ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- SmoothAddMonoidMorphism.smooth_toFun
structure SmoothAddMonoidMorphism (I : ModelWithCorners 𝕜 E H) (I' : ModelWithCorners 𝕜 E' H')
    (G : Type*) [TopologicalSpace G] [ChartedSpace H G] [AddMonoid G] [SmoothAdd I G]
    (G' : Type*) [TopologicalSpace G'] [ChartedSpace H' G'] [AddMonoid G']
    [SmoothAdd I' G'] extends G →+ G' where
  smooth_toFun : Smooth I I' toFun
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {H : Type u_2} [TopologicalSpace H] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H' : Type u_5} [TopologicalSpace H'] {E' : Type u_6} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {I : ModelWithCorners 𝕜 E H} {I' : ModelWithCorners 𝕜 E' H'} {G : Type u_8} [TopologicalSpace G] [ChartedSpace H G] [AddMonoid G] [SmoothAdd I G] {G' : Type u_9} [TopologicalSpace G'] [ChartedSpace H' G'] [AddMonoid G'] [SmoothAdd I' G'] (self : SmoothAddMonoidMorphism I I' G G')
Goal: Smooth I I' (↑self.toAddMonoidHom).toFun
-- SmoothMonoidMorphism.smooth_toFun
@[to_additive]
structure SmoothMonoidMorphism (I : ModelWithCorners 𝕜 E H) (I' : ModelWithCorners 𝕜 E' H')
    (G : Type*) [TopologicalSpace G] [ChartedSpace H G] [Monoid G] [SmoothMul I G] (G' : Type*)
    [TopologicalSpace G'] [ChartedSpace H' G'] [Monoid G'] [SmoothMul I' G'] extends
    G →* G' where
  smooth_toFun : Smooth I I' toFun
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {H : Type u_2} [TopologicalSpace H] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H' : Type u_5} [TopologicalSpace H'] {E' : Type u_6} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {I : ModelWithCorners 𝕜 E H} {I' : ModelWithCorners 𝕜 E' H'} {G : Type u_8} [TopologicalSpace G] [ChartedSpace H G] [Monoid G] [SmoothMul I G] {G' : Type u_9} [TopologicalSpace G'] [ChartedSpace H' G'] [Monoid G'] [SmoothMul I' G'] (self : SmoothMonoidMorphism I I' G G')
Goal: Smooth I I' (↑self.toMonoidHom).toFun
-- ContMDiffWithinAt.prod
@[to_additive]
theorem ContMDiffWithinAt.prod (h : ∀ i ∈ t, ContMDiffWithinAt I' I n (f i) s x₀) :
    ContMDiffWithinAt I' I n (fun x ↦ ∏ i ∈ t, f i x) s x₀ 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {H : Type u_3} [TopologicalSpace H] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_5} [CommMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothMul I G] {E' : Type u_6} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_7} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_8} [TopologicalSpace M] [ChartedSpace H' M] {s : Set M} {x₀ : M} {t : Finset ι} {f : ι → M → G} {n : ℕ∞} (h : ∀ i ∈ t, ContMDiffWithinAt I' I n (f i) s x₀)
Goal: ContMDiffWithinAt I' I n (fun x => ∏ i ∈ t, f i x) s x₀
-- ContMDiffWithinAt.sum
@[to_additive]
theorem ContMDiffWithinAt.prod (h : ∀ i ∈ t, ContMDiffWithinAt I' I n (f i) s x₀) :
    ContMDiffWithinAt I' I n (fun x ↦ ∏ i ∈ t, f i x) s x₀ 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {H : Type u_3} [TopologicalSpace H] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_5} [AddCommMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothAdd I G] {E' : Type u_6} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_7} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_8} [TopologicalSpace M] [ChartedSpace H' M] {s : Set M} {x₀ : M} {t : Finset ι} {f : ι → M → G} {n : ℕ∞} (h : ∀ i ∈ t, ContMDiffWithinAt I' I n (f i) s x₀)
Goal: ContMDiffWithinAt I' I n (fun x => ∑ i ∈ t, f i x) s x₀
-- contMDiffWithinAt_finsum
@[to_additive]
theorem contMDiffWithinAt_finprod (lf : LocallyFinite fun i ↦ mulSupport <| f i) {x₀ : M}
    (h : ∀ i, ContMDiffWithinAt I' I n (f i) s x₀) :
