ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- DifferentiableWithinAt.clm_apply
@[fun_prop]
theorem DifferentiableWithinAt.clm_apply (hc : DifferentiableWithinAt 𝕜 c s x)
    (hu : DifferentiableWithinAt 𝕜 u s x) : DifferentiableWithinAt 𝕜 (fun y => (c y) (u y)) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {x : E} {s : Set E} {H : Type u_6} [NormedAddCommGroup H] [NormedSpace 𝕜 H] {c : E → G →L[𝕜] H} {u : E → G} (hc : DifferentiableWithinAt 𝕜 c s x) (hu : DifferentiableWithinAt 𝕜 u s x)
Goal: DifferentiableWithinAt 𝕜 (fun y => (c y) (u y)) s x
-- DifferentiableAt.clm_apply
@[fun_prop]
theorem DifferentiableAt.clm_apply (hc : DifferentiableAt 𝕜 c x) (hu : DifferentiableAt 𝕜 u x) :
    DifferentiableAt 𝕜 (fun y => (c y) (u y)) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {x : E} {H : Type u_6} [NormedAddCommGroup H] [NormedSpace 𝕜 H] {c : E → G →L[𝕜] H} {u : E → G} (hc : DifferentiableAt 𝕜 c x) (hu : DifferentiableAt 𝕜 u x)
Goal: DifferentiableAt 𝕜 (fun y => (c y) (u y)) x
-- DifferentiableOn.clm_apply
@[fun_prop]
theorem DifferentiableOn.clm_apply (hc : DifferentiableOn 𝕜 c s) (hu : DifferentiableOn 𝕜 u s) :
    DifferentiableOn 𝕜 (fun y => (c y) (u y)) s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {s : Set E} {H : Type u_6} [NormedAddCommGroup H] [NormedSpace 𝕜 H] {c : E → G →L[𝕜] H} {u : E → G} (hc : DifferentiableOn 𝕜 c s) (hu : DifferentiableOn 𝕜 u s)
Goal: DifferentiableOn 𝕜 (fun y => (c y) (u y)) s
-- Differentiable.clm_apply
@[fun_prop]
theorem Differentiable.clm_apply (hc : Differentiable 𝕜 c) (hu : Differentiable 𝕜 u) :
    Differentiable 𝕜 fun y => (c y) (u y) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {H : Type u_6} [NormedAddCommGroup H] [NormedSpace 𝕜 H] {c : E → G →L[𝕜] H} {u : E → G} (hc : Differentiable 𝕜 c) (hu : Differentiable 𝕜 u)
Goal: Differentiable 𝕜 fun y => (c y) (u y)
-- fderivWithin_clm_apply
theorem fderivWithin_clm_apply (hxs : UniqueDiffWithinAt 𝕜 s x)
    (hc : DifferentiableWithinAt 𝕜 c s x) (hu : DifferentiableWithinAt 𝕜 u s x) :
    fderivWithin 𝕜 (fun y => (c y) (u y)) s x =
      (c x).comp (fderivWithin 𝕜 u s x) + (fderivWithin 𝕜 c s x).flip (u x) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {x : E} {s : Set E} {H : Type u_6} [NormedAddCommGroup H] [NormedSpace 𝕜 H] {c : E → G →L[𝕜] H} {u : E → G} (hxs : UniqueDiffWithinAt 𝕜 s x) (hc : DifferentiableWithinAt 𝕜 c s x) (hu : DifferentiableWithinAt 𝕜 u s x)
Goal: fderivWithin 𝕜 (fun y => (c y) (u y)) s x = (c x).comp (fderivWithin 𝕜 u s x) + (fderivWithin 𝕜 c s x).flip (u x)
-- fderiv_clm_apply
theorem fderiv_clm_apply (hc : DifferentiableAt 𝕜 c x) (hu : DifferentiableAt 𝕜 u x) :
    fderiv 𝕜 (fun y => (c y) (u y)) x = (c x).comp (fderiv 𝕜 u x) + (fderiv 𝕜 c x).flip (u x) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {x : E} {H : Type u_6} [NormedAddCommGroup H] [NormedSpace 𝕜 H] {c : E → G →L[𝕜] H} {u : E → G} (hc : DifferentiableAt 𝕜 c x) (hu : DifferentiableAt 𝕜 u x)
