File: Mathlib/Analysis/Calculus/FDeriv/Bilinear.lean

full_name: IsBoundedBilinearMap.hasStrictFDerivAt
code: @[fun_prop]
theorem IsBoundedBilinearMap.hasStrictFDerivAt (h : IsBoundedBilinearMap 𝕜 b) (p : E × F) :
    HasStrictFDerivAt b (h.deriv p) p 
goal: HasStrictFDerivAt b (h.deriv p) p

full_name: IsBoundedBilinearMap.hasFDerivAt
code: @[fun_prop]
theorem IsBoundedBilinearMap.hasFDerivAt (h : IsBoundedBilinearMap 𝕜 b) (p : E × F) :
    HasFDerivAt b (h.deriv p) p 
goal: HasFDerivAt b (h.deriv p) p

full_name: IsBoundedBilinearMap.hasFDerivWithinAt
code: @[fun_prop]
theorem IsBoundedBilinearMap.hasFDerivWithinAt (h : IsBoundedBilinearMap 𝕜 b) (p : E × F) :
    HasFDerivWithinAt b (h.deriv p) u p 
goal: HasFDerivWithinAt b (h.deriv p) u p

full_name: IsBoundedBilinearMap.differentiableAt
code: @[fun_prop]
theorem IsBoundedBilinearMap.differentiableAt (h : IsBoundedBilinearMap 𝕜 b) (p : E × F) :
    DifferentiableAt 𝕜 b p 
goal: DifferentiableAt 𝕜 b p

full_name: IsBoundedBilinearMap.differentiableWithinAt
code: @[fun_prop]
theorem IsBoundedBilinearMap.differentiableWithinAt (h : IsBoundedBilinearMap 𝕜 b) (p : E × F) :
    DifferentiableWithinAt 𝕜 b u p 
goal: DifferentiableWithinAt 𝕜 b u p

full_name: IsBoundedBilinearMap.fderiv
code: protected theorem IsBoundedBilinearMap.fderiv (h : IsBoundedBilinearMap 𝕜 b) (p : E × F) :
    fderiv 𝕜 b p = h.deriv p 
goal: fderiv 𝕜 b p = h.deriv p

full_name: IsBoundedBilinearMap.fderivWithin
code: protected theorem IsBoundedBilinearMap.fderivWithin (h : IsBoundedBilinearMap 𝕜 b) (p : E × F)
    (hxs : UniqueDiffWithinAt 𝕜 u p) : fderivWithin 𝕜 b u p = h.deriv p 
goal: fderivWithin 𝕜 b u p = h.deriv p

full_name: IsBoundedBilinearMap.differentiable
code: @[fun_prop]
theorem IsBoundedBilinearMap.differentiable (h : IsBoundedBilinearMap 𝕜 b) : Differentiable 𝕜 b 
goal: Differentiable 𝕜 b

full_name: IsBoundedBilinearMap.differentiableOn
code: @[fun_prop]
theorem IsBoundedBilinearMap.differentiableOn (h : IsBoundedBilinearMap 𝕜 b) :
    DifferentiableOn 𝕜 b u 
goal: DifferentiableOn 𝕜 b u

full_name: ContinuousLinearMap.hasFDerivWithinAt_of_bilinear
code: @[fun_prop]
theorem ContinuousLinearMap.hasFDerivWithinAt_of_bilinear {f : G' → E} {g : G' → F}
    {f' : G' →L[𝕜] E} {g' : G' →L[𝕜] F} {x : G'} {s : Set G'} (hf : HasFDerivWithinAt f f' s x)
    (hg : HasFDerivWithinAt g g' s x) :
    HasFDerivWithinAt (fun y => B (f y) (g y))
      (B.precompR G' (f x) g' + B.precompL G' f' (g x)) s x 
goal: HasFDerivWithinAt (fun y => (B (f y)) (g y))
  (((ContinuousLinearMap.precompR G' B) (f x)) g' + ((ContinuousLinearMap.precompL G' B) f') (g x)) s x

full_name: ContinuousLinearMap.hasFDerivAt_of_bilinear
code: @[fun_prop]
theorem ContinuousLinearMap.hasFDerivAt_of_bilinear {f : G' → E} {g : G' → F} {f' : G' →L[𝕜] E}
    {g' : G' →L[𝕜] F} {x : G'} (hf : HasFDerivAt f f' x) (hg : HasFDerivAt g g' x) :
    HasFDerivAt (fun y => B (f y) (g y)) (B.precompR G' (f x) g' + B.precompL G' f' (g x)) x 
goal: HasFDerivAt (fun y => (B (f y)) (g y))
  (((ContinuousLinearMap.precompR G' B) (f x)) g' + ((ContinuousLinearMap.precompL G' B) f') (g x)) x

full_name: ContinuousLinearMap.hasStrictFDerivAt_of_bilinear
code: @[fun_prop]
theorem ContinuousLinearMap.hasStrictFDerivAt_of_bilinear
    {f : G' → E} {g : G' → F} {f' : G' →L[𝕜] E}
    {g' : G' →L[𝕜] F} {x : G'} (hf : HasStrictFDerivAt f f' x) (hg : HasStrictFDerivAt g g' x) :
    HasStrictFDerivAt (fun y => B (f y) (g y))
      (B.precompR G' (f x) g' + B.precompL G' f' (g x)) x 
goal: HasStrictFDerivAt (fun y => (B (f y)) (g y))
  (((ContinuousLinearMap.precompR G' B) (f x)) g' + ((ContinuousLinearMap.precompL G' B) f') (g x)) x

full_name: ContinuousLinearMap.fderivWithin_of_bilinear
code: theorem ContinuousLinearMap.fderivWithin_of_bilinear {f : G' → E} {g : G' → F} {x : G'} {s : Set G'}
    (hf : DifferentiableWithinAt 𝕜 f s x) (hg : DifferentiableWithinAt 𝕜 g s x)
    (hs : UniqueDiffWithinAt 𝕜 s x) :
    fderivWithin 𝕜 (fun y => B (f y) (g y)) s x =
      B.precompR G' (f x) (fderivWithin 𝕜 g s x) + B.precompL G' (fderivWithin 𝕜 f s x) (g x) 
goal: fderivWithin 𝕜 (fun y => (B (f y)) (g y)) s x =
  ((ContinuousLinearMap.precompR G' B) (f x)) (fderivWithin 𝕜 g s x) +
    ((ContinuousLinearMap.precompL G' B) (fderivWithin 𝕜 f s x)) (g x)

full_name: ContinuousLinearMap.fderiv_of_bilinear
code: theorem ContinuousLinearMap.fderiv_of_bilinear {f : G' → E} {g : G' → F} {x : G'}
    (hf : DifferentiableAt 𝕜 f x) (hg : DifferentiableAt 𝕜 g x) :
    fderiv 𝕜 (fun y => B (f y) (g y)) x =
      B.precompR G' (f x) (fderiv 𝕜 g x) + B.precompL G' (fderiv 𝕜 f x) (g x) 
goal: fderiv 𝕜 (fun y => (B (f y)) (g y)) x =
  ((ContinuousLinearMap.precompR G' B) (f x)) (fderiv 𝕜 g x) +
    ((ContinuousLinearMap.precompL G' B) (fderiv 𝕜 f x)) (g x)