    ContMDiffWithinAt I' I n (fun x ↦ ∏ᶠ i, f i x) s x₀ 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {H : Type u_3} [TopologicalSpace H] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_5} [AddCommMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothAdd I G] {E' : Type u_6} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_7} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_8} [TopologicalSpace M] [ChartedSpace H' M] {s : Set M} {f : ι → M → G} {n : ℕ∞} (lf : LocallyFinite fun i => Function.support (f i)) {x₀ : M} (h : ∀ (i : ι), ContMDiffWithinAt I' I n (f i) s x₀)
Goal: ContMDiffWithinAt I' I n (fun x => ∑ᶠ (i : ι), f i x) s x₀
-- contMDiffWithinAt_finprod
@[to_additive]
theorem contMDiffWithinAt_finprod (lf : LocallyFinite fun i ↦ mulSupport <| f i) {x₀ : M}
    (h : ∀ i, ContMDiffWithinAt I' I n (f i) s x₀) :
    ContMDiffWithinAt I' I n (fun x ↦ ∏ᶠ i, f i x) s x₀ 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {H : Type u_3} [TopologicalSpace H] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_5} [CommMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothMul I G] {E' : Type u_6} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_7} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_8} [TopologicalSpace M] [ChartedSpace H' M] {s : Set M} {f : ι → M → G} {n : ℕ∞} (lf : LocallyFinite fun i => Function.mulSupport (f i)) {x₀ : M} (h : ∀ (i : ι), ContMDiffWithinAt I' I n (f i) s x₀)
Goal: ContMDiffWithinAt I' I n (fun x => ∏ᶠ (i : ι), f i x) s x₀
-- contMDiffWithinAt_finset_prod'
@[to_additive]
theorem contMDiffWithinAt_finset_prod' (h : ∀ i ∈ t, ContMDiffWithinAt I' I n (f i) s x) :
    ContMDiffWithinAt I' I n (∏ i ∈ t, f i) s x 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {H : Type u_3} [TopologicalSpace H] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_5} [CommMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothMul I G] {E' : Type u_6} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_7} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_8} [TopologicalSpace M] [ChartedSpace H' M] {s : Set M} {x : M} {t : Finset ι} {f : ι → M → G} {n : ℕ∞} (h : ∀ i ∈ t, ContMDiffWithinAt I' I n (f i) s x)
Goal: ContMDiffWithinAt I' I n (∏ i ∈ t, f i) s x
-- contMDiffWithinAt_finset_sum'
@[to_additive]
theorem contMDiffWithinAt_finset_prod' (h : ∀ i ∈ t, ContMDiffWithinAt I' I n (f i) s x) :
    ContMDiffWithinAt I' I n (∏ i ∈ t, f i) s x 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {H : Type u_3} [TopologicalSpace H] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_5} [AddCommMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothAdd I G] {E' : Type u_6} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_7} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_8} [TopologicalSpace M] [ChartedSpace H' M] {s : Set M} {x : M} {t : Finset ι} {f : ι → M → G} {n : ℕ∞} (h : ∀ i ∈ t, ContMDiffWithinAt I' I n (f i) s x)
Goal: ContMDiffWithinAt I' I n (∑ i ∈ t, f i) s x
-- contMDiffWithinAt_finset_sum
@[to_additive]
theorem contMDiffWithinAt_finset_prod (h : ∀ i ∈ t, ContMDiffWithinAt I' I n (f i) s x) :
    ContMDiffWithinAt I' I n (fun x => ∏ i ∈ t, f i x) s x 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {H : Type u_3} [TopologicalSpace H] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_5} [AddCommMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothAdd I G] {E' : Type u_6} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_7} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_8} [TopologicalSpace M] [ChartedSpace H' M] {s : Set M} {x : M} {t : Finset ι} {f : ι → M → G} {n : ℕ∞} (h : ∀ i ∈ t, ContMDiffWithinAt I' I n (f i) s x)
Goal: ContMDiffWithinAt I' I n (fun x => ∑ i ∈ t, f i x) s x
-- contMDiffWithinAt_finset_prod
