ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- differentiableOn_const_sub_iff
@[deprecated DifferentiableOn.sub_iff_right (since := "2024-07-11")]
theorem differentiableOn_const_sub_iff (c : F) :
    DifferentiableOn 𝕜 (fun y => c - f y) s ↔ DifferentiableOn 𝕜 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] {f : E → F} {s : Set E} (c : F)
Goal: DifferentiableOn 𝕜 (fun y => c - f y) s ↔ DifferentiableOn 𝕜 f s
-- Differentiable.const_sub
@[fun_prop]
theorem Differentiable.const_sub (hf : Differentiable 𝕜 f) (c : F) :
    Differentiable 𝕜 fun y => c - 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} (hf : Differentiable 𝕜 f) (c : F)
Goal: Differentiable 𝕜 fun y => c - f y
-- differentiable_const_sub_iff
@[deprecated Differentiable.sub_iff_right (since := "2024-07-11")]
theorem differentiable_const_sub_iff (c : F) :
    (Differentiable 𝕜 fun y => c - f y) ↔ Differentiable 𝕜 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] {f : E → F} (c : F)
Goal: (Differentiable 𝕜 fun y => c - f y) ↔ Differentiable 𝕜 f
-- fderivWithin_const_sub
theorem fderivWithin_const_sub (hxs : UniqueDiffWithinAt 𝕜 s x) (c : F) :
    fderivWithin 𝕜 (fun y => c - f y) s x = -fderivWithin 𝕜 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] {f : E → F} {x : E} {s : Set E} (hxs : UniqueDiffWithinAt 𝕜 s x) (c : F)
Goal: fderivWithin 𝕜 (fun y => c - f y) s x = -fderivWithin 𝕜 f s x
-- fderiv_const_sub
theorem fderiv_const_sub (c : F) : fderiv 𝕜 (fun y => c - f y) x = -fderiv 𝕜 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} (c : F)
Goal: fderiv 𝕜 (fun y => c - f y) x = -fderiv 𝕜 f x
-- TietzeExtension.of_tvs
theorem TietzeExtension.of_tvs (𝕜 : Type v) [NontriviallyNormedField 𝕜] {E : Type w}
    [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousSMul 𝕜 E]
    [T2Space E] [FiniteDimensional 𝕜 E] [CompleteSpace 𝕜] [TietzeExtension.{u, v} 𝕜] :
    TietzeExtension.{u, w} E 
Arguments: (𝕜 : Type v) [NontriviallyNormedField 𝕜] {E : Type w} [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousSMul 𝕜 E] [T2Space E] [FiniteDimensional 𝕜 E] [CompleteSpace 𝕜] [TietzeExtension 𝕜]
Goal: TietzeExtension E
-- Metric.instTietzeExtensionBall
theorem Metric.instTietzeExtensionBall {𝕜 : Type v} [RCLike 𝕜] {E : Type w}
    [NormedAddCommGroup E] [NormedSpace 𝕜 E] [FiniteDimensional 𝕜 E] {r : ℝ} (hr : 0 < r) :
    TietzeExtension.{u, w} (Metric.ball (0 : E) r) 
Arguments: {𝕜 : Type v} [RCLike 𝕜] {E : Type w} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [FiniteDimensional 𝕜 E] {r : ℝ} (hr : 0 < r)
Goal: TietzeExtension ↑(Metric.ball 0 r)
-- Metric.instTietzeExtensionClosedBall
theorem Metric.instTietzeExtensionClosedBall (𝕜 : Type v) [RCLike 𝕜] {E : Type w}
    [NormedAddCommGroup E] [NormedSpace 𝕜 E] [FiniteDimensional 𝕜 E] (y : E) {r : ℝ} (hr : 0 < r) :
    TietzeExtension.{u, w} (Metric.closedBall y r) 
Arguments: (𝕜 : Type v) [RCLike 𝕜] {E : Type w} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [FiniteDimensional 𝕜 E] (y : E) {r : ℝ} (hr : 0 < r)
Goal: TietzeExtension ↑(Metric.closedBall y r)
