ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- DifferentiableWithinAt.fst
@[fun_prop]
protected theorem DifferentiableWithinAt.fst (h : DifferentiableWithinAt 𝕜 f₂ s x) :
    DifferentiableWithinAt 𝕜 (fun x => (f₂ x).1) 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] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {x : E} {s : Set E} {f₂ : E → F × G} (h : DifferentiableWithinAt 𝕜 f₂ s x)
Goal: DifferentiableWithinAt 𝕜 (fun x => (f₂ x).1) s x
-- differentiableOn_fst
@[fun_prop]
theorem differentiableOn_fst {s : Set (E × F)} : DifferentiableOn 𝕜 Prod.fst 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 × F)}
Goal: DifferentiableOn 𝕜 Prod.fst s
-- DifferentiableOn.fst
@[fun_prop]
protected theorem DifferentiableOn.fst (h : DifferentiableOn 𝕜 f₂ s) :
    DifferentiableOn 𝕜 (fun x => (f₂ x).1) 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] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {s : Set E} {f₂ : E → F × G} (h : DifferentiableOn 𝕜 f₂ s)
Goal: DifferentiableOn 𝕜 (fun x => (f₂ x).1) s
-- fderiv_fst
theorem fderiv_fst : fderiv 𝕜 Prod.fst p = fst 𝕜 E 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] {p : E × F}
Goal: fderiv 𝕜 Prod.fst p = ContinuousLinearMap.fst 𝕜 E F
-- fderiv.fst
theorem fderiv.fst (h : DifferentiableAt 𝕜 f₂ x) :
    fderiv 𝕜 (fun x => (f₂ x).1) x = (fst 𝕜 F G).comp (fderiv 𝕜 f₂ 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] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {x : E} {f₂ : E → F × G} (h : DifferentiableAt 𝕜 f₂ x)
Goal: fderiv 𝕜 (fun x => (f₂ x).1) x = (ContinuousLinearMap.fst 𝕜 F G).comp (fderiv 𝕜 f₂ x)
-- fderivWithin_fst
theorem fderivWithin_fst {s : Set (E × F)} (hs : UniqueDiffWithinAt 𝕜 s p) :
    fderivWithin 𝕜 Prod.fst s p = fst 𝕜 E 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] {p : E × F} {s : Set (E × F)} (hs : UniqueDiffWithinAt 𝕜 s p)
Goal: fderivWithin 𝕜 Prod.fst s p = ContinuousLinearMap.fst 𝕜 E F
-- fderivWithin.fst
theorem fderivWithin.fst (hs : UniqueDiffWithinAt 𝕜 s x) (h : DifferentiableWithinAt 𝕜 f₂ s x) :
    fderivWithin 𝕜 (fun x => (f₂ x).1) s x = (fst 𝕜 F G).comp (fderivWithin 𝕜 f₂ 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] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {x : E} {s : Set E} {f₂ : E → F × G} (hs : UniqueDiffWithinAt 𝕜 s x) (h : DifferentiableWithinAt 𝕜 f₂ s x)
Goal: fderivWithin 𝕜 (fun x => (f₂ x).1) s x = (ContinuousLinearMap.fst 𝕜 F G).comp (fderivWithin 𝕜 f₂ s x)
-- hasStrictFDerivAt_snd
@[fun_prop]
theorem hasStrictFDerivAt_snd : HasStrictFDerivAt (@Prod.snd E F) (snd 𝕜 E F) p 
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] {p : E × F}
Goal: HasStrictFDerivAt Prod.snd (ContinuousLinearMap.snd 𝕜 E F) p
-- HasStrictFDerivAt.snd
@[fun_prop]