Goal: fderiv 𝕜 (fun y => (c y) (u y)) x = (c x).comp (fderiv 𝕜 u x) + (fderiv 𝕜 c x).flip (u x)
-- HasStrictFDerivAt.continuousMultilinear_apply_const
@[fun_prop]
theorem HasStrictFDerivAt.continuousMultilinear_apply_const (hc : HasStrictFDerivAt c c' x)
    (u : ∀ i, M i) : HasStrictFDerivAt (fun y ↦ (c y) u) (c'.flipMultilinear u) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {ι : Type u_6} [Fintype ι] {M : ι → Type u_7} [(i : ι) → NormedAddCommGroup (M i)] [(i : ι) → NormedSpace 𝕜 (M i)] {H : Type u_8} [NormedAddCommGroup H] [NormedSpace 𝕜 H] {c : E → ContinuousMultilinearMap 𝕜 M H} {c' : E →L[𝕜] ContinuousMultilinearMap 𝕜 M H} (hc : HasStrictFDerivAt c c' x) (u : (i : ι) → M i)
Goal: HasStrictFDerivAt (fun y => (c y) u) (c'.flipMultilinear u) x
-- HasFDerivWithinAt.continuousMultilinear_apply_const
@[fun_prop]
theorem HasFDerivWithinAt.continuousMultilinear_apply_const (hc : HasFDerivWithinAt c c' s x)
    (u : ∀ i, M i) :
    HasFDerivWithinAt (fun y ↦ (c y) u) (c'.flipMultilinear u) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {s : Set E} {ι : Type u_6} [Fintype ι] {M : ι → Type u_7} [(i : ι) → NormedAddCommGroup (M i)] [(i : ι) → NormedSpace 𝕜 (M i)] {H : Type u_8} [NormedAddCommGroup H] [NormedSpace 𝕜 H] {c : E → ContinuousMultilinearMap 𝕜 M H} {c' : E →L[𝕜] ContinuousMultilinearMap 𝕜 M H} (hc : HasFDerivWithinAt c c' s x) (u : (i : ι) → M i)
Goal: HasFDerivWithinAt (fun y => (c y) u) (c'.flipMultilinear u) s x
-- HasFDerivAt.continuousMultilinear_apply_const
@[fun_prop]
theorem HasFDerivAt.continuousMultilinear_apply_const (hc : HasFDerivAt c c' x) (u : ∀ i, M i) :
    HasFDerivAt (fun y ↦ (c y) u) (c'.flipMultilinear u) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {ι : Type u_6} [Fintype ι] {M : ι → Type u_7} [(i : ι) → NormedAddCommGroup (M i)] [(i : ι) → NormedSpace 𝕜 (M i)] {H : Type u_8} [NormedAddCommGroup H] [NormedSpace 𝕜 H] {c : E → ContinuousMultilinearMap 𝕜 M H} {c' : E →L[𝕜] ContinuousMultilinearMap 𝕜 M H} (hc : HasFDerivAt c c' x) (u : (i : ι) → M i)
Goal: HasFDerivAt (fun y => (c y) u) (c'.flipMultilinear u) x
-- DifferentiableWithinAt.continuousMultilinear_apply_const
@[fun_prop]
theorem DifferentiableWithinAt.continuousMultilinear_apply_const
    (hc : DifferentiableWithinAt 𝕜 c s x) (u : ∀ i, M i) :
    DifferentiableWithinAt 𝕜 (fun y ↦ (c y) u) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {s : Set E} {ι : Type u_6} [Fintype ι] {M : ι → Type u_7} [(i : ι) → NormedAddCommGroup (M i)] [(i : ι) → NormedSpace 𝕜 (M i)] {H : Type u_8} [NormedAddCommGroup H] [NormedSpace 𝕜 H] {c : E → ContinuousMultilinearMap 𝕜 M H} (hc : DifferentiableWithinAt 𝕜 c s x) (u : (i : ι) → M i)
Goal: DifferentiableWithinAt 𝕜 (fun y => (c y) u) s x
