ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- smooth_prod_assoc
theorem smooth_prod_assoc :
    Smooth ((I.prod I').prod J) (I.prod (I'.prod J)) fun x : (M × M') × N => (x.1.1, x.1.2, x.2) 
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 : Type u_11} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_12} [TopologicalSpace G] {J : ModelWithCorners 𝕜 F G} {N : Type u_13} [TopologicalSpace N] [ChartedSpace G N]
Goal: Smooth ((I.prod I').prod J) (I.prod (I'.prod J)) fun x => (x.1.1, x.1.2, x.2)
-- ContMDiffWithinAt.prod_map'
theorem ContMDiffWithinAt.prod_map' {p : M × N} (hf : ContMDiffWithinAt I I' n f s p.1)
    (hg : ContMDiffWithinAt J J' n g r p.2) :
    ContMDiffWithinAt (I.prod J) (I'.prod J') n (Prod.map f g) (s ×ˢ r) p 
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 : Type u_11} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_12} [TopologicalSpace G] {J : ModelWithCorners 𝕜 F G} {N : Type u_13} [TopologicalSpace N] [ChartedSpace G N] {F' : Type u_14} [NormedAddCommGroup F'] [NormedSpace 𝕜 F'] {G' : Type u_15} [TopologicalSpace G'] {J' : ModelWithCorners 𝕜 F' G'} {N' : Type u_16} [TopologicalSpace N'] [ChartedSpace G' N'] {f : M → M'} {s : Set M} {n : ℕ∞} {g : N → N'} {r : Set N} {p : M × N} (hf : ContMDiffWithinAt I I' n f s p.1) (hg : ContMDiffWithinAt J J' n g r p.2)
Goal: ContMDiffWithinAt (I.prod J) (I'.prod J') n (Prod.map f g) (s ×ˢ r) p
-- ContMDiffWithinAt.prod_map
theorem ContMDiffWithinAt.prod_map (hf : ContMDiffWithinAt I I' n f s x)
    (hg : ContMDiffWithinAt J J' n g r y) :
    ContMDiffWithinAt (I.prod J) (I'.prod J') n (Prod.map f g) (s ×ˢ r) (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'] {F : Type u_11} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_12} [TopologicalSpace G] {J : ModelWithCorners 𝕜 F G} {N : Type u_13} [TopologicalSpace N] [ChartedSpace G N] {F' : Type u_14} [NormedAddCommGroup F'] [NormedSpace 𝕜 F'] {G' : Type u_15} [TopologicalSpace G'] {J' : ModelWithCorners 𝕜 F' G'} {N' : Type u_16} [TopologicalSpace N'] [ChartedSpace G' N'] {f : M → M'} {s : Set M} {x : M} {n : ℕ∞} {g : N → N'} {r : Set N} {y : N} (hf : ContMDiffWithinAt I I' n f s x) (hg : ContMDiffWithinAt J J' n g r y)
Goal: ContMDiffWithinAt (I.prod J) (I'.prod J') n (Prod.map f g) (s ×ˢ r) (x, y)
-- ContMDiffAt.prod_map
theorem ContMDiffAt.prod_map (hf : ContMDiffAt I I' n f x) (hg : ContMDiffAt J J' n g y) :
    ContMDiffAt (I.prod J) (I'.prod J') n (Prod.map f g) (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'] {F : Type u_11} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_12} [TopologicalSpace G] {J : ModelWithCorners 𝕜 F G} {N : Type u_13} [TopologicalSpace N] [ChartedSpace G N] {F' : Type u_14} [NormedAddCommGroup F'] [NormedSpace 𝕜 F'] {G' : Type u_15} [TopologicalSpace G'] {J' : ModelWithCorners 𝕜 F' G'} {N' : Type u_16} [TopologicalSpace N'] [ChartedSpace G' N'] {f : M → M'} {x : M} {n : ℕ∞} {g : N → N'} {y : N} (hf : ContMDiffAt I I' n f x) (hg : ContMDiffAt J J' n g y)