-- BoundedContinuousFunction.exists_norm_eq_restrict_eq
theorem exists_norm_eq_restrict_eq (f : s →ᵇ E) :
    ∃ g : X →ᵇ E, ‖g‖ = ‖f‖ ∧ g.restrict s = f 
Arguments: {X : Type u} [TopologicalSpace X] [NormalSpace X] {s : Set X} (hs : IsClosed s) (𝕜 : Type v) [RCLike 𝕜] {E : Type w} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [FiniteDimensional 𝕜 E] (f : BoundedContinuousFunction (↑s) E)
Goal: ∃ g, ‖g‖ = ‖f‖ ∧ g.restrict s = f
-- UpperHalfPlane.smooth_coe
theorem smooth_coe : Smooth 𝓘(ℂ) 𝓘(ℂ) ((↑) : ℍ → ℂ) 
Goal: Smooth 𝓘(ℂ, ℂ) 𝓘(ℂ, ℂ) UpperHalfPlane.coe
-- UpperHalfPlane.mdifferentiable_coe
theorem mdifferentiable_coe : MDifferentiable 𝓘(ℂ) 𝓘(ℂ) ((↑) : ℍ → ℂ) 
Goal: MDifferentiable 𝓘(ℂ, ℂ) 𝓘(ℂ, ℂ) UpperHalfPlane.coe
-- UpperHalfPlane.abs_exp_two_pi_I_lt_one
theorem UpperHalfPlane.abs_exp_two_pi_I_lt_one (z : ℍ) :
    ‖(Complex.exp (2 * π * Complex.I * z))‖ < 1 
Arguments: (z : ℍ)
Goal: ‖cexp (2 * ↑π * Complex.I * ↑z)‖ < 1
-- ContinuousLinearEquiv.hasStrictFDerivAt
@[fun_prop]
protected theorem hasStrictFDerivAt : HasStrictFDerivAt iso (iso : E →L[𝕜] 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} (iso : E ≃L[𝕜] F)
Goal: HasStrictFDerivAt (⇑iso) (↑iso) x
-- ContinuousLinearEquiv.hasFDerivWithinAt
@[fun_prop]
protected theorem hasFDerivWithinAt : HasFDerivWithinAt iso (iso : E →L[𝕜] 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} (iso : E ≃L[𝕜] F)
Goal: HasFDerivWithinAt (⇑iso) (↑iso) s x
-- ContinuousLinearEquiv.hasFDerivAt
@[fun_prop]
protected theorem hasFDerivAt : HasFDerivAt iso (iso : E →L[𝕜] 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} (iso : E ≃L[𝕜] F)
Goal: HasFDerivAt (⇑iso) (↑iso) x
-- ContinuousLinearEquiv.differentiableAt
@[fun_prop]
protected theorem differentiableAt : DifferentiableAt 𝕜 iso 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} (iso : E ≃L[𝕜] F)
Goal: DifferentiableAt 𝕜 (⇑iso) x
-- ContinuousLinearEquiv.differentiableWithinAt
@[fun_prop]
protected theorem differentiableWithinAt : DifferentiableWithinAt 𝕜 iso 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} (iso : E ≃L[𝕜] F)
Goal: DifferentiableWithinAt 𝕜 (⇑iso) s x
-- ContinuousLinearEquiv.fderiv
protected theorem fderiv : fderiv 𝕜 iso x = iso 
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} (iso : E ≃L[𝕜] F)
Goal: fderiv 𝕜 (⇑iso) x = ↑iso
-- ContinuousLinearEquiv.fderivWithin
protected theorem fderivWithin (hxs : UniqueDiffWithinAt 𝕜 s x) : fderivWithin 𝕜 iso s x = iso 
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} (iso : E ≃L[𝕜] F) (hxs : UniqueDiffWithinAt 𝕜 s x)
Goal: fderivWithin 𝕜 (⇑iso) s x = ↑iso
-- ContinuousLinearEquiv.differentiable
@[fun_prop]
protected theorem differentiable : Differentiable 𝕜 iso 
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] (iso : E ≃L[𝕜] F)
Goal: Differentiable 𝕜 ⇑iso
-- ContinuousLinearEquiv.differentiableOn
@[fun_prop]
protected theorem differentiableOn : DifferentiableOn 𝕜 iso 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} (iso : E ≃L[𝕜] F)
Goal: DifferentiableOn 𝕜 (⇑iso) s
-- ContinuousLinearEquiv.comp_differentiableWithinAt_iff
theorem comp_differentiableWithinAt_iff {f : G → E} {s : Set G} {x : G} :
