l3lab/ntp-mathlib huggingface dataset:
instance instCstarRing : CstarRing 𝓜(𝕜, A) where
  norm_star_mul_self := @fun (a : 𝓜(𝕜, A)) => congr_arg ((↑) : ℝ≥0 → ℝ) <|
    show ‖star a * a‖₊ = ‖a‖₊ * ‖a‖₊ by
    /- The essence of the argument is this: let `a = (L,R)` and recall `‖a‖ = ‖L‖`.
    `star a = (star ∘ R ∘ star, star ∘ L ∘ star)`. Then for any `x y : A`, we have
    `‖star a * a‖ = ‖(star a * a).snd‖ = ‖R (star (L (star x))) * y‖ = ‖star (L (star x)) * L y‖`
    Now, on the one hand,
    `‖star (L (star x)) * L y‖ ≤ ‖star (L (star x))‖ * ‖L y‖ = ‖L (star x)‖ * ‖L y‖ ≤ ‖L‖ ^ 2`
    whenever `‖x‖, ‖y‖ ≤ 1`, so the supremum over all such `x, y` is at most `‖L‖ ^ 2`.
    On the other hand, for any `‖z‖ ≤ 1`, we may choose `x := star z` and `y := z` to get:
    `‖star (L (star x)) * L y‖ = ‖star (L z) * (L z)‖ = ‖L z‖ ^ 2`, and taking the supremum over
    all such `z` yields that the supremum is at least `‖L‖ ^ 2`. It is the latter part of the
    argument where `DenselyNormedField 𝕜` is required (for `sSup_closed_unit_ball_eq_nnnorm`). -/
      have hball : (Metric.closedBall (0 : A) 1).Nonempty :=
        Metric.nonempty_closedBall.2 zero_le_one
      have key :
        ∀ x y, ‖x‖₊ ≤ 1 → ‖y‖₊ ≤ 1 → ‖a.snd (star (a.fst (star x))) * y‖₊ ≤ ‖a‖₊ * ‖a‖₊ := by
        intro x y hx hy
        

Proof state:
𝕜 : Type u_1
A : Type u_2
inst✝⁸ : DenselyNormedField 𝕜
inst✝⁷ : StarRing 𝕜
inst✝⁶ : NonUnitalNormedRing A
inst✝⁵ : StarRing A
inst✝⁴ : CstarRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : SMulCommClass 𝕜 A A
inst✝¹ : IsScalarTower 𝕜 A A
inst✝ : StarModule 𝕜 A
a : 𝓜(𝕜, A)
hball : Set.Nonempty (Metric.closedBall 0 1)
x y : A
hx : ‖x‖₊ ≤ 1
hy : ‖y‖₊ ≤ 1
⊢ ‖a.toProd.2 (star (a.toProd.1 (star x))) * y‖₊ ≤ ‖a‖₊ * ‖a‖₊

Next tactic:
rw [a.central]
instance instCstarRing : CstarRing 𝓜(𝕜, A) where
  norm_star_mul_self := @fun (a : 𝓜(𝕜, A)) => congr_arg ((↑) : ℝ≥0 → ℝ) <|
    show ‖star a * a‖₊ = ‖a‖₊ * ‖a‖₊ by
    /- The essence of the argument is this: let `a = (L,R)` and recall `‖a‖ = ‖L‖`.
    `star a = (star ∘ R ∘ star, star ∘ L ∘ star)`. Then for any `x y : A`, we have
    `‖star a * a‖ = ‖(star a * a).snd‖ = ‖R (star (L (star x))) * y‖ = ‖star (L (star x)) * L y‖`
    Now, on the one hand,
    `‖star (L (star x)) * L y‖ ≤ ‖star (L (star x))‖ * ‖L y‖ = ‖L (star x)‖ * ‖L y‖ ≤ ‖L‖ ^ 2`
    whenever `‖x‖, ‖y‖ ≤ 1`, so the supremum over all such `x, y` is at most `‖L‖ ^ 2`.
    On the other hand, for any `‖z‖ ≤ 1`, we may choose `x := star z` and `y := z` to get:
    `‖star (L (star x)) * L y‖ = ‖star (L z) * (L z)‖ = ‖L z‖ ^ 2`, and taking the supremum over
    all such `z` yields that the supremum is at least `‖L‖ ^ 2`. It is the latter part of the
    argument where `DenselyNormedField 𝕜` is required (for `sSup_closed_unit_ball_eq_nnnorm`). -/
      have hball : (Metric.closedBall (0 : A) 1).Nonempty :=
        Metric.nonempty_closedBall.2 zero_le_one
      have key :
        ∀ x y, ‖x‖₊ ≤ 1 → ‖y‖₊ ≤ 1 → ‖a.snd (star (a.fst (star x))) * y‖₊ ≤ ‖a‖₊ * ‖a‖₊ := by
        intro x y hx hy
        rw [a.central]
        

