Theorem: hasFDerivAt_exp_smul_const_of_mem_ball
File: Mathlib/Analysis/SpecialFunctions/Exponential.lean

Proof steps:
state_before: 𝕂 : Type u_1
𝕊 : Type u_2
𝔸 : Type u_3
inst✝⁹ : NontriviallyNormedField 𝕂
inst✝⁸ : CharZero 𝕂
inst✝⁷ : NormedCommRing 𝕊
inst✝⁶ : NormedRing 𝔸
inst✝⁵ : NormedSpace 𝕂 𝕊
inst✝⁴ : NormedAlgebra 𝕂 𝔸
inst✝³ : Algebra 𝕊 𝔸
inst✝² : ContinuousSMul 𝕊 𝔸
inst✝¹ : IsScalarTower 𝕂 𝕊 𝔸
inst✝ : CompleteSpace 𝔸
x : 𝔸
t : 𝕊
htx : t • x ∈ EMetric.ball 0 (expSeries 𝕂 𝔸).radius
⊢ HasFDerivAt (fun u => exp 𝕂 (u • x)) (exp 𝕂 (t • x) • ContinuousLinearMap.smulRight 1 x) t
state_after: 𝕂 : Type u_1
𝕊 : Type u_2
𝔸 : Type u_3
inst✝⁹ : NontriviallyNormedField 𝕂
inst✝⁸ : CharZero 𝕂
inst✝⁷ : NormedCommRing 𝕊
inst✝⁶ : NormedRing 𝔸
inst✝⁵ : NormedSpace 𝕂 𝕊
inst✝⁴ : NormedAlgebra 𝕂 𝔸
inst✝³ : Algebra 𝕊 𝔸
inst✝² : ContinuousSMul 𝕊 𝔸
inst✝¹ : IsScalarTower 𝕂 𝕊 𝔸
inst✝ : CompleteSpace 𝔸
x : 𝔸
t : 𝕊
htx : t • x ∈ EMetric.ball 0 (expSeries 𝕂 𝔸).radius
hpos : 0 < (expSeries 𝕂 𝔸).radius
⊢ HasFDerivAt (fun u => exp 𝕂 (u • x)) (exp 𝕂 (t • x) • ContinuousLinearMap.smulRight 1 x) t
tactic: have hpos : 0 < (expSeries 𝕂 𝔸).radius := (zero_le _).trans_lt htx
state_before: 𝕂 : Type u_1
𝕊 : Type u_2
𝔸 : Type u_3
inst✝⁹ : NontriviallyNormedField 𝕂
inst✝⁸ : CharZero 𝕂
inst✝⁷ : NormedCommRing 𝕊
inst✝⁶ : NormedRing 𝔸
inst✝⁵ : NormedSpace 𝕂 𝕊
inst✝⁴ : NormedAlgebra 𝕂 𝔸
inst✝³ : Algebra 𝕊 𝔸
inst✝² : ContinuousSMul 𝕊 𝔸
inst✝¹ : IsScalarTower 𝕂 𝕊 𝔸
inst✝ : CompleteSpace 𝔸
x : 𝔸
t : 𝕊
htx : t • x ∈ EMetric.ball 0 (expSeries 𝕂 𝔸).radius
hpos : 0 < (expSeries 𝕂 𝔸).radius
⊢ HasFDerivAt (fun u => exp 𝕂 (u • x)) (exp 𝕂 (t • x) • ContinuousLinearMap.smulRight 1 x) t
state_after: 𝕂 : Type u_1
𝕊 : Type u_2
𝔸 : Type u_3
inst✝⁹ : NontriviallyNormedField 𝕂
inst✝⁸ : CharZero 𝕂
inst✝⁷ : NormedCommRing 𝕊
inst✝⁶ : NormedRing 𝔸
inst✝⁵ : NormedSpace 𝕂 𝕊
inst✝⁴ : NormedAlgebra 𝕂 𝔸
inst✝³ : Algebra 𝕊 𝔸
inst✝² : ContinuousSMul 𝕊 𝔸
