ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- PartialHomeomorph.tendsto_extend_comp_iff
theorem tendsto_extend_comp_iff {α : Type*} {l : Filter α} {g : α → M}
    (hg : ∀ᶠ z in l, g z ∈ f.source) {y : M} (hy : y ∈ f.source) :
    Tendsto (f.extend I ∘ g) l (𝓝 (f.extend I y)) ↔ Tendsto g l (𝓝 y) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (f : PartialHomeomorph M H) (I : ModelWithCorners 𝕜 E H) {α : Type u_8} {l : Filter α} {g : α → M} (hg : ∀ᶠ (z : α) in l, g z ∈ f.source) {y : M} (hy : y ∈ f.source)
Goal: Filter.Tendsto (↑(f.extend I) ∘ g) l (𝓝 (↑(f.extend I) y)) ↔ Filter.Tendsto g l (𝓝 y)
-- PartialHomeomorph.continuousWithinAt_writtenInExtend_iff
theorem continuousWithinAt_writtenInExtend_iff {f' : PartialHomeomorph M' H'} {g : M → M'} {y : M}
    (hy : y ∈ f.source) (hgy : g y ∈ f'.source) (hmaps : MapsTo g s f'.source) :
    ContinuousWithinAt (f'.extend I' ∘ g ∘ (f.extend I).symm)
      ((f.extend I).symm ⁻¹' s ∩ range I) (f.extend I y) ↔ ContinuousWithinAt g s y 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} {E' : Type u_5} {M' : Type u_6} {H' : Type u_7} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (f : PartialHomeomorph M H) (I : ModelWithCorners 𝕜 E H) [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] [TopologicalSpace H'] [TopologicalSpace M'] (I' : ModelWithCorners 𝕜 E' H') {s : Set M} {f' : PartialHomeomorph M' H'} {g : M → M'} {y : M} (hy : y ∈ f.source) (hgy : g y ∈ f'.source) (hmaps : Set.MapsTo g s f'.source)
Goal: ContinuousWithinAt (↑(f'.extend I') ∘ g ∘ ↑(f.extend I).symm) (↑(f.extend I).symm ⁻¹' s ∩ Set.range ↑I)
    (↑(f.extend I) y) ↔
  ContinuousWithinAt g s y
-- PartialHomeomorph.continuousOn_writtenInExtend_iff
theorem continuousOn_writtenInExtend_iff {f' : PartialHomeomorph M' H'} {g : M → M'}
    (hs : s ⊆ f.source) (hmaps : MapsTo g s f'.source) :
    ContinuousOn (f'.extend I' ∘ g ∘ (f.extend I).symm) (f.extend I '' s) ↔ ContinuousOn g s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} {E' : Type u_5} {M' : Type u_6} {H' : Type u_7} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (f : PartialHomeomorph M H) (I : ModelWithCorners 𝕜 E H) [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] [TopologicalSpace H'] [TopologicalSpace M'] (I' : ModelWithCorners 𝕜 E' H') {s : Set M} {f' : PartialHomeomorph M' H'} {g : M → M'} (hs : s ⊆ f.source) (hmaps : Set.MapsTo g s f'.source)
Goal: ContinuousOn (↑(f'.extend I') ∘ g ∘ ↑(f.extend I).symm) (↑(f.extend I) '' s) ↔ ContinuousOn g s
-- PartialHomeomorph.extend_preimage_mem_nhdsWithin
theorem extend_preimage_mem_nhdsWithin {x : M} (h : x ∈ f.source) (ht : t ∈ 𝓝[s] x) :
    (f.extend I).symm ⁻¹' t ∈ 𝓝[(f.extend I).symm ⁻¹' s ∩ range I] f.extend I x 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (f : PartialHomeomorph M H) (I : ModelWithCorners 𝕜 E H) {s t : Set M} {x : M} (h : x ∈ f.source) (ht : t ∈ 𝓝[s] x)
Goal: ↑(f.extend I).symm ⁻¹' t ∈ 𝓝[↑(f.extend I).symm ⁻¹' s ∩ Set.range ↑I] ↑(f.extend I) x
-- PartialHomeomorph.extend_preimage_mem_nhds
theorem extend_preimage_mem_nhds {x : M} (h : x ∈ f.source) (ht : t ∈ 𝓝 x) :