Proof state:
𝕜 : Type u_1
A : Type u_2
inst✝⁸ : DenselyNormedField 𝕜
inst✝⁷ : StarRing 𝕜
inst✝⁶ : NonUnitalNormedRing A
inst✝⁵ : StarRing A
inst✝⁴ : CstarRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : SMulCommClass 𝕜 A A
inst✝¹ : IsScalarTower 𝕜 A A
inst✝ : StarModule 𝕜 A
a : 𝓜(𝕜, A)
hball : Set.Nonempty (Metric.closedBall 0 1)
x y : A
hx : ‖x‖₊ ≤ 1
hy : ‖y‖₊ ≤ 1
⊢ ‖star (a.toProd.1 (star x)) * a.toProd.1 y‖₊ ≤ ‖a‖₊ * ‖a‖₊

Next tactic:
calc
          ‖star (a.fst (star x)) * a.fst y‖₊ ≤ ‖a.fst (star x)‖₊ * ‖a.fst y‖₊ :=
            nnnorm_star (a.fst (star x)) ▸ nnnorm_mul_le _ _
          _ ≤ ‖a.fst‖₊ * 1 * (‖a.fst‖₊ * 1) :=
            (mul_le_mul' (a.fst.le_op_norm_of_le ((nnnorm_star x).trans_le hx))
              (a.fst.le_op_norm_of_le hy))
          _ ≤ ‖a‖₊ * ‖a‖₊ := by simp only [mul_one, nnnorm_fst, le_rfl]
instance instCstarRing : CstarRing 𝓜(𝕜, A) where
  norm_star_mul_self := @fun (a : 𝓜(𝕜, A)) => congr_arg ((↑) : ℝ≥0 → ℝ) <|
    show ‖star a * a‖₊ = ‖a‖₊ * ‖a‖₊ by
    /- The essence of the argument is this: let `a = (L,R)` and recall `‖a‖ = ‖L‖`.
    `star a = (star ∘ R ∘ star, star ∘ L ∘ star)`. Then for any `x y : A`, we have
    `‖star a * a‖ = ‖(star a * a).snd‖ = ‖R (star (L (star x))) * y‖ = ‖star (L (star x)) * L y‖`
    Now, on the one hand,
    `‖star (L (star x)) * L y‖ ≤ ‖star (L (star x))‖ * ‖L y‖ = ‖L (star x)‖ * ‖L y‖ ≤ ‖L‖ ^ 2`
    whenever `‖x‖, ‖y‖ ≤ 1`, so the supremum over all such `x, y` is at most `‖L‖ ^ 2`.
    On the other hand, for any `‖z‖ ≤ 1`, we may choose `x := star z` and `y := z` to get:
    `‖star (L (star x)) * L y‖ = ‖star (L z) * (L z)‖ = ‖L z‖ ^ 2`, and taking the supremum over
    all such `z` yields that the supremum is at least `‖L‖ ^ 2`. It is the latter part of the
    argument where `DenselyNormedField 𝕜` is required (for `sSup_closed_unit_ball_eq_nnnorm`). -/
      have hball : (Metric.closedBall (0 : A) 1).Nonempty :=
        Metric.nonempty_closedBall.2 zero_le_one
      have key :
        ∀ x y, ‖x‖₊ ≤ 1 → ‖y‖₊ ≤ 1 → ‖a.snd (star (a.fst (star x))) * y‖₊ ≤ ‖a‖₊ * ‖a‖₊ := by
        intro x y hx hy
        rw [a.central]
        calc
          ‖star (a.fst (star x)) * a.fst y‖₊ ≤ ‖a.fst (star x)‖₊ * ‖a.fst y‖₊ :=
            nnnorm_star (a.fst (star x)) ▸ nnnorm_mul_le _ _
          _ ≤ ‖a.fst‖₊ * 1 * (‖a.fst‖₊ * 1) :=
            (mul_le_mul' (a.fst.le_op_norm_of_le ((nnnorm_star x).trans_le hx))
              (a.fst.le_op_norm_of_le hy))
          _ ≤ ‖a‖₊ * ‖a‖₊ := by 

Proof state:
𝕜 : Type u_1
A : Type u_2
inst✝⁸ : DenselyNormedField 𝕜
inst✝⁷ : StarRing 𝕜
inst✝⁶ : NonUnitalNormedRing A
inst✝⁵ : StarRing A
inst✝⁴ : CstarRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : SMulCommClass 𝕜 A A
inst✝¹ : IsScalarTower 𝕜 A A
inst✝ : StarModule 𝕜 A
a : 𝓜(𝕜, A)
hball : Set.Nonempty (Metric.closedBall 0 1)
x y : A
hx : ‖x‖₊ ≤ 1
hy : ‖y‖₊ ≤ 1
⊢ ‖a.toProd.1‖₊ * 1 * (‖a.toProd.1‖₊ * 1) ≤ ‖a‖₊ * ‖a‖₊

