ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- uniqueMDiffWithinAt_iff
theorem uniqueMDiffWithinAt_iff {s : Set M} {x : M} :
    UniqueMDiffWithinAt I s x ↔
      UniqueDiffWithinAt 𝕜 ((extChartAt I x).symm ⁻¹' s ∩ (extChartAt I x).target)
        ((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] {s : Set M} {x : M}
Goal: UniqueMDiffWithinAt I s x ↔
  UniqueDiffWithinAt 𝕜 (↑(extChartAt I x).symm ⁻¹' s ∩ (extChartAt I x).target) (↑(extChartAt I x) x)
-- UniqueMDiffWithinAt.mono_nhds
nonrec theorem UniqueMDiffWithinAt.mono_nhds {s t : Set M} {x : M} (hs : UniqueMDiffWithinAt I s x)
    (ht : 𝓝[s] x ≤ 𝓝[t] x) : UniqueMDiffWithinAt I 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] {s t : Set M} {x : M} (hs : UniqueMDiffWithinAt I s x) (ht : 𝓝[s] x ≤ 𝓝[t] x)
Goal: UniqueMDiffWithinAt I t x
-- UniqueMDiffWithinAt.mono_of_mem
theorem UniqueMDiffWithinAt.mono_of_mem {s t : Set M} {x : M} (hs : UniqueMDiffWithinAt I s x)
    (ht : t ∈ 𝓝[s] x) : UniqueMDiffWithinAt I 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] {s t : Set M} {x : M} (hs : UniqueMDiffWithinAt I s x) (ht : t ∈ 𝓝[s] x)
Goal: UniqueMDiffWithinAt I t x
-- UniqueMDiffWithinAt.mono
theorem UniqueMDiffWithinAt.mono (h : UniqueMDiffWithinAt I s x) (st : s ⊆ t) :
    UniqueMDiffWithinAt I 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] {x : M} {s t : Set M} (h : UniqueMDiffWithinAt I s x) (st : s ⊆ t)
Goal: UniqueMDiffWithinAt I t x
-- UniqueMDiffWithinAt.inter'
theorem UniqueMDiffWithinAt.inter' (hs : UniqueMDiffWithinAt I s x) (ht : t ∈ 𝓝[s] x) :
    UniqueMDiffWithinAt I (s ∩ 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] {x : M} {s t : Set M} (hs : UniqueMDiffWithinAt I s x) (ht : t ∈ 𝓝[s] x)
Goal: UniqueMDiffWithinAt I (s ∩ t) x
-- UniqueMDiffWithinAt.inter
theorem UniqueMDiffWithinAt.inter (hs : UniqueMDiffWithinAt I s x) (ht : t ∈ 𝓝 x) :
    UniqueMDiffWithinAt I (s ∩ 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] {x : M} {s t : Set M} (hs : UniqueMDiffWithinAt I s x) (ht : t ∈ 𝓝 x)
Goal: UniqueMDiffWithinAt I (s ∩ t) x
-- IsOpen.uniqueMDiffWithinAt
theorem IsOpen.uniqueMDiffWithinAt (hs : IsOpen s) (xs : x ∈ s) : UniqueMDiffWithinAt I 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] {x : M} {s : Set M} (hs : IsOpen s) (xs : x ∈ s)
Goal: UniqueMDiffWithinAt I s x
-- UniqueMDiffOn.inter
theorem UniqueMDiffOn.inter (hs : UniqueMDiffOn I s) (ht : IsOpen t) : UniqueMDiffOn I (s ∩ t) 
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] {s t : Set M} (hs : UniqueMDiffOn I s) (ht : IsOpen t)
Goal: UniqueMDiffOn I (s ∩ t)
-- IsOpen.uniqueMDiffOn
theorem IsOpen.uniqueMDiffOn (hs : IsOpen s) : UniqueMDiffOn I 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] {s : Set M} (hs : IsOpen s)
Goal: UniqueMDiffOn I s
-- uniqueMDiffOn_univ
theorem uniqueMDiffOn_univ : UniqueMDiffOn I (univ : Set M) 
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]
Goal: UniqueMDiffOn I Set.univ
-- UniqueMDiffWithinAt.eq