@[to_additive]
theorem contMDiffWithinAt_finset_prod (h : ∀ i ∈ t, ContMDiffWithinAt I' I n (f i) s x) :
    ContMDiffWithinAt I' I n (fun x => ∏ i ∈ t, f i x) s x 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {H : Type u_3} [TopologicalSpace H] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_5} [CommMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothMul I G] {E' : Type u_6} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_7} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_8} [TopologicalSpace M] [ChartedSpace H' M] {s : Set M} {x : M} {t : Finset ι} {f : ι → M → G} {n : ℕ∞} (h : ∀ i ∈ t, ContMDiffWithinAt I' I n (f i) s x)
Goal: ContMDiffWithinAt I' I n (fun x => ∏ i ∈ t, f i x) s x
-- ContMDiffAt.prod
@[to_additive]
theorem ContMDiffAt.prod (h : ∀ i ∈ t, ContMDiffAt I' I n (f i) x₀) :
    ContMDiffAt I' I n (fun x ↦ ∏ i ∈ t, f i x) x₀ 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {H : Type u_3} [TopologicalSpace H] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_5} [CommMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothMul I G] {E' : Type u_6} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_7} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_8} [TopologicalSpace M] [ChartedSpace H' M] {x₀ : M} {t : Finset ι} {f : ι → M → G} {n : ℕ∞} (h : ∀ i ∈ t, ContMDiffAt I' I n (f i) x₀)
Goal: ContMDiffAt I' I n (fun x => ∏ i ∈ t, f i x) x₀
-- ContMDiffAt.sum
@[to_additive]
theorem ContMDiffAt.prod (h : ∀ i ∈ t, ContMDiffAt I' I n (f i) x₀) :
    ContMDiffAt I' I n (fun x ↦ ∏ i ∈ t, f i x) x₀ 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {H : Type u_3} [TopologicalSpace H] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_5} [AddCommMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothAdd I G] {E' : Type u_6} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_7} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_8} [TopologicalSpace M] [ChartedSpace H' M] {x₀ : M} {t : Finset ι} {f : ι → M → G} {n : ℕ∞} (h : ∀ i ∈ t, ContMDiffAt I' I n (f i) x₀)
Goal: ContMDiffAt I' I n (fun x => ∑ i ∈ t, f i x) x₀
-- contMDiffAt_finsum
@[to_additive]
theorem contMDiffAt_finprod
    (lf : LocallyFinite fun i ↦ mulSupport <| f i) (h : ∀ i, ContMDiffAt I' I n (f i) x₀) :
    ContMDiffAt I' I n (fun x ↦ ∏ᶠ i, f i x) x₀ 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {H : Type u_3} [TopologicalSpace H] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_5} [AddCommMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothAdd I G] {E' : Type u_6} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_7} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_8} [TopologicalSpace M] [ChartedSpace H' M] {x₀ : M} {f : ι → M → G} {n : ℕ∞} (lf : LocallyFinite fun i => Function.support (f i)) (h : ∀ (i : ι), ContMDiffAt I' I n (f i) x₀)
Goal: ContMDiffAt I' I n (fun x => ∑ᶠ (i : ι), f i x) x₀
-- contMDiffAt_finprod
@[to_additive]
theorem contMDiffAt_finprod
    (lf : LocallyFinite fun i ↦ mulSupport <| f i) (h : ∀ i, ContMDiffAt I' I n (f i) x₀) :
    ContMDiffAt I' I n (fun x ↦ ∏ᶠ i, f i x) x₀ 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {H : Type u_3} [TopologicalSpace H] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_5} [CommMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothMul I G] {E' : Type u_6} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_7} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_8} [TopologicalSpace M] [ChartedSpace H' M] {x₀ : M} {f : ι → M → G} {n : ℕ∞} (lf : LocallyFinite fun i => Function.mulSupport (f i)) (h : ∀ (i : ι), ContMDiffAt I' I n (f i) x₀)
Goal: ContMDiffAt I' I n (fun x => ∏ᶠ (i : ι), f i x) x₀