Next tactic:
simp only [mul_one, nnnorm_fst, le_rfl]
instance instCstarRing : CstarRing 𝓜(𝕜, A) where
  norm_star_mul_self := @fun (a : 𝓜(𝕜, A)) => congr_arg ((↑) : ℝ≥0 → ℝ) <|
    show ‖star a * a‖₊ = ‖a‖₊ * ‖a‖₊ by
    /- The essence of the argument is this: let `a = (L,R)` and recall `‖a‖ = ‖L‖`.
    `star a = (star ∘ R ∘ star, star ∘ L ∘ star)`. Then for any `x y : A`, we have
    `‖star a * a‖ = ‖(star a * a).snd‖ = ‖R (star (L (star x))) * y‖ = ‖star (L (star x)) * L y‖`
    Now, on the one hand,
    `‖star (L (star x)) * L y‖ ≤ ‖star (L (star x))‖ * ‖L y‖ = ‖L (star x)‖ * ‖L y‖ ≤ ‖L‖ ^ 2`
    whenever `‖x‖, ‖y‖ ≤ 1`, so the supremum over all such `x, y` is at most `‖L‖ ^ 2`.
    On the other hand, for any `‖z‖ ≤ 1`, we may choose `x := star z` and `y := z` to get:
    `‖star (L (star x)) * L y‖ = ‖star (L z) * (L z)‖ = ‖L z‖ ^ 2`, and taking the supremum over
    all such `z` yields that the supremum is at least `‖L‖ ^ 2`. It is the latter part of the
    argument where `DenselyNormedField 𝕜` is required (for `sSup_closed_unit_ball_eq_nnnorm`). -/
      have hball : (Metric.closedBall (0 : A) 1).Nonempty :=
        Metric.nonempty_closedBall.2 zero_le_one
      have key :
        ∀ x y, ‖x‖₊ ≤ 1 → ‖y‖₊ ≤ 1 → ‖a.snd (star (a.fst (star x))) * y‖₊ ≤ ‖a‖₊ * ‖a‖₊ := by
        intro x y hx hy
        rw [a.central]
        calc
          ‖star (a.fst (star x)) * a.fst y‖₊ ≤ ‖a.fst (star x)‖₊ * ‖a.fst y‖₊ :=
            nnnorm_star (a.fst (star x)) ▸ nnnorm_mul_le _ _
          _ ≤ ‖a.fst‖₊ * 1 * (‖a.fst‖₊ * 1) :=
            (mul_le_mul' (a.fst.le_op_norm_of_le ((nnnorm_star x).trans_le hx))
              (a.fst.le_op_norm_of_le hy))
          _ ≤ ‖a‖₊ * ‖a‖₊ := by simp only [mul_one, nnnorm_fst, le_rfl]
      

Proof state:
𝕜 : Type u_1
A : Type u_2
inst✝⁸ : DenselyNormedField 𝕜
inst✝⁷ : StarRing 𝕜
inst✝⁶ : NonUnitalNormedRing A
inst✝⁵ : StarRing A
inst✝⁴ : CstarRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : SMulCommClass 𝕜 A A
inst✝¹ : IsScalarTower 𝕜 A A
inst✝ : StarModule 𝕜 A
a : 𝓜(𝕜, A)
hball : Set.Nonempty (Metric.closedBall 0 1)
key : ∀ (x y : A), ‖x‖₊ ≤ 1 → ‖y‖₊ ≤ 1 → ‖a.toProd.2 (star (a.toProd.1 (star x))) * y‖₊ ≤ ‖a‖₊ * ‖a‖₊
⊢ ‖star a * a‖₊ = ‖a‖₊ * ‖a‖₊