inst✝¹ : IsScalarTower 𝕂 𝕊 𝔸
inst✝ : CompleteSpace 𝔸
x : 𝔸
t : 𝕊
htx : t • x ∈ EMetric.ball 0 (expSeries 𝕂 𝔸).radius
hpos : 0 < (expSeries 𝕂 𝔸).radius
⊢ (fun h => exp 𝕂 ((t + h) • x) - exp 𝕂 (t • x) - (exp 𝕂 (t • x) • ContinuousLinearMap.smulRight 1 x) h) =o[𝓝 0]
    fun h => h
tactic: rw [hasFDerivAt_iff_isLittleO_nhds_zero]
state_before: 𝕂 : Type u_1
𝕊 : Type u_2
𝔸 : Type u_3
inst✝⁹ : NontriviallyNormedField 𝕂
inst✝⁸ : CharZero 𝕂
inst✝⁷ : NormedCommRing 𝕊
inst✝⁶ : NormedRing 𝔸
inst✝⁵ : NormedSpace 𝕂 𝕊
inst✝⁴ : NormedAlgebra 𝕂 𝔸
inst✝³ : Algebra 𝕊 𝔸
inst✝² : ContinuousSMul 𝕊 𝔸
inst✝¹ : IsScalarTower 𝕂 𝕊 𝔸
inst✝ : CompleteSpace 𝔸
x : 𝔸
t : 𝕊
htx : t • x ∈ EMetric.ball 0 (expSeries 𝕂 𝔸).radius
hpos : 0 < (expSeries 𝕂 𝔸).radius
⊢ (fun h => exp 𝕂 ((t + h) • x) - exp 𝕂 (t • x) - (exp 𝕂 (t • x) • ContinuousLinearMap.smulRight 1 x) h) =o[𝓝 0]
    fun h => h
state_after: 𝕂 : Type u_1
𝕊 : Type u_2
𝔸 : Type u_3
inst✝⁹ : NontriviallyNormedField 𝕂
inst✝⁸ : CharZero 𝕂
inst✝⁷ : NormedCommRing 𝕊
inst✝⁶ : NormedRing 𝔸
inst✝⁵ : NormedSpace 𝕂 𝕊
inst✝⁴ : NormedAlgebra 𝕂 𝔸
inst✝³ : Algebra 𝕊 𝔸
inst✝² : ContinuousSMul 𝕊 𝔸
inst✝¹ : IsScalarTower 𝕂 𝕊 𝔸
inst✝ : CompleteSpace 𝔸
x : 𝔸
t : 𝕊
htx : t • x ∈ EMetric.ball 0 (expSeries 𝕂 𝔸).radius
hpos : 0 < (expSeries 𝕂 𝔸).radius
⊢ (fun h => exp 𝕂 (t • x) * (exp 𝕂 ((0 + h) • x) - exp 𝕂 (0 • x) - (ContinuousLinearMap.smulRight 1 x) h)) =ᶠ[𝓝 0]
    fun h => exp 𝕂 ((t + h) • x) - exp 𝕂 (t • x) - (exp 𝕂 (t • x) • ContinuousLinearMap.smulRight 1 x) h
tactic: suffices (fun (h : 𝕊) => exp 𝕂 (t • x) *
      (exp 𝕂 ((0 + h) • x) - exp 𝕂 ((0 : 𝕊) • x) - ((1 : 𝕊 →L[𝕂] 𝕊).smulRight x) h)) =ᶠ[𝓝 0]
        fun h =>
          exp 𝕂 ((t + h) • x) - exp 𝕂 (t • x) - (exp 𝕂 (t • x) • (1 : 𝕊 →L[𝕂] 𝕊).smulRight x) h by
    apply (IsLittleO.const_mul_left _ _).congr' this (EventuallyEq.refl _ _)
    rw [← hasFDerivAt_iff_isLittleO_nhds_zero (f := fun u => exp 𝕂 (u • x))
      (f' := (1 : 𝕊 →L[𝕂] 𝕊).smulRight x) (x := 0)]