    DifferentiableWithinAt 𝕜 (iso ∘ f) s x ↔ DifferentiableWithinAt 𝕜 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] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] (iso : E ≃L[𝕜] F) {f : G → E} {s : Set G} {x : G}
Goal: DifferentiableWithinAt 𝕜 (⇑iso ∘ f) s x ↔ DifferentiableWithinAt 𝕜 f s x
-- ContinuousLinearEquiv.comp_differentiableAt_iff
theorem comp_differentiableAt_iff {f : G → E} {x : G} :
    DifferentiableAt 𝕜 (iso ∘ f) x ↔ DifferentiableAt 𝕜 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] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] (iso : E ≃L[𝕜] F) {f : G → E} {x : G}
Goal: DifferentiableAt 𝕜 (⇑iso ∘ f) x ↔ DifferentiableAt 𝕜 f x
-- ContinuousLinearEquiv.comp_differentiableOn_iff
theorem comp_differentiableOn_iff {f : G → E} {s : Set G} :
    DifferentiableOn 𝕜 (iso ∘ f) s ↔ DifferentiableOn 𝕜 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] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] (iso : E ≃L[𝕜] F) {f : G → E} {s : Set G}
Goal: DifferentiableOn 𝕜 (⇑iso ∘ f) s ↔ DifferentiableOn 𝕜 f s
-- ContinuousLinearEquiv.comp_differentiable_iff
theorem comp_differentiable_iff {f : G → E} : Differentiable 𝕜 (iso ∘ f) ↔ Differentiable 𝕜 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] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] (iso : E ≃L[𝕜] F) {f : G → E}
Goal: Differentiable 𝕜 (⇑iso ∘ f) ↔ Differentiable 𝕜 f
-- ContinuousLinearEquiv.comp_hasFDerivWithinAt_iff
theorem comp_hasFDerivWithinAt_iff {f : G → E} {s : Set G} {x : G} {f' : G →L[𝕜] E} :
    HasFDerivWithinAt (iso ∘ f) ((iso : E →L[𝕜] F).comp f') s x ↔ HasFDerivWithinAt f 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] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] (iso : E ≃L[𝕜] F) {f : G → E} {s : Set G} {x : G} {f' : G →L[𝕜] E}
Goal: HasFDerivWithinAt (⇑iso ∘ f) ((↑iso).comp f') s x ↔ HasFDerivWithinAt f f' s x
-- ContinuousLinearEquiv.comp_hasStrictFDerivAt_iff
theorem comp_hasStrictFDerivAt_iff {f : G → E} {x : G} {f' : G →L[𝕜] E} :
    HasStrictFDerivAt (iso ∘ f) ((iso : E →L[𝕜] F).comp f') x ↔ HasStrictFDerivAt f 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] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] (iso : E ≃L[𝕜] F) {f : G → E} {x : G} {f' : G →L[𝕜] E}
Goal: HasStrictFDerivAt (⇑iso ∘ f) ((↑iso).comp f') x ↔ HasStrictFDerivAt f f' x
-- ContinuousLinearEquiv.comp_hasFDerivAt_iff
theorem comp_hasFDerivAt_iff {f : G → E} {x : G} {f' : G →L[𝕜] E} :
    HasFDerivAt (iso ∘ f) ((iso : E →L[𝕜] F).comp f') x ↔ HasFDerivAt f 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] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] (iso : E ≃L[𝕜] F) {f : G → E} {x : G} {f' : G →L[𝕜] E}
Goal: HasFDerivAt (⇑iso ∘ f) ((↑iso).comp f') x ↔ HasFDerivAt f f' x
-- ContinuousLinearEquiv.comp_hasFDerivWithinAt_iff'
theorem comp_hasFDerivWithinAt_iff' {f : G → E} {s : Set G} {x : G} {f' : G →L[𝕜] F} :
    HasFDerivWithinAt (iso ∘ f) f' s x ↔