Goal: ContMDiffAt (I.prod J) (I'.prod J') n (Prod.map f g) (x, y)
-- ContMDiffAt.prod_map'
theorem ContMDiffAt.prod_map' {p : M × N} (hf : ContMDiffAt I I' n f p.1)
    (hg : ContMDiffAt J J' n g p.2) : ContMDiffAt (I.prod J) (I'.prod J') n (Prod.map f g) p 
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 : Type u_11} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_12} [TopologicalSpace G] {J : ModelWithCorners 𝕜 F G} {N : Type u_13} [TopologicalSpace N] [ChartedSpace G N] {F' : Type u_14} [NormedAddCommGroup F'] [NormedSpace 𝕜 F'] {G' : Type u_15} [TopologicalSpace G'] {J' : ModelWithCorners 𝕜 F' G'} {N' : Type u_16} [TopologicalSpace N'] [ChartedSpace G' N'] {f : M → M'} {n : ℕ∞} {g : N → N'} {p : M × N} (hf : ContMDiffAt I I' n f p.1) (hg : ContMDiffAt J J' n g p.2)
Goal: ContMDiffAt (I.prod J) (I'.prod J') n (Prod.map f g) p
-- ContMDiffOn.prod_map
theorem ContMDiffOn.prod_map (hf : ContMDiffOn I I' n f s) (hg : ContMDiffOn J J' n g r) :
    ContMDiffOn (I.prod J) (I'.prod J') n (Prod.map f g) (s ×ˢ r) 
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 : Type u_11} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_12} [TopologicalSpace G] {J : ModelWithCorners 𝕜 F G} {N : Type u_13} [TopologicalSpace N] [ChartedSpace G N] {F' : Type u_14} [NormedAddCommGroup F'] [NormedSpace 𝕜 F'] {G' : Type u_15} [TopologicalSpace G'] {J' : ModelWithCorners 𝕜 F' G'} {N' : Type u_16} [TopologicalSpace N'] [ChartedSpace G' N'] {f : M → M'} {s : Set M} {n : ℕ∞} {g : N → N'} {r : Set N} (hf : ContMDiffOn I I' n f s) (hg : ContMDiffOn J J' n g r)
Goal: ContMDiffOn (I.prod J) (I'.prod J') n (Prod.map f g) (s ×ˢ r)
-- ContMDiff.prod_map
theorem ContMDiff.prod_map (hf : ContMDiff I I' n f) (hg : ContMDiff J J' n g) :
    ContMDiff (I.prod J) (I'.prod J') n (Prod.map f g) 
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 : Type u_11} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_12} [TopologicalSpace G] {J : ModelWithCorners 𝕜 F G} {N : Type u_13} [TopologicalSpace N] [ChartedSpace G N] {F' : Type u_14} [NormedAddCommGroup F'] [NormedSpace 𝕜 F'] {G' : Type u_15} [TopologicalSpace G'] {J' : ModelWithCorners 𝕜 F' G'} {N' : Type u_16} [TopologicalSpace N'] [ChartedSpace G' N'] {f : M → M'} {n : ℕ∞} {g : N → N'} (hf : ContMDiff I I' n f) (hg : ContMDiff J J' n g)
Goal: ContMDiff (I.prod J) (I'.prod J') n (Prod.map f g)
-- SmoothWithinAt.prod_map
nonrec theorem SmoothWithinAt.prod_map (hf : SmoothWithinAt I I' f s x)
    (hg : SmoothWithinAt J J' g r y) :