-- contMDiffAt_finset_sum'
@[to_additive]
theorem contMDiffAt_finset_prod' (h : ∀ i ∈ t, ContMDiffAt I' I n (f i) x) :
    ContMDiffAt I' I n (∏ i ∈ t, f i) x 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {H : Type u_3} [TopologicalSpace H] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_5} [AddCommMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothAdd I G] {E' : Type u_6} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_7} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_8} [TopologicalSpace M] [ChartedSpace H' M] {x : M} {t : Finset ι} {f : ι → M → G} {n : ℕ∞} (h : ∀ i ∈ t, ContMDiffAt I' I n (f i) x)
Goal: ContMDiffAt I' I n (∑ i ∈ t, f i) x
-- contMDiffAt_finset_prod'
@[to_additive]
theorem contMDiffAt_finset_prod' (h : ∀ i ∈ t, ContMDiffAt I' I n (f i) x) :
    ContMDiffAt I' I n (∏ i ∈ t, f i) x 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {H : Type u_3} [TopologicalSpace H] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_5} [CommMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothMul I G] {E' : Type u_6} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_7} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_8} [TopologicalSpace M] [ChartedSpace H' M] {x : M} {t : Finset ι} {f : ι → M → G} {n : ℕ∞} (h : ∀ i ∈ t, ContMDiffAt I' I n (f i) x)
Goal: ContMDiffAt I' I n (∏ i ∈ t, f i) x
-- contMDiffAt_finset_prod
@[to_additive]
theorem contMDiffAt_finset_prod (h : ∀ i ∈ t, ContMDiffAt I' I n (f i) x) :
    ContMDiffAt I' I n (fun x => ∏ i ∈ t, f i x) x 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {H : Type u_3} [TopologicalSpace H] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_5} [CommMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothMul I G] {E' : Type u_6} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_7} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_8} [TopologicalSpace M] [ChartedSpace H' M] {x : M} {t : Finset ι} {f : ι → M → G} {n : ℕ∞} (h : ∀ i ∈ t, ContMDiffAt I' I n (f i) x)
Goal: ContMDiffAt I' I n (fun x => ∏ i ∈ t, f i x) x
-- contMDiffAt_finset_sum
@[to_additive]
theorem contMDiffAt_finset_prod (h : ∀ i ∈ t, ContMDiffAt I' I n (f i) x) :
    ContMDiffAt I' I n (fun x => ∏ i ∈ t, f i x) x 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {H : Type u_3} [TopologicalSpace H] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_5} [AddCommMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothAdd I G] {E' : Type u_6} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_7} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_8} [TopologicalSpace M] [ChartedSpace H' M] {x : M} {t : Finset ι} {f : ι → M → G} {n : ℕ∞} (h : ∀ i ∈ t, ContMDiffAt I' I n (f i) x)
Goal: ContMDiffAt I' I n (fun x => ∑ i ∈ t, f i x) x
-- contMDiffOn_finsum
@[to_additive]
theorem contMDiffOn_finprod
    (lf : LocallyFinite fun i ↦ Function.mulSupport <| f i) (h : ∀ i, ContMDiffOn I' I n (f i) s) :
    ContMDiffOn I' I n (fun x ↦ ∏ᶠ i, f i x) s 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {H : Type u_3} [TopologicalSpace H] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_5} [AddCommMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothAdd I G] {E' : Type u_6} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_7} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_8} [TopologicalSpace M] [ChartedSpace H' M] {s : Set M} {f : ι → M → G} {n : ℕ∞} (lf : LocallyFinite fun i => Function.support (f i)) (h : ∀ (i : ι), ContMDiffOn I' I n (f i) s)
Goal: ContMDiffOn I' I n (fun x => ∑ᶠ (i : ι), f i x) s
-- contMDiffOn_finprod
@[to_additive]
theorem contMDiffOn_finprod
    (lf : LocallyFinite fun i ↦ Function.mulSupport <| f i) (h : ∀ i, ContMDiffOn I' I n (f i) s) :