      HasFDerivWithinAt f ((iso.symm : F →L[𝕜] E).comp 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] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] (iso : E ≃L[𝕜] F) {f : G → E} {s : Set G} {x : G} {f' : G →L[𝕜] F}
Goal: HasFDerivWithinAt (⇑iso ∘ f) f' s x ↔ HasFDerivWithinAt f ((↑iso.symm).comp f') s x
-- ContinuousLinearEquiv.comp_hasFDerivAt_iff'
theorem comp_hasFDerivAt_iff' {f : G → E} {x : G} {f' : G →L[𝕜] F} :
    HasFDerivAt (iso ∘ f) f' x ↔ HasFDerivAt f ((iso.symm : F →L[𝕜] E).comp 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] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] (iso : E ≃L[𝕜] F) {f : G → E} {x : G} {f' : G →L[𝕜] F}
Goal: HasFDerivAt (⇑iso ∘ f) f' x ↔ HasFDerivAt f ((↑iso.symm).comp f') x
-- ContinuousLinearEquiv.comp_fderivWithin
theorem comp_fderivWithin {f : G → E} {s : Set G} {x : G} (hxs : UniqueDiffWithinAt 𝕜 s x) :
    fderivWithin 𝕜 (iso ∘ f) s x = (iso : E →L[𝕜] F).comp (fderivWithin 𝕜 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] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] (iso : E ≃L[𝕜] F) {f : G → E} {s : Set G} {x : G} (hxs : UniqueDiffWithinAt 𝕜 s x)
Goal: fderivWithin 𝕜 (⇑iso ∘ f) s x = (↑iso).comp (fderivWithin 𝕜 f s x)
-- ContinuousLinearEquiv.comp_fderiv
theorem comp_fderiv {f : G → E} {x : G} :
    fderiv 𝕜 (iso ∘ f) x = (iso : E →L[𝕜] F).comp (fderiv 𝕜 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] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] (iso : E ≃L[𝕜] F) {f : G → E} {x : G}
Goal: fderiv 𝕜 (⇑iso ∘ f) x = (↑iso).comp (fderiv 𝕜 f x)
-- fderivWithin_continuousLinearEquiv_comp
lemma _root_.fderivWithin_continuousLinearEquiv_comp (L : G ≃L[𝕜] G') (f : E → (F →L[𝕜] G))
    (hs : UniqueDiffWithinAt 𝕜 s x) :
    fderivWithin 𝕜 (fun x ↦ (L : G →L[𝕜] G').comp (f x)) s x =
      (((ContinuousLinearEquiv.refl 𝕜 F).arrowCongr L)) ∘L (fderivWithin 𝕜 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] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {G' : Type u_5} [NormedAddCommGroup G'] [NormedSpace 𝕜 G'] {x : E} {s : Set E} (L : G ≃L[𝕜] G') (f : E → F →L[𝕜] G) (hs : UniqueDiffWithinAt 𝕜 s x)
Goal: fderivWithin 𝕜 (fun x => (↑L).comp (f x)) s x =
  (↑((ContinuousLinearEquiv.refl 𝕜 F).arrowCongr L)).comp (fderivWithin 𝕜 f s x)
-- fderiv_continuousLinearEquiv_comp
lemma _root_.fderiv_continuousLinearEquiv_comp (L : G ≃L[𝕜] G') (f : E → (F →L[𝕜] G)) (x : E) :
    fderiv 𝕜 (fun x ↦ (L : G →L[𝕜] G').comp (f x)) x =
      (((ContinuousLinearEquiv.refl 𝕜 F).arrowCongr L)) ∘L (fderiv 𝕜 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] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {G' : Type u_5} [NormedAddCommGroup G'] [NormedSpace 𝕜 G'] (L : G ≃L[𝕜] G') (f : E → F →L[𝕜] G) (x : E)
Goal: fderiv 𝕜 (fun x => (↑L).comp (f x)) x = (↑((ContinuousLinearEquiv.refl 𝕜 F).arrowCongr L)).comp (fderiv 𝕜 f x)
-- fderiv_continuousLinearEquiv_comp'
lemma _root_.fderiv_continuousLinearEquiv_comp' (L : G ≃L[𝕜] G') (f : E → (F →L[𝕜] G)) :
    fderiv 𝕜 (fun x ↦ (L : G →L[𝕜] G').comp (f x)) =
      fun x ↦ (((ContinuousLinearEquiv.refl 𝕜 F).arrowCongr L)) ∘L (fderiv 𝕜 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] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {G' : Type u_5} [NormedAddCommGroup G'] [NormedSpace 𝕜 G'] (L : G ≃L[𝕜] G') (f : E → F →L[𝕜] G)
