ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- mfderivWithin_congr
theorem mfderivWithin_congr (hs : UniqueMDiffWithinAt I s x) (hL : ∀ x ∈ s, f₁ x = f x)
    (hx : f₁ x = f x) : mfderivWithin I I' f₁ s x = (mfderivWithin I I' 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 f₁ : M → M'} {x : M} {s : Set M} [Is : SmoothManifoldWithCorners I M] [I's : SmoothManifoldWithCorners I' M'] (hs : UniqueMDiffWithinAt I s x) (hL : ∀ x ∈ s, f₁ x = f x) (hx : f₁ x = f x)
Goal: mfderivWithin I I' f₁ s x = mfderivWithin I I' f s x
-- tangentMapWithin_congr
theorem tangentMapWithin_congr (h : ∀ x ∈ s, f x = f₁ x) (p : TangentBundle I M) (hp : p.1 ∈ s)
    (hs : UniqueMDiffWithinAt I s p.1) :
    tangentMapWithin I I' f s p = tangentMapWithin I I' f₁ 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] {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 f₁ : M → M'} {s : Set M} [Is : SmoothManifoldWithCorners I M] [I's : SmoothManifoldWithCorners I' M'] (h : ∀ x ∈ s, f x = f₁ x) (p : TangentBundle I M) (hp : p.proj ∈ s) (hs : UniqueMDiffWithinAt I s p.proj)
Goal: tangentMapWithin I I' f s p = tangentMapWithin I I' f₁ s p
-- Filter.EventuallyEq.mfderiv_eq
theorem Filter.EventuallyEq.mfderiv_eq (hL : f₁ =ᶠ[𝓝 x] f) :
    mfderiv I I' f₁ x = (mfderiv I I' 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 f₁ : M → M'} {x : M} [Is : SmoothManifoldWithCorners I M] [I's : SmoothManifoldWithCorners I' M'] (hL : f₁ =ᶠ[𝓝 x] f)
Goal: mfderiv I I' f₁ x = mfderiv I I' f x
-- mfderiv_congr_point
theorem mfderiv_congr_point {x' : M} (h : x = x') :
    @Eq (E →L[𝕜] E') (mfderiv I I' f x) (mfderiv I I' 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 : M → M'} {x : M} [Is : SmoothManifoldWithCorners I M] [I's : SmoothManifoldWithCorners I' M'] {x' : M} (h : x = x')
Goal: mfderiv I I' f x = mfderiv I I' f x'
-- mfderiv_congr
theorem mfderiv_congr {f' : M → M'} (h : f = f') :
    @Eq (E →L[𝕜] E') (mfderiv I I' f x) (mfderiv I I' 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 : M → M'} {x : M} [Is : SmoothManifoldWithCorners I M] [I's : SmoothManifoldWithCorners I' M'] {f' : M → M'} (h : f = f')
Goal: mfderiv I I' f x = mfderiv I I' f' x
-- writtenInExtChartAt_comp
theorem writtenInExtChartAt_comp (h : ContinuousWithinAt f s x) :
    {y | writtenInExtChartAt I I'' x (g ∘ f) y =
          (writtenInExtChartAt I' I'' (f x) g ∘ writtenInExtChartAt I I' x f) y} ∈
      𝓝[(extChartAt I x).symm ⁻¹' s ∩ range I] (extChartAt I x) 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'] {E'' : Type u_8} [NormedAddCommGroup E''] [NormedSpace 𝕜 E''] {H'' : Type u_9} [TopologicalSpace H''] {I'' : ModelWithCorners 𝕜 E'' H''} {M'' : Type u_10} [TopologicalSpace M''] [ChartedSpace H'' M''] {f : M → M'} {x : M} {s : Set M} {g : M' → M''} (h : ContinuousWithinAt f s x)