    ContMDiffOn I' I n (fun x ↦ ∏ᶠ i, f i x) s 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {H : Type u_3} [TopologicalSpace H] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_5} [CommMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothMul I G] {E' : Type u_6} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_7} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_8} [TopologicalSpace M] [ChartedSpace H' M] {s : Set M} {f : ι → M → G} {n : ℕ∞} (lf : LocallyFinite fun i => Function.mulSupport (f i)) (h : ∀ (i : ι), ContMDiffOn I' I n (f i) s)
Goal: ContMDiffOn I' I n (fun x => ∏ᶠ (i : ι), f i x) s
-- contMDiffOn_finset_sum'
@[to_additive]
theorem contMDiffOn_finset_prod' (h : ∀ i ∈ t, ContMDiffOn I' I n (f i) s) :
    ContMDiffOn I' I n (∏ i ∈ t, f i) s 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {H : Type u_3} [TopologicalSpace H] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_5} [AddCommMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothAdd I G] {E' : Type u_6} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_7} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_8} [TopologicalSpace M] [ChartedSpace H' M] {s : Set M} {t : Finset ι} {f : ι → M → G} {n : ℕ∞} (h : ∀ i ∈ t, ContMDiffOn I' I n (f i) s)
Goal: ContMDiffOn I' I n (∑ i ∈ t, f i) s
-- contMDiffOn_finset_prod'
@[to_additive]
theorem contMDiffOn_finset_prod' (h : ∀ i ∈ t, ContMDiffOn I' I n (f i) s) :
    ContMDiffOn I' I n (∏ i ∈ t, f i) s 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {H : Type u_3} [TopologicalSpace H] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_5} [CommMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothMul I G] {E' : Type u_6} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_7} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_8} [TopologicalSpace M] [ChartedSpace H' M] {s : Set M} {t : Finset ι} {f : ι → M → G} {n : ℕ∞} (h : ∀ i ∈ t, ContMDiffOn I' I n (f i) s)
Goal: ContMDiffOn I' I n (∏ i ∈ t, f i) s
-- contMDiffOn_finset_sum
@[to_additive]
theorem contMDiffOn_finset_prod (h : ∀ i ∈ t, ContMDiffOn I' I n (f i) s) :
    ContMDiffOn I' I n (fun x => ∏ i ∈ t, f i x) s 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {H : Type u_3} [TopologicalSpace H] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_5} [AddCommMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothAdd I G] {E' : Type u_6} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_7} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_8} [TopologicalSpace M] [ChartedSpace H' M] {s : Set M} {t : Finset ι} {f : ι → M → G} {n : ℕ∞} (h : ∀ i ∈ t, ContMDiffOn I' I n (f i) s)
Goal: ContMDiffOn I' I n (fun x => ∑ i ∈ t, f i x) s
-- contMDiffOn_finset_prod
@[to_additive]
theorem contMDiffOn_finset_prod (h : ∀ i ∈ t, ContMDiffOn I' I n (f i) s) :
    ContMDiffOn I' I n (fun x => ∏ i ∈ t, f i x) s 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {H : Type u_3} [TopologicalSpace H] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_5} [CommMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothMul I G] {E' : Type u_6} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_7} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_8} [TopologicalSpace M] [ChartedSpace H' M] {s : Set M} {t : Finset ι} {f : ι → M → G} {n : ℕ∞} (h : ∀ i ∈ t, ContMDiffOn I' I n (f i) s)
Goal: ContMDiffOn I' I n (fun x => ∏ i ∈ t, f i x) s
-- ContMDiff.sum
@[to_additive]
theorem ContMDiff.prod (h : ∀ i ∈ t, ContMDiff I' I n (f i)) :
