File: Mathlib/Analysis/SpecialFunctions/Exponential.lean

full_name: hasStrictFDerivAt_exp_zero_of_radius_pos
code: /-- The exponential in a Banach algebra `𝔸` over a normed field `𝕂` has strict Fréchet derivative
`1 : 𝔸 →L[𝕂] 𝔸` at zero, as long as it converges on a neighborhood of zero. -/
theorem hasStrictFDerivAt_exp_zero_of_radius_pos (h : 0 < (expSeries 𝕂 𝔸).radius) :
    HasStrictFDerivAt (exp 𝕂) (1 : 𝔸 →L[𝕂] 𝔸) 0 
goal: HasStrictFDerivAt (NormedSpace.exp 𝕂) 1 0

full_name: hasFDerivAt_exp_zero_of_radius_pos
code: /-- The exponential in a Banach algebra `𝔸` over a normed field `𝕂` has Fréchet derivative
`1 : 𝔸 →L[𝕂] 𝔸` at zero, as long as it converges on a neighborhood of zero. -/
theorem hasFDerivAt_exp_zero_of_radius_pos (h : 0 < (expSeries 𝕂 𝔸).radius) :
    HasFDerivAt (exp 𝕂) (1 : 𝔸 →L[𝕂] 𝔸) 0 
goal: HasFDerivAt (NormedSpace.exp 𝕂) 1 0

full_name: hasFDerivAt_exp_of_mem_ball
code: /-- The exponential map in a commutative Banach algebra `𝔸` over a normed field `𝕂` of
characteristic zero has Fréchet derivative `exp 𝕂 x • 1 : 𝔸 →L[𝕂] 𝔸` at any point `x` in the
disk of convergence. -/
theorem hasFDerivAt_exp_of_mem_ball [CharZero 𝕂] {x : 𝔸}
    (hx : x ∈ EMetric.ball (0 : 𝔸) (expSeries 𝕂 𝔸).radius) :
    HasFDerivAt (exp 𝕂) (exp 𝕂 x • (1 : 𝔸 →L[𝕂] 𝔸)) x 
goal: HasFDerivAt (NormedSpace.exp 𝕂) (NormedSpace.exp 𝕂 x • 1) x

full_name: hasStrictFDerivAt_exp_of_mem_ball
code: /-- The exponential map in a commutative Banach algebra `𝔸` over a normed field `𝕂` of
characteristic zero has strict Fréchet derivative `exp 𝕂 x • 1 : 𝔸 →L[𝕂] 𝔸` at any point `x` in
the disk of convergence. -/
theorem hasStrictFDerivAt_exp_of_mem_ball [CharZero 𝕂] {x : 𝔸}
    (hx : x ∈ EMetric.ball (0 : 𝔸) (expSeries 𝕂 𝔸).radius) :
    HasStrictFDerivAt (exp 𝕂) (exp 𝕂 x • (1 : 𝔸 →L[𝕂] 𝔸)) x 
goal: HasStrictFDerivAt (NormedSpace.exp 𝕂) (NormedSpace.exp 𝕂 x • 1) x

full_name: hasStrictDerivAt_exp_of_mem_ball
code: /-- The exponential map in a complete normed field `𝕂` of characteristic zero has strict derivative
`exp 𝕂 x` at any point `x` in the disk of convergence. -/
theorem hasStrictDerivAt_exp_of_mem_ball [CharZero 𝕂] {x : 𝕂}
    (hx : x ∈ EMetric.ball (0 : 𝕂) (expSeries 𝕂 𝕂).radius) :
    HasStrictDerivAt (exp 𝕂) (exp 𝕂 x) x 
goal: HasStrictDerivAt (NormedSpace.exp 𝕂) (NormedSpace.exp 𝕂 x) x

