ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- continuousAt_extChartAt_symm'
theorem continuousAt_extChartAt_symm' {x x' : M} (h : x' ∈ (extChartAt I x).source) :
    ContinuousAt (extChartAt I x).symm (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).symm) (↑(extChartAt I x) x')
-- continuousAt_extChartAt_symm
theorem continuousAt_extChartAt_symm (x : M) :
    ContinuousAt (extChartAt I x).symm ((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).symm) (↑(extChartAt I x) x)
-- continuousOn_extChartAt_symm
theorem continuousOn_extChartAt_symm (x : M) :
    ContinuousOn (extChartAt I x).symm (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: ContinuousOn (↑(extChartAt I x).symm) (extChartAt I x).target
-- isOpen_extChartAt_preimage'
theorem isOpen_extChartAt_preimage' (x : M) {s : Set E} (hs : IsOpen s) :
    IsOpen ((extChartAt I x).source ∩ extChartAt I x ⁻¹' 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] (I : ModelWithCorners 𝕜 E H) [ChartedSpace H M] (x : M) {s : Set E} (hs : IsOpen s)
Goal: IsOpen ((extChartAt I x).source ∩ ↑(extChartAt I x) ⁻¹' s)
-- isOpen_extChartAt_preimage
theorem isOpen_extChartAt_preimage (x : M) {s : Set E} (hs : IsOpen s) :
    IsOpen ((chartAt H x).source ∩ extChartAt I x ⁻¹' 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] (I : ModelWithCorners 𝕜 E H) [ChartedSpace H M] (x : M) {s : Set E} (hs : IsOpen s)
Goal: IsOpen ((chartAt H x).source ∩ ↑(extChartAt I x) ⁻¹' s)
-- map_extChartAt_nhdsWithin_eq_image'
theorem map_extChartAt_nhdsWithin_eq_image' {x y : M} (hy : y ∈ (extChartAt I x).source) :
    map (extChartAt I x) (𝓝[s] y) =
      𝓝[extChartAt I x '' ((extChartAt I x).source ∩ s)] 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) {s : Set M} [ChartedSpace H M] {x y : M} (hy : y ∈ (extChartAt I x).source)
Goal: Filter.map (↑(extChartAt I x)) (𝓝[s] y) = 𝓝[↑(extChartAt I x) '' ((extChartAt I x).source ∩ s)] ↑(extChartAt I x) y
-- map_extChartAt_nhdsWithin_eq_image
theorem map_extChartAt_nhdsWithin_eq_image (x : M) :
    map (extChartAt I x) (𝓝[s] x) =
      𝓝[extChartAt I x '' ((extChartAt I x).source ∩ s)] 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] (x : M)
Goal: Filter.map (↑(extChartAt I x)) (𝓝[s] x) = 𝓝[↑(extChartAt I x) '' ((extChartAt I x).source ∩ s)] ↑(extChartAt I x) x
-- map_extChartAt_nhdsWithin'
theorem map_extChartAt_nhdsWithin' {x y : M} (hy : y ∈ (extChartAt I x).source) :
    map (extChartAt I x) (𝓝[s] y) = 𝓝[(extChartAt I x).symm ⁻¹' s ∩ 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) {s : Set M} [ChartedSpace H M] {x y : M} (hy : y ∈ (extChartAt I x).source)
Goal: Filter.map (↑(extChartAt I x)) (𝓝[s] y) = 𝓝[↑(extChartAt I x).symm ⁻¹' s ∩ Set.range ↑I] ↑(extChartAt I x) y
-- map_extChartAt_nhdsWithin
theorem map_extChartAt_nhdsWithin (x : M) :
    map (extChartAt I x) (𝓝[s] x) = 𝓝[(extChartAt I x).symm ⁻¹' s ∩ 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) {s : Set M} [ChartedSpace H M] (x : M)
Goal: Filter.map (↑(extChartAt I x)) (𝓝[s] x) = 𝓝[↑(extChartAt I x).symm ⁻¹' s ∩ Set.range ↑I] ↑(extChartAt I x) x
-- map_extChartAt_symm_nhdsWithin'
theorem map_extChartAt_symm_nhdsWithin' {x y : M} (hy : y ∈ (extChartAt I x).source) :
    map (extChartAt I x).symm (𝓝[(extChartAt I x).symm ⁻¹' s ∩ range I] extChartAt I x y) =
      𝓝[s] 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) {s : Set M} [ChartedSpace H M] {x y : M} (hy : y ∈ (extChartAt I x).source)
