l3lab/ntp-mathlib huggingface dataset:
theorem eventually_isUnit_resolvent (a : A) : ∀ᶠ z in cobounded 𝕜, IsUnit (resolvent a z) := by
  

Proof state:
𝕜 : Type u_1
A : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a : A
⊢ ∀ᶠ (z : 𝕜) in cobounded 𝕜, IsUnit (resolvent a z)

Next tactic:
rw [← comap_norm_atTop, atTop_basis.comap (‖·‖) |>.eventually_iff]
theorem eventually_isUnit_resolvent (a : A) : ∀ᶠ z in cobounded 𝕜, IsUnit (resolvent a z) := by
  rw [← comap_norm_atTop, atTop_basis.comap (‖·‖) |>.eventually_iff]
  

Proof state:
𝕜 : Type u_1
A : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a : A
⊢ ∃ i, True ∧ ∀ ⦃x : 𝕜⦄, x ∈ (fun x => ‖x‖) ⁻¹' Set.Ici i → IsUnit (resolvent a x)

Next tactic:
refine ⟨‖a‖ * ‖(1 : A)‖ + 1, by trivial, fun z hz ↦ ?_⟩
theorem eventually_isUnit_resolvent (a : A) : ∀ᶠ z in cobounded 𝕜, IsUnit (resolvent a z) := by
  rw [← comap_norm_atTop, atTop_basis.comap (‖·‖) |>.eventually_iff]
  refine ⟨‖a‖ * ‖(1 : A)‖ + 1, by 

Proof state:
𝕜 : Type u_1
A : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a : A
⊢ True

Next tactic:
trivial
theorem eventually_isUnit_resolvent (a : A) : ∀ᶠ z in cobounded 𝕜, IsUnit (resolvent a z) := by
  rw [← comap_norm_atTop, atTop_basis.comap (‖·‖) |>.eventually_iff]
  refine ⟨‖a‖ * ‖(1 : A)‖ + 1, by trivial, fun z hz ↦ ?_⟩
  

Proof state:
𝕜 : Type u_1
A : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a : A
z : 𝕜
hz : z ∈ (fun x => ‖x‖) ⁻¹' Set.Ici (‖a‖ * ‖1‖ + 1)
⊢ IsUnit (resolvent a z)

Next tactic:
exact isUnit_resolvent.mp <| mem_resolventSet_of_norm_lt_mul <| (lt_add_one (‖a‖ * _)).trans_le hz
theorem resolvent_isBigO_inv (a : A) : resolvent a =O[cobounded 𝕜] Inv.inv :=
  have h : (fun z ↦ resolvent (z⁻¹ • a) (1 : 𝕜)) =O[cobounded 𝕜] (fun _ ↦ (1 : ℝ)) := by
    

Proof state:
𝕜 : Type u_1
A : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a : A
⊢ (fun z => resolvent (z⁻¹ • a) 1) =O[cobounded 𝕜] fun x => 1

Next tactic:
simpa [Function.comp_def, resolvent] using
      (NormedRing.inverse_one_sub_norm (R := A)).comp_tendsto
        (by simpa using (tendsto_inv₀_cobounded (α := 𝕜)).smul_const a)
theorem resolvent_isBigO_inv (a : A) : resolvent a =O[cobounded 𝕜] Inv.inv :=
  have h : (fun z ↦ resolvent (z⁻¹ • a) (1 : 𝕜)) =O[cobounded 𝕜] (fun _ ↦ (1 : ℝ)) := by
    simpa [Function.comp_def, resolvent] using
      (NormedRing.inverse_one_sub_norm (R := A)).comp_tendsto
        (by 

Proof state:
𝕜 : Type u_1
A : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a : A
⊢ Tendsto (fun x => x⁻¹ • a) (cobounded 𝕜) (𝓝 0)