    (f.extend I).symm ⁻¹' t ∈ 𝓝 (f.extend I x) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (f : PartialHomeomorph M H) (I : ModelWithCorners 𝕜 E H) {t : Set M} {x : M} (h : x ∈ f.source) (ht : t ∈ 𝓝 x)
Goal: ↑(f.extend I).symm ⁻¹' t ∈ 𝓝 (↑(f.extend I) x)
-- PartialHomeomorph.extend_preimage_inter_eq
theorem extend_preimage_inter_eq :
    (f.extend I).symm ⁻¹' (s ∩ t) ∩ range I =
      (f.extend I).symm ⁻¹' s ∩ range I ∩ (f.extend I).symm ⁻¹' t 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (f : PartialHomeomorph M H) (I : ModelWithCorners 𝕜 E H) {s t : Set M}
Goal: ↑(f.extend I).symm ⁻¹' (s ∩ t) ∩ Set.range ↑I = ↑(f.extend I).symm ⁻¹' s ∩ Set.range ↑I ∩ ↑(f.extend I).symm ⁻¹' t
-- PartialHomeomorph.extend_symm_preimage_inter_range_eventuallyEq_aux
theorem extend_symm_preimage_inter_range_eventuallyEq_aux {s : Set M} {x : M} (hx : x ∈ f.source) :
    ((f.extend I).symm ⁻¹' s ∩ range I : Set _) =ᶠ[𝓝 (f.extend I x)]
      ((f.extend I).target ∩ (f.extend I).symm ⁻¹' s : Set _) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (f : PartialHomeomorph M H) (I : ModelWithCorners 𝕜 E H) {s : Set M} {x : M} (hx : x ∈ f.source)
Goal: ↑(f.extend I).symm ⁻¹' s ∩ Set.range ↑I =ᶠ[𝓝 (↑(f.extend I) x)] (f.extend I).target ∩ ↑(f.extend I).symm ⁻¹' s
-- PartialHomeomorph.extend_symm_preimage_inter_range_eventuallyEq
theorem extend_symm_preimage_inter_range_eventuallyEq {s : Set M} {x : M} (hs : s ⊆ f.source)
    (hx : x ∈ f.source) :
    ((f.extend I).symm ⁻¹' s ∩ range I : Set _) =ᶠ[𝓝 (f.extend I x)] f.extend I '' s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (f : PartialHomeomorph M H) (I : ModelWithCorners 𝕜 E H) {s : Set M} {x : M} (hs : s ⊆ f.source) (hx : x ∈ f.source)
Goal: ↑(f.extend I).symm ⁻¹' s ∩ Set.range ↑I =ᶠ[𝓝 (↑(f.extend I) x)] ↑(f.extend I) '' s
-- PartialHomeomorph.extend_coord_change_source
theorem extend_coord_change_source :
    ((f.extend I).symm ≫ f'.extend I).source = I '' (f.symm ≫ₕ f').source 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (f f' : PartialHomeomorph M H) (I : ModelWithCorners 𝕜 E H)
Goal: ((f.extend I).symm ≫ f'.extend I).source = ↑I '' (f.symm ≫ₕ f').source
-- PartialHomeomorph.extend_image_source_inter
theorem extend_image_source_inter :
    f.extend I '' (f.source ∩ f'.source) = ((f.extend I).symm ≫ f'.extend I).source 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (f f' : PartialHomeomorph M H) (I : ModelWithCorners 𝕜 E H)
Goal: ↑(f.extend I) '' (f.source ∩ f'.source) = ((f.extend I).symm ≫ f'.extend I).source
-- PartialHomeomorph.extend_coord_change_source_mem_nhdsWithin
theorem extend_coord_change_source_mem_nhdsWithin {x : E}
    (hx : x ∈ ((f.extend I).symm ≫ f'.extend I).source) :
    ((f.extend I).symm ≫ f'.extend I).source ∈ 𝓝[range I] x 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (f f' : PartialHomeomorph M H) (I : ModelWithCorners 𝕜 E H) {x : E} (hx : x ∈ ((f.extend I).symm ≫ f'.extend I).source)
Goal: ((f.extend I).symm ≫ f'.extend I).source ∈ 𝓝[Set.range ↑I] x
-- PartialHomeomorph.extend_coord_change_source_mem_nhdsWithin'