Goal: Filter.map (↑(extChartAt I x).symm) (𝓝[↑(extChartAt I x).symm ⁻¹' s ∩ Set.range ↑I] ↑(extChartAt I x) y) = 𝓝[s] y
-- map_extChartAt_symm_nhdsWithin_range'
theorem map_extChartAt_symm_nhdsWithin_range' {x y : M} (hy : y ∈ (extChartAt I x).source) :
    map (extChartAt I x).symm (𝓝[range I] extChartAt I x y) = 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] {x y : M} (hy : y ∈ (extChartAt I x).source)
Goal: Filter.map (↑(extChartAt I x).symm) (𝓝[Set.range ↑I] ↑(extChartAt I x) y) = 𝓝 y
-- map_extChartAt_symm_nhdsWithin
theorem map_extChartAt_symm_nhdsWithin (x : M) :
    map (extChartAt I x).symm (𝓝[(extChartAt I x).symm ⁻¹' s ∩ range I] extChartAt I x x) =
      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: Filter.map (↑(extChartAt I x).symm) (𝓝[↑(extChartAt I x).symm ⁻¹' s ∩ Set.range ↑I] ↑(extChartAt I x) x) = 𝓝[s] x
-- map_extChartAt_symm_nhdsWithin_range
theorem map_extChartAt_symm_nhdsWithin_range (x : M) :
    map (extChartAt I x).symm (𝓝[range I] extChartAt I x x) = 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: Filter.map (↑(extChartAt I x).symm) (𝓝[Set.range ↑I] ↑(extChartAt I x) x) = 𝓝 x
-- extChartAt_preimage_mem_nhdsWithin'
theorem extChartAt_preimage_mem_nhdsWithin' {x x' : M} (h : x' ∈ (extChartAt I x).source)
    (ht : t ∈ 𝓝[s] x') :
    (extChartAt I x).symm ⁻¹' t ∈ 𝓝[(extChartAt I x).symm ⁻¹' s ∩ 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) {s t : Set M} [ChartedSpace H M] {x x' : M} (h : x' ∈ (extChartAt I x).source) (ht : t ∈ 𝓝[s] x')
Goal: ↑(extChartAt I x).symm ⁻¹' t ∈ 𝓝[↑(extChartAt I x).symm ⁻¹' s ∩ Set.range ↑I] ↑(extChartAt I x) x'
-- extChartAt_preimage_mem_nhdsWithin
theorem extChartAt_preimage_mem_nhdsWithin {x : M} (ht : t ∈ 𝓝[s] x) :
    (extChartAt I x).symm ⁻¹' t ∈ 𝓝[(extChartAt I x).symm ⁻¹' s ∩ 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) {s t : Set M} [ChartedSpace H M] {x : M} (ht : t ∈ 𝓝[s] x)
Goal: ↑(extChartAt I x).symm ⁻¹' t ∈ 𝓝[↑(extChartAt I x).symm ⁻¹' s ∩ Set.range ↑I] ↑(extChartAt I x) x
-- extChartAt_preimage_mem_nhds'
theorem extChartAt_preimage_mem_nhds' {x x' : M} (h : x' ∈ (extChartAt I x).source)
    (ht : t ∈ 𝓝 x') : (extChartAt I x).symm ⁻¹' t ∈ 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) {t : Set M} [ChartedSpace H M] {x x' : M} (h : x' ∈ (extChartAt I x).source) (ht : t ∈ 𝓝 x')
Goal: ↑(extChartAt I x).symm ⁻¹' t ∈ 𝓝 (↑(extChartAt I x) x')
-- extChartAt_preimage_mem_nhds
theorem extChartAt_preimage_mem_nhds {x : M} (ht : t ∈ 𝓝 x) :
    (extChartAt I x).symm ⁻¹' t ∈ 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) {t : Set M} [ChartedSpace H M] {x : M} (ht : t ∈ 𝓝 x)
Goal: ↑(extChartAt I x).symm ⁻¹' t ∈ 𝓝 (↑(extChartAt I x) x)
-- extChartAt_preimage_inter_eq
theorem extChartAt_preimage_inter_eq (x : M) :
    (extChartAt I x).symm ⁻¹' (s ∩ t) ∩ range I =
      (extChartAt I x).symm ⁻¹' s ∩ range I ∩ (extChartAt I x).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] (I : ModelWithCorners 𝕜 E H) {s t : Set M} [ChartedSpace H M] (x : M)
Goal: ↑(extChartAt I x).symm ⁻¹' (s ∩ t) ∩ Set.range ↑I =
  ↑(extChartAt I x).symm ⁻¹' s ∩ Set.range ↑I ∩ ↑(extChartAt I x).symm ⁻¹' t
-- ContinuousWithinAt.nhdsWithin_extChartAt_symm_preimage_inter_range
theorem ContinuousWithinAt.nhdsWithin_extChartAt_symm_preimage_inter_range
    {f : M → M'} {x : M} (hc : ContinuousWithinAt f s x) :
    𝓝[(extChartAt I x).symm ⁻¹' s ∩ range I] (extChartAt I x x) =