-- DifferentiableAt.continuousMultilinear_apply_const
@[fun_prop]
theorem DifferentiableAt.continuousMultilinear_apply_const (hc : DifferentiableAt 𝕜 c x)
    (u : ∀ i, M i) :
    DifferentiableAt 𝕜 (fun y ↦ (c y) u) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {ι : Type u_6} [Fintype ι] {M : ι → Type u_7} [(i : ι) → NormedAddCommGroup (M i)] [(i : ι) → NormedSpace 𝕜 (M i)] {H : Type u_8} [NormedAddCommGroup H] [NormedSpace 𝕜 H] {c : E → ContinuousMultilinearMap 𝕜 M H} (hc : DifferentiableAt 𝕜 c x) (u : (i : ι) → M i)
Goal: DifferentiableAt 𝕜 (fun y => (c y) u) x
-- DifferentiableOn.continuousMultilinear_apply_const
@[fun_prop]
theorem DifferentiableOn.continuousMultilinear_apply_const (hc : DifferentiableOn 𝕜 c s)
    (u : ∀ i, M i) : DifferentiableOn 𝕜 (fun y ↦ (c y) u) s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {s : Set E} {ι : Type u_6} [Fintype ι] {M : ι → Type u_7} [(i : ι) → NormedAddCommGroup (M i)] [(i : ι) → NormedSpace 𝕜 (M i)] {H : Type u_8} [NormedAddCommGroup H] [NormedSpace 𝕜 H] {c : E → ContinuousMultilinearMap 𝕜 M H} (hc : DifferentiableOn 𝕜 c s) (u : (i : ι) → M i)
Goal: DifferentiableOn 𝕜 (fun y => (c y) u) s
-- Differentiable.continuousMultilinear_apply_const
@[fun_prop]
theorem Differentiable.continuousMultilinear_apply_const (hc : Differentiable 𝕜 c) (u : ∀ i, M i) :
    Differentiable 𝕜 fun y ↦ (c y) u 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {ι : Type u_6} [Fintype ι] {M : ι → Type u_7} [(i : ι) → NormedAddCommGroup (M i)] [(i : ι) → NormedSpace 𝕜 (M i)] {H : Type u_8} [NormedAddCommGroup H] [NormedSpace 𝕜 H] {c : E → ContinuousMultilinearMap 𝕜 M H} (hc : Differentiable 𝕜 c) (u : (i : ι) → M i)
Goal: Differentiable 𝕜 fun y => (c y) u
-- fderivWithin_continuousMultilinear_apply_const
theorem fderivWithin_continuousMultilinear_apply_const (hxs : UniqueDiffWithinAt 𝕜 s x)
    (hc : DifferentiableWithinAt 𝕜 c s x) (u : ∀ i, M i) :
    fderivWithin 𝕜 (fun y ↦ (c y) u) s x = ((fderivWithin 𝕜 c s x).flipMultilinear u) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {s : Set E} {ι : Type u_6} [Fintype ι] {M : ι → Type u_7} [(i : ι) → NormedAddCommGroup (M i)] [(i : ι) → NormedSpace 𝕜 (M i)] {H : Type u_8} [NormedAddCommGroup H] [NormedSpace 𝕜 H] {c : E → ContinuousMultilinearMap 𝕜 M H} (hxs : UniqueDiffWithinAt 𝕜 s x) (hc : DifferentiableWithinAt 𝕜 c s x) (u : (i : ι) → M i)
Goal: fderivWithin 𝕜 (fun y => (c y) u) s x = (fderivWithin 𝕜 c s x).flipMultilinear u
-- fderiv_continuousMultilinear_apply_const
theorem fderiv_continuousMultilinear_apply_const (hc : DifferentiableAt 𝕜 c x) (u : ∀ i, M i) :
    (fderiv 𝕜 (fun y ↦ (c y) u) x) = (fderiv 𝕜 c x).flipMultilinear u 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {ι : Type u_6} [Fintype ι] {M : ι → Type u_7} [(i : ι) → NormedAddCommGroup (M i)] [(i : ι) → NormedSpace 𝕜 (M i)] {H : Type u_8} [NormedAddCommGroup H] [NormedSpace 𝕜 H] {c : E → ContinuousMultilinearMap 𝕜 M H} (hc : DifferentiableAt 𝕜 c x) (u : (i : ι) → M i)