theorem extend_coord_change_source_mem_nhdsWithin' {x : M} (hxf : x ∈ f.source)
    (hxf' : x ∈ f'.source) :
    ((f.extend I).symm ≫ f'.extend I).source ∈ 𝓝[range I] f.extend I x 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (f f' : PartialHomeomorph M H) (I : ModelWithCorners 𝕜 E H) {x : M} (hxf : x ∈ f.source) (hxf' : x ∈ f'.source)
Goal: ((f.extend I).symm ≫ f'.extend I).source ∈ 𝓝[Set.range ↑I] ↑(f.extend I) x
-- PartialHomeomorph.contDiffOn_extend_coord_change
theorem contDiffOn_extend_coord_change [ChartedSpace H M] (hf : f ∈ maximalAtlas I M)
    (hf' : f' ∈ maximalAtlas I M) :
    ContDiffOn 𝕜 ⊤ (f.extend I ∘ (f'.extend I).symm) ((f'.extend I).symm ≫ f.extend I).source 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] {f f' : PartialHomeomorph M H} (I : ModelWithCorners 𝕜 E H) [ChartedSpace H M] (hf : f ∈ SmoothManifoldWithCorners.maximalAtlas I M) (hf' : f' ∈ SmoothManifoldWithCorners.maximalAtlas I M)
Goal: ContDiffOn 𝕜 ⊤ (↑(f.extend I) ∘ ↑(f'.extend I).symm) ((f'.extend I).symm ≫ f.extend I).source
-- PartialHomeomorph.contDiffWithinAt_extend_coord_change
theorem contDiffWithinAt_extend_coord_change [ChartedSpace H M] (hf : f ∈ maximalAtlas I M)
    (hf' : f' ∈ maximalAtlas I M) {x : E} (hx : x ∈ ((f'.extend I).symm ≫ f.extend I).source) :
    ContDiffWithinAt 𝕜 ⊤ (f.extend I ∘ (f'.extend I).symm) (range I) x 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] {f f' : PartialHomeomorph M H} (I : ModelWithCorners 𝕜 E H) [ChartedSpace H M] (hf : f ∈ SmoothManifoldWithCorners.maximalAtlas I M) (hf' : f' ∈ SmoothManifoldWithCorners.maximalAtlas I M) {x : E} (hx : x ∈ ((f'.extend I).symm ≫ f.extend I).source)
Goal: ContDiffWithinAt 𝕜 ⊤ (↑(f.extend I) ∘ ↑(f'.extend I).symm) (Set.range ↑I) x
-- PartialHomeomorph.contDiffWithinAt_extend_coord_change'
theorem contDiffWithinAt_extend_coord_change' [ChartedSpace H M] (hf : f ∈ maximalAtlas I M)
    (hf' : f' ∈ maximalAtlas I M) {x : M} (hxf : x ∈ f.source) (hxf' : x ∈ f'.source) :
    ContDiffWithinAt 𝕜 ⊤ (f.extend I ∘ (f'.extend I).symm) (range I) (f'.extend I x) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] {f f' : PartialHomeomorph M H} (I : ModelWithCorners 𝕜 E H) [ChartedSpace H M] (hf : f ∈ SmoothManifoldWithCorners.maximalAtlas I M) (hf' : f' ∈ SmoothManifoldWithCorners.maximalAtlas I M) {x : M} (hxf : x ∈ f.source) (hxf' : x ∈ f'.source)
Goal: ContDiffWithinAt 𝕜 ⊤ (↑(f.extend I) ∘ ↑(f'.extend I).symm) (Set.range ↑I) (↑(f'.extend I) x)
-- extChartAt_coe
theorem extChartAt_coe (x : M) : ⇑(extChartAt I x) = I ∘ chartAt H x 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (I : ModelWithCorners 𝕜 E H) [ChartedSpace H M] (x : M)
Goal: ↑(extChartAt I x) = ↑I ∘ ↑(chartAt H x)
-- extChartAt_coe_symm
theorem extChartAt_coe_symm (x : M) : ⇑(extChartAt I x).symm = (chartAt H x).symm ∘ I.symm 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (I : ModelWithCorners 𝕜 E H) [ChartedSpace H M] (x : M)
Goal: ↑(extChartAt I x).symm = ↑(chartAt H x).symm ∘ ↑I.symm
-- extChartAt_source