    SmoothWithinAt (I.prod J) (I'.prod J') (Prod.map f g) (s ×ˢ r) (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'] {F : Type u_11} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_12} [TopologicalSpace G] {J : ModelWithCorners 𝕜 F G} {N : Type u_13} [TopologicalSpace N] [ChartedSpace G N] {F' : Type u_14} [NormedAddCommGroup F'] [NormedSpace 𝕜 F'] {G' : Type u_15} [TopologicalSpace G'] {J' : ModelWithCorners 𝕜 F' G'} {N' : Type u_16} [TopologicalSpace N'] [ChartedSpace G' N'] {f : M → M'} {s : Set M} {x : M} {g : N → N'} {r : Set N} {y : N} (hf : SmoothWithinAt I I' f s x) (hg : SmoothWithinAt J J' g r y)
Goal: SmoothWithinAt (I.prod J) (I'.prod J') (Prod.map f g) (s ×ˢ r) (x, y)
-- SmoothAt.prod_map
nonrec theorem SmoothAt.prod_map (hf : SmoothAt I I' f x) (hg : SmoothAt J J' g y) :
    SmoothAt (I.prod J) (I'.prod J') (Prod.map f g) (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'] {F : Type u_11} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_12} [TopologicalSpace G] {J : ModelWithCorners 𝕜 F G} {N : Type u_13} [TopologicalSpace N] [ChartedSpace G N] {F' : Type u_14} [NormedAddCommGroup F'] [NormedSpace 𝕜 F'] {G' : Type u_15} [TopologicalSpace G'] {J' : ModelWithCorners 𝕜 F' G'} {N' : Type u_16} [TopologicalSpace N'] [ChartedSpace G' N'] {f : M → M'} {x : M} {g : N → N'} {y : N} (hf : SmoothAt I I' f x) (hg : SmoothAt J J' g y)
Goal: SmoothAt (I.prod J) (I'.prod J') (Prod.map f g) (x, y)
-- SmoothOn.prod_map
nonrec theorem SmoothOn.prod_map (hf : SmoothOn I I' f s) (hg : SmoothOn J J' g r) :
    SmoothOn (I.prod J) (I'.prod J') (Prod.map f g) (s ×ˢ r) 
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 : Type u_11} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_12} [TopologicalSpace G] {J : ModelWithCorners 𝕜 F G} {N : Type u_13} [TopologicalSpace N] [ChartedSpace G N] {F' : Type u_14} [NormedAddCommGroup F'] [NormedSpace 𝕜 F'] {G' : Type u_15} [TopologicalSpace G'] {J' : ModelWithCorners 𝕜 F' G'} {N' : Type u_16} [TopologicalSpace N'] [ChartedSpace G' N'] {f : M → M'} {s : Set M} {g : N → N'} {r : Set N} (hf : SmoothOn I I' f s) (hg : SmoothOn J J' g r)
Goal: SmoothOn (I.prod J) (I'.prod J') (Prod.map f g) (s ×ˢ r)
-- Smooth.prod_map
nonrec theorem Smooth.prod_map (hf : Smooth I I' f) (hg : Smooth J J' g) :
    Smooth (I.prod J) (I'.prod J') (Prod.map f g) 
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 : Type u_11} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_12} [TopologicalSpace G] {J : ModelWithCorners 𝕜 F G} {N : Type u_13} [TopologicalSpace N] [ChartedSpace G N] {F' : Type u_14} [NormedAddCommGroup F'] [NormedSpace 𝕜 F'] {G' : Type u_15} [TopologicalSpace G'] {J' : ModelWithCorners 𝕜 F' G'} {N' : Type u_16} [TopologicalSpace N'] [ChartedSpace G' N'] {f : M → M'} {g : N → N'} (hf : Smooth I I' f) (hg : Smooth J J' g)
