ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- SmoothWithinAt.inv₀
theorem SmoothWithinAt.inv₀ (hf : SmoothWithinAt I' I f s a) (ha : f a ≠ 0) :
    SmoothWithinAt I' I (fun x => (f x)⁻¹) s a 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {H : Type u_2} [TopologicalSpace H] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [Inv G] [Zero G] [SmoothInv₀ I G] {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 → G} {s : Set M} {a : M} (hf : SmoothWithinAt I' I f s a) (ha : f a ≠ 0)
Goal: SmoothWithinAt I' I (fun x => (f x)⁻¹) s a
-- SmoothAt.inv₀
theorem SmoothAt.inv₀ (hf : SmoothAt I' I f a) (ha : f a ≠ 0) :
    SmoothAt I' I (fun x => (f x)⁻¹) a 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {H : Type u_2} [TopologicalSpace H] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [Inv G] [Zero G] [SmoothInv₀ I G] {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 → G} {a : M} (hf : SmoothAt I' I f a) (ha : f a ≠ 0)
Goal: SmoothAt I' I (fun x => (f x)⁻¹) a
-- Smooth.inv₀
theorem Smooth.inv₀ (hf : Smooth I' I f) (h0 : ∀ x, f x ≠ 0) : Smooth I' I fun x => (f x)⁻¹ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {H : Type u_2} [TopologicalSpace H] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [Inv G] [Zero G] [SmoothInv₀ I G] {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 → G} (hf : Smooth I' I f) (h0 : ∀ (x : M), f x ≠ 0)
Goal: Smooth I' I fun x => (f x)⁻¹
-- SmoothOn.inv₀
theorem SmoothOn.inv₀ (hf : SmoothOn I' I f s) (h0 : ∀ x ∈ s, f x ≠ 0) :
    SmoothOn I' I (fun x => (f x)⁻¹) s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {H : Type u_2} [TopologicalSpace H] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [Inv G] [Zero G] [SmoothInv₀ I G] {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 → G} {s : Set M} (hf : SmoothOn I' I f s) (h0 : ∀ x ∈ s, f x ≠ 0)
Goal: SmoothOn I' I (fun x => (f x)⁻¹) s
-- ContMDiffWithinAt.div₀
theorem ContMDiffWithinAt.div₀
    (hf : ContMDiffWithinAt I' I n f s a) (hg : ContMDiffWithinAt I' I n g s a) (h₀ : g a ≠ 0) :
    ContMDiffWithinAt I' I n (f / g) s a 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {H : Type u_2} [TopologicalSpace H] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [GroupWithZero G] [SmoothInv₀ I G] [SmoothMul I G] {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 g : M → G} {s : Set M} {a : M} {n : ℕ∞} (hf : ContMDiffWithinAt I' I n f s a) (hg : ContMDiffWithinAt I' I n g s a) (h₀ : g a ≠ 0)
Goal: ContMDiffWithinAt I' I n (f / g) s a
-- ContMDiffOn.div₀
theorem ContMDiffOn.div₀ (hf : ContMDiffOn I' I n f s) (hg : ContMDiffOn I' I n g s)
    (h₀ : ∀ x ∈ s, g x ≠ 0) : ContMDiffOn I' I n (f / g) s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {H : Type u_2} [TopologicalSpace H] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [GroupWithZero G] [SmoothInv₀ I G] [SmoothMul I G] {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 g : M → G} {s : Set M} {n : ℕ∞} (hf : ContMDiffOn I' I n f s) (hg : ContMDiffOn I' I n g s) (h₀ : ∀ x ∈ s, g x ≠ 0)
Goal: ContMDiffOn I' I n (f / g) s
-- ContMDiffAt.div₀
theorem ContMDiffAt.div₀ (hf : ContMDiffAt I' I n f a) (hg : ContMDiffAt I' I n g a)
    (h₀ : g a ≠ 0) : ContMDiffAt I' I n (f / g) a 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {H : Type u_2} [TopologicalSpace H] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [GroupWithZero G] [SmoothInv₀ I G] [SmoothMul I G] {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 g : M → G} {a : M} {n : ℕ∞} (hf : ContMDiffAt I' I n f a) (hg : ContMDiffAt I' I n g a) (h₀ : g a ≠ 0)
Goal: ContMDiffAt I' I n (f / g) a
-- ContMDiff.div₀
theorem ContMDiff.div₀ (hf : ContMDiff I' I n f) (hg : ContMDiff I' I n g) (h₀ : ∀ x, g x ≠ 0) :
    ContMDiff I' I n (f / g) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {H : Type u_2} [TopologicalSpace H] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [GroupWithZero G] [SmoothInv₀ I G] [SmoothMul I G] {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 g : M → G} {n : ℕ∞} (hf : ContMDiff I' I n f) (hg : ContMDiff I' I n g) (h₀ : ∀ (x : M), g x ≠ 0)