Goal: {y | writtenInExtChartAt I I'' x (g ∘ f) y = (writtenInExtChartAt I' I'' (f x) g ∘ writtenInExtChartAt I I' x f) y} ∈
  𝓝[↑(extChartAt I x).symm ⁻¹' s ∩ Set.range ↑I] ↑(extChartAt I x) x
-- HasMFDerivWithinAt.comp
theorem HasMFDerivWithinAt.comp (hg : HasMFDerivWithinAt I' I'' g u (f x) g')
    (hf : HasMFDerivWithinAt I I' f s x f') (hst : s ⊆ f ⁻¹' u) :
    HasMFDerivWithinAt I I'' (g ∘ f) s x (g'.comp 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'] {E'' : Type u_8} [NormedAddCommGroup E''] [NormedSpace 𝕜 E''] {H'' : Type u_9} [TopologicalSpace H''] {I'' : ModelWithCorners 𝕜 E'' H''} {M'' : Type u_10} [TopologicalSpace M''] [ChartedSpace H'' M''] {f : M → M'} (x : M) {s : Set M} {g : M' → M''} {u : Set M'} [Is : SmoothManifoldWithCorners I M] [I's : SmoothManifoldWithCorners I' M'] [I''s : SmoothManifoldWithCorners I'' M''] {f' : TangentSpace I x →L[𝕜] TangentSpace I' (f x)} {g' : TangentSpace I' (f x) →L[𝕜] TangentSpace I'' (g (f x))} (hg : HasMFDerivWithinAt I' I'' g u (f x) g') (hf : HasMFDerivWithinAt I I' f s x f') (hst : s ⊆ f ⁻¹' u)
Goal: HasMFDerivWithinAt I I'' (g ∘ f) s x (g'.comp f')
-- HasMFDerivAt.comp
theorem HasMFDerivAt.comp (hg : HasMFDerivAt I' I'' g (f x) g') (hf : HasMFDerivAt I I' f x f') :
    HasMFDerivAt I I'' (g ∘ f) x (g'.comp 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'] {E'' : Type u_8} [NormedAddCommGroup E''] [NormedSpace 𝕜 E''] {H'' : Type u_9} [TopologicalSpace H''] {I'' : ModelWithCorners 𝕜 E'' H''} {M'' : Type u_10} [TopologicalSpace M''] [ChartedSpace H'' M''] {f : M → M'} (x : M) {g : M' → M''} [Is : SmoothManifoldWithCorners I M] [I's : SmoothManifoldWithCorners I' M'] [I''s : SmoothManifoldWithCorners I'' M''] {f' : TangentSpace I x →L[𝕜] TangentSpace I' (f x)} {g' : TangentSpace I' (f x) →L[𝕜] TangentSpace I'' (g (f x))} (hg : HasMFDerivAt I' I'' g (f x) g') (hf : HasMFDerivAt I I' f x f')
Goal: HasMFDerivAt I I'' (g ∘ f) x (g'.comp f')
-- HasMFDerivAt.comp_hasMFDerivWithinAt
theorem HasMFDerivAt.comp_hasMFDerivWithinAt (hg : HasMFDerivAt I' I'' g (f x) g')
    (hf : HasMFDerivWithinAt I I' f s x f') :
    HasMFDerivWithinAt I I'' (g ∘ f) s x (g'.comp 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'] {E'' : Type u_8} [NormedAddCommGroup E''] [NormedSpace 𝕜 E''] {H'' : Type u_9} [TopologicalSpace H''] {I'' : ModelWithCorners 𝕜 E'' H''} {M'' : Type u_10} [TopologicalSpace M''] [ChartedSpace H'' M''] {f : M → M'} (x : M) {s : Set M} {g : M' → M''} [Is : SmoothManifoldWithCorners I M] [I's : SmoothManifoldWithCorners I' M'] [I''s : SmoothManifoldWithCorners I'' M''] {f' : TangentSpace I x →L[𝕜] TangentSpace I' (f x)} {g' : TangentSpace I' (f x) →L[𝕜] TangentSpace I'' (g (f x))} (hg : HasMFDerivAt I' I'' g (f x) g') (hf : HasMFDerivWithinAt I I' f s x f')