Goal: fderiv 𝕜 (fun y => (c y) u) x = (fderiv 𝕜 c x).flipMultilinear u
-- fderivWithin_continuousMultilinear_apply_const_apply
theorem fderivWithin_continuousMultilinear_apply_const_apply (hxs : UniqueDiffWithinAt 𝕜 s x)
    (hc : DifferentiableWithinAt 𝕜 c s x) (u : ∀ i, M i) (m : E) :
    (fderivWithin 𝕜 (fun y ↦ (c y) u) s x) m = (fderivWithin 𝕜 c s x) m u 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {s : Set E} {ι : Type u_6} [Fintype ι] {M : ι → Type u_7} [(i : ι) → NormedAddCommGroup (M i)] [(i : ι) → NormedSpace 𝕜 (M i)] {H : Type u_8} [NormedAddCommGroup H] [NormedSpace 𝕜 H] {c : E → ContinuousMultilinearMap 𝕜 M H} (hxs : UniqueDiffWithinAt 𝕜 s x) (hc : DifferentiableWithinAt 𝕜 c s x) (u : (i : ι) → M i) (m : E)
Goal: (fderivWithin 𝕜 (fun y => (c y) u) s x) m = ((fderivWithin 𝕜 c s x) m) u
-- fderiv_continuousMultilinear_apply_const_apply
theorem fderiv_continuousMultilinear_apply_const_apply (hc : DifferentiableAt 𝕜 c x)
    (u : ∀ i, M i) (m : E) :
    (fderiv 𝕜 (fun y ↦ (c y) u) x) m = (fderiv 𝕜 c x) m u 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {ι : Type u_6} [Fintype ι] {M : ι → Type u_7} [(i : ι) → NormedAddCommGroup (M i)] [(i : ι) → NormedSpace 𝕜 (M i)] {H : Type u_8} [NormedAddCommGroup H] [NormedSpace 𝕜 H] {c : E → ContinuousMultilinearMap 𝕜 M H} (hc : DifferentiableAt 𝕜 c x) (u : (i : ι) → M i) (m : E)
Goal: (fderiv 𝕜 (fun y => (c y) u) x) m = ((fderiv 𝕜 c x) m) u
-- HasStrictFDerivAt.smul
@[fun_prop]
theorem HasStrictFDerivAt.smul (hc : HasStrictFDerivAt c c' x) (hf : HasStrictFDerivAt f f' x) :
    HasStrictFDerivAt (fun y => c y • f y) (c x • f' + c'.smulRight (f x)) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {f' : E →L[𝕜] F} {x : E} {𝕜' : Type u_6} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F] {c : E → 𝕜'} {c' : E →L[𝕜] 𝕜'} (hc : HasStrictFDerivAt c c' x) (hf : HasStrictFDerivAt f f' x)
Goal: HasStrictFDerivAt (fun y => c y • f y) (c x • f' + c'.smulRight (f x)) x
-- HasFDerivWithinAt.smul
@[fun_prop]
theorem HasFDerivWithinAt.smul (hc : HasFDerivWithinAt c c' s x) (hf : HasFDerivWithinAt f f' s x) :
    HasFDerivWithinAt (fun y => c y • f y) (c x • f' + c'.smulRight (f x)) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {f' : E →L[𝕜] F} {x : E} {s : Set E} {𝕜' : Type u_6} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F] {c : E → 𝕜'} {c' : E →L[𝕜] 𝕜'} (hc : HasFDerivWithinAt c c' s x) (hf : HasFDerivWithinAt f f' s x)
Goal: HasFDerivWithinAt (fun y => c y • f y) (c x • f' + c'.smulRight (f x)) s x
-- HasFDerivAt.smul
@[fun_prop]
theorem HasFDerivAt.smul (hc : HasFDerivAt c c' x) (hf : HasFDerivAt f f' x) :