    have : HasFDerivAt (exp 𝕂) (1 : 𝔸 →L[𝕂] 𝔸) ((1 : 𝕊 →L[𝕂] 𝕊).smulRight x 0) := by
      rw [ContinuousLinearMap.smulRight_apply, ContinuousLinearMap.one_apply, zero_smul]
      exact hasFDerivAt_exp_zero_of_radius_pos hpos
    exact this.comp 0 ((1 : 𝕊 →L[𝕂] 𝕊).smulRight x).hasFDerivAt
state_before: 𝕂 : Type u_1
𝕊 : Type u_2
𝔸 : Type u_3
inst✝⁹ : NontriviallyNormedField 𝕂
inst✝⁸ : CharZero 𝕂
inst✝⁷ : NormedCommRing 𝕊
inst✝⁶ : NormedRing 𝔸
inst✝⁵ : NormedSpace 𝕂 𝕊
inst✝⁴ : NormedAlgebra 𝕂 𝔸
inst✝³ : Algebra 𝕊 𝔸
inst✝² : ContinuousSMul 𝕊 𝔸
inst✝¹ : IsScalarTower 𝕂 𝕊 𝔸
inst✝ : CompleteSpace 𝔸
x : 𝔸
t : 𝕊
htx : t • x ∈ EMetric.ball 0 (expSeries 𝕂 𝔸).radius
hpos : 0 < (expSeries 𝕂 𝔸).radius
⊢ (fun h => exp 𝕂 (t • x) * (exp 𝕂 ((0 + h) • x) - exp 𝕂 (0 • x) - (ContinuousLinearMap.smulRight 1 x) h)) =ᶠ[𝓝 0]
    fun h => exp 𝕂 ((t + h) • x) - exp 𝕂 (t • x) - (exp 𝕂 (t • x) • ContinuousLinearMap.smulRight 1 x) h
state_after: 𝕂 : Type u_1
𝕊 : Type u_2
𝔸 : Type u_3
inst✝⁹ : NontriviallyNormedField 𝕂
inst✝⁸ : CharZero 𝕂
inst✝⁷ : NormedCommRing 𝕊
inst✝⁶ : NormedRing 𝔸
inst✝⁵ : NormedSpace 𝕂 𝕊
inst✝⁴ : NormedAlgebra 𝕂 𝔸
inst✝³ : Algebra 𝕊 𝔸
inst✝² : ContinuousSMul 𝕊 𝔸
inst✝¹ : IsScalarTower 𝕂 𝕊 𝔸
inst✝ : CompleteSpace 𝔸
x : 𝔸
t : 𝕊
htx : t • x ∈ EMetric.ball 0 (expSeries 𝕂 𝔸).radius
hpos : 0 < (expSeries 𝕂 𝔸).radius
this : Tendsto (fun h => h • x) (𝓝 0) (𝓝 0)
⊢ (fun h => exp 𝕂 (t • x) * (exp 𝕂 ((0 + h) • x) - exp 𝕂 (0 • x) - (ContinuousLinearMap.smulRight 1 x) h)) =ᶠ[𝓝 0]
    fun h => exp 𝕂 ((t + h) • x) - exp 𝕂 (t • x) - (exp 𝕂 (t • x) • ContinuousLinearMap.smulRight 1 x) h
tactic: have : Tendsto (fun h : 𝕊 => h • x) (𝓝 0) (𝓝 0) := by
    rw [← zero_smul 𝕊 x]
    exact tendsto_id.smul_const x
state_before: 𝕂 : Type u_1
𝕊 : Type u_2
𝔸 : Type u_3
inst✝⁹ : NontriviallyNormedField 𝕂
inst✝⁸ : CharZero 𝕂
inst✝⁷ : NormedCommRing 𝕊
inst✝⁶ : NormedRing 𝔸
inst✝⁵ : NormedSpace 𝕂 𝕊
inst✝⁴ : NormedAlgebra 𝕂 𝔸