      𝓝[(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} {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] (I : ModelWithCorners 𝕜 E H) [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] [TopologicalSpace H'] [TopologicalSpace M'] (I' : ModelWithCorners 𝕜 E' H') {s : Set M} [ChartedSpace H M] [ChartedSpace H' M'] {f : M → M'} {x : M} (hc : ContinuousWithinAt f s x)
Goal: 𝓝[↑(extChartAt I x).symm ⁻¹' s ∩ Set.range ↑I] ↑(extChartAt I x) x =
  𝓝[(extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' (s ∩ f ⁻¹' (extChartAt I' (f x)).source)] ↑(extChartAt I x) x
-- ext_coord_change_source
theorem ext_coord_change_source (x x' : M) :
    ((extChartAt I x').symm ≫ extChartAt I x).source =
      I '' ((chartAt H x').symm ≫ₕ 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 x' : M)
Goal: ((extChartAt I x').symm ≫ extChartAt I x).source = ↑I '' ((chartAt H x').symm ≫ₕ chartAt H x).source
-- contDiffOn_ext_coord_change
theorem contDiffOn_ext_coord_change [SmoothManifoldWithCorners I M] (x x' : M) :
    ContDiffOn 𝕜 ⊤ (extChartAt I x ∘ (extChartAt I x').symm)
      ((extChartAt I x').symm ≫ 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] [SmoothManifoldWithCorners I M] (x x' : M)
Goal: ContDiffOn 𝕜 ⊤ (↑(extChartAt I x) ∘ ↑(extChartAt I x').symm) ((extChartAt I x').symm ≫ extChartAt I x).source
-- contDiffWithinAt_ext_coord_change
theorem contDiffWithinAt_ext_coord_change [SmoothManifoldWithCorners I M] (x x' : M) {y : E}
    (hy : y ∈ ((extChartAt I x').symm ≫ extChartAt I x).source) :
    ContDiffWithinAt 𝕜 ⊤ (extChartAt I x ∘ (extChartAt I x').symm) (range I) 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] [SmoothManifoldWithCorners I M] (x x' : M) {y : E} (hy : y ∈ ((extChartAt I x').symm ≫ extChartAt I x).source)
Goal: ContDiffWithinAt 𝕜 ⊤ (↑(extChartAt I x) ∘ ↑(extChartAt I x').symm) (Set.range ↑I) y
-- writtenInExtChartAt_chartAt
theorem writtenInExtChartAt_chartAt {x : M} {y : E} (h : y ∈ (extChartAt I x).target) :
    writtenInExtChartAt I I x (chartAt H x) y = 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: writtenInExtChartAt I I x (↑(chartAt H x)) y = y
-- writtenInExtChartAt_chartAt_symm
theorem writtenInExtChartAt_chartAt_symm {x : M} {y : E} (h : y ∈ (extChartAt I x).target) :
    writtenInExtChartAt I I (chartAt H x x) (chartAt H x).symm y = 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: writtenInExtChartAt I I (↑(chartAt H x) x) (↑(chartAt H x).symm) y = y
-- writtenInExtChartAt_extChartAt
theorem writtenInExtChartAt_extChartAt {x : M} {y : E} (h : y ∈ (extChartAt I x).target) :
    writtenInExtChartAt I 𝓘(𝕜, E) x (extChartAt I x) y = 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: writtenInExtChartAt I 𝓘(𝕜, E) x (↑(extChartAt I x)) y = y
-- writtenInExtChartAt_extChartAt_symm
theorem writtenInExtChartAt_extChartAt_symm {x : M} {y : E} (h : y ∈ (extChartAt I x).target) :
    writtenInExtChartAt 𝓘(𝕜, E) I (extChartAt I x x) (extChartAt I x).symm y = 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: writtenInExtChartAt 𝓘(𝕜, E) I (↑(extChartAt I x) x) (↑(extChartAt I x).symm) y = y
-- extChartAt_self_eq
theorem extChartAt_self_eq {x : H} : ⇑(extChartAt I x) = I 
Arguments: (𝕜 : Type u_1) {E : Type u_2} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {x : H}
Goal: ↑(extChartAt I x) = ↑I
-- extChartAt_self_apply
theorem extChartAt_self_apply {x y : H} : extChartAt I x y = I y 
Arguments: (𝕜 : Type u_1) {E : Type u_2} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {x y : H}
Goal: ↑(extChartAt I x) y = ↑I y
-- extChartAt_model_space_eq_id