Goal: (fderiv 𝕜 fun x => (↑L).comp (f x)) = fun x => (↑((ContinuousLinearEquiv.refl 𝕜 F).arrowCongr L)).comp (fderiv 𝕜 f x)
-- ContinuousLinearEquiv.comp_right_differentiableWithinAt_iff
theorem comp_right_differentiableWithinAt_iff {f : F → G} {s : Set F} {x : E} :
    DifferentiableWithinAt 𝕜 (f ∘ iso) (iso ⁻¹' s) x ↔ DifferentiableWithinAt 𝕜 f s (iso 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] (iso : E ≃L[𝕜] F) {f : F → G} {s : Set F} {x : E}
Goal: DifferentiableWithinAt 𝕜 (f ∘ ⇑iso) (⇑iso ⁻¹' s) x ↔ DifferentiableWithinAt 𝕜 f s (iso x)
-- ContinuousLinearEquiv.comp_right_differentiableAt_iff
theorem comp_right_differentiableAt_iff {f : F → G} {x : E} :
    DifferentiableAt 𝕜 (f ∘ iso) x ↔ DifferentiableAt 𝕜 f (iso 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] (iso : E ≃L[𝕜] F) {f : F → G} {x : E}
Goal: DifferentiableAt 𝕜 (f ∘ ⇑iso) x ↔ DifferentiableAt 𝕜 f (iso x)
-- ContinuousLinearEquiv.comp_right_differentiableOn_iff
theorem comp_right_differentiableOn_iff {f : F → G} {s : Set F} :
    DifferentiableOn 𝕜 (f ∘ iso) (iso ⁻¹' s) ↔ DifferentiableOn 𝕜 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] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] (iso : E ≃L[𝕜] F) {f : F → G} {s : Set F}
Goal: DifferentiableOn 𝕜 (f ∘ ⇑iso) (⇑iso ⁻¹' s) ↔ DifferentiableOn 𝕜 f s
-- ContinuousLinearEquiv.comp_right_differentiable_iff
theorem comp_right_differentiable_iff {f : F → G} :
    Differentiable 𝕜 (f ∘ iso) ↔ Differentiable 𝕜 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] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] (iso : E ≃L[𝕜] F) {f : F → G}
Goal: Differentiable 𝕜 (f ∘ ⇑iso) ↔ Differentiable 𝕜 f
-- ContinuousLinearEquiv.comp_right_hasFDerivWithinAt_iff
theorem comp_right_hasFDerivWithinAt_iff {f : F → G} {s : Set F} {x : E} {f' : F →L[𝕜] G} :
    HasFDerivWithinAt (f ∘ iso) (f'.comp (iso : E →L[𝕜] F)) (iso ⁻¹' s) x ↔
      HasFDerivWithinAt f f' s (iso 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] (iso : E ≃L[𝕜] F) {f : F → G} {s : Set F} {x : E} {f' : F →L[𝕜] G}
Goal: HasFDerivWithinAt (f ∘ ⇑iso) (f'.comp ↑iso) (⇑iso ⁻¹' s) x ↔ HasFDerivWithinAt f f' s (iso x)
-- ContinuousLinearEquiv.comp_right_hasFDerivAt_iff
theorem comp_right_hasFDerivAt_iff {f : F → G} {x : E} {f' : F →L[𝕜] G} :
    HasFDerivAt (f ∘ iso) (f'.comp (iso : E →L[𝕜] F)) x ↔ HasFDerivAt f f' (iso 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] (iso : E ≃L[𝕜] F) {f : F → G} {x : E} {f' : F →L[𝕜] G}
Goal: HasFDerivAt (f ∘ ⇑iso) (f'.comp ↑iso) x ↔ HasFDerivAt f f' (iso x)
-- ContinuousLinearEquiv.comp_right_hasFDerivWithinAt_iff'
theorem comp_right_hasFDerivWithinAt_iff' {f : F → G} {s : Set F} {x : E} {f' : E →L[𝕜] G} :
    HasFDerivWithinAt (f ∘ iso) f' (iso ⁻¹' s) x ↔
      HasFDerivWithinAt f (f'.comp (iso.symm : F →L[𝕜] E)) s (iso 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] (iso : E ≃L[𝕜] F) {f : F → G} {s : Set F} {x : E} {f' : E →L[𝕜] G}