protected theorem HasStrictFDerivAt.snd (h : HasStrictFDerivAt f₂ f₂' x) :
    HasStrictFDerivAt (fun x => (f₂ x).2) ((snd 𝕜 F G).comp f₂') 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] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {x : E} {f₂ : E → F × G} {f₂' : E →L[𝕜] F × G} (h : HasStrictFDerivAt f₂ f₂' x)
Goal: HasStrictFDerivAt (fun x => (f₂ x).2) ((ContinuousLinearMap.snd 𝕜 F G).comp f₂') x
-- hasFDerivAtFilter_snd
theorem hasFDerivAtFilter_snd {L : Filter (E × F)} :
    HasFDerivAtFilter (@Prod.snd E F) (snd 𝕜 E F) p L 
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] {p : E × F} {L : Filter (E × F)}
Goal: HasFDerivAtFilter Prod.snd (ContinuousLinearMap.snd 𝕜 E F) p L
-- HasFDerivAtFilter.snd
protected theorem HasFDerivAtFilter.snd (h : HasFDerivAtFilter f₂ f₂' x L) :
    HasFDerivAtFilter (fun x => (f₂ x).2) ((snd 𝕜 F G).comp f₂') x L 
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] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {x : E} {L : Filter E} {f₂ : E → F × G} {f₂' : E →L[𝕜] F × G} (h : HasFDerivAtFilter f₂ f₂' x L)
Goal: HasFDerivAtFilter (fun x => (f₂ x).2) ((ContinuousLinearMap.snd 𝕜 F G).comp f₂') x L
-- hasFDerivAt_snd
@[fun_prop]
theorem hasFDerivAt_snd : HasFDerivAt (@Prod.snd E F) (snd 𝕜 E F) p 
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] {p : E × F}
Goal: HasFDerivAt Prod.snd (ContinuousLinearMap.snd 𝕜 E F) p
-- HasFDerivAt.snd
@[fun_prop]
protected nonrec theorem HasFDerivAt.snd (h : HasFDerivAt f₂ f₂' x) :
    HasFDerivAt (fun x => (f₂ x).2) ((snd 𝕜 F G).comp f₂') 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] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {x : E} {f₂ : E → F × G} {f₂' : E →L[𝕜] F × G} (h : HasFDerivAt f₂ f₂' x)
Goal: HasFDerivAt (fun x => (f₂ x).2) ((ContinuousLinearMap.snd 𝕜 F G).comp f₂') x
-- hasFDerivWithinAt_snd
@[fun_prop]
theorem hasFDerivWithinAt_snd {s : Set (E × F)} :
    HasFDerivWithinAt (@Prod.snd E F) (snd 𝕜 E F) s p 
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] {p : E × F} {s : Set (E × F)}
Goal: HasFDerivWithinAt Prod.snd (ContinuousLinearMap.snd 𝕜 E F) s p
-- HasFDerivWithinAt.snd
@[fun_prop]
protected nonrec theorem HasFDerivWithinAt.snd (h : HasFDerivWithinAt f₂ f₂' s x) :
    HasFDerivWithinAt (fun x => (f₂ x).2) ((snd 𝕜 F G).comp f₂') 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] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {x : E} {s : Set E} {f₂ : E → F × G} {f₂' : E →L[𝕜] F × G} (h : HasFDerivWithinAt f₂ f₂' s x)
Goal: HasFDerivWithinAt (fun x => (f₂ x).2) ((ContinuousLinearMap.snd 𝕜 F G).comp f₂') s x
-- differentiableAt_snd
@[fun_prop]
theorem differentiableAt_snd : DifferentiableAt 𝕜 Prod.snd p 
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] {p : E × F}