theorem extChartAt_model_space_eq_id (x : E) : extChartAt 𝓘(𝕜, E) x = PartialEquiv.refl E 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] (x : E)
Goal: extChartAt 𝓘(𝕜, E) x = PartialEquiv.refl E
-- ext_chart_model_space_apply
theorem ext_chart_model_space_apply {x y : E} : extChartAt 𝓘(𝕜, E) x y = y 
Arguments: (𝕜 : Type u_1) {E : Type u_2} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x y : E}
Goal: ↑(extChartAt 𝓘(𝕜, E) x) y = y
-- extChartAt_prod
theorem extChartAt_prod (x : M × M') :
    extChartAt (I.prod I') x = (extChartAt I x.1).prod (extChartAt I' x.2) 
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] (I : ModelWithCorners 𝕜 E H) [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] [TopologicalSpace H'] [TopologicalSpace M'] (I' : ModelWithCorners 𝕜 E' H') [ChartedSpace H M] [ChartedSpace H' M'] (x : M × M')
Goal: extChartAt (I.prod I') x = (extChartAt I x.1).prod (extChartAt I' x.2)
-- extChartAt_comp
theorem extChartAt_comp [ChartedSpace H H'] (x : M') :
    (letI := ChartedSpace.comp H H' M'; extChartAt I x) =
      (chartAt H' x).toPartialEquiv ≫ extChartAt I (chartAt H' x x) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {H : Type u_4} {M' : Type u_6} {H' : Type u_7} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) [TopologicalSpace H'] [TopologicalSpace M'] [ChartedSpace H' M'] [ChartedSpace H H'] (x : M')
Goal: extChartAt I x = (chartAt H' x).toPartialEquiv ≫ extChartAt I (↑(chartAt H' x) x)
-- writtenInExtChartAt_chartAt_comp
theorem writtenInExtChartAt_chartAt_comp [ChartedSpace H H'] (x : M') {y}
    (hy : y ∈ letI := ChartedSpace.comp H H' M'; (extChartAt I x).target) :
    (letI := ChartedSpace.comp H H' M'; writtenInExtChartAt I I x (chartAt H' x) y) = y 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {H : Type u_4} {M' : Type u_6} {H' : Type u_7} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) [TopologicalSpace H'] [TopologicalSpace M'] [ChartedSpace H' M'] [ChartedSpace H H'] (x : M') {y : E} (hy : y ∈ (extChartAt I x).target)
Goal: writtenInExtChartAt I I x (↑(chartAt H' x)) y = y
-- writtenInExtChartAt_chartAt_symm_comp
theorem writtenInExtChartAt_chartAt_symm_comp [ChartedSpace H H'] (x : M') {y}
    (hy : y ∈ letI := ChartedSpace.comp H H' M'; (extChartAt I x).target) :
    ( letI := ChartedSpace.comp H H' M'
      writtenInExtChartAt I I (chartAt H' x x) (chartAt H' x).symm y) = y 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {H : Type u_4} {M' : Type u_6} {H' : Type u_7} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) [TopologicalSpace H'] [TopologicalSpace M'] [ChartedSpace H' M'] [ChartedSpace H H'] (x : M') {y : E} (hy : y ∈ (extChartAt I x).target)
Goal: writtenInExtChartAt I I (↑(chartAt H' x) x) (↑(chartAt H' x).symm) y = y
-- Manifold.locallyCompact_of_finiteDimensional
lemma Manifold.locallyCompact_of_finiteDimensional [LocallyCompactSpace 𝕜]
    [FiniteDimensional 𝕜 E] : LocallyCompactSpace M 
Arguments: {E : Type u_1} {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] [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] [LocallyCompactSpace 𝕜] [FiniteDimensional 𝕜 E]
Goal: LocallyCompactSpace M
-- SmoothMap.coe_add
@[to_additive (attr := simp)]
theorem coe_mul {G : Type*} [Mul G] [TopologicalSpace G] [ChartedSpace H' G] [SmoothMul I' G]