Goal: Smooth (I.prod J) (I'.prod J') (Prod.map f g)
-- contMDiffWithinAt_pi_space
theorem contMDiffWithinAt_pi_space :
    ContMDiffWithinAt I 𝓘(𝕜, ∀ i, Fi i) n φ s x ↔
      ∀ i, ContMDiffWithinAt I 𝓘(𝕜, Fi i) n (fun x => φ x 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] {s : Set M} {x : M} {n : ℕ∞} {ι : Type u_21} [Fintype ι] {Fi : ι → Type u_22} [(i : ι) → NormedAddCommGroup (Fi i)] [(i : ι) → NormedSpace 𝕜 (Fi i)] {φ : M → (i : ι) → Fi i}
Goal: ContMDiffWithinAt I 𝓘(𝕜, (i : ι) → Fi i) n φ s x ↔ ∀ (i : ι), ContMDiffWithinAt I 𝓘(𝕜, Fi i) n (fun x => φ x i) s x
-- contMDiffOn_pi_space
theorem contMDiffOn_pi_space :
    ContMDiffOn I 𝓘(𝕜, ∀ i, Fi i) n φ s ↔ ∀ i, ContMDiffOn I 𝓘(𝕜, Fi i) n (fun x => φ x 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} {n : ℕ∞} {ι : Type u_21} [Fintype ι] {Fi : ι → Type u_22} [(i : ι) → NormedAddCommGroup (Fi i)] [(i : ι) → NormedSpace 𝕜 (Fi i)] {φ : M → (i : ι) → Fi i}
Goal: ContMDiffOn I 𝓘(𝕜, (i : ι) → Fi i) n φ s ↔ ∀ (i : ι), ContMDiffOn I 𝓘(𝕜, Fi i) n (fun x => φ x i) s
-- contMDiffAt_pi_space
theorem contMDiffAt_pi_space :
    ContMDiffAt I 𝓘(𝕜, ∀ i, Fi i) n φ x ↔ ∀ i, ContMDiffAt I 𝓘(𝕜, Fi i) n (fun x => φ x 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] {x : M} {n : ℕ∞} {ι : Type u_21} [Fintype ι] {Fi : ι → Type u_22} [(i : ι) → NormedAddCommGroup (Fi i)] [(i : ι) → NormedSpace 𝕜 (Fi i)] {φ : M → (i : ι) → Fi i}
Goal: ContMDiffAt I 𝓘(𝕜, (i : ι) → Fi i) n φ x ↔ ∀ (i : ι), ContMDiffAt I 𝓘(𝕜, Fi i) n (fun x => φ x i) x
-- contMDiff_pi_space
theorem contMDiff_pi_space :
    ContMDiff I 𝓘(𝕜, ∀ i, Fi i) n φ ↔ ∀ i, ContMDiff I 𝓘(𝕜, Fi i) n fun x => φ x i 
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] {n : ℕ∞} {ι : Type u_21} [Fintype ι] {Fi : ι → Type u_22} [(i : ι) → NormedAddCommGroup (Fi i)] [(i : ι) → NormedSpace 𝕜 (Fi i)] {φ : M → (i : ι) → Fi i}
Goal: ContMDiff I 𝓘(𝕜, (i : ι) → Fi i) n φ ↔ ∀ (i : ι), ContMDiff I 𝓘(𝕜, Fi i) n fun x => φ x i
-- smoothWithinAt_pi_space
theorem smoothWithinAt_pi_space :
    SmoothWithinAt I 𝓘(𝕜, ∀ i, Fi i) φ s x ↔
      ∀ i, SmoothWithinAt I 𝓘(𝕜, Fi i) (fun x => φ x 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] {s : Set M} {x : M} {ι : Type u_21} [Fintype ι] {Fi : ι → Type u_22} [(i : ι) → NormedAddCommGroup (Fi i)] [(i : ι) → NormedSpace 𝕜 (Fi i)] {φ : M → (i : ι) → Fi i}
Goal: SmoothWithinAt I 𝓘(𝕜, (i : ι) → Fi i) φ s x ↔ ∀ (i : ι), SmoothWithinAt I 𝓘(𝕜, Fi i) (fun x => φ x i) s x
-- smoothOn_pi_space
theorem smoothOn_pi_space :
    SmoothOn I 𝓘(𝕜, ∀ i, Fi i) φ s ↔ ∀ i, SmoothOn I 𝓘(𝕜, Fi i) (fun x => φ x 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} {ι : Type u_21} [Fintype ι] {Fi : ι → Type u_22} [(i : ι) → NormedAddCommGroup (Fi i)] [(i : ι) → NormedSpace 𝕜 (Fi i)] {φ : M → (i : ι) → Fi i}
