ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Smooth.prod_mk
nonrec theorem Smooth.prod_mk {f : M → M'} {g : M → N'} (hf : Smooth I I' f) (hg : Smooth I J' g) :
    Smooth I (I'.prod J') fun x => (f x, g 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' : 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 : M → N'} (hf : Smooth I I' f) (hg : Smooth I J' g)
Goal: Smooth I (I'.prod J') fun x => (f x, g x)
-- Smooth.prod_mk_space
nonrec theorem Smooth.prod_mk_space {f : M → E'} {g : M → F'} (hf : Smooth I 𝓘(𝕜, E') f)
    (hg : Smooth I 𝓘(𝕜, F') g) : Smooth I 𝓘(𝕜, E' × F') fun x => (f x, g 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'] {F' : Type u_14} [NormedAddCommGroup F'] [NormedSpace 𝕜 F'] {f : M → E'} {g : M → F'} (hf : Smooth I 𝓘(𝕜, E') f) (hg : Smooth I 𝓘(𝕜, F') g)
Goal: Smooth I 𝓘(𝕜, E' × F') fun x => (f x, g x)
-- contMDiffWithinAt_fst
theorem contMDiffWithinAt_fst {s : Set (M × N)} {p : M × N} :
    ContMDiffWithinAt (I.prod J) I n Prod.fst s 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] {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] {n : ℕ∞} {s : Set (M × N)} {p : M × N}
Goal: ContMDiffWithinAt (I.prod J) I n Prod.fst s p
-- ContMDiffWithinAt.fst
theorem ContMDiffWithinAt.fst {f : N → M × M'} {s : Set N} {x : N}
    (hf : ContMDiffWithinAt J (I.prod I') n f s x) :
    ContMDiffWithinAt J I n (fun x => (f x).1) 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 : 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] {n : ℕ∞} {f : N → M × M'} {s : Set N} {x : N} (hf : ContMDiffWithinAt J (I.prod I') n f s x)
Goal: ContMDiffWithinAt J I n (fun x => (f x).1) s x
-- contMDiffAt_fst
theorem contMDiffAt_fst {p : M × N} : ContMDiffAt (I.prod J) I n Prod.fst 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] {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] {n : ℕ∞} {p : M × N}
Goal: ContMDiffAt (I.prod J) I n Prod.fst p
-- contMDiffOn_fst
theorem contMDiffOn_fst {s : Set (M × N)} : ContMDiffOn (I.prod J) I n Prod.fst 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] {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] {n : ℕ∞} {s : Set (M × N)}
Goal: ContMDiffOn (I.prod J) I n Prod.fst s
-- contMDiff_fst
theorem contMDiff_fst : ContMDiff (I.prod J) I n (@Prod.fst M N) 
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] {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] {n : ℕ∞}
Goal: ContMDiff (I.prod J) I n Prod.fst
-- smoothWithinAt_fst
theorem smoothWithinAt_fst {s : Set (M × N)} {p : M × N} :
    SmoothWithinAt (I.prod J) I Prod.fst s 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] {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] {s : Set (M × N)} {p : M × N}
Goal: SmoothWithinAt (I.prod J) I Prod.fst s p
-- smoothAt_fst
theorem smoothAt_fst {p : M × N} : SmoothAt (I.prod J) I Prod.fst 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] {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] {p : M × N}
Goal: SmoothAt (I.prod J) I Prod.fst p
-- smoothOn_fst
theorem smoothOn_fst {s : Set (M × N)} : SmoothOn (I.prod J) I Prod.fst 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] {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] {s : Set (M × N)}
Goal: SmoothOn (I.prod J) I Prod.fst s
-- smooth_fst
theorem smooth_fst : Smooth (I.prod J) I (@Prod.fst M N) 
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] {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 J) I Prod.fst
-- ContMDiffAt.fst
theorem ContMDiffAt.fst {f : N → M × M'} {x : N} (hf : ContMDiffAt J (I.prod I') n f x) :