nonrec theorem UniqueMDiffWithinAt.eq (U : UniqueMDiffWithinAt I s x)
    (h : HasMFDerivWithinAt I I' f s x f') (h₁ : HasMFDerivWithinAt I I' f s x f₁') : f' = f₁' 
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' f₁' : TangentSpace I x →L[𝕜] TangentSpace I' (f x)} (U : UniqueMDiffWithinAt I s x) (h : HasMFDerivWithinAt I I' f s x f') (h₁ : HasMFDerivWithinAt I I' f s x f₁')
Goal: f' = f₁'
-- UniqueMDiffOn.eq
theorem UniqueMDiffOn.eq (U : UniqueMDiffOn I s) (hx : x ∈ s) (h : HasMFDerivWithinAt I I' f s x f')
    (h₁ : HasMFDerivWithinAt I I' f s x f₁') : f' = f₁' 
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' f₁' : TangentSpace I x →L[𝕜] TangentSpace I' (f x)} (U : UniqueMDiffOn I s) (hx : x ∈ s) (h : HasMFDerivWithinAt I I' f s x f') (h₁ : HasMFDerivWithinAt I I' f s x f₁')
Goal: f' = f₁'
-- UniqueMDiffWithinAt.prod
nonrec theorem UniqueMDiffWithinAt.prod {x : M} {y : M'} {s t} (hs : UniqueMDiffWithinAt I s x)
    (ht : UniqueMDiffWithinAt I' t y) : UniqueMDiffWithinAt (I.prod I') (s ×ˢ t) (x, y) 
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'] {x : M} {y : M'} {s : Set M} {t : Set M'} (hs : UniqueMDiffWithinAt I s x) (ht : UniqueMDiffWithinAt I' t y)
Goal: UniqueMDiffWithinAt (I.prod I') (s ×ˢ t) (x, y)
-- UniqueMDiffOn.prod
theorem UniqueMDiffOn.prod {s : Set M} {t : Set M'} (hs : UniqueMDiffOn I s)
    (ht : UniqueMDiffOn I' t) : UniqueMDiffOn (I.prod I') (s ×ˢ t) 
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'] {s : Set M} {t : Set M'} (hs : UniqueMDiffOn I s) (ht : UniqueMDiffOn I' t)
Goal: UniqueMDiffOn (I.prod I') (s ×ˢ t)
-- mdifferentiableWithinAt_iff
theorem mdifferentiableWithinAt_iff {f : M → M'} {s : Set M} {x : M} :
    MDifferentiableWithinAt I I' f s x ↔
      ContinuousWithinAt f s x ∧
        DifferentiableWithinAt 𝕜 (writtenInExtChartAt I I' x f)
          ((extChartAt I x).target ∩ (extChartAt I x).symm ⁻¹' s) ((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'} {s : Set M} {x : M}
Goal: MDifferentiableWithinAt I I' f s x ↔
  ContinuousWithinAt f s x ∧
    DifferentiableWithinAt 𝕜 (writtenInExtChartAt I I' x f) ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' s)
      (↑(extChartAt I x) x)
-- mdifferentiableWithinAt_iff_of_mem_source
theorem mdifferentiableWithinAt_iff_of_mem_source {x' : M} {y : M'}
    (hx : x' ∈ (chartAt H x).source) (hy : f x' ∈ (chartAt H' y).source) :
    MDifferentiableWithinAt I I' f s x' ↔
      ContinuousWithinAt f s x' ∧
        DifferentiableWithinAt 𝕜 (extChartAt I' y ∘ f ∘ (extChartAt I x).symm)
          ((extChartAt I x).symm ⁻¹' s ∩ 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} {s : Set 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: MDifferentiableWithinAt I I' f s x' ↔
  ContinuousWithinAt f s x' ∧
    DifferentiableWithinAt 𝕜 (↑(extChartAt I' y) ∘ f ∘ ↑(extChartAt I x).symm)
      (↑(extChartAt I x).symm ⁻¹' s ∩ Set.range ↑I) (↑(extChartAt I x) x')
-- mfderivWithin_zero_of_not_mdifferentiableWithinAt