full_name: hasDerivAt_exp_of_mem_ball
code: /-- The exponential map in a complete normed field `𝕂` of characteristic zero has derivative
`exp 𝕂 x` at any point `x` in the disk of convergence. -/
theorem hasDerivAt_exp_of_mem_ball [CharZero 𝕂] {x : 𝕂}
    (hx : x ∈ EMetric.ball (0 : 𝕂) (expSeries 𝕂 𝕂).radius) : HasDerivAt (exp 𝕂) (exp 𝕂 x) x 
goal: HasDerivAt (NormedSpace.exp 𝕂) (NormedSpace.exp 𝕂 x) x

full_name: hasStrictDerivAt_exp_zero_of_radius_pos
code: /-- The exponential map in a complete normed field `𝕂` of characteristic zero has strict derivative
`1` at zero, as long as it converges on a neighborhood of zero. -/
theorem hasStrictDerivAt_exp_zero_of_radius_pos (h : 0 < (expSeries 𝕂 𝕂).radius) :
    HasStrictDerivAt (exp 𝕂) (1 : 𝕂) 0 
goal: HasStrictDerivAt (NormedSpace.exp 𝕂) 1 0

full_name: hasDerivAt_exp_zero_of_radius_pos
code: /-- The exponential map in a complete normed field `𝕂` of characteristic zero has derivative
`1` at zero, as long as it converges on a neighborhood of zero. -/
theorem hasDerivAt_exp_zero_of_radius_pos (h : 0 < (expSeries 𝕂 𝕂).radius) :
    HasDerivAt (exp 𝕂) (1 : 𝕂) 0 
goal: HasDerivAt (NormedSpace.exp 𝕂) 1 0

full_name: hasStrictFDerivAt_exp_zero
code: /-- The exponential in a Banach algebra `𝔸` over `𝕂 = ℝ` or `𝕂 = ℂ` has strict Fréchet derivative
`1 : 𝔸 →L[𝕂] 𝔸` at zero. -/
theorem hasStrictFDerivAt_exp_zero : HasStrictFDerivAt (exp 𝕂) (1 : 𝔸 →L[𝕂] 𝔸) 0 
goal: HasStrictFDerivAt (NormedSpace.exp 𝕂) 1 0

full_name: hasFDerivAt_exp_zero
code: /-- The exponential in a Banach algebra `𝔸` over `𝕂 = ℝ` or `𝕂 = ℂ` has Fréchet derivative
`1 : 𝔸 →L[𝕂] 𝔸` at zero. -/
theorem hasFDerivAt_exp_zero : HasFDerivAt (exp 𝕂) (1 : 𝔸 →L[𝕂] 𝔸) 0 
goal: HasFDerivAt (NormedSpace.exp 𝕂) 1 0

full_name: hasStrictFDerivAt_exp
code: /-- The exponential map in a commutative Banach algebra `𝔸` over `𝕂 = ℝ` or `𝕂 = ℂ` has strict
Fréchet derivative `exp 𝕂 x • 1 : 𝔸 →L[𝕂] 𝔸` at any point `x`. -/
theorem hasStrictFDerivAt_exp {x : 𝔸} : HasStrictFDerivAt (exp 𝕂) (exp 𝕂 x • (1 : 𝔸 →L[𝕂] 𝔸)) x 
goal: HasStrictFDerivAt (NormedSpace.exp 𝕂) (NormedSpace.exp 𝕂 x • 1) x

full_name: hasFDerivAt_exp
code: /-- The exponential map in a commutative Banach algebra `𝔸` over `𝕂 = ℝ` or `𝕂 = ℂ` has
Fréchet derivative `exp 𝕂 x • 1 : 𝔸 →L[𝕂] 𝔸` at any point `x`. -/
theorem hasFDerivAt_exp {x : 𝔸} : HasFDerivAt (exp 𝕂) (exp 𝕂 x • (1 : 𝔸 →L[𝕂] 𝔸)) x 
goal: HasFDerivAt (NormedSpace.exp 𝕂) (NormedSpace.exp 𝕂 x • 1) x

full_name: hasStrictDerivAt_exp
code: /-- The exponential map in `𝕂 = ℝ` or `𝕂 = ℂ` has strict derivative `exp 𝕂 x` at any point
`x`. -/
theorem hasStrictDerivAt_exp {x : 𝕂} : HasStrictDerivAt (exp 𝕂) (exp 𝕂 x) x 
goal: HasStrictDerivAt (NormedSpace.exp 𝕂) (NormedSpace.exp 𝕂 x) x