    HasFDerivAt (fun y => c y • f y) (c x • f' + c'.smulRight (f x)) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {f' : E →L[𝕜] F} {x : E} {𝕜' : Type u_6} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F] {c : E → 𝕜'} {c' : E →L[𝕜] 𝕜'} (hc : HasFDerivAt c c' x) (hf : HasFDerivAt f f' x)
Goal: HasFDerivAt (fun y => c y • f y) (c x • f' + c'.smulRight (f x)) x
-- DifferentiableWithinAt.smul
@[fun_prop]
theorem DifferentiableWithinAt.smul (hc : DifferentiableWithinAt 𝕜 c s x)
    (hf : DifferentiableWithinAt 𝕜 f s x) : DifferentiableWithinAt 𝕜 (fun y => c y • f y) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {x : E} {s : Set E} {𝕜' : Type u_6} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F] {c : E → 𝕜'} (hc : DifferentiableWithinAt 𝕜 c s x) (hf : DifferentiableWithinAt 𝕜 f s x)
Goal: DifferentiableWithinAt 𝕜 (fun y => c y • f y) s x
-- DifferentiableAt.smul
@[simp, fun_prop]
theorem DifferentiableAt.smul (hc : DifferentiableAt 𝕜 c x) (hf : DifferentiableAt 𝕜 f x) :
    DifferentiableAt 𝕜 (fun y => c y • f y) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {x : E} {𝕜' : Type u_6} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F] {c : E → 𝕜'} (hc : DifferentiableAt 𝕜 c x) (hf : DifferentiableAt 𝕜 f x)
Goal: DifferentiableAt 𝕜 (fun y => c y • f y) x
-- DifferentiableOn.smul
@[fun_prop]
theorem DifferentiableOn.smul (hc : DifferentiableOn 𝕜 c s) (hf : DifferentiableOn 𝕜 f s) :
    DifferentiableOn 𝕜 (fun y => c y • f y) s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {s : Set E} {𝕜' : Type u_6} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F] {c : E → 𝕜'} (hc : DifferentiableOn 𝕜 c s) (hf : DifferentiableOn 𝕜 f s)
Goal: DifferentiableOn 𝕜 (fun y => c y • f y) s
-- Differentiable.smul
@[simp, fun_prop]
theorem Differentiable.smul (hc : Differentiable 𝕜 c) (hf : Differentiable 𝕜 f) :
    Differentiable 𝕜 fun y => c y • f y 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {𝕜' : Type u_6} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F] {c : E → 𝕜'} (hc : Differentiable 𝕜 c) (hf : Differentiable 𝕜 f)
Goal: Differentiable 𝕜 fun y => c y • f y
-- fderivWithin_smul
theorem fderivWithin_smul (hxs : UniqueDiffWithinAt 𝕜 s x) (hc : DifferentiableWithinAt 𝕜 c s x)
    (hf : DifferentiableWithinAt 𝕜 f s x) :
    fderivWithin 𝕜 (fun y => c y • f y) s x =
      c x • fderivWithin 𝕜 f s x + (fderivWithin 𝕜 c s x).smulRight (f x) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {x : E} {s : Set E} {𝕜' : Type u_6} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F] {c : E → 𝕜'} (hxs : UniqueDiffWithinAt 𝕜 s x) (hc : DifferentiableWithinAt 𝕜 c s x) (hf : DifferentiableWithinAt 𝕜 f s x)
Goal: fderivWithin 𝕜 (fun y => c y • f y) s x = c x • fderivWithin 𝕜 f s x + (fderivWithin 𝕜 c s x).smulRight (f x)
-- fderiv_smul
theorem fderiv_smul (hc : DifferentiableAt 𝕜 c x) (hf : DifferentiableAt 𝕜 f x) :