Next tactic:
rw [← nnnorm_snd]
instance instCstarRing : CstarRing 𝓜(𝕜, A) where
  norm_star_mul_self := @fun (a : 𝓜(𝕜, A)) => congr_arg ((↑) : ℝ≥0 → ℝ) <|
    show ‖star a * a‖₊ = ‖a‖₊ * ‖a‖₊ by
    /- The essence of the argument is this: let `a = (L,R)` and recall `‖a‖ = ‖L‖`.
    `star a = (star ∘ R ∘ star, star ∘ L ∘ star)`. Then for any `x y : A`, we have
    `‖star a * a‖ = ‖(star a * a).snd‖ = ‖R (star (L (star x))) * y‖ = ‖star (L (star x)) * L y‖`
    Now, on the one hand,
    `‖star (L (star x)) * L y‖ ≤ ‖star (L (star x))‖ * ‖L y‖ = ‖L (star x)‖ * ‖L y‖ ≤ ‖L‖ ^ 2`
    whenever `‖x‖, ‖y‖ ≤ 1`, so the supremum over all such `x, y` is at most `‖L‖ ^ 2`.
    On the other hand, for any `‖z‖ ≤ 1`, we may choose `x := star z` and `y := z` to get:
    `‖star (L (star x)) * L y‖ = ‖star (L z) * (L z)‖ = ‖L z‖ ^ 2`, and taking the supremum over
    all such `z` yields that the supremum is at least `‖L‖ ^ 2`. It is the latter part of the
    argument where `DenselyNormedField 𝕜` is required (for `sSup_closed_unit_ball_eq_nnnorm`). -/
      have hball : (Metric.closedBall (0 : A) 1).Nonempty :=
        Metric.nonempty_closedBall.2 zero_le_one
      have key :
        ∀ x y, ‖x‖₊ ≤ 1 → ‖y‖₊ ≤ 1 → ‖a.snd (star (a.fst (star x))) * y‖₊ ≤ ‖a‖₊ * ‖a‖₊ := by
        intro x y hx hy
        rw [a.central]
        calc
          ‖star (a.fst (star x)) * a.fst y‖₊ ≤ ‖a.fst (star x)‖₊ * ‖a.fst y‖₊ :=
            nnnorm_star (a.fst (star x)) ▸ nnnorm_mul_le _ _
          _ ≤ ‖a.fst‖₊ * 1 * (‖a.fst‖₊ * 1) :=
            (mul_le_mul' (a.fst.le_op_norm_of_le ((nnnorm_star x).trans_le hx))
              (a.fst.le_op_norm_of_le hy))
          _ ≤ ‖a‖₊ * ‖a‖₊ := by simp only [mul_one, nnnorm_fst, le_rfl]
      rw [← nnnorm_snd]
      

Proof state:
𝕜 : Type u_1
A : Type u_2
inst✝⁸ : DenselyNormedField 𝕜
inst✝⁷ : StarRing 𝕜
inst✝⁶ : NonUnitalNormedRing A
inst✝⁵ : StarRing A
inst✝⁴ : CstarRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : SMulCommClass 𝕜 A A
inst✝¹ : IsScalarTower 𝕜 A A
inst✝ : StarModule 𝕜 A
a : 𝓜(𝕜, A)
hball : Set.Nonempty (Metric.closedBall 0 1)
key : ∀ (x y : A), ‖x‖₊ ≤ 1 → ‖y‖₊ ≤ 1 → ‖a.toProd.2 (star (a.toProd.1 (star x))) * y‖₊ ≤ ‖a‖₊ * ‖a‖₊
⊢ ‖(star a * a).toProd.2‖₊ = ‖a‖₊ * ‖a‖₊

Next tactic:
simp only [mul_snd, ← sSup_closed_unit_ball_eq_nnnorm, star_snd, mul_apply]
instance instCstarRing : CstarRing 𝓜(𝕜, A) where
  norm_star_mul_self := @fun (a : 𝓜(𝕜, A)) => congr_arg ((↑) : ℝ≥0 → ℝ) <|
    show ‖star a * a‖₊ = ‖a‖₊ * ‖a‖₊ by
    /- The essence of the argument is this: let `a = (L,R)` and recall `‖a‖ = ‖L‖`.
    `star a = (star ∘ R ∘ star, star ∘ L ∘ star)`. Then for any `x y : A`, we have
    `‖star a * a‖ = ‖(star a * a).snd‖ = ‖R (star (L (star x))) * y‖ = ‖star (L (star x)) * L y‖`
    Now, on the one hand,
    `‖star (L (star x)) * L y‖ ≤ ‖star (L (star x))‖ * ‖L y‖ = ‖L (star x)‖ * ‖L y‖ ≤ ‖L‖ ^ 2`
    whenever `‖x‖, ‖y‖ ≤ 1`, so the supremum over all such `x, y` is at most `‖L‖ ^ 2`.
    On the other hand, for any `‖z‖ ≤ 1`, we may choose `x := star z` and `y := z` to get:
    `‖star (L (star x)) * L y‖ = ‖star (L z) * (L z)‖ = ‖L z‖ ^ 2`, and taking the supremum over
    all such `z` yields that the supremum is at least `‖L‖ ^ 2`. It is the latter part of the
    argument where `DenselyNormedField 𝕜` is required (for `sSup_closed_unit_ball_eq_nnnorm`). -/
      have hball : (Metric.closedBall (0 : A) 1).Nonempty :=
        Metric.nonempty_closedBall.2 zero_le_one
      have key :
        ∀ x y, ‖x‖₊ ≤ 1 → ‖y‖₊ ≤ 1 → ‖a.snd (star (a.fst (star x))) * y‖₊ ≤ ‖a‖₊ * ‖a‖₊ := by
        intro x y hx hy
        rw [a.central]
        calc
          ‖star (a.fst (star x)) * a.fst y‖₊ ≤ ‖a.fst (star x)‖₊ * ‖a.fst y‖₊ :=
            nnnorm_star (a.fst (star x)) ▸ nnnorm_mul_le _ _
          _ ≤ ‖a.fst‖₊ * 1 * (‖a.fst‖₊ * 1) :=
            (mul_le_mul' (a.fst.le_op_norm_of_le ((nnnorm_star x).trans_le hx))
              (a.fst.le_op_norm_of_le hy))
          _ ≤ ‖a‖₊ * ‖a‖₊ := by simp only [mul_one, nnnorm_fst, le_rfl]
      rw [← nnnorm_snd]
      simp only [mul_snd, ← sSup_closed_unit_ball_eq_nnnorm, star_snd, mul_apply]
      

