ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- HasMFDerivAt.mfderiv
protected theorem HasMFDerivAt.mfderiv (h : HasMFDerivAt I I' f x f') : mfderiv I I' f x = f' 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} {x : M} [Is : SmoothManifoldWithCorners I M] [I's : SmoothManifoldWithCorners I' M'] {f' : TangentSpace I x →L[𝕜] TangentSpace I' (f x)} (h : HasMFDerivAt I I' f x f')
Goal: mfderiv I I' f x = f'
-- HasMFDerivWithinAt.mfderivWithin
theorem HasMFDerivWithinAt.mfderivWithin (h : HasMFDerivWithinAt I I' f s x f')
    (hxs : UniqueMDiffWithinAt I s x) : mfderivWithin I I' f s x = f' 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} {x : M} {s : Set M} [Is : SmoothManifoldWithCorners I M] [I's : SmoothManifoldWithCorners I' M'] {f' : TangentSpace I x →L[𝕜] TangentSpace I' (f x)} (h : HasMFDerivWithinAt I I' f s x f') (hxs : UniqueMDiffWithinAt I s x)
Goal: mfderivWithin I I' f s x = f'
-- MDifferentiable.mfderivWithin
theorem MDifferentiable.mfderivWithin (h : MDifferentiableAt I I' f x)
    (hxs : UniqueMDiffWithinAt I s x) : mfderivWithin I I' f s x = mfderiv I I' f x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} {x : M} {s : Set M} [Is : SmoothManifoldWithCorners I M] [I's : SmoothManifoldWithCorners I' M'] (h : MDifferentiableAt I I' f x) (hxs : UniqueMDiffWithinAt I s x)
Goal: mfderivWithin I I' f s x = mfderiv I I' f x
-- mfderivWithin_subset
theorem mfderivWithin_subset (st : s ⊆ t) (hs : UniqueMDiffWithinAt I s x)
    (h : MDifferentiableWithinAt I I' f t x) :
    mfderivWithin I I' f s x = mfderivWithin I I' f t x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} {x : M} {s t : Set M} [Is : SmoothManifoldWithCorners I M] [I's : SmoothManifoldWithCorners I' M'] (st : s ⊆ t) (hs : UniqueMDiffWithinAt I s x) (h : MDifferentiableWithinAt I I' f t x)
Goal: mfderivWithin I I' f s x = mfderivWithin I I' f t x
-- MDifferentiableWithinAt.mono
theorem MDifferentiableWithinAt.mono (hst : s ⊆ t) (h : MDifferentiableWithinAt I I' f t x) :
    MDifferentiableWithinAt I I' f s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} {x : M} {s t : Set M} (hst : s ⊆ t) (h : MDifferentiableWithinAt I I' f t x)
Goal: MDifferentiableWithinAt I I' f s x
-- mdifferentiableWithinAt_univ
theorem mdifferentiableWithinAt_univ :
    MDifferentiableWithinAt I I' f univ x ↔ MDifferentiableAt I I' f x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} {x : M}
Goal: MDifferentiableWithinAt I I' f Set.univ x ↔ MDifferentiableAt I I' f x
-- mdifferentiableWithinAt_inter
theorem mdifferentiableWithinAt_inter (ht : t ∈ 𝓝 x) :
    MDifferentiableWithinAt I I' f (s ∩ t) x ↔ MDifferentiableWithinAt I I' f s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} {x : M} {s t : Set M} (ht : t ∈ 𝓝 x)
Goal: MDifferentiableWithinAt I I' f (s ∩ t) x ↔ MDifferentiableWithinAt I I' f s x
-- mdifferentiableWithinAt_inter'
theorem mdifferentiableWithinAt_inter' (ht : t ∈ 𝓝[s] x) :
    MDifferentiableWithinAt I I' f (s ∩ t) x ↔ MDifferentiableWithinAt I I' f s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} {x : M} {s t : Set M} (ht : t ∈ 𝓝[s] x)
Goal: MDifferentiableWithinAt I I' f (s ∩ t) x ↔ MDifferentiableWithinAt I I' f s x
-- MDifferentiableAt.mdifferentiableWithinAt
theorem MDifferentiableAt.mdifferentiableWithinAt (h : MDifferentiableAt I I' f x) :
    MDifferentiableWithinAt I I' f s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} {x : M} {s : Set M} (h : MDifferentiableAt I I' f x)
Goal: MDifferentiableWithinAt I I' f s x
-- MDifferentiableWithinAt.mdifferentiableAt
theorem MDifferentiableWithinAt.mdifferentiableAt (h : MDifferentiableWithinAt I I' f s x)
    (hs : s ∈ 𝓝 x) : MDifferentiableAt I I' f x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} {x : M} {s : Set M} (h : MDifferentiableWithinAt I I' f s x) (hs : s ∈ 𝓝 x)