Goal: DifferentiableAt 𝕜 Prod.snd p
-- DifferentiableAt.snd
@[simp, fun_prop]
protected theorem DifferentiableAt.snd (h : DifferentiableAt 𝕜 f₂ x) :
    DifferentiableAt 𝕜 (fun x => (f₂ x).2) 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] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {x : E} {f₂ : E → F × G} (h : DifferentiableAt 𝕜 f₂ x)
Goal: DifferentiableAt 𝕜 (fun x => (f₂ x).2) x
-- differentiable_snd
@[fun_prop]
theorem differentiable_snd : Differentiable 𝕜 (Prod.snd : E × F → 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]
Goal: Differentiable 𝕜 Prod.snd
-- Differentiable.snd
@[simp, fun_prop]
protected theorem Differentiable.snd (h : Differentiable 𝕜 f₂) :
    Differentiable 𝕜 fun x => (f₂ x).2 
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] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f₂ : E → F × G} (h : Differentiable 𝕜 f₂)
Goal: Differentiable 𝕜 fun x => (f₂ x).2
-- differentiableWithinAt_snd
@[fun_prop]
theorem differentiableWithinAt_snd {s : Set (E × F)} : DifferentiableWithinAt 𝕜 Prod.snd s p 
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] {p : E × F} {s : Set (E × F)}
Goal: DifferentiableWithinAt 𝕜 Prod.snd s p
-- DifferentiableWithinAt.snd
@[fun_prop]
protected theorem DifferentiableWithinAt.snd (h : DifferentiableWithinAt 𝕜 f₂ s x) :
    DifferentiableWithinAt 𝕜 (fun x => (f₂ x).2) 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] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {x : E} {s : Set E} {f₂ : E → F × G} (h : DifferentiableWithinAt 𝕜 f₂ s x)
Goal: DifferentiableWithinAt 𝕜 (fun x => (f₂ x).2) s x
-- differentiableOn_snd
@[fun_prop]
theorem differentiableOn_snd {s : Set (E × F)} : DifferentiableOn 𝕜 Prod.snd 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 × F)}
Goal: DifferentiableOn 𝕜 Prod.snd s
-- DifferentiableOn.snd
@[fun_prop]
protected theorem DifferentiableOn.snd (h : DifferentiableOn 𝕜 f₂ s) :
    DifferentiableOn 𝕜 (fun x => (f₂ x).2) 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] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {s : Set E} {f₂ : E → F × G} (h : DifferentiableOn 𝕜 f₂ s)
Goal: DifferentiableOn 𝕜 (fun x => (f₂ x).2) s
-- fderiv_snd
theorem fderiv_snd : fderiv 𝕜 Prod.snd p = snd 𝕜 E 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] {p : E × F}
Goal: fderiv 𝕜 Prod.snd p = ContinuousLinearMap.snd 𝕜 E F
-- fderiv.snd
theorem fderiv.snd (h : DifferentiableAt 𝕜 f₂ x) :
    fderiv 𝕜 (fun x => (f₂ x).2) x = (snd 𝕜 F G).comp (fderiv 𝕜 f₂ 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] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {x : E} {f₂ : E → F × G} (h : DifferentiableAt 𝕜 f₂ x)
Goal: fderiv 𝕜 (fun x => (f₂ x).2) x = (ContinuousLinearMap.snd 𝕜 F G).comp (fderiv 𝕜 f₂ x)
