ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- contMDiffWithinAt_iff'
theorem contMDiffWithinAt_iff' :
    ContMDiffWithinAt I I' n f s x ↔
      ContinuousWithinAt f s x ∧
        ContDiffWithinAt 𝕜 n (extChartAt I' (f x) ∘ f ∘ (extChartAt I x).symm)
          ((extChartAt I x).target ∩
            (extChartAt I x).symm ⁻¹' (s ∩ f ⁻¹' (extChartAt I' (f x)).source))
          (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} {n : ℕ∞}
Goal: ContMDiffWithinAt I I' n f s x ↔
  ContinuousWithinAt f s x ∧
    ContDiffWithinAt 𝕜 n (↑(extChartAt I' (f x)) ∘ f ∘ ↑(extChartAt I x).symm)
      ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' (s ∩ f ⁻¹' (extChartAt I' (f x)).source))
      (↑(extChartAt I x) x)
-- contMDiffWithinAt_iff_target
theorem contMDiffWithinAt_iff_target :
    ContMDiffWithinAt I I' n f s x ↔
      ContinuousWithinAt f s x ∧ ContMDiffWithinAt I 𝓘(𝕜, E') n (extChartAt I' (f x) ∘ 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'} {s : Set M} {x : M} {n : ℕ∞}
Goal: ContMDiffWithinAt I I' n f s x ↔
  ContinuousWithinAt f s x ∧ ContMDiffWithinAt I 𝓘(𝕜, E') n (↑(extChartAt I' (f x)) ∘ f) s x
-- smoothWithinAt_iff
theorem smoothWithinAt_iff :
    SmoothWithinAt I I' f s x ↔
      ContinuousWithinAt f s x ∧
        ContDiffWithinAt 𝕜 ∞ (extChartAt I' (f x) ∘ f ∘ (extChartAt I x).symm)
          ((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'} {s : Set M} {x : M}
Goal: SmoothWithinAt I I' f s x ↔
  ContinuousWithinAt f s x ∧
    ContDiffWithinAt 𝕜 ⊤ (↑(extChartAt I' (f x)) ∘ f ∘ ↑(extChartAt I x).symm)
      (↑(extChartAt I x).symm ⁻¹' s ∩ Set.range ↑I) (↑(extChartAt I x) x)
-- smoothWithinAt_iff_target
theorem smoothWithinAt_iff_target :
    SmoothWithinAt I I' f s x ↔
      ContinuousWithinAt f s x ∧ SmoothWithinAt I 𝓘(𝕜, E') (extChartAt I' (f x) ∘ 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'} {s : Set M} {x : M}
Goal: SmoothWithinAt I I' f s x ↔ ContinuousWithinAt f s x ∧ SmoothWithinAt I 𝓘(𝕜, E') (↑(extChartAt I' (f x)) ∘ f) s x
-- contMDiffAt_iff_target
theorem contMDiffAt_iff_target {x : M} :
    ContMDiffAt I I' n f x ↔
      ContinuousAt f x ∧ ContMDiffAt I 𝓘(𝕜, E') n (extChartAt I' (f x) ∘ 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'} {n : ℕ∞} {x : M}
Goal: ContMDiffAt I I' n f x ↔ ContinuousAt f x ∧ ContMDiffAt I 𝓘(𝕜, E') n (↑(extChartAt I' (f x)) ∘ f) x
-- smoothAt_iff_target
theorem smoothAt_iff_target {x : M} :
    SmoothAt I I' f x ↔ ContinuousAt f x ∧ SmoothAt I 𝓘(𝕜, E') (extChartAt I' (f x) ∘ 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: SmoothAt I I' f x ↔ ContinuousAt f x ∧ SmoothAt I 𝓘(𝕜, E') (↑(extChartAt I' (f x)) ∘ f) x
-- contMDiffWithinAt_iff_of_mem_maximalAtlas
theorem contMDiffWithinAt_iff_of_mem_maximalAtlas {x : M} (he : e ∈ maximalAtlas I M)
    (he' : e' ∈ maximalAtlas I' M') (hx : x ∈ e.source) (hy : f x ∈ e'.source) :
    ContMDiffWithinAt I I' n f s x ↔
      ContinuousWithinAt f s x ∧
        ContDiffWithinAt 𝕜 n (e'.extend I' ∘ f ∘ (e.extend I).symm)