theorem mfderivWithin_zero_of_not_mdifferentiableWithinAt
    (h : ¬MDifferentiableWithinAt I I' f s x) : mfderivWithin I I' f s x = 0 
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 : ¬MDifferentiableWithinAt I I' f s x)
Goal: mfderivWithin I I' f s x = 0
-- mfderiv_zero_of_not_mdifferentiableAt
theorem mfderiv_zero_of_not_mdifferentiableAt (h : ¬MDifferentiableAt I I' f x) :
    mfderiv I I' f x = 0 
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'] (h : ¬MDifferentiableAt I I' f x)
Goal: mfderiv I I' f x = 0
-- HasMFDerivWithinAt.mono
theorem HasMFDerivWithinAt.mono (h : HasMFDerivWithinAt I I' f t x f') (hst : s ⊆ t) :
    HasMFDerivWithinAt 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 t : 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 t x f') (hst : s ⊆ t)
Goal: HasMFDerivWithinAt I I' f s x f'
-- HasMFDerivAt.hasMFDerivWithinAt
theorem HasMFDerivAt.hasMFDerivWithinAt (h : HasMFDerivAt I I' f x f') :
    HasMFDerivWithinAt 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 : HasMFDerivAt I I' f x f')
Goal: HasMFDerivWithinAt I I' f s x f'
-- HasMFDerivWithinAt.mdifferentiableWithinAt
theorem HasMFDerivWithinAt.mdifferentiableWithinAt (h : HasMFDerivWithinAt I I' f s x 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} [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: MDifferentiableWithinAt I I' f s x
-- HasMFDerivAt.mdifferentiableAt
theorem HasMFDerivAt.mdifferentiableAt (h : HasMFDerivAt I I' f x 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} [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: MDifferentiableAt I I' f x
-- hasMFDerivWithinAt_univ
@[simp, mfld_simps]
theorem hasMFDerivWithinAt_univ :
    HasMFDerivWithinAt I I' f univ x f' ↔ HasMFDerivAt 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)}
Goal: HasMFDerivWithinAt I I' f Set.univ x f' ↔ HasMFDerivAt I I' f x f'
-- hasMFDerivAt_unique
theorem hasMFDerivAt_unique (h₀ : HasMFDerivAt I I' f x f₀') (h₁ : HasMFDerivAt I I' f x f₁') :
    f₀' = f₁' 
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'] {f₀' f₁' : TangentSpace I x →L[𝕜] TangentSpace I' (f x)} (h₀ : HasMFDerivAt I I' f x f₀') (h₁ : HasMFDerivAt I I' f x f₁')
Goal: f₀' = f₁'
-- hasMFDerivWithinAt_inter'
theorem hasMFDerivWithinAt_inter' (h : t ∈ 𝓝[s] x) :
    HasMFDerivWithinAt I I' f (s ∩ t) x f' ↔ HasMFDerivWithinAt 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 t : Set M} [Is : SmoothManifoldWithCorners I M] [I's : SmoothManifoldWithCorners I' M'] {f' : TangentSpace I x →L[𝕜] TangentSpace I' (f x)} (h : t ∈ 𝓝[s] x)
Goal: HasMFDerivWithinAt I I' f (s ∩ t) x f' ↔ HasMFDerivWithinAt I I' f s x f'
-- hasMFDerivWithinAt_inter
theorem hasMFDerivWithinAt_inter (h : t ∈ 𝓝 x) :
    HasMFDerivWithinAt I I' f (s ∩ t) x f' ↔ HasMFDerivWithinAt 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 t : Set M} [Is : SmoothManifoldWithCorners I M] [I's : SmoothManifoldWithCorners I' M'] {f' : TangentSpace I x →L[𝕜] TangentSpace I' (f x)} (h : t ∈ 𝓝 x)