-- fderivWithin_snd
theorem fderivWithin_snd {s : Set (E × F)} (hs : UniqueDiffWithinAt 𝕜 s p) :
    fderivWithin 𝕜 Prod.snd s p = snd 𝕜 E 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] {p : E × F} {s : Set (E × F)} (hs : UniqueDiffWithinAt 𝕜 s p)
Goal: fderivWithin 𝕜 Prod.snd s p = ContinuousLinearMap.snd 𝕜 E F
-- fderivWithin.snd
theorem fderivWithin.snd (hs : UniqueDiffWithinAt 𝕜 s x) (h : DifferentiableWithinAt 𝕜 f₂ s x) :
    fderivWithin 𝕜 (fun x => (f₂ x).2) s x = (snd 𝕜 F G).comp (fderivWithin 𝕜 f₂ 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] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {x : E} {s : Set E} {f₂ : E → F × G} (hs : UniqueDiffWithinAt 𝕜 s x) (h : DifferentiableWithinAt 𝕜 f₂ s x)
Goal: fderivWithin 𝕜 (fun x => (f₂ x).2) s x = (ContinuousLinearMap.snd 𝕜 F G).comp (fderivWithin 𝕜 f₂ s x)
-- HasStrictFDerivAt.prodMap
@[fun_prop]
protected theorem HasStrictFDerivAt.prodMap (hf : HasStrictFDerivAt f f' p.1)
    (hf₂ : HasStrictFDerivAt f₂ f₂' p.2) : HasStrictFDerivAt (Prod.map f f₂) (f'.prodMap f₂') p 
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] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {G' : Type u_5} [NormedAddCommGroup G'] [NormedSpace 𝕜 G'] {f : E → F} {f' : E →L[𝕜] F} {f₂ : G → G'} {f₂' : G →L[𝕜] G'} (p : E × G) (hf : HasStrictFDerivAt f f' p.1) (hf₂ : HasStrictFDerivAt f₂ f₂' p.2)
Goal: HasStrictFDerivAt (Prod.map f f₂) (f'.prodMap f₂') p
-- HasFDerivAt.prodMap
@[fun_prop]
protected theorem HasFDerivAt.prodMap (hf : HasFDerivAt f f' p.1) (hf₂ : HasFDerivAt f₂ f₂' p.2) :
    HasFDerivAt (Prod.map f f₂) (f'.prodMap f₂') p 
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] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {G' : Type u_5} [NormedAddCommGroup G'] [NormedSpace 𝕜 G'] {f : E → F} {f' : E →L[𝕜] F} {f₂ : G → G'} {f₂' : G →L[𝕜] G'} (p : E × G) (hf : HasFDerivAt f f' p.1) (hf₂ : HasFDerivAt f₂ f₂' p.2)
Goal: HasFDerivAt (Prod.map f f₂) (f'.prodMap f₂') p
-- DifferentiableAt.prod_map
@[simp, fun_prop]
protected theorem DifferentiableAt.prod_map (hf : DifferentiableAt 𝕜 f p.1)
    (hf₂ : DifferentiableAt 𝕜 f₂ p.2) : DifferentiableAt 𝕜 (fun p : E × G => (f p.1, f₂ p.2)) p 
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] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {G' : Type u_5} [NormedAddCommGroup G'] [NormedSpace 𝕜 G'] {f : E → F} {f₂ : G → G'} (p : E × G) (hf : DifferentiableAt 𝕜 f p.1) (hf₂ : DifferentiableAt 𝕜 f₂ p.2)
Goal: DifferentiableAt 𝕜 (fun p => (f p.1, f₂ p.2)) p
-- hasStrictFDerivAt_pi'
@[simp]
theorem hasStrictFDerivAt_pi' :
    HasStrictFDerivAt Φ Φ' x ↔ ∀ i, HasStrictFDerivAt (fun x => Φ x i) ((proj i).comp Φ') 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] {F' : ι → Type u_7} [(i : ι) → NormedAddCommGroup (F' i)] [(i : ι) → NormedSpace 𝕜 (F' i)] {Φ : E → (i : ι) → F' i} {Φ' : E →L[𝕜] (i : ι) → F' i}