Next tactic:
simpa using (tendsto_inv₀_cobounded (α := 𝕜)).smul_const a
theorem resolvent_isBigO_inv (a : A) : resolvent a =O[cobounded 𝕜] Inv.inv :=
  have h : (fun z ↦ resolvent (z⁻¹ • a) (1 : 𝕜)) =O[cobounded 𝕜] (fun _ ↦ (1 : ℝ)) := by
    simpa [Function.comp_def, resolvent] using
      (NormedRing.inverse_one_sub_norm (R := A)).comp_tendsto
        (by simpa using (tendsto_inv₀_cobounded (α := 𝕜)).smul_const a)
  calc
    resolvent a =ᶠ[cobounded 𝕜] fun z ↦ z⁻¹ • resolvent (z⁻¹ • a) (1 : 𝕜) := by
      

Proof state:
𝕜 : Type u_1
A : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a : A
h : (fun z => resolvent (z⁻¹ • a) 1) =O[cobounded 𝕜] fun x => 1
⊢ resolvent a =ᶠ[cobounded 𝕜] fun z => z⁻¹ • resolvent (z⁻¹ • a) 1

Next tactic:
filter_upwards [isBounded_singleton (x := 0)] with z hz
theorem resolvent_isBigO_inv (a : A) : resolvent a =O[cobounded 𝕜] Inv.inv :=
  have h : (fun z ↦ resolvent (z⁻¹ • a) (1 : 𝕜)) =O[cobounded 𝕜] (fun _ ↦ (1 : ℝ)) := by
    simpa [Function.comp_def, resolvent] using
      (NormedRing.inverse_one_sub_norm (R := A)).comp_tendsto
        (by simpa using (tendsto_inv₀_cobounded (α := 𝕜)).smul_const a)
  calc
    resolvent a =ᶠ[cobounded 𝕜] fun z ↦ z⁻¹ • resolvent (z⁻¹ • a) (1 : 𝕜) := by
      filter_upwards [isBounded_singleton (x := 0)] with z hz
      

Proof state:
case h
𝕜 : Type u_1
A : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a : A
h : (fun z => resolvent (z⁻¹ • a) 1) =O[cobounded 𝕜] fun x => 1
z : 𝕜
hz : z ∈ {0}ᶜ
⊢ resolvent a z = z⁻¹ • resolvent (z⁻¹ • a) 1

Next tactic:
lift z to 𝕜ˣ using Ne.isUnit hz
theorem resolvent_isBigO_inv (a : A) : resolvent a =O[cobounded 𝕜] Inv.inv :=
  have h : (fun z ↦ resolvent (z⁻¹ • a) (1 : 𝕜)) =O[cobounded 𝕜] (fun _ ↦ (1 : ℝ)) := by
    simpa [Function.comp_def, resolvent] using
      (NormedRing.inverse_one_sub_norm (R := A)).comp_tendsto
        (by simpa using (tendsto_inv₀_cobounded (α := 𝕜)).smul_const a)
  calc
    resolvent a =ᶠ[cobounded 𝕜] fun z ↦ z⁻¹ • resolvent (z⁻¹ • a) (1 : 𝕜) := by
      filter_upwards [isBounded_singleton (x := 0)] with z hz
      lift z to 𝕜ˣ using Ne.isUnit hz
      

Proof state:
case h.intro
𝕜 : Type u_1
A : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a : A
h : (fun z => resolvent (z⁻¹ • a) 1) =O[cobounded 𝕜] fun x => 1
z : 𝕜ˣ
hz : ↑z ∈ {0}ᶜ
⊢ resolvent a ↑z = (↑z)⁻¹ • resolvent ((↑z)⁻¹ • a) 1