full_name: hasDerivAt_exp
code: /-- The exponential map in `𝕂 = ℝ` or `𝕂 = ℂ` has derivative `exp 𝕂 x` at any point `x`. -/
theorem hasDerivAt_exp {x : 𝕂} : HasDerivAt (exp 𝕂) (exp 𝕂 x) x 
goal: HasDerivAt (NormedSpace.exp 𝕂) (NormedSpace.exp 𝕂 x) x

full_name: hasStrictDerivAt_exp_zero
code: /-- The exponential map in `𝕂 = ℝ` or `𝕂 = ℂ` has strict derivative `1` at zero. -/
theorem hasStrictDerivAt_exp_zero : HasStrictDerivAt (exp 𝕂) (1 : 𝕂) 0 
goal: HasStrictDerivAt (NormedSpace.exp 𝕂) 1 0

full_name: hasDerivAt_exp_zero
code: /-- The exponential map in `𝕂 = ℝ` or `𝕂 = ℂ` has derivative `1` at zero. -/
theorem hasDerivAt_exp_zero : HasDerivAt (exp 𝕂) (1 : 𝕂) 0 
goal: HasDerivAt (NormedSpace.exp 𝕂) 1 0

full_name: Complex.exp_eq_exp_ℂ
code: theorem Complex.exp_eq_exp_ℂ : Complex.exp = NormedSpace.exp ℂ 
goal: Complex.exp = NormedSpace.exp ℂ

full_name: Real.exp_eq_exp_ℝ
code: theorem Real.exp_eq_exp_ℝ : Real.exp = NormedSpace.exp ℝ 
goal: Real.exp = NormedSpace.exp ℝ

full_name: hasFDerivAt_exp_smul_const_of_mem_ball
code: theorem hasFDerivAt_exp_smul_const_of_mem_ball (x : 𝔸) (t : 𝕊)
    (htx : t • x ∈ EMetric.ball (0 : 𝔸) (expSeries 𝕂 𝔸).radius) :
    HasFDerivAt (fun u : 𝕊 => exp 𝕂 (u • x)) (exp 𝕂 (t • x) • (1 : 𝕊 →L[𝕂] 𝕊).smulRight x) t 
goal: HasFDerivAt (fun u => NormedSpace.exp 𝕂 (u • x)) (NormedSpace.exp 𝕂 (t • x) • ContinuousLinearMap.smulRight 1 x) t

full_name: hasFDerivAt_exp_smul_const_of_mem_ball'
code: theorem hasFDerivAt_exp_smul_const_of_mem_ball' (x : 𝔸) (t : 𝕊)
    (htx : t • x ∈ EMetric.ball (0 : 𝔸) (expSeries 𝕂 𝔸).radius) :
    HasFDerivAt (fun u : 𝕊 => exp 𝕂 (u • x))
      (((1 : 𝕊 →L[𝕂] 𝕊).smulRight x).smulRight (exp 𝕂 (t • x))) t 
goal: HasFDerivAt (fun u => NormedSpace.exp 𝕂 (u • x))
  ((ContinuousLinearMap.smulRight 1 x).smulRight (NormedSpace.exp 𝕂 (t • x))) t

full_name: hasStrictFDerivAt_exp_smul_const_of_mem_ball
code: theorem hasStrictFDerivAt_exp_smul_const_of_mem_ball (x : 𝔸) (t : 𝕊)
    (htx : t • x ∈ EMetric.ball (0 : 𝔸) (expSeries 𝕂 𝔸).radius) :
    HasStrictFDerivAt (fun u : 𝕊 => exp 𝕂 (u • x))
      (exp 𝕂 (t • x) • (1 : 𝕊 →L[𝕂] 𝕊).smulRight x) t 
goal: HasStrictFDerivAt (fun u => NormedSpace.exp 𝕂 (u • x)) (NormedSpace.exp 𝕂 (t • x) • ContinuousLinearMap.smulRight 1 x) t