Goal: HasStrictFDerivAt Φ Φ' x ↔ ∀ (i : ι), HasStrictFDerivAt (fun x => Φ x i) ((ContinuousLinearMap.proj i).comp Φ') x
-- hasStrictFDerivAt_pi''
@[fun_prop]
theorem hasStrictFDerivAt_pi'' (hφ : ∀ i, HasStrictFDerivAt (fun x => Φ x i) ((proj i).comp Φ') x) :
    HasStrictFDerivAt Φ Φ' 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] {F' : ι → Type u_7} [(i : ι) → NormedAddCommGroup (F' i)] [(i : ι) → NormedSpace 𝕜 (F' i)] {Φ : E → (i : ι) → F' i} {Φ' : E →L[𝕜] (i : ι) → F' i} (hφ : ∀ (i : ι), HasStrictFDerivAt (fun x => Φ x i) ((ContinuousLinearMap.proj i).comp Φ') x)
Goal: HasStrictFDerivAt Φ Φ' x
-- hasStrictFDerivAt_apply
@[fun_prop]
theorem hasStrictFDerivAt_apply (i : ι) (f : ∀ i, F' i) :
    HasStrictFDerivAt (𝕜 := 𝕜) (fun f : ∀ i, F' i => f i) (proj i) f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {ι : Type u_6} [Fintype ι] {F' : ι → Type u_7} [(i : ι) → NormedAddCommGroup (F' i)] [(i : ι) → NormedSpace 𝕜 (F' i)] (i : ι) (f : (i : ι) → F' i)
Goal: HasStrictFDerivAt (fun f => f i) (ContinuousLinearMap.proj i) f
-- hasStrictFDerivAt_pi
@[simp 1100] -- Porting note: increased priority to make lint happy
theorem hasStrictFDerivAt_pi :
    HasStrictFDerivAt (fun x i => φ i x) (ContinuousLinearMap.pi φ') x ↔
      ∀ i, HasStrictFDerivAt (φ i) (φ' i) 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] {F' : ι → Type u_7} [(i : ι) → NormedAddCommGroup (F' i)] [(i : ι) → NormedSpace 𝕜 (F' i)] {φ : (i : ι) → E → F' i} {φ' : (i : ι) → E →L[𝕜] F' i}
Goal: HasStrictFDerivAt (fun x i => φ i x) (ContinuousLinearMap.pi φ') x ↔ ∀ (i : ι), HasStrictFDerivAt (φ i) (φ' i) x
-- hasFDerivAtFilter_pi'
@[simp]
theorem hasFDerivAtFilter_pi' :
    HasFDerivAtFilter Φ Φ' x L ↔
      ∀ i, HasFDerivAtFilter (fun x => Φ x i) ((proj i).comp Φ') x L 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {L : Filter E} {ι : Type u_6} [Fintype ι] {F' : ι → Type u_7} [(i : ι) → NormedAddCommGroup (F' i)] [(i : ι) → NormedSpace 𝕜 (F' i)] {Φ : E → (i : ι) → F' i} {Φ' : E →L[𝕜] (i : ι) → F' i}
Goal: HasFDerivAtFilter Φ Φ' x L ↔ ∀ (i : ι), HasFDerivAtFilter (fun x => Φ x i) ((ContinuousLinearMap.proj i).comp Φ') x L
-- hasFDerivAtFilter_pi
theorem hasFDerivAtFilter_pi :
    HasFDerivAtFilter (fun x i => φ i x) (ContinuousLinearMap.pi φ') x L ↔
      ∀ i, HasFDerivAtFilter (φ i) (φ' i) x L 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {L : Filter E} {ι : Type u_6} [Fintype ι] {F' : ι → Type u_7} [(i : ι) → NormedAddCommGroup (F' i)] [(i : ι) → NormedSpace 𝕜 (F' i)] {φ : (i : ι) → E → F' i} {φ' : (i : ι) → E →L[𝕜] F' i}
Goal: HasFDerivAtFilter (fun x i => φ i x) (ContinuousLinearMap.pi φ') x L ↔ ∀ (i : ι), HasFDerivAtFilter (φ i) (φ' i) x L