Next tactic:
simpa [Units.smul_def] using congr(z⁻¹ • $(units_smul_resolvent_self (r := z) (a := a)))
theorem resolvent_isBigO_inv (a : A) : resolvent a =O[cobounded 𝕜] Inv.inv :=
  have h : (fun z ↦ resolvent (z⁻¹ • a) (1 : 𝕜)) =O[cobounded 𝕜] (fun _ ↦ (1 : ℝ)) := by
    simpa [Function.comp_def, resolvent] using
      (NormedRing.inverse_one_sub_norm (R := A)).comp_tendsto
        (by simpa using (tendsto_inv₀_cobounded (α := 𝕜)).smul_const a)
  calc
    resolvent a =ᶠ[cobounded 𝕜] fun z ↦ z⁻¹ • resolvent (z⁻¹ • a) (1 : 𝕜) := by
      filter_upwards [isBounded_singleton (x := 0)] with z hz
      lift z to 𝕜ˣ using Ne.isUnit hz
      simpa [Units.smul_def] using congr(z⁻¹ • $(units_smul_resolvent_self (r := z) (a := a)))
    _ =O[cobounded 𝕜] (· ⁻¹) := IsBigO.of_norm_right <| by
      

Proof state:
𝕜 : Type u_1
A : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a : A
h : (fun z => resolvent (z⁻¹ • a) 1) =O[cobounded 𝕜] fun x => 1
⊢ (fun z => z⁻¹ • resolvent (z⁻¹ • a) 1) =O[cobounded 𝕜] fun x => ‖x⁻¹‖

Next tactic:
simpa using (isBigO_refl (· ⁻¹) (cobounded 𝕜)).norm_right.smul h
/-- In a Banach algebra `A` over a nontrivially normed field `𝕜`, for any `a : A` the
power series with coefficients `a ^ n` represents the function `(1 - z • a)⁻¹` in a disk of
radius `‖a‖₊⁻¹`. -/
theorem hasFPowerSeriesOnBall_inverse_one_sub_smul [CompleteSpace A] (a : A) :
    HasFPowerSeriesOnBall (fun z : 𝕜 => Ring.inverse (1 - z • a))
      (fun n => ContinuousMultilinearMap.mkPiField 𝕜 (Fin n) (a ^ n)) 0 ‖a‖₊⁻¹ :=
  { r_le := by
      

Proof state:
𝕜 : Type u_1
A : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a : A
⊢ (↑‖a‖₊)⁻¹ ≤ radius fun n => ContinuousMultilinearMap.mkPiField 𝕜 (Fin n) (a ^ n)

Next tactic:
refine'
        le_of_forall_nnreal_lt fun r hr => le_radius_of_bound_nnreal _ (max 1 ‖(1 : A)‖₊) fun n => _
/-- In a Banach algebra `A` over a nontrivially normed field `𝕜`, for any `a : A` the
power series with coefficients `a ^ n` represents the function `(1 - z • a)⁻¹` in a disk of
radius `‖a‖₊⁻¹`. -/
theorem hasFPowerSeriesOnBall_inverse_one_sub_smul [CompleteSpace A] (a : A) :
    HasFPowerSeriesOnBall (fun z : 𝕜 => Ring.inverse (1 - z • a))
      (fun n => ContinuousMultilinearMap.mkPiField 𝕜 (Fin n) (a ^ n)) 0 ‖a‖₊⁻¹ :=
  { r_le := by
      refine'
        le_of_forall_nnreal_lt fun r hr => le_radius_of_bound_nnreal _ (max 1 ‖(1 : A)‖₊) fun n => _
      

Proof state:
𝕜 : Type u_1
A : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a : A
r : ℝ≥0
hr : ↑r < (↑‖a‖₊)⁻¹
n : ℕ
⊢ ‖ContinuousMultilinearMap.mkPiField 𝕜 (Fin n) (a ^ n)‖₊ * r ^ n ≤ max 1 ‖1‖₊

Next tactic:
rw [← norm_toNNReal, norm_mkPiField, norm_toNNReal]
/-- In a Banach algebra `A` over a nontrivially normed field `𝕜`, for any `a : A` the
power series with coefficients `a ^ n` represents the function `(1 - z • a)⁻¹` in a disk of
radius `‖a‖₊⁻¹`. -/
theorem hasFPowerSeriesOnBall_inverse_one_sub_smul [CompleteSpace A] (a : A) :
    HasFPowerSeriesOnBall (fun z : 𝕜 => Ring.inverse (1 - z • a))
      (fun n => ContinuousMultilinearMap.mkPiField 𝕜 (Fin n) (a ^ n)) 0 ‖a‖₊⁻¹ :=
  { r_le := by
      refine'
        le_of_forall_nnreal_lt fun r hr => le_radius_of_bound_nnreal _ (max 1 ‖(1 : A)‖₊) fun n => _
      rw [← norm_toNNReal, norm_mkPiField, norm_toNNReal]
      