Goal: MDifferentiableAt I I' f x
-- MDifferentiableOn.mdifferentiableAt
theorem MDifferentiableOn.mdifferentiableAt (h : MDifferentiableOn I I' f s) (hx : s ∈ 𝓝 x) :
    MDifferentiableAt I I' f x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} {x : M} {s : Set M} (h : MDifferentiableOn I I' f s) (hx : s ∈ 𝓝 x)
Goal: MDifferentiableAt I I' f x
-- MDifferentiableOn.mono
theorem MDifferentiableOn.mono (h : MDifferentiableOn I I' f t) (st : s ⊆ t) :
    MDifferentiableOn I I' f s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} {s t : Set M} (h : MDifferentiableOn I I' f t) (st : s ⊆ t)
Goal: MDifferentiableOn I I' f s
-- mdifferentiableOn_univ
theorem mdifferentiableOn_univ : MDifferentiableOn I I' f univ ↔ MDifferentiable I I' f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'}
Goal: MDifferentiableOn I I' f Set.univ ↔ MDifferentiable I I' f
-- MDifferentiable.mdifferentiableOn
theorem MDifferentiable.mdifferentiableOn (h : MDifferentiable I I' f) :
    MDifferentiableOn I I' f s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} {s : Set M} (h : MDifferentiable I I' f)
Goal: MDifferentiableOn I I' f s
-- mdifferentiableOn_of_locally_mdifferentiableOn
theorem mdifferentiableOn_of_locally_mdifferentiableOn
    (h : ∀ x ∈ s, ∃ u, IsOpen u ∧ x ∈ u ∧ MDifferentiableOn I I' f (s ∩ u)) :
    MDifferentiableOn I I' f s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} {s : Set M} (h : ∀ x ∈ s, ∃ u, IsOpen u ∧ x ∈ u ∧ MDifferentiableOn I I' f (s ∩ u))
Goal: MDifferentiableOn I I' f s
-- mfderivWithin_univ
@[simp, mfld_simps]
theorem mfderivWithin_univ : mfderivWithin I I' f univ = mfderiv I I' f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} [Is : SmoothManifoldWithCorners I M] [I's : SmoothManifoldWithCorners I' M']
Goal: mfderivWithin I I' f Set.univ = mfderiv I I' f
-- mfderivWithin_inter
theorem mfderivWithin_inter (ht : t ∈ 𝓝 x) :
    mfderivWithin I I' f (s ∩ t) x = mfderivWithin I I' f s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} {x : M} {s t : Set M} [Is : SmoothManifoldWithCorners I M] [I's : SmoothManifoldWithCorners I' M'] (ht : t ∈ 𝓝 x)
Goal: mfderivWithin I I' f (s ∩ t) x = mfderivWithin I I' f s x
-- mfderivWithin_of_mem_nhds
theorem mfderivWithin_of_mem_nhds (h : s ∈ 𝓝 x) : mfderivWithin I I' f s x = mfderiv I I' f x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} {x : M} {s : Set M} [Is : SmoothManifoldWithCorners I M] [I's : SmoothManifoldWithCorners I' M'] (h : s ∈ 𝓝 x)
Goal: mfderivWithin I I' f s x = mfderiv I I' f x
-- mfderivWithin_of_isOpen
lemma mfderivWithin_of_isOpen (hs : IsOpen s) (hx : x ∈ s) :
    mfderivWithin I I' f s x = mfderiv I I' f x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} {x : M} {s : Set M} [Is : SmoothManifoldWithCorners I M] [I's : SmoothManifoldWithCorners I' M'] (hs : IsOpen s) (hx : x ∈ s)
Goal: mfderivWithin I I' f s x = mfderiv I I' f x
-- mfderivWithin_eq_mfderiv
theorem mfderivWithin_eq_mfderiv (hs : UniqueMDiffWithinAt I s x) (h : MDifferentiableAt I I' f x) :
    mfderivWithin I I' f s x = mfderiv I I' f x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} {x : M} {s : Set M} [Is : SmoothManifoldWithCorners I M] [I's : SmoothManifoldWithCorners I' M'] (hs : UniqueMDiffWithinAt I s x) (h : MDifferentiableAt I I' f x)
Goal: mfderivWithin I I' f s x = mfderiv I I' f x
-- mdifferentiableAt_iff_of_mem_source
theorem mdifferentiableAt_iff_of_mem_source {x' : M} {y : M'}
    (hx : x' ∈ (chartAt H x).source) (hy : f x' ∈ (chartAt H' y).source) :
    MDifferentiableAt I I' f x' ↔
      ContinuousAt f x' ∧
        DifferentiableWithinAt 𝕜 (extChartAt I' y ∘ f ∘ (extChartAt I x).symm) (Set.range I)
          ((extChartAt I x) x') 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} {x : M} [Is : SmoothManifoldWithCorners I M] [I's : SmoothManifoldWithCorners I' M'] {x' : M} {y : M'} (hx : x' ∈ (chartAt H x).source) (hy : f x' ∈ (chartAt H' y).source)