Proof state:
𝕜 : Type u_1
A : Type u_2
inst✝⁸ : DenselyNormedField 𝕜
inst✝⁷ : StarRing 𝕜
inst✝⁶ : NonUnitalNormedRing A
inst✝⁵ : StarRing A
inst✝⁴ : CstarRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : SMulCommClass 𝕜 A A
inst✝¹ : IsScalarTower 𝕜 A A
inst✝ : StarModule 𝕜 A
a : 𝓜(𝕜, A)
hball : Set.Nonempty (Metric.closedBall 0 1)
key : ∀ (x y : A), ‖x‖₊ ≤ 1 → ‖y‖₊ ≤ 1 → ‖a.toProd.2 (star (a.toProd.1 (star x))) * y‖₊ ≤ ‖a‖₊ * ‖a‖₊
⊢ sSup ((fun a_1 => ‖a.toProd.2 (star (a.toProd.1 (star a_1)))‖₊) '' Metric.closedBall 0 1) = ‖a‖₊ * ‖a‖₊

Next tactic:
simp only [← @op_nnnorm_mul_apply 𝕜 _ A]
instance instCstarRing : CstarRing 𝓜(𝕜, A) where
  norm_star_mul_self := @fun (a : 𝓜(𝕜, A)) => congr_arg ((↑) : ℝ≥0 → ℝ) <|
    show ‖star a * a‖₊ = ‖a‖₊ * ‖a‖₊ by
    /- The essence of the argument is this: let `a = (L,R)` and recall `‖a‖ = ‖L‖`.
    `star a = (star ∘ R ∘ star, star ∘ L ∘ star)`. Then for any `x y : A`, we have
    `‖star a * a‖ = ‖(star a * a).snd‖ = ‖R (star (L (star x))) * y‖ = ‖star (L (star x)) * L y‖`
    Now, on the one hand,
    `‖star (L (star x)) * L y‖ ≤ ‖star (L (star x))‖ * ‖L y‖ = ‖L (star x)‖ * ‖L y‖ ≤ ‖L‖ ^ 2`
    whenever `‖x‖, ‖y‖ ≤ 1`, so the supremum over all such `x, y` is at most `‖L‖ ^ 2`.
    On the other hand, for any `‖z‖ ≤ 1`, we may choose `x := star z` and `y := z` to get:
    `‖star (L (star x)) * L y‖ = ‖star (L z) * (L z)‖ = ‖L z‖ ^ 2`, and taking the supremum over
    all such `z` yields that the supremum is at least `‖L‖ ^ 2`. It is the latter part of the
    argument where `DenselyNormedField 𝕜` is required (for `sSup_closed_unit_ball_eq_nnnorm`). -/
      have hball : (Metric.closedBall (0 : A) 1).Nonempty :=
        Metric.nonempty_closedBall.2 zero_le_one
      have key :
        ∀ x y, ‖x‖₊ ≤ 1 → ‖y‖₊ ≤ 1 → ‖a.snd (star (a.fst (star x))) * y‖₊ ≤ ‖a‖₊ * ‖a‖₊ := by
        intro x y hx hy
        rw [a.central]
        calc
          ‖star (a.fst (star x)) * a.fst y‖₊ ≤ ‖a.fst (star x)‖₊ * ‖a.fst y‖₊ :=
            nnnorm_star (a.fst (star x)) ▸ nnnorm_mul_le _ _
          _ ≤ ‖a.fst‖₊ * 1 * (‖a.fst‖₊ * 1) :=
            (mul_le_mul' (a.fst.le_op_norm_of_le ((nnnorm_star x).trans_le hx))
              (a.fst.le_op_norm_of_le hy))
          _ ≤ ‖a‖₊ * ‖a‖₊ := by simp only [mul_one, nnnorm_fst, le_rfl]
      rw [← nnnorm_snd]
      simp only [mul_snd, ← sSup_closed_unit_ball_eq_nnnorm, star_snd, mul_apply]
      simp only [← @op_nnnorm_mul_apply 𝕜 _ A]
      