Goal: SmoothOn I 𝓘(𝕜, (i : ι) → Fi i) φ s ↔ ∀ (i : ι), SmoothOn I 𝓘(𝕜, Fi i) (fun x => φ x i) s
-- smoothAt_pi_space
theorem smoothAt_pi_space :
    SmoothAt I 𝓘(𝕜, ∀ i, Fi i) φ x ↔ ∀ i, SmoothAt I 𝓘(𝕜, Fi i) (fun x => φ x 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] {x : M} {ι : Type u_21} [Fintype ι] {Fi : ι → Type u_22} [(i : ι) → NormedAddCommGroup (Fi i)] [(i : ι) → NormedSpace 𝕜 (Fi i)] {φ : M → (i : ι) → Fi i}
Goal: SmoothAt I 𝓘(𝕜, (i : ι) → Fi i) φ x ↔ ∀ (i : ι), SmoothAt I 𝓘(𝕜, Fi i) (fun x => φ x i) x
-- smooth_pi_space
theorem smooth_pi_space : Smooth I 𝓘(𝕜, ∀ i, Fi i) φ ↔ ∀ i, Smooth I 𝓘(𝕜, Fi i) fun x => φ x i 
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] {ι : Type u_21} [Fintype ι] {Fi : ι → Type u_22} [(i : ι) → NormedAddCommGroup (Fi i)] [(i : ι) → NormedSpace 𝕜 (Fi i)] {φ : M → (i : ι) → Fi i}
Goal: Smooth I 𝓘(𝕜, (i : ι) → Fi i) φ ↔ ∀ (i : ι), Smooth I 𝓘(𝕜, Fi i) fun x => φ x i
-- FiberwiseLinear.trans_partialHomeomorph_apply
theorem trans_partialHomeomorph_apply (hU : IsOpen U)
    (hφ : ContinuousOn (fun x => φ x : B → F →L[𝕜] F) U)
    (h2φ : ContinuousOn (fun x => (φ x).symm : B → F →L[𝕜] F) U) (hU' : IsOpen U')
    (hφ' : ContinuousOn (fun x => φ' x : B → F →L[𝕜] F) U')
    (h2φ' : ContinuousOn (fun x => (φ' x).symm : B → F →L[𝕜] F) U') (b : B) (v : F) :
    (FiberwiseLinear.partialHomeomorph φ hU hφ h2φ ≫ₕ
      FiberwiseLinear.partialHomeomorph φ' hU' hφ' h2φ')
        ⟨b, v⟩ =
      ⟨b, φ' b (φ b v)⟩ 
Arguments: {𝕜 : Type u_1} {B : Type u_2} {F : Type u_3} [TopologicalSpace B] [NontriviallyNormedField 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {φ φ' : B → F ≃L[𝕜] F} {U U' : Set B} (hU : IsOpen U) (hφ : ContinuousOn (fun x => ↑(φ x)) U) (h2φ : ContinuousOn (fun x => ↑(φ x).symm) U) (hU' : IsOpen U') (hφ' : ContinuousOn (fun x => ↑(φ' x)) U') (h2φ' : ContinuousOn (fun x => ↑(φ' x).symm) U') (b : B) (v : F)
Goal: ↑(FiberwiseLinear.partialHomeomorph φ hU hφ h2φ ≫ₕ FiberwiseLinear.partialHomeomorph φ' hU' hφ' h2φ') (b, v) =
  (b, (φ' b) ((φ b) v))
-- FiberwiseLinear.source_trans_partialHomeomorph
theorem source_trans_partialHomeomorph (hU : IsOpen U)
    (hφ : ContinuousOn (fun x => φ x : B → F →L[𝕜] F) U)
    (h2φ : ContinuousOn (fun x => (φ x).symm : B → F →L[𝕜] F) U) (hU' : IsOpen U')
    (hφ' : ContinuousOn (fun x => φ' x : B → F →L[𝕜] F) U')
    (h2φ' : ContinuousOn (fun x => (φ' x).symm : B → F →L[𝕜] F) U') :
    (FiberwiseLinear.partialHomeomorph φ hU hφ h2φ ≫ₕ
          FiberwiseLinear.partialHomeomorph φ' hU' hφ' h2φ').source =
      (U ∩ U') ×ˢ univ 