Goal: HasMFDerivWithinAt I I'' (g ∘ f) s x (g'.comp f')
-- MDifferentiableWithinAt.comp
theorem MDifferentiableWithinAt.comp (hg : MDifferentiableWithinAt I' I'' g u (f x))
    (hf : MDifferentiableWithinAt I I' f s x) (h : s ⊆ f ⁻¹' u) :
    MDifferentiableWithinAt I I'' (g ∘ 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'] {E'' : Type u_8} [NormedAddCommGroup E''] [NormedSpace 𝕜 E''] {H'' : Type u_9} [TopologicalSpace H''] {I'' : ModelWithCorners 𝕜 E'' H''} {M'' : Type u_10} [TopologicalSpace M''] [ChartedSpace H'' M''] {f : M → M'} (x : M) {s : Set M} {g : M' → M''} {u : Set M'} [Is : SmoothManifoldWithCorners I M] [I's : SmoothManifoldWithCorners I' M'] [I''s : SmoothManifoldWithCorners I'' M''] (hg : MDifferentiableWithinAt I' I'' g u (f x)) (hf : MDifferentiableWithinAt I I' f s x) (h : s ⊆ f ⁻¹' u)
Goal: MDifferentiableWithinAt I I'' (g ∘ f) s x
-- MDifferentiableAt.comp
theorem MDifferentiableAt.comp (hg : MDifferentiableAt I' I'' g (f x))
    (hf : MDifferentiableAt I I' f x) : MDifferentiableAt I I'' (g ∘ 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'] {E'' : Type u_8} [NormedAddCommGroup E''] [NormedSpace 𝕜 E''] {H'' : Type u_9} [TopologicalSpace H''] {I'' : ModelWithCorners 𝕜 E'' H''} {M'' : Type u_10} [TopologicalSpace M''] [ChartedSpace H'' M''] {f : M → M'} (x : M) {g : M' → M''} [Is : SmoothManifoldWithCorners I M] [I's : SmoothManifoldWithCorners I' M'] [I''s : SmoothManifoldWithCorners I'' M''] (hg : MDifferentiableAt I' I'' g (f x)) (hf : MDifferentiableAt I I' f x)
Goal: MDifferentiableAt I I'' (g ∘ f) x
-- mfderivWithin_comp
theorem mfderivWithin_comp (hg : MDifferentiableWithinAt I' I'' g u (f x))
    (hf : MDifferentiableWithinAt I I' f s x) (h : s ⊆ f ⁻¹' u) (hxs : UniqueMDiffWithinAt I s x) :
    mfderivWithin I I'' (g ∘ f) s x =
      (mfderivWithin I' I'' g u (f x)).comp (mfderivWithin I I' 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'] {E'' : Type u_8} [NormedAddCommGroup E''] [NormedSpace 𝕜 E''] {H'' : Type u_9} [TopologicalSpace H''] {I'' : ModelWithCorners 𝕜 E'' H''} {M'' : Type u_10} [TopologicalSpace M''] [ChartedSpace H'' M''] {f : M → M'} (x : M) {s : Set M} {g : M' → M''} {u : Set M'} [Is : SmoothManifoldWithCorners I M] [I's : SmoothManifoldWithCorners I' M'] [I''s : SmoothManifoldWithCorners I'' M''] (hg : MDifferentiableWithinAt I' I'' g u (f x)) (hf : MDifferentiableWithinAt I I' f s x) (h : s ⊆ f ⁻¹' u) (hxs : UniqueMDiffWithinAt I s x)
Goal: mfderivWithin I I'' (g ∘ f) s x = (mfderivWithin I' I'' g u (f x)).comp (mfderivWithin I I' f s x)
-- mfderiv_comp
theorem mfderiv_comp (hg : MDifferentiableAt I' I'' g (f x)) (hf : MDifferentiableAt I I' f x) :