          ((e.extend I).symm ⁻¹' s ∩ range I) (e.extend I 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] [SmoothManifoldWithCorners I 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'] [SmoothManifoldWithCorners I' M'] {e : PartialHomeomorph M H} {e' : PartialHomeomorph M' H'} {f : M → M'} {s : Set M} {n : ℕ∞} {x : M} (he : e ∈ SmoothManifoldWithCorners.maximalAtlas I M) (he' : e' ∈ SmoothManifoldWithCorners.maximalAtlas I' M') (hx : x ∈ e.source) (hy : f x ∈ e'.source)
Goal: ContMDiffWithinAt I I' n f s x ↔
  ContinuousWithinAt f s x ∧
    ContDiffWithinAt 𝕜 n (↑(e'.extend I') ∘ f ∘ ↑(e.extend I).symm) (↑(e.extend I).symm ⁻¹' s ∩ Set.range ↑I)
      (↑(e.extend I) x)
-- contMDiffWithinAt_iff_image
theorem contMDiffWithinAt_iff_image {x : M} (he : e ∈ maximalAtlas I M)
    (he' : e' ∈ maximalAtlas I' M') (hs : s ⊆ e.source) (hx : x ∈ e.source) (hy : f x ∈ e'.source) :
    ContMDiffWithinAt I I' n f s x ↔
      ContinuousWithinAt f s x ∧
        ContDiffWithinAt 𝕜 n (e'.extend I' ∘ f ∘ (e.extend I).symm) (e.extend I '' s)
          (e.extend I 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] [SmoothManifoldWithCorners I 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'] [SmoothManifoldWithCorners I' M'] {e : PartialHomeomorph M H} {e' : PartialHomeomorph M' H'} {f : M → M'} {s : Set M} {n : ℕ∞} {x : M} (he : e ∈ SmoothManifoldWithCorners.maximalAtlas I M) (he' : e' ∈ SmoothManifoldWithCorners.maximalAtlas I' M') (hs : s ⊆ e.source) (hx : x ∈ e.source) (hy : f x ∈ e'.source)
Goal: ContMDiffWithinAt I I' n f s x ↔
  ContinuousWithinAt f s x ∧
    ContDiffWithinAt 𝕜 n (↑(e'.extend I') ∘ f ∘ ↑(e.extend I).symm) (↑(e.extend I) '' s) (↑(e.extend I) x)
-- contMDiffWithinAt_iff_of_mem_source
theorem contMDiffWithinAt_iff_of_mem_source {x' : M} {y : M'} (hx : x' ∈ (chartAt H x).source)
    (hy : f x' ∈ (chartAt H' y).source) :
    ContMDiffWithinAt I I' n f s x' ↔
      ContinuousWithinAt f s x' ∧
        ContDiffWithinAt 𝕜 n (extChartAt I' y ∘ f ∘ (extChartAt I x).symm)
          ((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] [SmoothManifoldWithCorners I 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'] [SmoothManifoldWithCorners I' M'] {f : M → M'} {s : Set M} {x : M} {n : ℕ∞} {x' : M} {y : M'} (hx : x' ∈ (chartAt H x).source) (hy : f x' ∈ (chartAt H' y).source)
Goal: ContMDiffWithinAt I I' n f s x' ↔
  ContinuousWithinAt f s x' ∧
    ContDiffWithinAt 𝕜 n (↑(extChartAt I' y) ∘ f ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x).symm ⁻¹' s ∩ Set.range ↑I)
      (↑(extChartAt I x) x')
-- contMDiffWithinAt_iff_of_mem_source'
theorem contMDiffWithinAt_iff_of_mem_source' {x' : M} {y : M'} (hx : x' ∈ (chartAt H x).source)
    (hy : f x' ∈ (chartAt H' y).source) :
    ContMDiffWithinAt I I' n f s x' ↔
      ContinuousWithinAt f s x' ∧
        ContDiffWithinAt 𝕜 n (extChartAt I' y ∘ f ∘ (extChartAt I x).symm)
          ((extChartAt I x).target ∩ (extChartAt I x).symm ⁻¹' (s ∩ f ⁻¹' (extChartAt I' y).source))
          (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] [SmoothManifoldWithCorners I 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'] [SmoothManifoldWithCorners I' M'] {f : M → M'} {s : Set M} {x : M} {n : ℕ∞} {x' : M} {y : M'} (hx : x' ∈ (chartAt H x).source) (hy : f x' ∈ (chartAt H' y).source)