Arguments: {𝕜 : Type u_1} {B : Type u_2} {F : Type u_3} [TopologicalSpace B] [NontriviallyNormedField 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {φ φ' : B → F ≃L[𝕜] F} {U U' : Set B} (hU : IsOpen U) (hφ : ContinuousOn (fun x => ↑(φ x)) U) (h2φ : ContinuousOn (fun x => ↑(φ x).symm) U) (hU' : IsOpen U') (hφ' : ContinuousOn (fun x => ↑(φ' x)) U') (h2φ' : ContinuousOn (fun x => ↑(φ' x).symm) U')
Goal: (FiberwiseLinear.partialHomeomorph φ hU hφ h2φ ≫ₕ FiberwiseLinear.partialHomeomorph φ' hU' hφ' h2φ').source =
  (U ∩ U') ×ˢ Set.univ
-- FiberwiseLinear.target_trans_partialHomeomorph
theorem target_trans_partialHomeomorph (hU : IsOpen U)
    (hφ : ContinuousOn (fun x => φ x : B → F →L[𝕜] F) U)
    (h2φ : ContinuousOn (fun x => (φ x).symm : B → F →L[𝕜] F) U) (hU' : IsOpen U')
    (hφ' : ContinuousOn (fun x => φ' x : B → F →L[𝕜] F) U')
    (h2φ' : ContinuousOn (fun x => (φ' x).symm : B → F →L[𝕜] F) U') :
    (FiberwiseLinear.partialHomeomorph φ hU hφ h2φ ≫ₕ
          FiberwiseLinear.partialHomeomorph φ' hU' hφ' h2φ').target =
      (U ∩ U') ×ˢ univ 
Arguments: {𝕜 : Type u_1} {B : Type u_2} {F : Type u_3} [TopologicalSpace B] [NontriviallyNormedField 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {φ φ' : B → F ≃L[𝕜] F} {U U' : Set B} (hU : IsOpen U) (hφ : ContinuousOn (fun x => ↑(φ x)) U) (h2φ : ContinuousOn (fun x => ↑(φ x).symm) U) (hU' : IsOpen U') (hφ' : ContinuousOn (fun x => ↑(φ' x)) U') (h2φ' : ContinuousOn (fun x => ↑(φ' x).symm) U')
Goal: (FiberwiseLinear.partialHomeomorph φ hU hφ h2φ ≫ₕ FiberwiseLinear.partialHomeomorph φ' hU' hφ' h2φ').target =
  (U ∩ U') ×ˢ Set.univ
-- SmoothFiberwiseLinear.locality_aux₁
theorem SmoothFiberwiseLinear.locality_aux₁ (e : PartialHomeomorph (B × F) (B × F))
    (h : ∀ p ∈ e.source, ∃ s : Set (B × F), IsOpen s ∧ p ∈ s ∧
      ∃ (φ : B → F ≃L[𝕜] F) (u : Set B) (hu : IsOpen u)
        (hφ : SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun x => (φ x : F →L[𝕜] F)) u)
        (h2φ : SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun x => ((φ x).symm : F →L[𝕜] F)) u),
          (e.restr s).EqOnSource
            (FiberwiseLinear.partialHomeomorph φ hu hφ.continuousOn h2φ.continuousOn)) :
    ∃ U : Set B, e.source = U ×ˢ univ ∧ ∀ x ∈ U,
        ∃ (φ : B → F ≃L[𝕜] F) (u : Set B) (hu : IsOpen u) (_huU : u ⊆ U) (_hux : x ∈ u),
          ∃ (hφ : SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun x => (φ x : F →L[𝕜] F)) u)
            (h2φ : SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun x => ((φ x).symm : F →L[𝕜] F)) u),
            (e.restr (u ×ˢ univ)).EqOnSource
              (FiberwiseLinear.partialHomeomorph φ hu hφ.continuousOn h2φ.continuousOn) 
Arguments: {𝕜 : Type u_1} {B : Type u_2} {F : Type u_3} [TopologicalSpace B] [NontriviallyNormedField 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {EB : Type u_4} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {HB : Type u_5} [TopologicalSpace HB] [ChartedSpace HB B] {IB : ModelWithCorners 𝕜 EB HB} (e : PartialHomeomorph (B × F) (B × F)) (h :