    fderiv 𝕜 (fun y => c y • f y) x = c x • fderiv 𝕜 f x + (fderiv 𝕜 c x).smulRight (f x) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {x : E} {𝕜' : Type u_6} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F] {c : E → 𝕜'} (hc : DifferentiableAt 𝕜 c x) (hf : DifferentiableAt 𝕜 f x)
Goal: fderiv 𝕜 (fun y => c y • f y) x = c x • fderiv 𝕜 f x + (fderiv 𝕜 c x).smulRight (f x)
-- HasStrictFDerivAt.smul_const
@[fun_prop]
theorem HasStrictFDerivAt.smul_const (hc : HasStrictFDerivAt c c' x) (f : F) :
    HasStrictFDerivAt (fun y => c y • f) (c'.smulRight f) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {x : E} {𝕜' : Type u_6} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F] {c : E → 𝕜'} {c' : E →L[𝕜] 𝕜'} (hc : HasStrictFDerivAt c c' x) (f : F)
Goal: HasStrictFDerivAt (fun y => c y • f) (c'.smulRight f) x
-- HasFDerivWithinAt.smul_const
@[fun_prop]
theorem HasFDerivWithinAt.smul_const (hc : HasFDerivWithinAt c c' s x) (f : F) :
    HasFDerivWithinAt (fun y => c y • f) (c'.smulRight f) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {x : E} {s : Set E} {𝕜' : Type u_6} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F] {c : E → 𝕜'} {c' : E →L[𝕜] 𝕜'} (hc : HasFDerivWithinAt c c' s x) (f : F)
Goal: HasFDerivWithinAt (fun y => c y • f) (c'.smulRight f) s x
-- HasFDerivAt.smul_const
@[fun_prop]
theorem HasFDerivAt.smul_const (hc : HasFDerivAt c c' x) (f : F) :
    HasFDerivAt (fun y => c y • f) (c'.smulRight f) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {x : E} {𝕜' : Type u_6} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F] {c : E → 𝕜'} {c' : E →L[𝕜] 𝕜'} (hc : HasFDerivAt c c' x) (f : F)
Goal: HasFDerivAt (fun y => c y • f) (c'.smulRight f) x
-- DifferentiableWithinAt.smul_const
@[fun_prop]
theorem DifferentiableWithinAt.smul_const (hc : DifferentiableWithinAt 𝕜 c s x) (f : F) :
    DifferentiableWithinAt 𝕜 (fun y => c y • f) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {x : E} {s : Set E} {𝕜' : Type u_6} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F] {c : E → 𝕜'} (hc : DifferentiableWithinAt 𝕜 c s x) (f : F)
Goal: DifferentiableWithinAt 𝕜 (fun y => c y • f) s x
-- DifferentiableAt.smul_const
@[fun_prop]
theorem DifferentiableAt.smul_const (hc : DifferentiableAt 𝕜 c x) (f : F) :
    DifferentiableAt 𝕜 (fun y => c y • f) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {x : E} {𝕜' : Type u_6} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F] {c : E → 𝕜'} (hc : DifferentiableAt 𝕜 c x) (f : F)
Goal: DifferentiableAt 𝕜 (fun y => c y • f) x
-- DifferentiableOn.smul_const
@[fun_prop]
theorem DifferentiableOn.smul_const (hc : DifferentiableOn 𝕜 c s) (f : F) :
    DifferentiableOn 𝕜 (fun y => c y • f) s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {s : Set E} {𝕜' : Type u_6} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F] {c : E → 𝕜'} (hc : DifferentiableOn 𝕜 c s) (f : F)