theorem extChartAt_source (x : M) : (extChartAt I x).source = (chartAt H x).source 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (I : ModelWithCorners 𝕜 E H) [ChartedSpace H M] (x : M)
Goal: (extChartAt I x).source = (chartAt H x).source
-- isOpen_extChartAt_source
theorem isOpen_extChartAt_source (x : M) : IsOpen (extChartAt I x).source 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (I : ModelWithCorners 𝕜 E H) [ChartedSpace H M] (x : M)
Goal: IsOpen (extChartAt I x).source
-- mem_extChartAt_source
theorem mem_extChartAt_source (x : M) : x ∈ (extChartAt I x).source 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (I : ModelWithCorners 𝕜 E H) [ChartedSpace H M] (x : M)
Goal: x ∈ (extChartAt I x).source
-- mem_extChartAt_target
theorem mem_extChartAt_target (x : M) : extChartAt I x x ∈ (extChartAt I x).target 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (I : ModelWithCorners 𝕜 E H) [ChartedSpace H M] (x : M)
Goal: ↑(extChartAt I x) x ∈ (extChartAt I x).target
-- extChartAt_target
theorem extChartAt_target (x : M) :
    (extChartAt I x).target = I.symm ⁻¹' (chartAt H x).target ∩ range I 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (I : ModelWithCorners 𝕜 E H) [ChartedSpace H M] (x : M)
Goal: (extChartAt I x).target = ↑I.symm ⁻¹' (chartAt H x).target ∩ Set.range ↑I
-- uniqueDiffOn_extChartAt_target
theorem uniqueDiffOn_extChartAt_target (x : M) : UniqueDiffOn 𝕜 (extChartAt I x).target 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (I : ModelWithCorners 𝕜 E H) [ChartedSpace H M] (x : M)
Goal: UniqueDiffOn 𝕜 (extChartAt I x).target
-- uniqueDiffWithinAt_extChartAt_target
theorem uniqueDiffWithinAt_extChartAt_target (x : M) :
    UniqueDiffWithinAt 𝕜 (extChartAt I x).target (extChartAt I x x) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (I : ModelWithCorners 𝕜 E H) [ChartedSpace H M] (x : M)
Goal: UniqueDiffWithinAt 𝕜 (extChartAt I x).target (↑(extChartAt I x) x)
-- extChartAt_to_inv
theorem extChartAt_to_inv (x : M) : (extChartAt I x).symm ((extChartAt I x) x) = x 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (I : ModelWithCorners 𝕜 E H) [ChartedSpace H M] (x : M)
Goal: ↑(extChartAt I x).symm (↑(extChartAt I x) x) = x
-- mapsTo_extChartAt
theorem mapsTo_extChartAt {x : M} (hs : s ⊆ (chartAt H x).source) :
    MapsTo (extChartAt I x) s ((extChartAt I x).symm ⁻¹' s ∩ range I) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (I : ModelWithCorners 𝕜 E H) {s : Set M} [ChartedSpace H M] {x : M} (hs : s ⊆ (chartAt H x).source)
Goal: Set.MapsTo (↑(extChartAt I x)) s (↑(extChartAt I x).symm ⁻¹' s ∩ Set.range ↑I)
-- extChartAt_source_mem_nhds'
theorem extChartAt_source_mem_nhds' {x x' : M} (h : x' ∈ (extChartAt I x).source) :
    (extChartAt I x).source ∈ 𝓝 x' 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (I : ModelWithCorners 𝕜 E H) [ChartedSpace H M] {x x' : M} (h : x' ∈ (extChartAt I x).source)
Goal: (extChartAt I x).source ∈ 𝓝 x'
-- extChartAt_source_mem_nhds
theorem extChartAt_source_mem_nhds (x : M) : (extChartAt I x).source ∈ 𝓝 x 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (I : ModelWithCorners 𝕜 E H) [ChartedSpace H M] (x : M)