Goal: ContMDiffWithinAt I I' n f s x' ↔
  ContinuousWithinAt f s x' ∧
    ContDiffWithinAt 𝕜 n (↑(extChartAt I' y) ∘ f ∘ ↑(extChartAt I x).symm)
      ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' (s ∩ f ⁻¹' (extChartAt I' y).source)) (↑(extChartAt I x) x')
-- contMDiffAt_iff_of_mem_source
theorem contMDiffAt_iff_of_mem_source {x' : M} {y : M'} (hx : x' ∈ (chartAt H x).source)
    (hy : f x' ∈ (chartAt H' y).source) :
    ContMDiffAt I I' n f x' ↔
      ContinuousAt f x' ∧
        ContDiffWithinAt 𝕜 n (extChartAt I' y ∘ f ∘ (extChartAt I x).symm) (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] [SmoothManifoldWithCorners I 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'] [SmoothManifoldWithCorners I' M'] {f : M → M'} {x : M} {n : ℕ∞} {x' : M} {y : M'} (hx : x' ∈ (chartAt H x).source) (hy : f x' ∈ (chartAt H' y).source)
Goal: ContMDiffAt I I' n f x' ↔
  ContinuousAt f x' ∧
    ContDiffWithinAt 𝕜 n (↑(extChartAt I' y) ∘ f ∘ ↑(extChartAt I x).symm) (Set.range ↑I) (↑(extChartAt I x) x')
-- contMDiffWithinAt_iff_target_of_mem_source
theorem contMDiffWithinAt_iff_target_of_mem_source {x : M} {y : M'}
    (hy : f x ∈ (chartAt H' y).source) :
    ContMDiffWithinAt I I' n f s x ↔
      ContinuousWithinAt f s x ∧ ContMDiffWithinAt I 𝓘(𝕜, E') n (extChartAt I' y ∘ 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'] [SmoothManifoldWithCorners I' M'] {f : M → M'} {s : Set M} {n : ℕ∞} {x : M} {y : M'} (hy : f x ∈ (chartAt H' y).source)
Goal: ContMDiffWithinAt I I' n f s x ↔ ContinuousWithinAt f s x ∧ ContMDiffWithinAt I 𝓘(𝕜, E') n (↑(extChartAt I' y) ∘ f) s x
-- contMDiffAt_iff_target_of_mem_source
theorem contMDiffAt_iff_target_of_mem_source {x : M} {y : M'} (hy : f x ∈ (chartAt H' y).source) :
    ContMDiffAt I I' n f x ↔
      ContinuousAt f x ∧ ContMDiffAt I 𝓘(𝕜, E') n (extChartAt I' y ∘ 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'] [SmoothManifoldWithCorners I' M'] {f : M → M'} {n : ℕ∞} {x : M} {y : M'} (hy : f x ∈ (chartAt H' y).source)
Goal: ContMDiffAt I I' n f x ↔ ContinuousAt f x ∧ ContMDiffAt I 𝓘(𝕜, E') n (↑(extChartAt I' y) ∘ f) x
-- contMDiffWithinAt_iff_source_of_mem_maximalAtlas
theorem contMDiffWithinAt_iff_source_of_mem_maximalAtlas (he : e ∈ maximalAtlas I M)
    (hx : x ∈ e.source) :
    ContMDiffWithinAt I I' n f s x ↔
      ContMDiffWithinAt 𝓘(𝕜, E) I' n (f ∘ (e.extend I).symm) ((e.extend I).symm ⁻¹' s ∩ range I)
        (e.extend I 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] [SmoothManifoldWithCorners I 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'] {e : PartialHomeomorph M H} {f : M → M'} {s : Set M} {x : M} {n : ℕ∞} (he : e ∈ SmoothManifoldWithCorners.maximalAtlas I M) (hx : x ∈ e.source)
Goal: ContMDiffWithinAt I I' n f s x ↔
  ContMDiffWithinAt 𝓘(𝕜, E) I' n (f ∘ ↑(e.extend I).symm) (↑(e.extend I).symm ⁻¹' s ∩ Set.range ↑I) (↑(e.extend I) x)
-- contMDiffWithinAt_iff_source_of_mem_source
theorem contMDiffWithinAt_iff_source_of_mem_source {x' : M} (hx' : x' ∈ (chartAt H x).source) :
    ContMDiffWithinAt I I' n f s x' ↔
      ContMDiffWithinAt 𝓘(𝕜, E) I' n (f ∘ (extChartAt I x).symm)