Proof state:
𝕜 : Type u_1
A : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a : A
r : ℝ≥0
hr : ↑r < (↑‖a‖₊)⁻¹
n : ℕ
⊢ ‖a ^ n‖₊ * r ^ n ≤ max 1 ‖1‖₊

Next tactic:
cases' n with n
/-- In a Banach algebra `A` over a nontrivially normed field `𝕜`, for any `a : A` the
power series with coefficients `a ^ n` represents the function `(1 - z • a)⁻¹` in a disk of
radius `‖a‖₊⁻¹`. -/
theorem hasFPowerSeriesOnBall_inverse_one_sub_smul [CompleteSpace A] (a : A) :
    HasFPowerSeriesOnBall (fun z : 𝕜 => Ring.inverse (1 - z • a))
      (fun n => ContinuousMultilinearMap.mkPiField 𝕜 (Fin n) (a ^ n)) 0 ‖a‖₊⁻¹ :=
  { r_le := by
      refine'
        le_of_forall_nnreal_lt fun r hr => le_radius_of_bound_nnreal _ (max 1 ‖(1 : A)‖₊) fun n => _
      rw [← norm_toNNReal, norm_mkPiField, norm_toNNReal]
      cases' n with n
      · 

Proof state:
case zero
𝕜 : Type u_1
A : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a : A
r : ℝ≥0
hr : ↑r < (↑‖a‖₊)⁻¹
⊢ ‖a ^ Nat.zero‖₊ * r ^ Nat.zero ≤ max 1 ‖1‖₊

Next tactic:
simp only [Nat.zero_eq, le_refl, mul_one, or_true_iff, le_max_iff, pow_zero]
/-- In a Banach algebra `A` over a nontrivially normed field `𝕜`, for any `a : A` the
power series with coefficients `a ^ n` represents the function `(1 - z • a)⁻¹` in a disk of
radius `‖a‖₊⁻¹`. -/
theorem hasFPowerSeriesOnBall_inverse_one_sub_smul [CompleteSpace A] (a : A) :
    HasFPowerSeriesOnBall (fun z : 𝕜 => Ring.inverse (1 - z • a))
      (fun n => ContinuousMultilinearMap.mkPiField 𝕜 (Fin n) (a ^ n)) 0 ‖a‖₊⁻¹ :=
  { r_le := by
      refine'
        le_of_forall_nnreal_lt fun r hr => le_radius_of_bound_nnreal _ (max 1 ‖(1 : A)‖₊) fun n => _
      rw [← norm_toNNReal, norm_mkPiField, norm_toNNReal]
      cases' n with n
      · simp only [Nat.zero_eq, le_refl, mul_one, or_true_iff, le_max_iff, pow_zero]
      · 

Proof state:
case succ
𝕜 : Type u_1
A : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a : A
r : ℝ≥0
hr : ↑r < (↑‖a‖₊)⁻¹
n : ℕ
⊢ ‖a ^ Nat.succ n‖₊ * r ^ Nat.succ n ≤ max 1 ‖1‖₊