Goal: HasMFDerivWithinAt I I' f (s ∩ t) x f' ↔ HasMFDerivWithinAt I I' f s x f'
-- HasMFDerivWithinAt.union
theorem HasMFDerivWithinAt.union (hs : HasMFDerivWithinAt I I' f s x f')
    (ht : HasMFDerivWithinAt I I' f t x f') : HasMFDerivWithinAt I I' f (s ∪ t) 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 t : Set M} [Is : SmoothManifoldWithCorners I M] [I's : SmoothManifoldWithCorners I' M'] {f' : TangentSpace I x →L[𝕜] TangentSpace I' (f x)} (hs : HasMFDerivWithinAt I I' f s x f') (ht : HasMFDerivWithinAt I I' f t x f')
Goal: HasMFDerivWithinAt I I' f (s ∪ t) x f'
-- HasMFDerivWithinAt.mono_of_mem
theorem HasMFDerivWithinAt.mono_of_mem (h : HasMFDerivWithinAt I I' f s x f') (ht : s ∈ 𝓝[t] x) :
    HasMFDerivWithinAt I I' f t 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 t : 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') (ht : s ∈ 𝓝[t] x)
Goal: HasMFDerivWithinAt I I' f t x f'
-- HasMFDerivWithinAt.hasMFDerivAt
theorem HasMFDerivWithinAt.hasMFDerivAt (h : HasMFDerivWithinAt I I' f s x f') (hs : s ∈ 𝓝 x) :
    HasMFDerivAt 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} {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') (hs : s ∈ 𝓝 x)
Goal: HasMFDerivAt I I' f x f'
-- MDifferentiableWithinAt.hasMFDerivWithinAt
theorem MDifferentiableWithinAt.hasMFDerivWithinAt (h : MDifferentiableWithinAt I I' f s x) :
    HasMFDerivWithinAt I I' f s 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 : Set M} [Is : SmoothManifoldWithCorners I M] [I's : SmoothManifoldWithCorners I' M'] (h : MDifferentiableWithinAt I I' f s x)
Goal: HasMFDerivWithinAt I I' f s x (mfderivWithin I I' f s x)
-- MDifferentiableWithinAt.mfderivWithin
protected theorem MDifferentiableWithinAt.mfderivWithin (h : MDifferentiableWithinAt I I' f s x) :
    mfderivWithin I I' f s x =
      fderivWithin 𝕜 (writtenInExtChartAt I I' x f : _) ((extChartAt I x).symm ⁻¹' s ∩ 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} {s : Set M} [Is : SmoothManifoldWithCorners I M] [I's : SmoothManifoldWithCorners I' M'] (h : MDifferentiableWithinAt I I' f s x)
Goal: mfderivWithin I I' f s x =
  fderivWithin 𝕜 (writtenInExtChartAt I I' x f) (↑(extChartAt I x).symm ⁻¹' s ∩ Set.range ↑I) (↑(extChartAt I x) x)
-- MDifferentiableAt.hasMFDerivAt
theorem MDifferentiableAt.hasMFDerivAt (h : MDifferentiableAt I I' f x) :
    HasMFDerivAt I I' f 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} [Is : SmoothManifoldWithCorners I M] [I's : SmoothManifoldWithCorners I' M'] (h : MDifferentiableAt I I' f x)
Goal: HasMFDerivAt I I' f x (mfderiv I I' f x)
-- MDifferentiableAt.mfderiv
protected theorem MDifferentiableAt.mfderiv (h : MDifferentiableAt I I' f x) :
    mfderiv I I' f x =
      fderivWithin 𝕜 (writtenInExtChartAt I I' x f : _) (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'] (h : MDifferentiableAt I I' f x)
Goal: mfderiv I I' f x = fderivWithin 𝕜 (writtenInExtChartAt I I' x f) (Set.range ↑I) (↑(extChartAt I x) x)