    ContMDiffAt J I n (fun x => (f x).1) 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 : 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] {n : ℕ∞} {f : N → M × M'} {x : N} (hf : ContMDiffAt J (I.prod I') n f x)
Goal: ContMDiffAt J I n (fun x => (f x).1) x
-- ContMDiff.fst
theorem ContMDiff.fst {f : N → M × M'} (hf : ContMDiff J (I.prod I') n f) :
    ContMDiff J I n fun x => (f x).1 
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] {n : ℕ∞} {f : N → M × M'} (hf : ContMDiff J (I.prod I') n f)
Goal: ContMDiff J I n fun x => (f x).1
-- SmoothAt.fst
theorem SmoothAt.fst {f : N → M × M'} {x : N} (hf : SmoothAt J (I.prod I') f x) :
    SmoothAt J I (fun x => (f x).1) 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 : 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 : N → M × M'} {x : N} (hf : SmoothAt J (I.prod I') f x)
Goal: SmoothAt J I (fun x => (f x).1) x
-- Smooth.fst
theorem Smooth.fst {f : N → M × M'} (hf : Smooth J (I.prod I') f) : Smooth J I fun x => (f x).1 
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 : N → M × M'} (hf : Smooth J (I.prod I') f)
Goal: Smooth J I fun x => (f x).1
-- contMDiffWithinAt_snd
theorem contMDiffWithinAt_snd {s : Set (M × N)} {p : M × N} :
    ContMDiffWithinAt (I.prod J) J n Prod.snd s 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] {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] {n : ℕ∞} {s : Set (M × N)} {p : M × N}
Goal: ContMDiffWithinAt (I.prod J) J n Prod.snd s p
-- ContMDiffWithinAt.snd
theorem ContMDiffWithinAt.snd {f : N → M × M'} {s : Set N} {x : N}
    (hf : ContMDiffWithinAt J (I.prod I') n f s x) :
    ContMDiffWithinAt J I' n (fun x => (f x).2) 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 : 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] {n : ℕ∞} {f : N → M × M'} {s : Set N} {x : N} (hf : ContMDiffWithinAt J (I.prod I') n f s x)
Goal: ContMDiffWithinAt J I' n (fun x => (f x).2) s x
-- contMDiffAt_snd
theorem contMDiffAt_snd {p : M × N} : ContMDiffAt (I.prod J) J n Prod.snd 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] {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] {n : ℕ∞} {p : M × N}
Goal: ContMDiffAt (I.prod J) J n Prod.snd p
-- contMDiffOn_snd
theorem contMDiffOn_snd {s : Set (M × N)} : ContMDiffOn (I.prod J) J n Prod.snd 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] {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] {n : ℕ∞} {s : Set (M × N)}
Goal: ContMDiffOn (I.prod J) J n Prod.snd s
-- contMDiff_snd
theorem contMDiff_snd : ContMDiff (I.prod J) J n (@Prod.snd M N) 
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] {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] {n : ℕ∞}
Goal: ContMDiff (I.prod J) J n Prod.snd
-- smoothWithinAt_snd
theorem smoothWithinAt_snd {s : Set (M × N)} {p : M × N} :
    SmoothWithinAt (I.prod J) J Prod.snd s 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] {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] {s : Set (M × N)} {p : M × N}
Goal: SmoothWithinAt (I.prod J) J Prod.snd s p
-- smoothAt_snd
theorem smoothAt_snd {p : M × N} : SmoothAt (I.prod J) J Prod.snd 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] {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] {p : M × N}
Goal: SmoothAt (I.prod J) J Prod.snd p
-- smoothOn_snd
theorem smoothOn_snd {s : Set (M × N)} : SmoothOn (I.prod J) J Prod.snd 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] {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] {s : Set (M × N)}
Goal: SmoothOn (I.prod J) J Prod.snd s
-- smooth_snd