Goal: DifferentiableOn 𝕜 (fun y => c y • f) s
-- Differentiable.smul_const
@[fun_prop]
theorem Differentiable.smul_const (hc : Differentiable 𝕜 c) (f : F) :
    Differentiable 𝕜 fun y => c y • f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {𝕜' : Type u_6} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F] {c : E → 𝕜'} (hc : Differentiable 𝕜 c) (f : F)
Goal: Differentiable 𝕜 fun y => c y • f
-- fderivWithin_smul_const
theorem fderivWithin_smul_const (hxs : UniqueDiffWithinAt 𝕜 s x)
    (hc : DifferentiableWithinAt 𝕜 c s x) (f : F) :
    fderivWithin 𝕜 (fun y => c y • f) s x = (fderivWithin 𝕜 c s x).smulRight f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {x : E} {s : Set E} {𝕜' : Type u_6} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F] {c : E → 𝕜'} (hxs : UniqueDiffWithinAt 𝕜 s x) (hc : DifferentiableWithinAt 𝕜 c s x) (f : F)
Goal: fderivWithin 𝕜 (fun y => c y • f) s x = (fderivWithin 𝕜 c s x).smulRight f
-- fderiv_smul_const
theorem fderiv_smul_const (hc : DifferentiableAt 𝕜 c x) (f : F) :
    fderiv 𝕜 (fun y => c y • f) x = (fderiv 𝕜 c x).smulRight f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {x : E} {𝕜' : Type u_6} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F] {c : E → 𝕜'} (hc : DifferentiableAt 𝕜 c x) (f : F)
Goal: fderiv 𝕜 (fun y => c y • f) x = (fderiv 𝕜 c x).smulRight f
-- HasStrictFDerivAt.mul'
@[fun_prop]
theorem HasStrictFDerivAt.mul' {x : E} (ha : HasStrictFDerivAt a a' x)
    (hb : HasStrictFDerivAt b b' x) :
    HasStrictFDerivAt (fun y => a y * b y) (a x • b' + a'.smulRight (b x)) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {𝔸 : Type u_6} [NormedRing 𝔸] [NormedAlgebra 𝕜 𝔸] {a b : E → 𝔸} {a' b' : E →L[𝕜] 𝔸} {x : E} (ha : HasStrictFDerivAt a a' x) (hb : HasStrictFDerivAt b b' x)
Goal: HasStrictFDerivAt (fun y => a y * b y) (a x • b' + a'.smulRight (b x)) x
-- HasStrictFDerivAt.mul
@[fun_prop]
theorem HasStrictFDerivAt.mul (hc : HasStrictFDerivAt c c' x) (hd : HasStrictFDerivAt d d' x) :
    HasStrictFDerivAt (fun y => c y * d y) (c x • d' + d x • c') x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {𝔸' : Type u_7} [NormedCommRing 𝔸'] [NormedAlgebra 𝕜 𝔸'] {c d : E → 𝔸'} {c' d' : E →L[𝕜] 𝔸'} (hc : HasStrictFDerivAt c c' x) (hd : HasStrictFDerivAt d d' x)
Goal: HasStrictFDerivAt (fun y => c y * d y) (c x • d' + d x • c') x
-- HasFDerivWithinAt.mul'
@[fun_prop]
theorem HasFDerivWithinAt.mul' (ha : HasFDerivWithinAt a a' s x) (hb : HasFDerivWithinAt b b' s x) :
    HasFDerivWithinAt (fun y => a y * b y) (a x • b' + a'.smulRight (b x)) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {s : Set E} {𝔸 : Type u_6} [NormedRing 𝔸] [NormedAlgebra 𝕜 𝔸] {a b : E → 𝔸} {a' b' : E →L[𝕜] 𝔸} (ha : HasFDerivWithinAt a a' s x) (hb : HasFDerivWithinAt b b' s x)
Goal: HasFDerivWithinAt (fun y => a y * b y) (a x • b' + a'.smulRight (b x)) s x