Next tactic:
refine'
          le_trans (le_trans (mul_le_mul_right' (nnnorm_pow_le' a n.succ_pos) (r ^ n.succ)) _)
            (le_max_left _ _)
/-- In a Banach algebra `A` over a nontrivially normed field `𝕜`, for any `a : A` the
power series with coefficients `a ^ n` represents the function `(1 - z • a)⁻¹` in a disk of
radius `‖a‖₊⁻¹`. -/
theorem hasFPowerSeriesOnBall_inverse_one_sub_smul [CompleteSpace A] (a : A) :
    HasFPowerSeriesOnBall (fun z : 𝕜 => Ring.inverse (1 - z • a))
      (fun n => ContinuousMultilinearMap.mkPiField 𝕜 (Fin n) (a ^ n)) 0 ‖a‖₊⁻¹ :=
  { r_le := by
      refine'
        le_of_forall_nnreal_lt fun r hr => le_radius_of_bound_nnreal _ (max 1 ‖(1 : A)‖₊) fun n => _
      rw [← norm_toNNReal, norm_mkPiField, norm_toNNReal]
      cases' n with n
      · simp only [Nat.zero_eq, le_refl, mul_one, or_true_iff, le_max_iff, pow_zero]
      · refine'
          le_trans (le_trans (mul_le_mul_right' (nnnorm_pow_le' a n.succ_pos) (r ^ n.succ)) _)
            (le_max_left _ _)
        · 

Proof state:
case succ
𝕜 : Type u_1
A : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a : A
r : ℝ≥0
hr : ↑r < (↑‖a‖₊)⁻¹
n : ℕ
⊢ ‖a‖₊ ^ Nat.succ n * r ^ Nat.succ n ≤ 1

Next tactic:
by_cases h : ‖a‖₊ = 0
/-- In a Banach algebra `A` over a nontrivially normed field `𝕜`, for any `a : A` the
power series with coefficients `a ^ n` represents the function `(1 - z • a)⁻¹` in a disk of
radius `‖a‖₊⁻¹`. -/
theorem hasFPowerSeriesOnBall_inverse_one_sub_smul [CompleteSpace A] (a : A) :
    HasFPowerSeriesOnBall (fun z : 𝕜 => Ring.inverse (1 - z • a))
      (fun n => ContinuousMultilinearMap.mkPiField 𝕜 (Fin n) (a ^ n)) 0 ‖a‖₊⁻¹ :=
  { r_le := by
      refine'
        le_of_forall_nnreal_lt fun r hr => le_radius_of_bound_nnreal _ (max 1 ‖(1 : A)‖₊) fun n => _
      rw [← norm_toNNReal, norm_mkPiField, norm_toNNReal]
      cases' n with n
      · simp only [Nat.zero_eq, le_refl, mul_one, or_true_iff, le_max_iff, pow_zero]
      · refine'
          le_trans (le_trans (mul_le_mul_right' (nnnorm_pow_le' a n.succ_pos) (r ^ n.succ)) _)
            (le_max_left _ _)
        · by_cases h : ‖a‖₊ = 0
          · 

Proof state:
case pos
𝕜 : Type u_1
A : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a : A
r : ℝ≥0
hr : ↑r < (↑‖a‖₊)⁻¹
n : ℕ
h : ‖a‖₊ = 0
⊢ ‖a‖₊ ^ Nat.succ n * r ^ Nat.succ n ≤ 1

Next tactic:
simp only [h, zero_mul, zero_le', pow_succ]
/-- In a Banach algebra `A` over a nontrivially normed field `𝕜`, for any `a : A` the
power series with coefficients `a ^ n` represents the function `(1 - z • a)⁻¹` in a disk of
radius `‖a‖₊⁻¹`. -/
theorem hasFPowerSeriesOnBall_inverse_one_sub_smul [CompleteSpace A] (a : A) :
    HasFPowerSeriesOnBall (fun z : 𝕜 => Ring.inverse (1 - z • a))
      (fun n => ContinuousMultilinearMap.mkPiField 𝕜 (Fin n) (a ^ n)) 0 ‖a‖₊⁻¹ :=
  { r_le := by
      refine'
        le_of_forall_nnreal_lt fun r hr => le_radius_of_bound_nnreal _ (max 1 ‖(1 : A)‖₊) fun n => _
      rw [← norm_toNNReal, norm_mkPiField, norm_toNNReal]
      cases' n with n
      · simp only [Nat.zero_eq, le_refl, mul_one, or_true_iff, le_max_iff, pow_zero]
      · refine'
          le_trans (le_trans (mul_le_mul_right' (nnnorm_pow_le' a n.succ_pos) (r ^ n.succ)) _)
            (le_max_left _ _)
        · by_cases h : ‖a‖₊ = 0
          · simp only [h, zero_mul, zero_le', pow_succ]
          · 