    mfderiv I I'' (g ∘ f) x = (mfderiv I' I'' g (f x)).comp (mfderiv I I' 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'] {E'' : Type u_8} [NormedAddCommGroup E''] [NormedSpace 𝕜 E''] {H'' : Type u_9} [TopologicalSpace H''] {I'' : ModelWithCorners 𝕜 E'' H''} {M'' : Type u_10} [TopologicalSpace M''] [ChartedSpace H'' M''] {f : M → M'} (x : M) {g : M' → M''} [Is : SmoothManifoldWithCorners I M] [I's : SmoothManifoldWithCorners I' M'] [I''s : SmoothManifoldWithCorners I'' M''] (hg : MDifferentiableAt I' I'' g (f x)) (hf : MDifferentiableAt I I' f x)
Goal: mfderiv I I'' (g ∘ f) x = (mfderiv I' I'' g (f x)).comp (mfderiv I I' f x)
-- mfderiv_comp_of_eq
theorem mfderiv_comp_of_eq {x : M} {y : M'} (hg : MDifferentiableAt I' I'' g y)
    (hf : MDifferentiableAt I I' f x) (hy : f x = y) :
    mfderiv I I'' (g ∘ f) x = (mfderiv I' I'' g (f x)).comp (mfderiv I I' 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'] {E'' : Type u_8} [NormedAddCommGroup E''] [NormedSpace 𝕜 E''] {H'' : Type u_9} [TopologicalSpace H''] {I'' : ModelWithCorners 𝕜 E'' H''} {M'' : Type u_10} [TopologicalSpace M''] [ChartedSpace H'' M''] {f : M → M'} {g : M' → M''} [Is : SmoothManifoldWithCorners I M] [I's : SmoothManifoldWithCorners I' M'] [I''s : SmoothManifoldWithCorners I'' M''] {x : M} {y : M'} (hg : MDifferentiableAt I' I'' g y) (hf : MDifferentiableAt I I' f x) (hy : f x = y)
Goal: mfderiv I I'' (g ∘ f) x = (mfderiv I' I'' g (f x)).comp (mfderiv I I' f x)
-- MDifferentiableOn.comp
theorem MDifferentiableOn.comp (hg : MDifferentiableOn I' I'' g u) (hf : MDifferentiableOn I I' f s)
    (st : s ⊆ f ⁻¹' u) : MDifferentiableOn I I'' (g ∘ f) 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] {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'] {E'' : Type u_8} [NormedAddCommGroup E''] [NormedSpace 𝕜 E''] {H'' : Type u_9} [TopologicalSpace H''] {I'' : ModelWithCorners 𝕜 E'' H''} {M'' : Type u_10} [TopologicalSpace M''] [ChartedSpace H'' M''] {f : M → M'} {s : Set M} {g : M' → M''} {u : Set M'} [Is : SmoothManifoldWithCorners I M] [I's : SmoothManifoldWithCorners I' M'] [I''s : SmoothManifoldWithCorners I'' M''] (hg : MDifferentiableOn I' I'' g u) (hf : MDifferentiableOn I I' f s) (st : s ⊆ f ⁻¹' u)
Goal: MDifferentiableOn I I'' (g ∘ f) s
-- MDifferentiable.comp
theorem MDifferentiable.comp (hg : MDifferentiable I' I'' g) (hf : MDifferentiable I I' f) :
    MDifferentiable I I'' (g ∘ 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'] {E'' : Type u_8} [NormedAddCommGroup E''] [NormedSpace 𝕜 E''] {H'' : Type u_9} [TopologicalSpace H''] {I'' : ModelWithCorners 𝕜 E'' H''} {M'' : Type u_10} [TopologicalSpace M''] [ChartedSpace H'' M''] {f : M → M'} {g : M' → M''} [Is : SmoothManifoldWithCorners I M] [I's : SmoothManifoldWithCorners I' M'] [I''s : SmoothManifoldWithCorners I'' M''] (hg : MDifferentiable I' I'' g) (hf : MDifferentiable I I' f)