inst✝³ : Algebra 𝕊 𝔸
inst✝² : ContinuousSMul 𝕊 𝔸
inst✝¹ : IsScalarTower 𝕂 𝕊 𝔸
inst✝ : CompleteSpace 𝔸
x : 𝔸
t : 𝕊
htx : t • x ∈ EMetric.ball 0 (expSeries 𝕂 𝔸).radius
hpos : 0 < (expSeries 𝕂 𝔸).radius
this : Tendsto (fun h => h • x) (𝓝 0) (𝓝 0)
⊢ (fun h => exp 𝕂 (t • x) * (exp 𝕂 ((0 + h) • x) - exp 𝕂 (0 • x) - (ContinuousLinearMap.smulRight 1 x) h)) =ᶠ[𝓝 0]
    fun h => exp 𝕂 ((t + h) • x) - exp 𝕂 (t • x) - (exp 𝕂 (t • x) • ContinuousLinearMap.smulRight 1 x) h
state_after: 𝕂 : Type u_1
𝕊 : Type u_2
𝔸 : Type u_3
inst✝⁹ : NontriviallyNormedField 𝕂
inst✝⁸ : CharZero 𝕂
inst✝⁷ : NormedCommRing 𝕊
inst✝⁶ : NormedRing 𝔸
inst✝⁵ : NormedSpace 𝕂 𝕊
inst✝⁴ : NormedAlgebra 𝕂 𝔸
inst✝³ : Algebra 𝕊 𝔸
inst✝² : ContinuousSMul 𝕊 𝔸
inst✝¹ : IsScalarTower 𝕂 𝕊 𝔸
inst✝ : CompleteSpace 𝔸
x : 𝔸
t : 𝕊
htx : t • x ∈ EMetric.ball 0 (expSeries 𝕂 𝔸).radius
hpos : 0 < (expSeries 𝕂 𝔸).radius
this✝ : Tendsto (fun h => h • x) (𝓝 0) (𝓝 0)
this : ∀ᶠ (h : 𝕊) in 𝓝 0, h • x ∈ EMetric.ball 0 (expSeries 𝕂 𝔸).radius
⊢ (fun h => exp 𝕂 (t • x) * (exp 𝕂 ((0 + h) • x) - exp 𝕂 (0 • x) - (ContinuousLinearMap.smulRight 1 x) h)) =ᶠ[𝓝 0]
    fun h => exp 𝕂 ((t + h) • x) - exp 𝕂 (t • x) - (exp 𝕂 (t • x) • ContinuousLinearMap.smulRight 1 x) h
tactic: have : ∀ᶠ h in 𝓝 (0 : 𝕊), h • x ∈ EMetric.ball (0 : 𝔸) (expSeries 𝕂 𝔸).radius :=
    this.eventually (EMetric.ball_mem_nhds _ hpos)
state_before: 𝕂 : Type u_1
𝕊 : Type u_2
𝔸 : Type u_3
inst✝⁹ : NontriviallyNormedField 𝕂
inst✝⁸ : CharZero 𝕂
inst✝⁷ : NormedCommRing 𝕊
inst✝⁶ : NormedRing 𝔸
inst✝⁵ : NormedSpace 𝕂 𝕊
inst✝⁴ : NormedAlgebra 𝕂 𝔸
inst✝³ : Algebra 𝕊 𝔸
inst✝² : ContinuousSMul 𝕊 𝔸
inst✝¹ : IsScalarTower 𝕂 𝕊 𝔸
inst✝ : CompleteSpace 𝔸
x : 𝔸
t : 𝕊
htx : t • x ∈ EMetric.ball 0 (expSeries 𝕂 𝔸).radius
hpos : 0 < (expSeries 𝕂 𝔸).radius
this✝ : Tendsto (fun h => h • x) (𝓝 0) (𝓝 0)
this : ∀ᶠ (h : 𝕊) in 𝓝 0, h • x ∈ EMetric.ball 0 (expSeries 𝕂 𝔸).radius
