤ C * ‖f‖₊ * ‖b‖₊ ^ 2
⊢ ‖f‖₊ ≤ C

Proof state:
𝕜 : Type u_1
A : Type u_2
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NonUnitalNormedRing A
inst✝⁴ : NormedSpace 𝕜 A
inst✝³ : SMulCommClass 𝕜 A A
inst✝² : IsScalarTower 𝕜 A A
inst✝¹ : StarRing A
inst✝ : CStarRing A
a : 𝓜(𝕜, A)
h0 : ∀ (f : A →L[𝕜] A) (C : ℝ≥0), (∀ (b : A), ‖f b‖₊ ^ 2 ≤ C * ‖f b‖₊ * ‖b‖₊) → ‖f‖₊ ≤ C
h1 : ∀ (b : A), ‖a.toProd.1 b‖₊ ^ 2 ≤ ‖a.toProd.2‖₊ * ‖a.toProd.1 b‖₊ * ‖b‖₊
⊢ ‖a.toProd.1‖ = ‖a.toProd.2‖

Tactic:
have h2 : ∀ b, ‖a.snd b‖₊ ^ 2 ≤ ‖a.fst‖₊ * ‖a.snd b‖₊ * ‖b‖₊ := by
  intro b
  calc
    ‖a.snd b‖₊ ^ 2 = ‖a.snd b * star (a.snd b)‖₊ := by
      simpa only [← sq] using CStarRing.nnnorm_self_mul_star.symm
    _ ≤ ‖b‖₊ * ‖a.fst (star (a.snd b))‖₊ :=
      ((a.central b (star (a.snd b))).symm ▸ nnnorm_mul_le _ _)
    _ = ‖a.fst (star (a.snd b))‖₊ * ‖b‖₊ := mul_comm _ _
    _ ≤ ‖a.fst‖₊ * ‖a.snd b‖₊ * ‖b‖₊ :=
      nnnorm_star (a.snd b) ▸ mul_le_mul_right' (a.fst.le_opNNNorm _) _

Result:
𝕜 : Type u_1
A : Type u_2
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NonUnitalNormedRing A
inst✝⁴ : NormedSpace 𝕜 A
inst✝³ : SMulCommClass 𝕜 A A
inst✝² : IsScalarTower 𝕜 A A
inst✝¹ : StarRing A
inst✝ : CStarRing A
a : 𝓜(𝕜, A)
h0 : ∀ (f : A →L[𝕜] A) (C : ℝ≥0), (∀ (b : A), ‖f b‖₊ ^ 2 ≤ C * ‖f b‖₊ * ‖b‖₊) → ‖f‖₊ ≤ C
h1 : ∀ (b : A), ‖a.toProd.1 b‖₊ ^ 2 ≤ ‖a.toProd.2‖₊ * ‖a.toProd.1 b‖₊ * ‖b‖₊
h2 : ∀ (b : A), ‖a.toProd.2 b‖₊ ^ 2 ≤ ‖a.toProd.1‖₊ * ‖a.toProd.2 b‖₊ * ‖b‖₊
⊢ ‖a.toProd.1‖ = ‖a.toProd.2‖

Proof state:
𝕜 : Type u_1
A : Type u_2
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NonUnitalNormedRing A
inst✝⁴ : NormedSpace 𝕜 A
inst✝³ : SMulCommClass 𝕜 A A
inst✝² : IsScalarTower 𝕜 A A
inst✝¹ : StarRing A
inst✝ : CStarRing A
a : 𝓜(𝕜, A)
h0 : ∀ (f : A →L[𝕜] A) (C : ℝ≥0), (∀ (b : A), ‖f b‖₊ ^ 2 ≤ C * ‖f b‖₊ * ‖b‖₊) → ‖f‖₊ ≤ C
⊢ ‖a.toProd.1‖ = ‖a.toProd.2‖

Tactic:
have h1 : ∀ b, ‖a.fst b‖₊ ^ 2 ≤ ‖a.snd‖₊ * ‖a.fst b‖₊ * ‖b‖₊ := by
  intro b
  calc
    ‖a.fst b‖₊ ^ 2 = ‖star (a.fst b) * a.fst b‖₊ := by
      simpa only [← sq] using CStarRing.nnnorm_star_mul_self.symm
    _ ≤ ‖a.snd (star (a.fst b))‖₊ * ‖b‖₊ := (a.central (star (a.fst b)) b ▸ nnnorm_mul_le _ _)
    _ ≤ ‖a.snd‖₊ * ‖a.fst b‖₊ * ‖b‖₊ :=
      nnnorm_star (a.fst b) ▸ mul_le_mul_right' (a.snd.le_opNNNorm _) _

Result:
𝕜 : Type u_1
A : Type u_2
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NonUnitalNormedRing A
inst✝⁴ : NormedSpace 𝕜 A
inst✝³ : SMulCommClass 𝕜 A A
inst✝² : IsScalarTower 𝕜 A A
inst✝¹ : StarRing A
inst✝ : CStarRing A
a : 𝓜(𝕜, A)
h0 : ∀ (f : A →L[𝕜] A) (C : ℝ≥0), (∀ (b : A), ‖f b‖₊ ^ 2 ≤ C * ‖f b‖₊ * ‖b‖₊) → ‖f‖₊ ≤ C
h1 : ∀ (b : A), ‖a.toProd.1 b‖₊ ^ 2 ≤ ‖a.toProd.2‖₊ * ‖a.toProd.1 b‖₊ * ‖b‖₊
⊢ ‖a.toProd.1‖ = ‖a.toProd.2‖