Goal: (extChartAt I x).source ∈ 𝓝 x
-- extChartAt_source_mem_nhdsWithin'
theorem extChartAt_source_mem_nhdsWithin' {x x' : M} (h : x' ∈ (extChartAt I x).source) :
    (extChartAt I x).source ∈ 𝓝[s] x' 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (I : ModelWithCorners 𝕜 E H) {s : Set M} [ChartedSpace H M] {x x' : M} (h : x' ∈ (extChartAt I x).source)
Goal: (extChartAt I x).source ∈ 𝓝[s] x'
-- extChartAt_source_mem_nhdsWithin
theorem extChartAt_source_mem_nhdsWithin (x : M) : (extChartAt I x).source ∈ 𝓝[s] x 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (I : ModelWithCorners 𝕜 E H) {s : Set M} [ChartedSpace H M] (x : M)
Goal: (extChartAt I x).source ∈ 𝓝[s] x
-- continuousOn_extChartAt
theorem continuousOn_extChartAt (x : M) : ContinuousOn (extChartAt I x) (extChartAt I x).source 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (I : ModelWithCorners 𝕜 E H) [ChartedSpace H M] (x : M)
Goal: ContinuousOn (↑(extChartAt I x)) (extChartAt I x).source
-- continuousAt_extChartAt'
theorem continuousAt_extChartAt' {x x' : M} (h : x' ∈ (extChartAt I x).source) :
    ContinuousAt (extChartAt I x) x' 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (I : ModelWithCorners 𝕜 E H) [ChartedSpace H M] {x x' : M} (h : x' ∈ (extChartAt I x).source)
Goal: ContinuousAt (↑(extChartAt I x)) x'
-- continuousAt_extChartAt
theorem continuousAt_extChartAt (x : M) : ContinuousAt (extChartAt I x) x 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (I : ModelWithCorners 𝕜 E H) [ChartedSpace H M] (x : M)
Goal: ContinuousAt (↑(extChartAt I x)) x
-- map_extChartAt_nhds'
theorem map_extChartAt_nhds' {x y : M} (hy : y ∈ (extChartAt I x).source) :
    map (extChartAt I x) (𝓝 y) = 𝓝[range I] extChartAt I x y 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (I : ModelWithCorners 𝕜 E H) [ChartedSpace H M] {x y : M} (hy : y ∈ (extChartAt I x).source)
Goal: Filter.map (↑(extChartAt I x)) (𝓝 y) = 𝓝[Set.range ↑I] ↑(extChartAt I x) y
-- map_extChartAt_nhds
theorem map_extChartAt_nhds (x : M) : map (extChartAt I x) (𝓝 x) = 𝓝[range I] extChartAt I x x 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (I : ModelWithCorners 𝕜 E H) [ChartedSpace H M] (x : M)
Goal: Filter.map (↑(extChartAt I x)) (𝓝 x) = 𝓝[Set.range ↑I] ↑(extChartAt I x) x
-- map_extChartAt_nhds_of_boundaryless
theorem map_extChartAt_nhds_of_boundaryless [I.Boundaryless] (x : M) :
    map (extChartAt I x) (𝓝 x) = 𝓝 (extChartAt I x x) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (I : ModelWithCorners 𝕜 E H) [ChartedSpace H M] [I.Boundaryless] (x : M)
Goal: Filter.map (↑(extChartAt I x)) (𝓝 x) = 𝓝 (↑(extChartAt I x) x)
-- extChartAt_image_nhd_mem_nhds_of_boundaryless
theorem extChartAt_image_nhd_mem_nhds_of_boundaryless [I.Boundaryless]
    {x : M} (hx : s ∈ 𝓝 x) : extChartAt I x '' s ∈ 𝓝 (extChartAt I x x) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (I : ModelWithCorners 𝕜 E H) {s : Set M} [ChartedSpace H M] [I.Boundaryless] {x : M} (hx : s ∈ 𝓝 x)
Goal: ↑(extChartAt I x) '' s ∈ 𝓝 (↑(extChartAt I x) x)
-- extChartAt_target_mem_nhdsWithin'