Proof state:
case neg
𝕜 : Type u_1
A : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a : A
r : ℝ≥0
hr : ↑r < (↑‖a‖₊)⁻¹
n : ℕ
h : ¬‖a‖₊ = 0
⊢ ‖a‖₊ ^ Nat.succ n * r ^ Nat.succ n ≤ 1

Next tactic:
rw [← coe_inv h, coe_lt_coe, NNReal.lt_inv_iff_mul_lt h] at hr
/-- In a Banach algebra `A` over a nontrivially normed field `𝕜`, for any `a : A` the
power series with coefficients `a ^ n` represents the function `(1 - z • a)⁻¹` in a disk of
radius `‖a‖₊⁻¹`. -/
theorem hasFPowerSeriesOnBall_inverse_one_sub_smul [CompleteSpace A] (a : A) :
    HasFPowerSeriesOnBall (fun z : 𝕜 => Ring.inverse (1 - z • a))
      (fun n => ContinuousMultilinearMap.mkPiField 𝕜 (Fin n) (a ^ n)) 0 ‖a‖₊⁻¹ :=
  { r_le := by
      refine'
        le_of_forall_nnreal_lt fun r hr => le_radius_of_bound_nnreal _ (max 1 ‖(1 : A)‖₊) fun n => _
      rw [← norm_toNNReal, norm_mkPiField, norm_toNNReal]
      cases' n with n
      · simp only [Nat.zero_eq, le_refl, mul_one, or_true_iff, le_max_iff, pow_zero]
      · refine'
          le_trans (le_trans (mul_le_mul_right' (nnnorm_pow_le' a n.succ_pos) (r ^ n.succ)) _)
            (le_max_left _ _)
        · by_cases h : ‖a‖₊ = 0
          · simp only [h, zero_mul, zero_le', pow_succ]
          · rw [← coe_inv h, coe_lt_coe, NNReal.lt_inv_iff_mul_lt h] at hr
            

Proof state:
case neg
𝕜 : Type u_1
A : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a : A
r : ℝ≥0
hr : r * ‖a‖₊ < 1
n : ℕ
h : ¬‖a‖₊ = 0
⊢ ‖a‖₊ ^ Nat.succ n * r ^ Nat.succ n ≤ 1

Next tactic:
simpa only [← mul_pow, mul_comm] using pow_le_one' hr.le n.succ
/-- In a Banach algebra `A` over a nontrivially normed field `𝕜`, for any `a : A` the
power series with coefficients `a ^ n` represents the function `(1 - z • a)⁻¹` in a disk of
radius `‖a‖₊⁻¹`. -/
theorem hasFPowerSeriesOnBall_inverse_one_sub_smul [CompleteSpace A] (a : A) :
    HasFPowerSeriesOnBall (fun z : 𝕜 => Ring.inverse (1 - z • a))
      (fun n => ContinuousMultilinearMap.mkPiField 𝕜 (Fin n) (a ^ n)) 0 ‖a‖₊⁻¹ :=
  { r_le := by
      refine'
        le_of_forall_nnreal_lt fun r hr => le_radius_of_bound_nnreal _ (max 1 ‖(1 : A)‖₊) fun n => _
      rw [← norm_toNNReal, norm_mkPiField, norm_toNNReal]
      cases' n with n
      · simp only [Nat.zero_eq, le_refl, mul_one, or_true_iff, le_max_iff, pow_zero]
      · refine'
          le_trans (le_trans (mul_le_mul_right' (nnnorm_pow_le' a n.succ_pos) (r ^ n.succ)) _)
            (le_max_left _ _)
        · by_cases h : ‖a‖₊ = 0
          · simp only [h, zero_mul, zero_le', pow_succ]
          · rw [← coe_inv h, coe_lt_coe, NNReal.lt_inv_iff_mul_lt h] at hr
            simpa only [← mul_pow, mul_comm] using pow_le_one' hr.le n.succ
    r_pos := ENNReal.inv_pos.mpr coe_ne_top
    hasSum := fun {y} hy => by
      