theorem smooth_snd : Smooth (I.prod J) J (@Prod.snd M N) 
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] {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 J) J Prod.snd
-- ContMDiffAt.snd
theorem ContMDiffAt.snd {f : N → M × M'} {x : N} (hf : ContMDiffAt J (I.prod I') n f x) :
    ContMDiffAt J I' n (fun x => (f x).2) 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 : 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] {n : ℕ∞} {f : N → M × M'} {x : N} (hf : ContMDiffAt J (I.prod I') n f x)
Goal: ContMDiffAt J I' n (fun x => (f x).2) x
-- ContMDiff.snd
theorem ContMDiff.snd {f : N → M × M'} (hf : ContMDiff J (I.prod I') n f) :
    ContMDiff J I' n fun x => (f 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] {n : ℕ∞} {f : N → M × M'} (hf : ContMDiff J (I.prod I') n f)
Goal: ContMDiff J I' n fun x => (f x).2
-- SmoothAt.snd
theorem SmoothAt.snd {f : N → M × M'} {x : N} (hf : SmoothAt J (I.prod I') f x) :
    SmoothAt J I' (fun x => (f x).2) 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 : 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 : N → M × M'} {x : N} (hf : SmoothAt J (I.prod I') f x)
Goal: SmoothAt J I' (fun x => (f x).2) x
-- Smooth.snd
theorem Smooth.snd {f : N → M × M'} (hf : Smooth J (I.prod I') f) : Smooth J I' fun x => (f 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] {f : N → M × M'} (hf : Smooth J (I.prod I') f)
Goal: Smooth J I' fun x => (f x).2
-- contMDiffWithinAt_prod_iff
theorem contMDiffWithinAt_prod_iff (f : M → M' × N') {s : Set M} {x : M} :
    ContMDiffWithinAt I (I'.prod J') n f s x ↔
      ContMDiffWithinAt I I' n (Prod.fst ∘ f) s x ∧ ContMDiffWithinAt I J' n (Prod.snd ∘ 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' : 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'] {n : ℕ∞} (f : M → M' × N') {s : Set M} {x : M}
Goal: ContMDiffWithinAt I (I'.prod J') n f s x ↔
  ContMDiffWithinAt I I' n (Prod.fst ∘ f) s x ∧ ContMDiffWithinAt I J' n (Prod.snd ∘ f) s x
-- contMDiffAt_prod_iff
theorem contMDiffAt_prod_iff (f : M → M' × N') {x : M} :
    ContMDiffAt I (I'.prod J') n f x ↔
      ContMDiffAt I I' n (Prod.fst ∘ f) x ∧ ContMDiffAt I J' n (Prod.snd ∘ 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' : 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'] {n : ℕ∞} (f : M → M' × N') {x : M}
Goal: ContMDiffAt I (I'.prod J') n f x ↔ ContMDiffAt I I' n (Prod.fst ∘ f) x ∧ ContMDiffAt I J' n (Prod.snd ∘ f) x
-- contMDiff_prod_iff
theorem contMDiff_prod_iff (f : M → M' × N') :
    ContMDiff I (I'.prod J') n f ↔
      ContMDiff I I' n (Prod.fst ∘ f) ∧ ContMDiff I J' n (Prod.snd ∘ f) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {F' : 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'] {n : ℕ∞} (f : M → M' × N')
Goal: ContMDiff I (I'.prod J') n f ↔ ContMDiff I I' n (Prod.fst ∘ f) ∧ ContMDiff I J' n (Prod.snd ∘ f)
-- smoothAt_prod_iff
theorem smoothAt_prod_iff (f : M → M' × N') {x : M} :
    SmoothAt I (I'.prod J') f x ↔ SmoothAt I I' (Prod.fst ∘ f) x ∧ SmoothAt I J' (Prod.snd ∘ 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' : 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') {x : M}
Goal: SmoothAt I (I'.prod J') f x ↔ SmoothAt I I' (Prod.fst ∘ f) x ∧ SmoothAt I J' (Prod.snd ∘ f) x
-- smooth_prod_iff
theorem smooth_prod_iff (f : M → M' × N') :
    Smooth I (I'.prod J') f ↔ Smooth I I' (Prod.fst ∘ f) ∧ Smooth I J' (Prod.snd ∘ f) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {F' : 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')
Goal: Smooth I (I'.prod J') f ↔ Smooth I I' (Prod.fst ∘ f) ∧ Smooth I J' (Prod.snd ∘ f)