Goal: MDifferentiable I I'' (g ∘ f)
-- tangentMapWithin_comp_at
theorem tangentMapWithin_comp_at (p : TangentBundle I M)
    (hg : MDifferentiableWithinAt I' I'' g u (f p.1)) (hf : MDifferentiableWithinAt I I' f s p.1)
    (h : s ⊆ f ⁻¹' u) (hps : UniqueMDiffWithinAt I s p.1) :
    tangentMapWithin I I'' (g ∘ f) s p =
      tangentMapWithin I' I'' g u (tangentMapWithin I I' f 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] {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'] {E'' : Type u_8} [NormedAddCommGroup E''] [NormedSpace 𝕜 E''] {H'' : Type u_9} [TopologicalSpace H''] {I'' : ModelWithCorners 𝕜 E'' H''} {M'' : Type u_10} [TopologicalSpace M''] [ChartedSpace H'' M''] {f : M → M'} {s : Set M} {g : M' → M''} {u : Set M'} [Is : SmoothManifoldWithCorners I M] [I's : SmoothManifoldWithCorners I' M'] [I''s : SmoothManifoldWithCorners I'' M''] (p : TangentBundle I M) (hg : MDifferentiableWithinAt I' I'' g u (f p.proj)) (hf : MDifferentiableWithinAt I I' f s p.proj) (h : s ⊆ f ⁻¹' u) (hps : UniqueMDiffWithinAt I s p.proj)
Goal: tangentMapWithin I I'' (g ∘ f) s p = tangentMapWithin I' I'' g u (tangentMapWithin I I' f s p)
-- tangentMap_comp_at
theorem tangentMap_comp_at (p : TangentBundle I M) (hg : MDifferentiableAt I' I'' g (f p.1))
    (hf : MDifferentiableAt I I' f p.1) :
    tangentMap I I'' (g ∘ f) p = tangentMap I' I'' g (tangentMap I I' f 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'] {E'' : Type u_8} [NormedAddCommGroup E''] [NormedSpace 𝕜 E''] {H'' : Type u_9} [TopologicalSpace H''] {I'' : ModelWithCorners 𝕜 E'' H''} {M'' : Type u_10} [TopologicalSpace M''] [ChartedSpace H'' M''] {f : M → M'} {g : M' → M''} [Is : SmoothManifoldWithCorners I M] [I's : SmoothManifoldWithCorners I' M'] [I''s : SmoothManifoldWithCorners I'' M''] (p : TangentBundle I M) (hg : MDifferentiableAt I' I'' g (f p.proj)) (hf : MDifferentiableAt I I' f p.proj)
Goal: tangentMap I I'' (g ∘ f) p = tangentMap I' I'' g (tangentMap I I' f p)
-- tangentMap_comp
theorem tangentMap_comp (hg : MDifferentiable I' I'' g) (hf : MDifferentiable I I' f) :
    tangentMap I I'' (g ∘ f) = tangentMap I' I'' g ∘ tangentMap I I' 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'] {E'' : Type u_8} [NormedAddCommGroup E''] [NormedSpace 𝕜 E''] {H'' : Type u_9} [TopologicalSpace H''] {I'' : ModelWithCorners 𝕜 E'' H''} {M'' : Type u_10} [TopologicalSpace M''] [ChartedSpace H'' M''] {f : M → M'} {g : M' → M''} [Is : SmoothManifoldWithCorners I M] [I's : SmoothManifoldWithCorners I' M'] [I''s : SmoothManifoldWithCorners I'' M''] (hg : MDifferentiable I' I'' g) (hf : MDifferentiable I I' f)
Goal: tangentMap I I'' (g ∘ f) = tangentMap I' I'' g ∘ tangentMap I I' f
-- InnerProductGeometry.IsConformalMap.preserves_angle