⊢ (fun h => exp 𝕂 (t • x) * (exp 𝕂 ((0 + h) • x) - exp 𝕂 (0 • x) - (ContinuousLinearMap.smulRight 1 x) h)) =ᶠ[𝓝 0]
    fun h => exp 𝕂 ((t + h) • x) - exp 𝕂 (t • x) - (exp 𝕂 (t • x) • ContinuousLinearMap.smulRight 1 x) h
state_after: case h
𝕂 : Type u_1
𝕊 : Type u_2
𝔸 : Type u_3
inst✝⁹ : NontriviallyNormedField 𝕂
inst✝⁸ : CharZero 𝕂
inst✝⁷ : NormedCommRing 𝕊
inst✝⁶ : NormedRing 𝔸
inst✝⁵ : NormedSpace 𝕂 𝕊
inst✝⁴ : NormedAlgebra 𝕂 𝔸
inst✝³ : Algebra 𝕊 𝔸
inst✝² : ContinuousSMul 𝕊 𝔸
inst✝¹ : IsScalarTower 𝕂 𝕊 𝔸
inst✝ : CompleteSpace 𝔸
x : 𝔸
t : 𝕊
htx : t • x ∈ EMetric.ball 0 (expSeries 𝕂 𝔸).radius
hpos : 0 < (expSeries 𝕂 𝔸).radius
this✝ : Tendsto (fun h => h • x) (𝓝 0) (𝓝 0)
this : ∀ᶠ (h : 𝕊) in 𝓝 0, h • x ∈ EMetric.ball 0 (expSeries 𝕂 𝔸).radius
h : 𝕊
hh : h • x ∈ EMetric.ball 0 (expSeries 𝕂 𝔸).radius
⊢ exp 𝕂 (t • x) * (exp 𝕂 ((0 + h) • x) - exp 𝕂 (0 • x) - (ContinuousLinearMap.smulRight 1 x) h) =
    exp 𝕂 ((t + h) • x) - exp 𝕂 (t • x) - (exp 𝕂 (t • x) • ContinuousLinearMap.smulRight 1 x) h
tactic: filter_upwards [this] with h hh
state_before: case h
𝕂 : Type u_1
𝕊 : Type u_2
𝔸 : Type u_3
inst✝⁹ : NontriviallyNormedField 𝕂
inst✝⁸ : CharZero 𝕂
inst✝⁷ : NormedCommRing 𝕊
inst✝⁶ : NormedRing 𝔸
inst✝⁵ : NormedSpace 𝕂 𝕊
inst✝⁴ : NormedAlgebra 𝕂 𝔸
inst✝³ : Algebra 𝕊 𝔸
inst✝² : ContinuousSMul 𝕊 𝔸
inst✝¹ : IsScalarTower 𝕂 𝕊 𝔸
inst✝ : CompleteSpace 𝔸
x : 𝔸
t : 𝕊
htx : t • x ∈ EMetric.ball 0 (expSeries 𝕂 𝔸).radius
hpos : 0 < (expSeries 𝕂 𝔸).radius
this✝ : Tendsto (fun h => h • x) (𝓝 0) (𝓝 0)
this : ∀ᶠ (h : 𝕊) in 𝓝 0, h • x ∈ EMetric.ball 0 (expSeries 𝕂 𝔸).radius
h : 𝕊
hh : h • x ∈ EMetric.ball 0 (expSeries 𝕂 𝔸).radius
⊢ exp 𝕂 (t • x) * (exp 𝕂 ((0 + h) • x) - exp 𝕂 (0 • x) - (ContinuousLinearMap.smulRight 1 x) h) =
    exp 𝕂 ((t + h) • x) - exp 𝕂 (t • x) - (exp 𝕂 (t • x) • ContinuousLinearMap.smulRight 1 x) h
state_after: case h
𝕂 : Type u_1
𝕊 : Type u_2
𝔸 : Type u_3
inst✝⁹ : NontriviallyNormedField 𝕂
inst✝⁸ : CharZero 𝕂
inst✝⁷ : NormedCommRing 𝕊
inst✝⁶ : NormedRing 𝔸