Goal: MDifferentiableAt I I' f x' ↔
  ContinuousAt f x' ∧
    DifferentiableWithinAt 𝕜 (↑(extChartAt I' y) ∘ f ∘ ↑(extChartAt I x).symm) (Set.range ↑I) (↑(extChartAt I x) x')
-- ContMDiffWithinAt.mdifferentiableWithinAt
theorem ContMDiffWithinAt.mdifferentiableWithinAt (hf : ContMDiffWithinAt I I' n f s x)
    (hn : 1 ≤ n) : MDifferentiableWithinAt I I' f s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} {x : M} {s : Set M} {n : ℕ∞} (hf : ContMDiffWithinAt I I' n f s x) (hn : 1 ≤ n)
Goal: MDifferentiableWithinAt I I' f s x
-- ContMDiffAt.mdifferentiableAt
theorem ContMDiffAt.mdifferentiableAt (hf : ContMDiffAt I I' n f x) (hn : 1 ≤ n) :
    MDifferentiableAt I I' f x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} {x : M} {n : ℕ∞} (hf : ContMDiffAt I I' n f x) (hn : 1 ≤ n)
Goal: MDifferentiableAt I I' f x
-- ContMDiffOn.mdifferentiableOn
theorem ContMDiffOn.mdifferentiableOn (hf : ContMDiffOn I I' n f s) (hn : 1 ≤ n) :
    MDifferentiableOn I I' f s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} {s : Set M} {n : ℕ∞} (hf : ContMDiffOn I I' n f s) (hn : 1 ≤ n)
Goal: MDifferentiableOn I I' f s
-- ContMDiff.mdifferentiable
theorem ContMDiff.mdifferentiable (hf : ContMDiff I I' n f) (hn : 1 ≤ n) : MDifferentiable I I' f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} {n : ℕ∞} (hf : ContMDiff I I' n f) (hn : 1 ≤ n)
Goal: MDifferentiable I I' f
-- SmoothWithinAt.mdifferentiableWithinAt
nonrec theorem SmoothWithinAt.mdifferentiableWithinAt (hf : SmoothWithinAt I I' f s x) :
    MDifferentiableWithinAt I I' f s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} {x : M} {s : Set M} (hf : SmoothWithinAt I I' f s x)
Goal: MDifferentiableWithinAt I I' f s x
-- SmoothAt.mdifferentiableAt
nonrec theorem SmoothAt.mdifferentiableAt (hf : SmoothAt I I' f x) : MDifferentiableAt I I' f x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} {x : M} (hf : SmoothAt I I' f x)
Goal: MDifferentiableAt I I' f x
-- SmoothOn.mdifferentiableOn
nonrec theorem SmoothOn.mdifferentiableOn (hf : SmoothOn I I' f s) : MDifferentiableOn I I' f s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} {s : Set M} (hf : SmoothOn I I' f s)
Goal: MDifferentiableOn I I' f s
-- Smooth.mdifferentiable
theorem Smooth.mdifferentiable (hf : Smooth I I' f) : MDifferentiable I I' f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} (hf : Smooth I I' f)
Goal: MDifferentiable I I' f
-- Smooth.mdifferentiableAt
theorem Smooth.mdifferentiableAt (hf : Smooth I I' f) : MDifferentiableAt I I' f x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} {x : M} (hf : Smooth I I' f)
Goal: MDifferentiableAt I I' f x
-- Smooth.mdifferentiableWithinAt
theorem Smooth.mdifferentiableWithinAt (hf : Smooth I I' f) : MDifferentiableWithinAt I I' f s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} {x : M} {s : Set M} (hf : Smooth I I' f)
Goal: MDifferentiableWithinAt I I' f s x
-- HasMFDerivWithinAt.continuousWithinAt
theorem HasMFDerivWithinAt.continuousWithinAt (h : HasMFDerivWithinAt I I' f s x f') :
    ContinuousWithinAt f s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} {x : M} {s : Set M} [Is : SmoothManifoldWithCorners I M] [I's : SmoothManifoldWithCorners I' M'] {f' : TangentSpace I x →L[𝕜] TangentSpace I' (f x)} (h : HasMFDerivWithinAt I I' f s x f')
Goal: ContinuousWithinAt f s x
-- HasMFDerivAt.continuousAt
theorem HasMFDerivAt.continuousAt (h : HasMFDerivAt I I' f x f') : ContinuousAt f x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} {x : M} [Is : SmoothManifoldWithCorners I M] [I's : SmoothManifoldWithCorners I' M'] {f' : TangentSpace I x →L[𝕜] TangentSpace I' (f x)} (h : HasMFDerivAt I I' f x f')
Goal: ContinuousAt f x