  ∀ p ∈ e.source,
    ∃ s,
      IsOpen s ∧
        p ∈ s ∧
          ∃ φ u,
            ∃ (hu : IsOpen u) (hφ : SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun x => ↑(φ x)) u) (h2φ :
              SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun x => ↑(φ x).symm) u),
              (e.restr s).EqOnSource (FiberwiseLinear.partialHomeomorph φ hu ⋯ ⋯))
Goal: ∃ U,
  e.source = U ×ˢ Set.univ ∧
    ∀ x ∈ U,
      ∃ φ u,
        ∃ (hu : IsOpen u) (_ : u ⊆ U) (_ : x ∈ u) (hφ : SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun x => ↑(φ x)) u) (h2φ :
          SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun x => ↑(φ x).symm) u),
          (e.restr (u ×ˢ Set.univ)).EqOnSource (FiberwiseLinear.partialHomeomorph φ hu ⋯ ⋯)
-- SmoothFiberwiseLinear.locality_aux₂
theorem SmoothFiberwiseLinear.locality_aux₂ (e : PartialHomeomorph (B × F) (B × F)) (U : Set B)
    (hU : e.source = U ×ˢ univ)
    (h : ∀ x ∈ U,
      ∃ (φ : B → F ≃L[𝕜] F) (u : Set B) (hu : IsOpen u) (_hUu : u ⊆ U) (_hux : x ∈ u)
        (hφ : SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun x => (φ x : F →L[𝕜] F)) u)
        (h2φ : SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun x => ((φ x).symm : F →L[𝕜] F)) u),
          (e.restr (u ×ˢ univ)).EqOnSource
            (FiberwiseLinear.partialHomeomorph φ hu hφ.continuousOn h2φ.continuousOn)) :
    ∃ (Φ : B → F ≃L[𝕜] F) (U : Set B) (hU₀ : IsOpen U) (hΦ :
      SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun x => (Φ x : F →L[𝕜] F)) U) (h2Φ :
      SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun x => ((Φ x).symm : F →L[𝕜] F)) U),
      e.EqOnSource (FiberwiseLinear.partialHomeomorph Φ hU₀ hΦ.continuousOn h2Φ.continuousOn) 
Arguments: {𝕜 : Type u_1} {B : Type u_2} {F : Type u_3} [TopologicalSpace B] [NontriviallyNormedField 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {EB : Type u_4} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {HB : Type u_5} [TopologicalSpace HB] [ChartedSpace HB B] {IB : ModelWithCorners 𝕜 EB HB} (e : PartialHomeomorph (B × F) (B × F)) (U : Set B) (hU : e.source = U ×ˢ Set.univ) (h :
  ∀ x ∈ U,
    ∃ φ u,
      ∃ (hu : IsOpen u) (_ : u ⊆ U) (_ : x ∈ u) (hφ : SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun x => ↑(φ x)) u) (h2φ :
        SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun x => ↑(φ x).symm) u),
        (e.restr (u ×ˢ Set.univ)).EqOnSource (FiberwiseLinear.partialHomeomorph φ hu ⋯ ⋯))
Goal: ∃ Φ U,
  ∃ (hU₀ : IsOpen U) (hΦ : SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun x => ↑(Φ x)) U) (h2Φ :
    SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun x => ↑(Φ x).symm) U), e.EqOnSource (FiberwiseLinear.partialHomeomorph Φ hU₀ ⋯ ⋯)