inst✝⁵ : NormedSpace 𝕂 𝕊
inst✝⁴ : NormedAlgebra 𝕂 𝔸
inst✝³ : Algebra 𝕊 𝔸
inst✝² : ContinuousSMul 𝕊 𝔸
inst✝¹ : IsScalarTower 𝕂 𝕊 𝔸
inst✝ : CompleteSpace 𝔸
x : 𝔸
t : 𝕊
htx : t • x ∈ EMetric.ball 0 (expSeries 𝕂 𝔸).radius
hpos : 0 < (expSeries 𝕂 𝔸).radius
this✝¹ : Tendsto (fun h => h • x) (𝓝 0) (𝓝 0)
this✝ : ∀ᶠ (h : 𝕊) in 𝓝 0, h • x ∈ EMetric.ball 0 (expSeries 𝕂 𝔸).radius
h : 𝕊
hh : h • x ∈ EMetric.ball 0 (expSeries 𝕂 𝔸).radius
this : Commute (t • x) (h • x)
⊢ exp 𝕂 (t • x) * (exp 𝕂 ((0 + h) • x) - exp 𝕂 (0 • x) - (ContinuousLinearMap.smulRight 1 x) h) =
    exp 𝕂 ((t + h) • x) - exp 𝕂 (t • x) - (exp 𝕂 (t • x) • ContinuousLinearMap.smulRight 1 x) h
tactic: have : Commute (t • x) (h • x) := ((Commute.refl x).smul_left t).smul_right h
state_before: case h
𝕂 : Type u_1
𝕊 : Type u_2
𝔸 : Type u_3
inst✝⁹ : NontriviallyNormedField 𝕂
inst✝⁸ : CharZero 𝕂
inst✝⁷ : NormedCommRing 𝕊
inst✝⁶ : NormedRing 𝔸
inst✝⁵ : NormedSpace 𝕂 𝕊
inst✝⁴ : NormedAlgebra 𝕂 𝔸
inst✝³ : Algebra 𝕊 𝔸
inst✝² : ContinuousSMul 𝕊 𝔸
inst✝¹ : IsScalarTower 𝕂 𝕊 𝔸
inst✝ : CompleteSpace 𝔸
x : 𝔸
t : 𝕊
htx : t • x ∈ EMetric.ball 0 (expSeries 𝕂 𝔸).radius
hpos : 0 < (expSeries 𝕂 𝔸).radius
this✝¹ : Tendsto (fun h => h • x) (𝓝 0) (𝓝 0)
this✝ : ∀ᶠ (h : 𝕊) in 𝓝 0, h • x ∈ EMetric.ball 0 (expSeries 𝕂 𝔸).radius
h : 𝕊
hh : h • x ∈ EMetric.ball 0 (expSeries 𝕂 𝔸).radius
this : Commute (t • x) (h • x)
⊢ exp 𝕂 (t • x) * (exp 𝕂 ((0 + h) • x) - exp 𝕂 (0 • x) - (ContinuousLinearMap.smulRight 1 x) h) =
    exp 𝕂 ((t + h) • x) - exp 𝕂 (t • x) - (exp 𝕂 (t • x) • ContinuousLinearMap.smulRight 1 x) h
state_after: no goals
tactic: rw [add_smul t h, exp_add_of_commute_of_mem_ball this htx hh, zero_add, zero_smul, exp_zero,
    ContinuousLinearMap.smulRight_apply, ContinuousLinearMap.one_apply,
    ContinuousLinearMap.smul_apply, ContinuousLinearMap.smulRight_apply,
    ContinuousLinearMap.one_apply, smul_eq_mul, mul_sub_left_distrib, mul_sub_left_distrib, mul_one]