theorem extChartAt_target_mem_nhdsWithin' {x y : M} (hy : y ∈ (extChartAt I x).source) :
    (extChartAt I x).target ∈ 𝓝[range I] extChartAt I x y 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (I : ModelWithCorners 𝕜 E H) [ChartedSpace H M] {x y : M} (hy : y ∈ (extChartAt I x).source)
Goal: (extChartAt I x).target ∈ 𝓝[Set.range ↑I] ↑(extChartAt I x) y
-- extChartAt_target_mem_nhdsWithin
theorem extChartAt_target_mem_nhdsWithin (x : M) :
    (extChartAt I x).target ∈ 𝓝[range I] extChartAt I x x 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (I : ModelWithCorners 𝕜 E H) [ChartedSpace H M] (x : M)
Goal: (extChartAt I x).target ∈ 𝓝[Set.range ↑I] ↑(extChartAt I x) x
-- isOpen_extChartAt_target
theorem isOpen_extChartAt_target [I.Boundaryless] (x : M) : IsOpen (extChartAt I x).target 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (I : ModelWithCorners 𝕜 E H) [ChartedSpace H M] [I.Boundaryless] (x : M)
Goal: IsOpen (extChartAt I x).target
-- extChartAt_target_mem_nhds
theorem extChartAt_target_mem_nhds [I.Boundaryless] (x : M) :
    (extChartAt I x).target ∈ 𝓝 (extChartAt I x x) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (I : ModelWithCorners 𝕜 E H) [ChartedSpace H M] [I.Boundaryless] (x : M)
Goal: (extChartAt I x).target ∈ 𝓝 (↑(extChartAt I x) x)
-- extChartAt_target_mem_nhds'
theorem extChartAt_target_mem_nhds' [I.Boundaryless] {x : M} {y : E}
    (m : y ∈ (extChartAt I x).target) : (extChartAt I x).target ∈ 𝓝 y 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (I : ModelWithCorners 𝕜 E H) [ChartedSpace H M] [I.Boundaryless] {x : M} {y : E} (m : y ∈ (extChartAt I x).target)
Goal: (extChartAt I x).target ∈ 𝓝 y
-- extChartAt_target_subset_range
theorem extChartAt_target_subset_range (x : M) : (extChartAt I x).target ⊆ range I 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (I : ModelWithCorners 𝕜 E H) [ChartedSpace H M] (x : M)
Goal: (extChartAt I x).target ⊆ Set.range ↑I
-- nhdsWithin_extChartAt_target_eq'
theorem nhdsWithin_extChartAt_target_eq' {x y : M} (hy : y ∈ (extChartAt I x).source) :
    𝓝[(extChartAt I x).target] extChartAt I x y = 𝓝[range I] extChartAt I x y 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (I : ModelWithCorners 𝕜 E H) [ChartedSpace H M] {x y : M} (hy : y ∈ (extChartAt I x).source)
Goal: 𝓝[(extChartAt I x).target] ↑(extChartAt I x) y = 𝓝[Set.range ↑I] ↑(extChartAt I x) y
-- nhdsWithin_extChartAt_target_eq
theorem nhdsWithin_extChartAt_target_eq (x : M) :
    𝓝[(extChartAt I x).target] (extChartAt I x) x = 𝓝[range I] (extChartAt I x) x 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (I : ModelWithCorners 𝕜 E H) [ChartedSpace H M] (x : M)
Goal: 𝓝[(extChartAt I x).target] ↑(extChartAt I x) x = 𝓝[Set.range ↑I] ↑(extChartAt I x) x
-- continuousAt_extChartAt_symm''
theorem continuousAt_extChartAt_symm'' {x : M} {y : E} (h : y ∈ (extChartAt I x).target) :
    ContinuousAt (extChartAt I x).symm y 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (I : ModelWithCorners 𝕜 E H) [ChartedSpace H M] {x : M} {y : E} (h : y ∈ (extChartAt I x).target)
Goal: ContinuousAt (↑(extChartAt I x).symm) y