-- hasFDerivAt_pi'
@[simp]
theorem hasFDerivAt_pi' :
    HasFDerivAt Φ Φ' x ↔ ∀ i, HasFDerivAt (fun x => Φ x i) ((proj i).comp Φ') 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] {F' : ι → Type u_7} [(i : ι) → NormedAddCommGroup (F' i)] [(i : ι) → NormedSpace 𝕜 (F' i)] {Φ : E → (i : ι) → F' i} {Φ' : E →L[𝕜] (i : ι) → F' i}
Goal: HasFDerivAt Φ Φ' x ↔ ∀ (i : ι), HasFDerivAt (fun x => Φ x i) ((ContinuousLinearMap.proj i).comp Φ') x
-- hasFDerivAt_pi''
@[fun_prop]
theorem hasFDerivAt_pi'' (hφ : ∀ i, HasFDerivAt (fun x => Φ x i) ((proj i).comp Φ') x) :
    HasFDerivAt Φ Φ' 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] {F' : ι → Type u_7} [(i : ι) → NormedAddCommGroup (F' i)] [(i : ι) → NormedSpace 𝕜 (F' i)] {Φ : E → (i : ι) → F' i} {Φ' : E →L[𝕜] (i : ι) → F' i} (hφ : ∀ (i : ι), HasFDerivAt (fun x => Φ x i) ((ContinuousLinearMap.proj i).comp Φ') x)
Goal: HasFDerivAt Φ Φ' x
-- hasFDerivAt_apply
@[fun_prop]
theorem hasFDerivAt_apply (i : ι) (f : ∀ i, F' i) :
    HasFDerivAt (𝕜 := 𝕜) (fun f : ∀ i, F' i => f i) (proj i) f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {ι : Type u_6} [Fintype ι] {F' : ι → Type u_7} [(i : ι) → NormedAddCommGroup (F' i)] [(i : ι) → NormedSpace 𝕜 (F' i)] (i : ι) (f : (i : ι) → F' i)
Goal: HasFDerivAt (fun f => f i) (ContinuousLinearMap.proj i) f
-- hasFDerivAt_pi
theorem hasFDerivAt_pi :
    HasFDerivAt (fun x i => φ i x) (ContinuousLinearMap.pi φ') x ↔
      ∀ i, HasFDerivAt (φ i) (φ' i) 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] {F' : ι → Type u_7} [(i : ι) → NormedAddCommGroup (F' i)] [(i : ι) → NormedSpace 𝕜 (F' i)] {φ : (i : ι) → E → F' i} {φ' : (i : ι) → E →L[𝕜] F' i}
Goal: HasFDerivAt (fun x i => φ i x) (ContinuousLinearMap.pi φ') x ↔ ∀ (i : ι), HasFDerivAt (φ i) (φ' i) x
-- hasFDerivWithinAt_pi'
@[simp]
theorem hasFDerivWithinAt_pi' :
    HasFDerivWithinAt Φ Φ' s x ↔ ∀ i, HasFDerivWithinAt (fun x => Φ x i) ((proj i).comp Φ') 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] {F' : ι → Type u_7} [(i : ι) → NormedAddCommGroup (F' i)] [(i : ι) → NormedSpace 𝕜 (F' i)] {Φ : E → (i : ι) → F' i} {Φ' : E →L[𝕜] (i : ι) → F' i}
Goal: HasFDerivWithinAt Φ Φ' s x ↔ ∀ (i : ι), HasFDerivWithinAt (fun x => Φ x i) ((ContinuousLinearMap.proj i).comp Φ') s x
-- hasFDerivWithinAt_pi''
@[fun_prop]
theorem hasFDerivWithinAt_pi''
    (hφ : ∀ i, HasFDerivWithinAt (fun x => Φ x i) ((proj i).comp Φ') s x) :
    HasFDerivWithinAt Φ Φ' 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] {F' : ι → Type u_7} [(i : ι) → NormedAddCommGroup (F' i)] [(i : ι) → NormedSpace 𝕜 (F' i)] {Φ : E → (i : ι) → F' i} {Φ' : E →L[𝕜] (i : ι) → F' i} (hφ : ∀ (i : ι), HasFDerivWithinAt (fun x => Φ x i) ((ContinuousLinearMap.proj i).comp Φ') s x)