full_name: hasStrictFDerivAt_exp_smul_const_of_mem_ball'
code: theorem hasStrictFDerivAt_exp_smul_const_of_mem_ball' (x : 𝔸) (t : 𝕊)
    (htx : t • x ∈ EMetric.ball (0 : 𝔸) (expSeries 𝕂 𝔸).radius) :
    HasStrictFDerivAt (fun u : 𝕊 => exp 𝕂 (u • x))
      (((1 : 𝕊 →L[𝕂] 𝕊).smulRight x).smulRight (exp 𝕂 (t • x))) t 
goal: HasStrictFDerivAt (fun u => NormedSpace.exp 𝕂 (u • x))
  ((ContinuousLinearMap.smulRight 1 x).smulRight (NormedSpace.exp 𝕂 (t • x))) t

full_name: hasStrictDerivAt_exp_smul_const_of_mem_ball
code: theorem hasStrictDerivAt_exp_smul_const_of_mem_ball (x : 𝔸) (t : 𝕂)
    (htx : t • x ∈ EMetric.ball (0 : 𝔸) (expSeries 𝕂 𝔸).radius) :
    HasStrictDerivAt (fun u : 𝕂 => exp 𝕂 (u • x)) (exp 𝕂 (t • x) * x) t 
goal: HasStrictDerivAt (fun u => NormedSpace.exp 𝕂 (u • x)) (NormedSpace.exp 𝕂 (t • x) * x) t

full_name: hasStrictDerivAt_exp_smul_const_of_mem_ball'
code: theorem hasStrictDerivAt_exp_smul_const_of_mem_ball' (x : 𝔸) (t : 𝕂)
    (htx : t • x ∈ EMetric.ball (0 : 𝔸) (expSeries 𝕂 𝔸).radius) :
    HasStrictDerivAt (fun u : 𝕂 => exp 𝕂 (u • x)) (x * exp 𝕂 (t • x)) t 
goal: HasStrictDerivAt (fun u => NormedSpace.exp 𝕂 (u • x)) (x * NormedSpace.exp 𝕂 (t • x)) t

full_name: hasDerivAt_exp_smul_const_of_mem_ball
code: theorem hasDerivAt_exp_smul_const_of_mem_ball (x : 𝔸) (t : 𝕂)
    (htx : t • x ∈ EMetric.ball (0 : 𝔸) (expSeries 𝕂 𝔸).radius) :
    HasDerivAt (fun u : 𝕂 => exp 𝕂 (u • x)) (exp 𝕂 (t • x) * x) t 
goal: HasDerivAt (fun u => NormedSpace.exp 𝕂 (u • x)) (NormedSpace.exp 𝕂 (t • x) * x) t

full_name: hasDerivAt_exp_smul_const_of_mem_ball'
code: theorem hasDerivAt_exp_smul_const_of_mem_ball' (x : 𝔸) (t : 𝕂)
    (htx : t • x ∈ EMetric.ball (0 : 𝔸) (expSeries 𝕂 𝔸).radius) :
    HasDerivAt (fun u : 𝕂 => exp 𝕂 (u • x)) (x * exp 𝕂 (t • x)) t 
goal: HasDerivAt (fun u => NormedSpace.exp 𝕂 (u • x)) (x * NormedSpace.exp 𝕂 (t • x)) t

full_name: hasFDerivAt_exp_smul_const
code: theorem hasFDerivAt_exp_smul_const (x : 𝔸) (t : 𝕊) :
    HasFDerivAt (fun u : 𝕊 => exp 𝕂 (u • x)) (exp 𝕂 (t • x) • (1 : 𝕊 →L[𝕂] 𝕊).smulRight x) t 
goal: HasFDerivAt (fun u => NormedSpace.exp 𝕂 (u • x)) (NormedSpace.exp 𝕂 (t • x) • ContinuousLinearMap.smulRight 1 x) t