Proof state:
𝕜 : Type u_1
A : Type u_2
inst✝⁸ : DenselyNormedField 𝕜
inst✝⁷ : StarRing 𝕜
inst✝⁶ : NonUnitalNormedRing A
inst✝⁵ : StarRing A
inst✝⁴ : CstarRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : SMulCommClass 𝕜 A A
inst✝¹ : IsScalarTower 𝕜 A A
inst✝ : StarModule 𝕜 A
a : 𝓜(𝕜, A)
hball : Set.Nonempty (Metric.closedBall 0 1)
key : ∀ (x y : A), ‖x‖₊ ≤ 1 → ‖y‖₊ ≤ 1 → ‖a.toProd.2 (star (a.toProd.1 (star x))) * y‖₊ ≤ ‖a‖₊ * ‖a‖₊
⊢ sSup
      ((fun a_1 => ‖(ContinuousLinearMap.mul 𝕜 A) (a.toProd.2 (star (a.toProd.1 (star a_1))))‖₊) ''
        Metric.closedBall 0 1) =
    ‖a‖₊ * ‖a‖₊

Next tactic:
simp only [← sSup_closed_unit_ball_eq_nnnorm, mul_apply']
instance instCstarRing : CstarRing 𝓜(𝕜, A) where
  norm_star_mul_self := @fun (a : 𝓜(𝕜, A)) => congr_arg ((↑) : ℝ≥0 → ℝ) <|
    show ‖star a * a‖₊ = ‖a‖₊ * ‖a‖₊ by
    /- The essence of the argument is this: let `a = (L,R)` and recall `‖a‖ = ‖L‖`.
    `star a = (star ∘ R ∘ star, star ∘ L ∘ star)`. Then for any `x y : A`, we have
    `‖star a * a‖ = ‖(star a * a).snd‖ = ‖R (star (L (star x))) * y‖ = ‖star (L (star x)) * L y‖`
    Now, on the one hand,
    `‖star (L (star x)) * L y‖ ≤ ‖star (L (star x))‖ * ‖L y‖ = ‖L (star x)‖ * ‖L y‖ ≤ ‖L‖ ^ 2`
    whenever `‖x‖, ‖y‖ ≤ 1`, so the supremum over all such `x, y` is at most `‖L‖ ^ 2`.
    On the other hand, for any `‖z‖ ≤ 1`, we may choose `x := star z` and `y := z` to get:
    `‖star (L (star x)) * L y‖ = ‖star (L z) * (L z)‖ = ‖L z‖ ^ 2`, and taking the supremum over
    all such `z` yields that the supremum is at least `‖L‖ ^ 2`. It is the latter part of the
    argument where `DenselyNormedField 𝕜` is required (for `sSup_closed_unit_ball_eq_nnnorm`). -/
      have hball : (Metric.closedBall (0 : A) 1).Nonempty :=
        Metric.nonempty_closedBall.2 zero_le_one
      have key :
        ∀ x y, ‖x‖₊ ≤ 1 → ‖y‖₊ ≤ 1 → ‖a.snd (star (a.fst (star x))) * y‖₊ ≤ ‖a‖₊ * ‖a‖₊ := by
        intro x y hx hy
        rw [a.central]
        calc
          ‖star (a.fst (star x)) * a.fst y‖₊ ≤ ‖a.fst (star x)‖₊ * ‖a.fst y‖₊ :=
            nnnorm_star (a.fst (star x)) ▸ nnnorm_mul_le _ _
          _ ≤ ‖a.fst‖₊ * 1 * (‖a.fst‖₊ * 1) :=
            (mul_le_mul' (a.fst.le_op_norm_of_le ((nnnorm_star x).trans_le hx))
              (a.fst.le_op_norm_of_le hy))
          _ ≤ ‖a‖₊ * ‖a‖₊ := by simp only [mul_one, nnnorm_fst, le_rfl]
      rw [← nnnorm_snd]
      simp only [mul_snd, ← sSup_closed_unit_ball_eq_nnnorm, star_snd, mul_apply]
      simp only [← @op_nnnorm_mul_apply 𝕜 _ A]
      simp only [← sSup_closed_unit_ball_eq_nnnorm, mul_apply']
      