    (f g : C^∞⟮I, N; I', G⟯) : ⇑(f * g) = f * g 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H : Type u_4} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {H' : Type u_5} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {N : Type u_6} [TopologicalSpace N] [ChartedSpace H N] {G : Type u_10} [Add G] [TopologicalSpace G] [ChartedSpace H' G] [SmoothAdd I' G] (f g : C^⊤⟮I, N; I', G⟯)
Goal: ⇑(f + g) = ⇑f + ⇑g
-- SmoothMap.coe_mul
@[to_additive (attr := simp)]
theorem coe_mul {G : Type*} [Mul G] [TopologicalSpace G] [ChartedSpace H' G] [SmoothMul I' G]
    (f g : C^∞⟮I, N; I', G⟯) : ⇑(f * g) = f * g 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H : Type u_4} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {H' : Type u_5} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {N : Type u_6} [TopologicalSpace N] [ChartedSpace H N] {G : Type u_10} [Mul G] [TopologicalSpace G] [ChartedSpace H' G] [SmoothMul I' G] (f g : C^⊤⟮I, N; I', G⟯)
Goal: ⇑(f * g) = ⇑f * ⇑g
-- SmoothMap.add_comp
@[to_additive (attr := simp)]
theorem mul_comp {G : Type*} [Mul G] [TopologicalSpace G] [ChartedSpace H' G] [SmoothMul I' G]
    (f g : C^∞⟮I'', N'; I', G⟯) (h : C^∞⟮I, N; I'', N'⟯) : (f * g).comp h = f.comp h * g.comp h 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H : Type u_4} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {H' : Type u_5} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {N : Type u_6} [TopologicalSpace N] [ChartedSpace H N] {E'' : Type u_7} [NormedAddCommGroup E''] [NormedSpace 𝕜 E''] {H'' : Type u_8} [TopologicalSpace H''] {I'' : ModelWithCorners 𝕜 E'' H''} {N' : Type u_9} [TopologicalSpace N'] [ChartedSpace H'' N'] {G : Type u_10} [Add G] [TopologicalSpace G] [ChartedSpace H' G] [SmoothAdd I' G] (f g : C^⊤⟮I'', N'; I', G⟯) (h : C^⊤⟮I, N; I'', N'⟯)
Goal: (f + g).comp h = f.comp h + g.comp h
-- SmoothMap.mul_comp
@[to_additive (attr := simp)]
theorem mul_comp {G : Type*} [Mul G] [TopologicalSpace G] [ChartedSpace H' G] [SmoothMul I' G]
    (f g : C^∞⟮I'', N'; I', G⟯) (h : C^∞⟮I, N; I'', N'⟯) : (f * g).comp h = f.comp h * g.comp h 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H : Type u_4} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {H' : Type u_5} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {N : Type u_6} [TopologicalSpace N] [ChartedSpace H N] {E'' : Type u_7} [NormedAddCommGroup E''] [NormedSpace 𝕜 E''] {H'' : Type u_8} [TopologicalSpace H''] {I'' : ModelWithCorners 𝕜 E'' H''} {N' : Type u_9} [TopologicalSpace N'] [ChartedSpace H'' N'] {G : Type u_10} [Mul G] [TopologicalSpace G] [ChartedSpace H' G] [SmoothMul I' G] (f g : C^⊤⟮I'', N'; I', G⟯) (h : C^⊤⟮I, N; I'', N'⟯)
Goal: (f * g).comp h = f.comp h * g.comp h
-- SmoothMap.coe_zero
@[to_additive (attr := simp)]
theorem coe_one {G : Type*} [One G] [TopologicalSpace G] [ChartedSpace H' G] :
    ⇑(1 : C^∞⟮I, N; I', G⟯) = 1 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H : Type u_4} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {H' : Type u_5} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {N : Type u_6} [TopologicalSpace N] [ChartedSpace H N] {G : Type u_10} [Zero G] [TopologicalSpace G] [ChartedSpace H' G]
Goal: ⇑0 = 0
-- SmoothMap.coe_one
@[to_additive (attr := simp)]
theorem coe_one {G : Type*} [One G] [TopologicalSpace G] [ChartedSpace H' G] :
    ⇑(1 : C^∞⟮I, N; I', G⟯) = 1 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H : Type u_4} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {H' : Type u_5} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {N : Type u_6} [TopologicalSpace N] [ChartedSpace H N] {G : Type u_10} [One G] [TopologicalSpace G] [ChartedSpace H' G]
Goal: ⇑1 = 1