full_name: hasFDerivAt_exp_smul_const'
code: theorem hasFDerivAt_exp_smul_const' (x : 𝔸) (t : 𝕊) :
    HasFDerivAt (fun u : 𝕊 => exp 𝕂 (u • x))
      (((1 : 𝕊 →L[𝕂] 𝕊).smulRight x).smulRight (exp 𝕂 (t • x))) t 
goal: HasFDerivAt (fun u => NormedSpace.exp 𝕂 (u • x))
  ((ContinuousLinearMap.smulRight 1 x).smulRight (NormedSpace.exp 𝕂 (t • x))) t

full_name: hasStrictFDerivAt_exp_smul_const
code: theorem hasStrictFDerivAt_exp_smul_const (x : 𝔸) (t : 𝕊) :
    HasStrictFDerivAt (fun u : 𝕊 => exp 𝕂 (u • x))
      (exp 𝕂 (t • x) • (1 : 𝕊 →L[𝕂] 𝕊).smulRight x) t 
goal: HasStrictFDerivAt (fun u => NormedSpace.exp 𝕂 (u • x)) (NormedSpace.exp 𝕂 (t • x) • ContinuousLinearMap.smulRight 1 x) t

full_name: hasStrictFDerivAt_exp_smul_const'
code: theorem hasStrictFDerivAt_exp_smul_const' (x : 𝔸) (t : 𝕊) :
    HasStrictFDerivAt (fun u : 𝕊 => exp 𝕂 (u • x))
      (((1 : 𝕊 →L[𝕂] 𝕊).smulRight x).smulRight (exp 𝕂 (t • x))) t 
goal: HasStrictFDerivAt (fun u => NormedSpace.exp 𝕂 (u • x))
  ((ContinuousLinearMap.smulRight 1 x).smulRight (NormedSpace.exp 𝕂 (t • x))) t

full_name: hasStrictDerivAt_exp_smul_const
code: theorem hasStrictDerivAt_exp_smul_const (x : 𝔸) (t : 𝕂) :
    HasStrictDerivAt (fun u : 𝕂 => exp 𝕂 (u • x)) (exp 𝕂 (t • x) * x) t 
goal: HasStrictDerivAt (fun u => NormedSpace.exp 𝕂 (u • x)) (NormedSpace.exp 𝕂 (t • x) * x) t

full_name: hasStrictDerivAt_exp_smul_const'
code: theorem hasStrictDerivAt_exp_smul_const' (x : 𝔸) (t : 𝕂) :
    HasStrictDerivAt (fun u : 𝕂 => exp 𝕂 (u • x)) (x * exp 𝕂 (t • x)) t 
goal: HasStrictDerivAt (fun u => NormedSpace.exp 𝕂 (u • x)) (x * NormedSpace.exp 𝕂 (t • x)) t

full_name: hasDerivAt_exp_smul_const
code: theorem hasDerivAt_exp_smul_const (x : 𝔸) (t : 𝕂) :
    HasDerivAt (fun u : 𝕂 => exp 𝕂 (u • x)) (exp 𝕂 (t • x) * x) t 
goal: HasDerivAt (fun u => NormedSpace.exp 𝕂 (u • x)) (NormedSpace.exp 𝕂 (t • x) * x) t

full_name: hasDerivAt_exp_smul_const'
code: theorem hasDerivAt_exp_smul_const' (x : 𝔸) (t : 𝕂) :
    HasDerivAt (fun u : 𝕂 => exp 𝕂 (u • x)) (x * exp 𝕂 (t • x)) t 
goal: HasDerivAt (fun u => NormedSpace.exp 𝕂 (u • x)) (x * NormedSpace.exp 𝕂 (t • x)) t

full_name: HasSum.exp
code: /-- If `f` has sum `a`, then `exp ∘ f` has product `exp a`. -/
lemma HasSum.exp {ι : Type*} {f : ι → 𝔸} {a : 𝔸} (h : HasSum f a) :
    HasProd (exp 𝕂 ∘ f) (exp 𝕂 a) 
goal: HasProd (NormedSpace.exp 𝕂 ∘ f) (NormedSpace.exp 𝕂 a)