Proof state:
𝕜 : Type u_1
A : Type u_2
inst✝⁸ : DenselyNormedField 𝕜
inst✝⁷ : StarRing 𝕜
inst✝⁶ : NonUnitalNormedRing A
inst✝⁵ : StarRing A
inst✝⁴ : CstarRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : SMulCommClass 𝕜 A A
inst✝¹ : IsScalarTower 𝕜 A A
inst✝ : StarModule 𝕜 A
a : 𝓜(𝕜, A)
hball : Set.Nonempty (Metric.closedBall 0 1)
key : ∀ (x y : A), ‖x‖₊ ≤ 1 → ‖y‖₊ ≤ 1 → ‖a.toProd.2 (star (a.toProd.1 (star x))) * y‖₊ ≤ ‖a‖₊ * ‖a‖₊
⊢ sSup
      ((fun a_1 => sSup ((fun a_2 => ‖a.toProd.2 (star (a.toProd.1 (star a_1))) * a_2‖₊) '' Metric.closedBall 0 1)) ''
        Metric.closedBall 0 1) =
    ‖a‖₊ * ‖a‖₊

Next tactic:
refine' csSup_eq_of_forall_le_of_forall_lt_exists_gt (hball.image _) _ fun r hr => _
instance instCstarRing : CstarRing 𝓜(𝕜, A) where
  norm_star_mul_self := @fun (a : 𝓜(𝕜, A)) => congr_arg ((↑) : ℝ≥0 → ℝ) <|
    show ‖star a * a‖₊ = ‖a‖₊ * ‖a‖₊ by
    /- The essence of the argument is this: let `a = (L,R)` and recall `‖a‖ = ‖L‖`.
    `star a = (star ∘ R ∘ star, star ∘ L ∘ star)`. Then for any `x y : A`, we have
    `‖star a * a‖ = ‖(star a * a).snd‖ = ‖R (star (L (star x))) * y‖ = ‖star (L (star x)) * L y‖`
    Now, on the one hand,
    `‖star (L (star x)) * L y‖ ≤ ‖star (L (star x))‖ * ‖L y‖ = ‖L (star x)‖ * ‖L y‖ ≤ ‖L‖ ^ 2`
    whenever `‖x‖, ‖y‖ ≤ 1`, so the supremum over all such `x, y` is at most `‖L‖ ^ 2`.
    On the other hand, for any `‖z‖ ≤ 1`, we may choose `x := star z` and `y := z` to get:
    `‖star (L (star x)) * L y‖ = ‖star (L z) * (L z)‖ = ‖L z‖ ^ 2`, and taking the supremum over
    all such `z` yields that the supremum is at least `‖L‖ ^ 2`. It is the latter part of the
    argument where `DenselyNormedField 𝕜` is required (for `sSup_closed_unit_ball_eq_nnnorm`). -/
      have hball : (Metric.closedBall (0 : A) 1).Nonempty :=
        Metric.nonempty_closedBall.2 zero_le_one
      have key :
        ∀ x y, ‖x‖₊ ≤ 1 → ‖y‖₊ ≤ 1 → ‖a.snd (star (a.fst (star x))) * y‖₊ ≤ ‖a‖₊ * ‖a‖₊ := by
        intro x y hx hy
        rw [a.central]
        calc
          ‖star (a.fst (star x)) * a.fst y‖₊ ≤ ‖a.fst (star x)‖₊ * ‖a.fst y‖₊ :=
            nnnorm_star (a.fst (star x)) ▸ nnnorm_mul_le _ _
          _ ≤ ‖a.fst‖₊ * 1 * (‖a.fst‖₊ * 1) :=
            (mul_le_mul' (a.fst.le_op_norm_of_le ((nnnorm_star x).trans_le hx))
              (a.fst.le_op_norm_of_le hy))
          _ ≤ ‖a‖₊ * ‖a‖₊ := by simp only [mul_one, nnnorm_fst, le_rfl]
      rw [← nnnorm_snd]
      simp only [mul_snd, ← sSup_closed_unit_ball_eq_nnnorm, star_snd, mul_apply]
      simp only [← @op_nnnorm_mul_apply 𝕜 _ A]
      simp only [← sSup_closed_unit_ball_eq_nnnorm, mul_apply']
      refine' csSup_eq_of_forall_le_of_forall_lt_exists_gt (hball.image _) _ fun r hr => _
      · 

Proof state:
case refine'_1
𝕜 : Type u_1
A : Type u_2
inst✝⁸ : DenselyNormedField 𝕜
inst✝⁷ : StarRing 𝕜
inst✝⁶ : NonUnitalNormedRing A
inst✝⁵ : StarRing A
inst✝⁴ : CstarRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : SMulCommClass 𝕜 A A
inst✝¹ : IsScalarTower 𝕜 A A
inst✝ : StarModule 𝕜 A
a : 𝓜(𝕜, A)
hball : Set.Nonempty (Metric.closedBall 0 1)
key : ∀ (x y : A), ‖x‖₊ ≤ 1 → ‖y‖₊ ≤ 1 → ‖a.toProd.2 (star (a.toProd.1 (star x))) * y‖₊ ≤ ‖a‖₊ * ‖a‖₊
⊢ ∀
    a_1 ∈
      (fun a_2 => sSup ((fun a_3 => ‖a.toProd.2 (star (a.toProd.1 (star a_2))) * a_3‖₊) '' Metric.closedBall 0 1)) ''
        Metric.closedBall 0 1,
    a_1 ≤ ‖a‖₊ * ‖a‖₊