Goal: HasFDerivWithinAt Φ Φ' s x
-- hasFDerivWithinAt_apply
@[fun_prop]
theorem hasFDerivWithinAt_apply (i : ι) (f : ∀ i, F' i) (s' : Set (∀ i, F' i)) :
    HasFDerivWithinAt (𝕜 := 𝕜) (fun f : ∀ i, F' i => f i) (proj i) s' f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {ι : Type u_6} [Fintype ι] {F' : ι → Type u_7} [(i : ι) → NormedAddCommGroup (F' i)] [(i : ι) → NormedSpace 𝕜 (F' i)] (i : ι) (f : (i : ι) → F' i) (s' : Set ((i : ι) → F' i))
Goal: HasFDerivWithinAt (fun f => f i) (ContinuousLinearMap.proj i) s' f
-- hasFDerivWithinAt_pi
theorem hasFDerivWithinAt_pi :
    HasFDerivWithinAt (fun x i => φ i x) (ContinuousLinearMap.pi φ') s x ↔
      ∀ i, HasFDerivWithinAt (φ i) (φ' i) 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] {F' : ι → Type u_7} [(i : ι) → NormedAddCommGroup (F' i)] [(i : ι) → NormedSpace 𝕜 (F' i)] {φ : (i : ι) → E → F' i} {φ' : (i : ι) → E →L[𝕜] F' i}
Goal: HasFDerivWithinAt (fun x i => φ i x) (ContinuousLinearMap.pi φ') s x ↔ ∀ (i : ι), HasFDerivWithinAt (φ i) (φ' i) s x
-- differentiableWithinAt_pi
@[simp]
theorem differentiableWithinAt_pi :
    DifferentiableWithinAt 𝕜 Φ s x ↔ ∀ i, DifferentiableWithinAt 𝕜 (fun x => Φ x i) 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] {F' : ι → Type u_7} [(i : ι) → NormedAddCommGroup (F' i)] [(i : ι) → NormedSpace 𝕜 (F' i)] {Φ : E → (i : ι) → F' i}
Goal: DifferentiableWithinAt 𝕜 Φ s x ↔ ∀ (i : ι), DifferentiableWithinAt 𝕜 (fun x => Φ x i) s x
-- differentiableWithinAt_pi''
@[fun_prop]
theorem differentiableWithinAt_pi'' (hφ : ∀ i, DifferentiableWithinAt 𝕜 (fun x => Φ x i) s x) :
    DifferentiableWithinAt 𝕜 Φ 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] {F' : ι → Type u_7} [(i : ι) → NormedAddCommGroup (F' i)] [(i : ι) → NormedSpace 𝕜 (F' i)] {Φ : E → (i : ι) → F' i} (hφ : ∀ (i : ι), DifferentiableWithinAt 𝕜 (fun x => Φ x i) s x)
Goal: DifferentiableWithinAt 𝕜 Φ s x
-- differentiableWithinAt_apply
@[fun_prop]
theorem differentiableWithinAt_apply (i : ι) (f : ∀ i, F' i) (s' : Set (∀ i, F' i)) :
    DifferentiableWithinAt (𝕜 := 𝕜) (fun f : ∀ i, F' i => f i) s' f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {ι : Type u_6} [Fintype ι] {F' : ι → Type u_7} [(i : ι) → NormedAddCommGroup (F' i)] [(i : ι) → NormedSpace 𝕜 (F' i)] (i : ι) (f : (i : ι) → F' i) (s' : Set ((i : ι) → F' i))
Goal: DifferentiableWithinAt 𝕜 (fun f => f i) s' f
-- differentiableAt_pi
@[simp]
theorem differentiableAt_pi : DifferentiableAt 𝕜 Φ x ↔ ∀ i, DifferentiableAt 𝕜 (fun x => Φ x i) 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] {F' : ι → Type u_7} [(i : ι) → NormedAddCommGroup (F' i)] [(i : ι) → NormedSpace 𝕜 (F' i)] {Φ : E → (i : ι) → F' i}
Goal: DifferentiableAt 𝕜 Φ x ↔ ∀ (i : ι), DifferentiableAt 𝕜 (fun x => Φ x i) x