theorem IsConformalMap.preserves_angle {f' : E →L[ℝ] F} (h : IsConformalMap f') (u v : E) :
    angle (f' u) (f' v) = angle u v 
Arguments: {E : Type u_1} {F : Type u_2} [NormedAddCommGroup E] [NormedAddCommGroup F] [InnerProductSpace ℝ E] [InnerProductSpace ℝ F] {f' : E →L[ℝ] F} (h : IsConformalMap f') (u v : E)
Goal: InnerProductGeometry.angle (f' u) (f' v) = InnerProductGeometry.angle u v
-- InnerProductGeometry.ConformalAt.preserves_angle
theorem ConformalAt.preserves_angle {f : E → F} {x : E} {f' : E →L[ℝ] F} (h : HasFDerivAt f f' x)
    (H : ConformalAt f x) (u v : E) : angle (f' u) (f' v) = angle u v 
Arguments: {E : Type u_1} {F : Type u_2} [NormedAddCommGroup E] [NormedAddCommGroup F] [InnerProductSpace ℝ E] [InnerProductSpace ℝ F] {f : E → F} {x : E} {f' : E →L[ℝ] F} (h : HasFDerivAt f f' x) (H : ConformalAt f x) (u v : E)
Goal: InnerProductGeometry.angle (f' u) (f' v) = InnerProductGeometry.angle u v
-- contMDiff_model
theorem contMDiff_model : ContMDiff I 𝓘(𝕜, E) n 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} {n : ℕ∞}
Goal: ContMDiff I 𝓘(𝕜, E) n ↑I
-- contMDiffOn_model_symm
theorem contMDiffOn_model_symm : ContMDiffOn 𝓘(𝕜, E) I n I.symm (range 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} {n : ℕ∞}
Goal: ContMDiffOn 𝓘(𝕜, E) I n (↑I.symm) (Set.range ↑I)
-- contMDiffOn_of_mem_maximalAtlas
theorem contMDiffOn_of_mem_maximalAtlas (h : e ∈ maximalAtlas I M) : ContMDiffOn I I n e e.source 
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] [SmoothManifoldWithCorners I M] {e : PartialHomeomorph M H} {n : ℕ∞} (h : e ∈ SmoothManifoldWithCorners.maximalAtlas I M)
Goal: ContMDiffOn I I n (↑e) e.source
-- contMDiffOn_symm_of_mem_maximalAtlas
theorem contMDiffOn_symm_of_mem_maximalAtlas (h : e ∈ maximalAtlas I M) :
    ContMDiffOn I I n e.symm e.target 
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] [SmoothManifoldWithCorners I M] {e : PartialHomeomorph M H} {n : ℕ∞} (h : e ∈ SmoothManifoldWithCorners.maximalAtlas I M)
Goal: ContMDiffOn I I n (↑e.symm) e.target
-- contMDiffAt_of_mem_maximalAtlas
theorem contMDiffAt_of_mem_maximalAtlas (h : e ∈ maximalAtlas I M) (hx : x ∈ e.source) :
    ContMDiffAt I I n e 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] [SmoothManifoldWithCorners I M] {e : PartialHomeomorph M H} {x : M} {n : ℕ∞} (h : e ∈ SmoothManifoldWithCorners.maximalAtlas I M) (hx : x ∈ e.source)
Goal: ContMDiffAt I I n (↑e) x
-- contMDiffAt_symm_of_mem_maximalAtlas
theorem contMDiffAt_symm_of_mem_maximalAtlas {x : H} (h : e ∈ maximalAtlas I M)
    (hx : x ∈ e.target) : ContMDiffAt I I n e.symm 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] [SmoothManifoldWithCorners I M] {e : PartialHomeomorph M H} {n : ℕ∞} {x : H} (h : e ∈ SmoothManifoldWithCorners.maximalAtlas I M) (hx : x ∈ e.target)
Goal: ContMDiffAt I I n (↑e.symm) x