Proof state:
𝕜 : Type u_1
A : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a : A
y : 𝕜
hy : y ∈ EMetric.ball 0 (↑‖a‖₊)⁻¹
⊢ HasSum (fun n => (ContinuousMultilinearMap.mkPiField 𝕜 (Fin n) (a ^ n)) fun x => y) (Ring.inverse (1 - (0 + y) • a))

Next tactic:
have norm_lt : ‖y • a‖ < 1 := by
        by_cases h : ‖a‖₊ = 0
        · simp only [nnnorm_eq_zero.mp h, norm_zero, zero_lt_one, smul_zero]
        · have nnnorm_lt : ‖y‖₊ < ‖a‖₊⁻¹ := by
            simpa only [← coe_inv h, mem_ball_zero_iff, Metric.emetric_ball_nnreal] using hy
          rwa [← coe_nnnorm, ← Real.lt_toNNReal_iff_coe_lt, Real.toNNReal_one, nnnorm_smul,
            ← NNReal.lt_inv_iff_mul_lt h]
/-- In a Banach algebra `A` over a nontrivially normed field `𝕜`, for any `a : A` the
power series with coefficients `a ^ n` represents the function `(1 - z • a)⁻¹` in a disk of
radius `‖a‖₊⁻¹`. -/
theorem hasFPowerSeriesOnBall_inverse_one_sub_smul [CompleteSpace A] (a : A) :
    HasFPowerSeriesOnBall (fun z : 𝕜 => Ring.inverse (1 - z • a))
      (fun n => ContinuousMultilinearMap.mkPiField 𝕜 (Fin n) (a ^ n)) 0 ‖a‖₊⁻¹ :=
  { r_le := by
      refine'
        le_of_forall_nnreal_lt fun r hr => le_radius_of_bound_nnreal _ (max 1 ‖(1 : A)‖₊) fun n => _
      rw [← norm_toNNReal, norm_mkPiField, norm_toNNReal]
      cases' n with n
      · simp only [Nat.zero_eq, le_refl, mul_one, or_true_iff, le_max_iff, pow_zero]
      · refine'
          le_trans (le_trans (mul_le_mul_right' (nnnorm_pow_le' a n.succ_pos) (r ^ n.succ)) _)
            (le_max_left _ _)
        · by_cases h : ‖a‖₊ = 0
          · simp only [h, zero_mul, zero_le', pow_succ]
          · rw [← coe_inv h, coe_lt_coe, NNReal.lt_inv_iff_mul_lt h] at hr
            simpa only [← mul_pow, mul_comm] using pow_le_one' hr.le n.succ
    r_pos := ENNReal.inv_pos.mpr coe_ne_top
    hasSum := fun {y} hy => by
      have norm_lt : ‖y • a‖ < 1 := by
        

Proof state:
𝕜 : Type u_1
A : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a : A
y : 𝕜
hy : y ∈ EMetric.ball 0 (↑‖a‖₊)⁻¹
⊢ ‖y • a‖ < 1