Next tactic:
rintro - ⟨x, hx, rfl⟩
instance instCstarRing : CstarRing 𝓜(𝕜, A) where
  norm_star_mul_self := @fun (a : 𝓜(𝕜, A)) => congr_arg ((↑) : ℝ≥0 → ℝ) <|
    show ‖star a * a‖₊ = ‖a‖₊ * ‖a‖₊ by
    /- The essence of the argument is this: let `a = (L,R)` and recall `‖a‖ = ‖L‖`.
    `star a = (star ∘ R ∘ star, star ∘ L ∘ star)`. Then for any `x y : A`, we have
    `‖star a * a‖ = ‖(star a * a).snd‖ = ‖R (star (L (star x))) * y‖ = ‖star (L (star x)) * L y‖`
    Now, on the one hand,
    `‖star (L (star x)) * L y‖ ≤ ‖star (L (star x))‖ * ‖L y‖ = ‖L (star x)‖ * ‖L y‖ ≤ ‖L‖ ^ 2`
    whenever `‖x‖, ‖y‖ ≤ 1`, so the supremum over all such `x, y` is at most `‖L‖ ^ 2`.
    On the other hand, for any `‖z‖ ≤ 1`, we may choose `x := star z` and `y := z` to get:
    `‖star (L (star x)) * L y‖ = ‖star (L z) * (L z)‖ = ‖L z‖ ^ 2`, and taking the supremum over
    all such `z` yields that the supremum is at least `‖L‖ ^ 2`. It is the latter part of the
    argument where `DenselyNormedField 𝕜` is required (for `sSup_closed_unit_ball_eq_nnnorm`). -/
      have hball : (Metric.closedBall (0 : A) 1).Nonempty :=
        Metric.nonempty_closedBall.2 zero_le_one
      have key :
        ∀ x y, ‖x‖₊ ≤ 1 → ‖y‖₊ ≤ 1 → ‖a.snd (star (a.fst (star x))) * y‖₊ ≤ ‖a‖₊ * ‖a‖₊ := by
        intro x y hx hy
        rw [a.central]
        calc
          ‖star (a.fst (star x)) * a.fst y‖₊ ≤ ‖a.fst (star x)‖₊ * ‖a.fst y‖₊ :=
            nnnorm_star (a.fst (star x)) ▸ nnnorm_mul_le _ _
          _ ≤ ‖a.fst‖₊ * 1 * (‖a.fst‖₊ * 1) :=
            (mul_le_mul' (a.fst.le_op_norm_of_le ((nnnorm_star x).trans_le hx))
              (a.fst.le_op_norm_of_le hy))
          _ ≤ ‖a‖₊ * ‖a‖₊ := by simp only [mul_one, nnnorm_fst, le_rfl]
      rw [← nnnorm_snd]
      simp only [mul_snd, ← sSup_closed_unit_ball_eq_nnnorm, star_snd, mul_apply]
      simp only [← @op_nnnorm_mul_apply 𝕜 _ A]
      simp only [← sSup_closed_unit_ball_eq_nnnorm, mul_apply']
      refine' csSup_eq_of_forall_le_of_forall_lt_exists_gt (hball.image _) _ fun r hr => _
      · rintro - ⟨x, hx, rfl⟩
        

Proof state:
case refine'_1.intro.intro
𝕜 : Type u_1
A : Type u_2
inst✝⁸ : DenselyNormedField 𝕜
inst✝⁷ : StarRing 𝕜
inst✝⁶ : NonUnitalNormedRing A
inst✝⁵ : StarRing A
inst✝⁴ : CstarRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : SMulCommClass 𝕜 A A
inst✝¹ : IsScalarTower 𝕜 A A
inst✝ : StarModule 𝕜 A
a : 𝓜(𝕜, A)
hball : Set.Nonempty (Metric.closedBall 0 1)
key : ∀ (x y : A), ‖x‖₊ ≤ 1 → ‖y‖₊ ≤ 1 → ‖a.toProd.2 (star (a.toProd.1 (star x))) * y‖₊ ≤ ‖a‖₊ * ‖a‖₊
x : A
hx : x ∈ Metric.closedBall 0 1
⊢ (fun a_1 => sSup ((fun a_2 => ‖a.toProd.2 (star (a.toProd.1 (star a_1))) * a_2‖₊) '' Metric.closedBall 0 1)) x ≤
    ‖a‖₊ * ‖a‖₊

Next tactic:
refine' csSup_le (hball.image _) _