        ((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] [SmoothManifoldWithCorners I 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} {n : ℕ∞} {x' : M} (hx' : x' ∈ (chartAt H x).source)
Goal: ContMDiffWithinAt I I' n f s x' ↔
  ContMDiffWithinAt 𝓘(𝕜, E) I' n (f ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x).symm ⁻¹' s ∩ Set.range ↑I)
    (↑(extChartAt I x) x')
-- contMDiffAt_iff_source_of_mem_source
theorem contMDiffAt_iff_source_of_mem_source {x' : M} (hx' : x' ∈ (chartAt H x).source) :
    ContMDiffAt I I' n f x' ↔
      ContMDiffWithinAt 𝓘(𝕜, E) I' n (f ∘ (extChartAt I x).symm) (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] [SmoothManifoldWithCorners I 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∞} {x' : M} (hx' : x' ∈ (chartAt H x).source)
Goal: ContMDiffAt I I' n f x' ↔
  ContMDiffWithinAt 𝓘(𝕜, E) I' n (f ∘ ↑(extChartAt I x).symm) (Set.range ↑I) (↑(extChartAt I x) x')
-- contMDiffOn_iff_of_mem_maximalAtlas
theorem contMDiffOn_iff_of_mem_maximalAtlas (he : e ∈ maximalAtlas I M)
    (he' : e' ∈ maximalAtlas I' M') (hs : s ⊆ e.source) (h2s : MapsTo f s e'.source) :
    ContMDiffOn I I' n f s ↔
      ContinuousOn f s ∧
        ContDiffOn 𝕜 n (e'.extend I' ∘ f ∘ (e.extend I).symm) (e.extend 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] [SmoothManifoldWithCorners I 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'] [SmoothManifoldWithCorners I' M'] {e : PartialHomeomorph M H} {e' : PartialHomeomorph M' H'} {f : M → M'} {s : Set M} {n : ℕ∞} (he : e ∈ SmoothManifoldWithCorners.maximalAtlas I M) (he' : e' ∈ SmoothManifoldWithCorners.maximalAtlas I' M') (hs : s ⊆ e.source) (h2s : Set.MapsTo f s e'.source)
Goal: ContMDiffOn I I' n f s ↔
  ContinuousOn f s ∧ ContDiffOn 𝕜 n (↑(e'.extend I') ∘ f ∘ ↑(e.extend I).symm) (↑(e.extend I) '' s)
-- contMDiffOn_iff_of_mem_maximalAtlas'
theorem contMDiffOn_iff_of_mem_maximalAtlas' (he : e ∈ maximalAtlas I M)
    (he' : e' ∈ maximalAtlas I' M') (hs : s ⊆ e.source) (h2s : MapsTo f s e'.source) :
    ContMDiffOn I I' n f s ↔
      ContDiffOn 𝕜 n (e'.extend I' ∘ f ∘ (e.extend I).symm) (e.extend 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] [SmoothManifoldWithCorners I 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'] [SmoothManifoldWithCorners I' M'] {e : PartialHomeomorph M H} {e' : PartialHomeomorph M' H'} {f : M → M'} {s : Set M} {n : ℕ∞} (he : e ∈ SmoothManifoldWithCorners.maximalAtlas I M) (he' : e' ∈ SmoothManifoldWithCorners.maximalAtlas I' M') (hs : s ⊆ e.source) (h2s : Set.MapsTo f s e'.source)
Goal: ContMDiffOn I I' n f s ↔ ContDiffOn 𝕜 n (↑(e'.extend I') ∘ f ∘ ↑(e.extend I).symm) (↑(e.extend I) '' s)
-- contMDiffOn_iff_of_subset_source
theorem contMDiffOn_iff_of_subset_source {x : M} {y : M'} (hs : s ⊆ (chartAt H x).source)
    (h2s : MapsTo f s (chartAt H' y).source) :
    ContMDiffOn I I' n f s ↔
      ContinuousOn f s ∧
        ContDiffOn 𝕜 n (extChartAt I' y ∘ f ∘ (extChartAt I x).symm) (extChartAt I x '' 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] [SmoothManifoldWithCorners I 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'] [SmoothManifoldWithCorners I' M'] {f : M → M'} {s : Set M} {n : ℕ∞} {x : M} {y : M'} (hs : s ⊆ (chartAt H x).source) (h2s : Set.MapsTo f s (chartAt H' y).source)