Next tactic:
by_cases h : ‖a‖₊ = 0
/-- In a Banach algebra `A` over a nontrivially normed field `𝕜`, for any `a : A` the
power series with coefficients `a ^ n` represents the function `(1 - z • a)⁻¹` in a disk of
radius `‖a‖₊⁻¹`. -/
theorem hasFPowerSeriesOnBall_inverse_one_sub_smul [CompleteSpace A] (a : A) :
    HasFPowerSeriesOnBall (fun z : 𝕜 => Ring.inverse (1 - z • a))
      (fun n => ContinuousMultilinearMap.mkPiField 𝕜 (Fin n) (a ^ n)) 0 ‖a‖₊⁻¹ :=
  { r_le := by
      refine'
        le_of_forall_nnreal_lt fun r hr => le_radius_of_bound_nnreal _ (max 1 ‖(1 : A)‖₊) fun n => _
      rw [← norm_toNNReal, norm_mkPiField, norm_toNNReal]
      cases' n with n
      · simp only [Nat.zero_eq, le_refl, mul_one, or_true_iff, le_max_iff, pow_zero]
      · refine'
          le_trans (le_trans (mul_le_mul_right' (nnnorm_pow_le' a n.succ_pos) (r ^ n.succ)) _)
            (le_max_left _ _)
        · by_cases h : ‖a‖₊ = 0
          · simp only [h, zero_mul, zero_le', pow_succ]
          · rw [← coe_inv h, coe_lt_coe, NNReal.lt_inv_iff_mul_lt h] at hr
            simpa only [← mul_pow, mul_comm] using pow_le_one' hr.le n.succ
    r_pos := ENNReal.inv_pos.mpr coe_ne_top
    hasSum := fun {y} hy => by
      have norm_lt : ‖y • a‖ < 1 := by
        by_cases h : ‖a‖₊ = 0
        · 

Proof state:
case pos
𝕜 : Type u_1
A : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a : A
y : 𝕜
hy : y ∈ EMetric.ball 0 (↑‖a‖₊)⁻¹
h : ‖a‖₊ = 0
⊢ ‖y • a‖ < 1

Next tactic:
simp only [nnnorm_eq_zero.mp h, norm_zero, zero_lt_one, smul_zero]
/-- In a Banach algebra `A` over a nontrivially normed field `𝕜`, for any `a : A` the
power series with coefficients `a ^ n` represents the function `(1 - z • a)⁻¹` in a disk of
radius `‖a‖₊⁻¹`. -/
theorem hasFPowerSeriesOnBall_inverse_one_sub_smul [CompleteSpace A] (a : A) :
    HasFPowerSeriesOnBall (fun z : 𝕜 => Ring.inverse (1 - z • a))
      (fun n => ContinuousMultilinearMap.mkPiField 𝕜 (Fin n) (a ^ n)) 0 ‖a‖₊⁻¹ :=
  { r_le := by
      refine'
        le_of_forall_nnreal_lt fun r hr => le_radius_of_bound_nnreal _ (max 1 ‖(1 : A)‖₊) fun n => _
      rw [← norm_toNNReal, norm_mkPiField, norm_toNNReal]
      cases' n with n
      · simp only [Nat.zero_eq, le_refl, mul_one, or_true_iff, le_max_iff, pow_zero]
      · refine'
          le_trans (le_trans (mul_le_mul_right' (nnnorm_pow_le' a n.succ_pos) (r ^ n.succ)) _)
            (le_max_left _ _)
        · by_cases h : ‖a‖₊ = 0
          · simp only [h, zero_mul, zero_le', pow_succ]
          · rw [← coe_inv h, coe_lt_coe, NNReal.lt_inv_iff_mul_lt h] at hr
            simpa only [← mul_pow, mul_comm] using pow_le_one' hr.le n.succ
    r_pos := ENNReal.inv_pos.mpr coe_ne_top
    hasSum := fun {y} hy => by
      have norm_lt : ‖y • a‖ < 1 := by
        by_cases h : ‖a‖₊ = 0
        · simp only [nnnorm_eq_zero.mp h, norm_zero, zero_lt_one, smul_zero]
        · 

Proof state:
case neg
𝕜 : Type u_1
A : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : CompleteSpace A
a : A
y : 𝕜
hy : y ∈ EMetric.ball 0 (↑‖a‖₊)⁻¹
h : ¬‖a‖₊ = 0
⊢ ‖y • a‖ < 1

Next tactic:
have nnnorm_lt : ‖y‖₊ < ‖a‖₊⁻¹ := by
            simpa only [← coe_inv h, mem_ball_zero_iff, Metric.emetric_ball_nnreal] using hy