Proof state:
case h.e'_4
𝕜 : Type u_1
A : Type u_2
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NonUnitalNormedRing A
inst✝⁴ : NormedSpace 𝕜 A
inst✝³ : SMulCommClass 𝕜 A A
inst✝² : IsScalarTower 𝕜 A A
inst✝¹ : StarRing A
inst✝ : CStarRing A
a : 𝓜(𝕜, A)
f : A →L[𝕜] A
C : ℝ≥0
h : ∀ (b : A), ‖f b‖₊ ^ 2 ≤ C * ‖f b‖₊ * ‖b‖₊
b : A
⊢ C * ‖f‖₊ * ‖b‖₊ ^ 2 = C * (‖f‖₊ * ‖b‖₊) * ‖b‖₊

Tactic:
ring

Result:
no goals

Proof state:
𝕜 : Type u_1
A : Type u_2
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NonUnitalNormedRing A
inst✝⁴ : NormedSpace 𝕜 A
inst✝³ : SMulCommClass 𝕜 A A
inst✝² : IsScalarTower 𝕜 A A
inst✝¹ : StarRing A
inst✝ : CStarRing A
a : 𝓜(𝕜, A)
h0 : ∀ (f : A →L[𝕜] A) (C : ℝ≥0), (∀ (b : A), ‖f b‖₊ ^ 2 ≤ C * ‖f b‖₊ * ‖b‖₊) → ‖f‖₊ ≤ C
h1 : ∀ (b : A), ‖a.toProd.1 b‖₊ ^ 2 ≤ ‖a.toProd.2‖₊ * ‖a.toProd.1 b‖₊ * ‖b‖₊
b : A
⊢ ‖a.toProd.2 b‖₊ ^ 2 = ‖a.toProd.2 b * star (a.toProd.2 b)‖₊

Tactic:
simpa only [← sq] using CStarRing.nnnorm_self_mul_star.symm

Result:
no goals

Proof state:
𝕜 : Type u_1
A : Type u_2
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NonUnitalNormedRing A
inst✝⁴ : NormedSpace 𝕜 A
inst✝³ : SMulCommClass 𝕜 A A
inst✝² : IsScalarTower 𝕜 A A
inst✝¹ : StarRing A
inst✝ : CStarRing A
a : 𝓜(𝕜, A)
h0 : ∀ (f : A →L[𝕜] A) (C : ℝ≥0), (∀ (b : A), ‖f b‖₊ ^ 2 ≤ C * ‖f b‖₊ * ‖b‖₊) → ‖f‖₊ ≤ C
h1 : ∀ (b : A), ‖a.toProd.1 b‖₊ ^ 2 ≤ ‖a.toProd.2‖₊ * ‖a.toProd.1 b‖₊ * ‖b‖₊
h2 : ∀ (b : A), ‖a.toProd.2 b‖₊ ^ 2 ≤ ‖a.toProd.1‖₊ * ‖a.toProd.2 b‖₊ * ‖b‖₊
⊢ ‖a.toProd.1‖ = ‖a.toProd.2‖

Tactic:
exact le_antisymm (h0 _ _ h1) (h0 _ _ h2)

Result:
no goals

Proof state:
𝕜 : Type u_1
A : Type u_2
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NonUnitalNormedRing A
inst✝⁴ : NormedSpace 𝕜 A
inst✝³ : SMulCommClass 𝕜 A A
inst✝² : IsScalarTower 𝕜 A A
inst✝¹ : StarRing A
inst✝ : CStarRing A
a : 𝓜(𝕜, A)
f : A →L[𝕜] A
C : ℝ≥0
h : ∀ (b : A), ‖f b‖₊ ^ 2 ≤ C * ‖f b‖₊ * ‖b‖₊
b : A
⊢ C * ‖f b‖₊ * ‖b‖₊ ≤ C * ‖f‖₊ * ‖b‖₊ ^ 2

Tactic:
convert mul_le_mul_right' (mul_le_mul_left' (f.le_opNNNorm b) C) ‖b‖₊ using 1

Result:
case h.e'_4
𝕜 : Type u_1
A : Type u_2
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NonUnitalNormedRing A
inst✝⁴ : NormedSpace 𝕜 A
inst✝³ : SMulCommClass 𝕜 A A
inst✝² : IsScalarTower 𝕜 A A
inst✝¹ : StarRing A
inst✝ : CStarRing A
a : 𝓜(𝕜, A)
f : A →L[𝕜] A
C : ℝ≥0
h : ∀ (b : A), ‖f b‖₊ ^ 2 ≤ C * ‖f b‖₊ * ‖b‖₊
b : A
⊢ C * ‖f‖₊ * ‖b‖₊ ^ 2 = C * (‖f‖₊ * ‖b‖₊) * ‖b‖₊