Goal: ContMDiffOn I I' n f s ↔
  ContinuousOn f s ∧ ContDiffOn 𝕜 n (↑(extChartAt I' y) ∘ f ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) '' s)
-- contMDiffOn_iff_of_subset_source'
theorem contMDiffOn_iff_of_subset_source' {x : M} {y : M'} (hs : s ⊆ (extChartAt I x).source)
    (h2s : MapsTo f s (extChartAt I' y).source) :
    ContMDiffOn I I' n f s ↔
        ContDiffOn 𝕜 n (extChartAt I' y ∘ f ∘ (extChartAt I x).symm) (extChartAt I x '' 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] [SmoothManifoldWithCorners I 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'] [SmoothManifoldWithCorners I' M'] {f : M → M'} {s : Set M} {n : ℕ∞} {x : M} {y : M'} (hs : s ⊆ (extChartAt I x).source) (h2s : Set.MapsTo f s (extChartAt I' y).source)
Goal: ContMDiffOn I I' n f s ↔ ContDiffOn 𝕜 n (↑(extChartAt I' y) ∘ f ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) '' s)
-- contMDiffOn_iff
theorem contMDiffOn_iff :
    ContMDiffOn I I' n f s ↔
      ContinuousOn f s ∧
        ∀ (x : M) (y : M'),
          ContDiffOn 𝕜 n (extChartAt I' y ∘ f ∘ (extChartAt I x).symm)
            ((extChartAt I x).target ∩
              (extChartAt I x).symm ⁻¹' (s ∩ f ⁻¹' (extChartAt I' y).source)) 
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] [SmoothManifoldWithCorners I 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'] [SmoothManifoldWithCorners I' M'] {f : M → M'} {s : Set M} {n : ℕ∞}
Goal: ContMDiffOn I I' n f s ↔
  ContinuousOn f s ∧
    ∀ (x : M) (y : M'),
      ContDiffOn 𝕜 n (↑(extChartAt I' y) ∘ f ∘ ↑(extChartAt I x).symm)
        ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' (s ∩ f ⁻¹' (extChartAt I' y).source))
-- contMDiffOn_iff_target
theorem contMDiffOn_iff_target :
    ContMDiffOn I I' n f s ↔
      ContinuousOn f s ∧
        ∀ y : M',
          ContMDiffOn I 𝓘(𝕜, E') n (extChartAt I' y ∘ f) (s ∩ f ⁻¹' (extChartAt I' y).source) 
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] [SmoothManifoldWithCorners I 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'] [SmoothManifoldWithCorners I' M'] {f : M → M'} {s : Set M} {n : ℕ∞}
Goal: ContMDiffOn I I' n f s ↔
  ContinuousOn f s ∧ ∀ (y : M'), ContMDiffOn I 𝓘(𝕜, E') n (↑(extChartAt I' y) ∘ f) (s ∩ f ⁻¹' (extChartAt I' y).source)
-- smoothOn_iff
theorem smoothOn_iff :
    SmoothOn I I' f s ↔
      ContinuousOn f s ∧
        ∀ (x : M) (y : M'),
          ContDiffOn 𝕜 ⊤ (extChartAt I' y ∘ f ∘ (extChartAt I x).symm)
            ((extChartAt I x).target ∩
              (extChartAt I x).symm ⁻¹' (s ∩ f ⁻¹' (extChartAt I' y).source)) 
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] [SmoothManifoldWithCorners I 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'] [SmoothManifoldWithCorners I' M'] {f : M → M'} {s : Set M}
Goal: SmoothOn I I' f s ↔
  ContinuousOn f s ∧
    ∀ (x : M) (y : M'),
      ContDiffOn 𝕜 ⊤ (↑(extChartAt I' y) ∘ f ∘ ↑(extChartAt I x).symm)
        ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' (s ∩ f ⁻¹' (extChartAt I' y).source))