    ContMDiff I' I n fun x ↦ ∏ i ∈ t, f i x 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {H : Type u_3} [TopologicalSpace H] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_5} [AddCommMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothAdd I G] {E' : Type u_6} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_7} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_8} [TopologicalSpace M] [ChartedSpace H' M] {t : Finset ι} {f : ι → M → G} {n : ℕ∞} (h : ∀ i ∈ t, ContMDiff I' I n (f i))
Goal: ContMDiff I' I n fun x => ∑ i ∈ t, f i x
-- ContMDiff.prod
@[to_additive]
theorem ContMDiff.prod (h : ∀ i ∈ t, ContMDiff I' I n (f i)) :
    ContMDiff I' I n fun x ↦ ∏ i ∈ t, f i x 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {H : Type u_3} [TopologicalSpace H] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_5} [CommMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothMul I G] {E' : Type u_6} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_7} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_8} [TopologicalSpace M] [ChartedSpace H' M] {t : Finset ι} {f : ι → M → G} {n : ℕ∞} (h : ∀ i ∈ t, ContMDiff I' I n (f i))
Goal: ContMDiff I' I n fun x => ∏ i ∈ t, f i x
-- contMDiff_finset_prod'
@[to_additive]
theorem contMDiff_finset_prod' (h : ∀ i ∈ t, ContMDiff I' I n (f i)) :
    ContMDiff I' I n (∏ i ∈ t, f i) 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {H : Type u_3} [TopologicalSpace H] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_5} [CommMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothMul I G] {E' : Type u_6} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_7} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_8} [TopologicalSpace M] [ChartedSpace H' M] {t : Finset ι} {f : ι → M → G} {n : ℕ∞} (h : ∀ i ∈ t, ContMDiff I' I n (f i))
Goal: ContMDiff I' I n (∏ i ∈ t, f i)
-- contMDiff_finset_sum'
@[to_additive]
theorem contMDiff_finset_prod' (h : ∀ i ∈ t, ContMDiff I' I n (f i)) :
    ContMDiff I' I n (∏ i ∈ t, f i) 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {H : Type u_3} [TopologicalSpace H] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_5} [AddCommMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothAdd I G] {E' : Type u_6} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_7} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_8} [TopologicalSpace M] [ChartedSpace H' M] {t : Finset ι} {f : ι → M → G} {n : ℕ∞} (h : ∀ i ∈ t, ContMDiff I' I n (f i))
Goal: ContMDiff I' I n (∑ i ∈ t, f i)
-- contMDiff_finset_sum
@[to_additive]
theorem contMDiff_finset_prod (h : ∀ i ∈ t, ContMDiff I' I n (f i)) :
    ContMDiff I' I n fun x => ∏ i ∈ t, f i x 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {H : Type u_3} [TopologicalSpace H] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_5} [AddCommMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothAdd I G] {E' : Type u_6} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_7} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_8} [TopologicalSpace M] [ChartedSpace H' M] {t : Finset ι} {f : ι → M → G} {n : ℕ∞} (h : ∀ i ∈ t, ContMDiff I' I n (f i))
Goal: ContMDiff I' I n fun x => ∑ i ∈ t, f i x
-- contMDiff_finset_prod
@[to_additive]
theorem contMDiff_finset_prod (h : ∀ i ∈ t, ContMDiff I' I n (f i)) :
    ContMDiff I' I n fun x => ∏ i ∈ t, f i x 
Arguments: {ι : Type u_1} {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {H : Type u_3} [TopologicalSpace H] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_5} [CommMonoid G] [TopologicalSpace G] [ChartedSpace H G] [SmoothMul I G] {E' : Type u_6} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_7} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_8} [TopologicalSpace M] [ChartedSpace H' M] {t : Finset ι} {f : ι → M → G} {n : ℕ∞} (h : ∀ i ∈ t, ContMDiff I' I n (f i))
Goal: ContMDiff I' I n fun x => ∏ i ∈ t, f i x