Goal: HasFDerivWithinAt (f ∘ ⇑iso) f' (⇑iso ⁻¹' s) x ↔ HasFDerivWithinAt f (f'.comp ↑iso.symm) s (iso x)
-- ContinuousLinearEquiv.comp_right_hasFDerivAt_iff'
theorem comp_right_hasFDerivAt_iff' {f : F → G} {x : E} {f' : E →L[𝕜] G} :
    HasFDerivAt (f ∘ iso) f' x ↔ HasFDerivAt f (f'.comp (iso.symm : F →L[𝕜] E)) (iso 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] (iso : E ≃L[𝕜] F) {f : F → G} {x : E} {f' : E →L[𝕜] G}
Goal: HasFDerivAt (f ∘ ⇑iso) f' x ↔ HasFDerivAt f (f'.comp ↑iso.symm) (iso x)
-- ContinuousLinearEquiv.comp_right_fderivWithin
theorem comp_right_fderivWithin {f : F → G} {s : Set F} {x : E}
    (hxs : UniqueDiffWithinAt 𝕜 (iso ⁻¹' s) x) :
    fderivWithin 𝕜 (f ∘ iso) (iso ⁻¹' s) x =
      (fderivWithin 𝕜 f s (iso x)).comp (iso : E →L[𝕜] 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] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] (iso : E ≃L[𝕜] F) {f : F → G} {s : Set F} {x : E} (hxs : UniqueDiffWithinAt 𝕜 (⇑iso ⁻¹' s) x)
Goal: fderivWithin 𝕜 (f ∘ ⇑iso) (⇑iso ⁻¹' s) x = (fderivWithin 𝕜 f s (iso x)).comp ↑iso
-- ContinuousLinearEquiv.comp_right_fderiv
theorem comp_right_fderiv {f : F → G} {x : E} :
    fderiv 𝕜 (f ∘ iso) x = (fderiv 𝕜 f (iso x)).comp (iso : E →L[𝕜] 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] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] (iso : E ≃L[𝕜] F) {f : F → G} {x : E}
Goal: fderiv 𝕜 (f ∘ ⇑iso) x = (fderiv 𝕜 f (iso x)).comp ↑iso
-- LinearIsometryEquiv.hasStrictFDerivAt
@[fun_prop]
protected theorem hasStrictFDerivAt : HasStrictFDerivAt iso (iso : E →L[𝕜] 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} (iso : E ≃ₗᵢ[𝕜] F)
Goal: HasStrictFDerivAt (⇑iso) (↑{ toLinearEquiv := iso.toLinearEquiv, continuous_toFun := ⋯, continuous_invFun := ⋯ }) x
-- LinearIsometryEquiv.hasFDerivWithinAt
@[fun_prop]
protected theorem hasFDerivWithinAt : HasFDerivWithinAt iso (iso : E →L[𝕜] 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} (iso : E ≃ₗᵢ[𝕜] F)
Goal: HasFDerivWithinAt (⇑iso) (↑{ toLinearEquiv := iso.toLinearEquiv, continuous_toFun := ⋯, continuous_invFun := ⋯ }) s x
-- LinearIsometryEquiv.hasFDerivAt
@[fun_prop]
protected theorem hasFDerivAt : HasFDerivAt iso (iso : E →L[𝕜] 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} (iso : E ≃ₗᵢ[𝕜] F)
Goal: HasFDerivAt (⇑iso) (↑{ toLinearEquiv := iso.toLinearEquiv, continuous_toFun := ⋯, continuous_invFun := ⋯ }) x
-- LinearIsometryEquiv.differentiableAt
@[fun_prop]
protected theorem differentiableAt : DifferentiableAt 𝕜 iso 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} (iso : E ≃ₗᵢ[𝕜] F)
Goal: DifferentiableAt 𝕜 (⇑iso) x
-- LinearIsometryEquiv.differentiableWithinAt
@[fun_prop]
protected theorem differentiableWithinAt : DifferentiableWithinAt 𝕜 iso 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} (iso : E ≃ₗᵢ[𝕜] F)
Goal: DifferentiableWithinAt 𝕜 (⇑iso) s x
-- LinearIsometryEquiv.fderiv
protected theorem fderiv : fderiv 𝕜 iso x = iso 
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} (iso : E ≃ₗᵢ[𝕜] F)
Goal: fderiv 𝕜 (⇑iso) x = ↑{ toLinearEquiv := iso.toLinearEquiv, continuous_toFun := ⋯, continuous_invFun := ⋯ }