Goal: ContMDiff I' I n (f / g)
-- SmoothWithinAt.div₀
theorem SmoothWithinAt.div₀ (hf : SmoothWithinAt I' I f s a)
    (hg : SmoothWithinAt I' I g s a) (h₀ : g a ≠ 0) : SmoothWithinAt I' I (f / g) s a 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {H : Type u_2} [TopologicalSpace H] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [GroupWithZero G] [SmoothInv₀ I G] [SmoothMul I G] {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 g : M → G} {s : Set M} {a : M} (hf : SmoothWithinAt I' I f s a) (hg : SmoothWithinAt I' I g s a) (h₀ : g a ≠ 0)
Goal: SmoothWithinAt I' I (f / g) s a
-- SmoothOn.div₀
theorem SmoothOn.div₀ (hf : SmoothOn I' I f s) (hg : SmoothOn I' I g s) (h₀ : ∀ x ∈ s, g x ≠ 0) :
    SmoothOn I' I (f / g) s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {H : Type u_2} [TopologicalSpace H] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [GroupWithZero G] [SmoothInv₀ I G] [SmoothMul I G] {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 g : M → G} {s : Set M} (hf : SmoothOn I' I f s) (hg : SmoothOn I' I g s) (h₀ : ∀ x ∈ s, g x ≠ 0)
Goal: SmoothOn I' I (f / g) s
-- SmoothAt.div₀
theorem SmoothAt.div₀ (hf : SmoothAt I' I f a) (hg : SmoothAt I' I g a) (h₀ : g a ≠ 0) :
    SmoothAt I' I (f / g) a 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {H : Type u_2} [TopologicalSpace H] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [GroupWithZero G] [SmoothInv₀ I G] [SmoothMul I G] {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 g : M → G} {a : M} (hf : SmoothAt I' I f a) (hg : SmoothAt I' I g a) (h₀ : g a ≠ 0)
Goal: SmoothAt I' I (f / g) a
-- Smooth.div₀
theorem Smooth.div₀ (hf : Smooth I' I f) (hg : Smooth I' I g) (h₀ : ∀ x, g x ≠ 0) :
    Smooth I' I (f / g) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {H : Type u_2} [TopologicalSpace H] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [GroupWithZero G] [SmoothInv₀ I G] [SmoothMul I G] {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 g : M → G} (hf : Smooth I' I f) (hg : Smooth I' I g) (h₀ : ∀ (x : M), g x ≠ 0)
Goal: Smooth I' I (f / g)
-- contDiffOn_fderiv_coord_change
theorem contDiffOn_fderiv_coord_change (i j : atlas H M) :
    ContDiffOn 𝕜 ∞ (fderivWithin 𝕜 (j.1.extend I ∘ (i.1.extend I).symm) (range I))
      ((i.1.extend I).symm ≫ j.1.extend I).source 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_4} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u_6} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] (i j : ↑(atlas H M))
Goal: ContDiffOn 𝕜 ⊤ (fderivWithin 𝕜 (↑((↑j).extend I) ∘ ↑((↑i).extend I).symm) (Set.range ↑I))
  (((↑i).extend I).symm ≫ (↑j).extend I).source
-- tangentBundleCore_coordChange
@[simps indexAt coordChange]
def tangentBundleCore : VectorBundleCore 𝕜 M E (atlas H M) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_4} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) (M : Type u_6) [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] (i j : ↑(atlas H M)) (x : M)
Goal: (tangentBundleCore I M).coordChange i j x =
  fderivWithin 𝕜 (↑((↑j).extend I) ∘ ↑((↑i).extend I).symm) (Set.range ↑I) (↑((↑i).extend I) x)
-- tangentBundleCore_indexAt
@[simps indexAt coordChange]
def tangentBundleCore : VectorBundleCore 𝕜 M E (atlas H M) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_4} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) (M : Type u_6) [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] (x : M)
Goal: (tangentBundleCore I M).indexAt x = achart H x
-- tangentBundleCore_baseSet
@[simp high]
theorem tangentBundleCore_baseSet (i) : (tangentBundleCore I M).baseSet i = i.1.source 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_4} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) (M : Type u_6) [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] (i : ↑(atlas H M))
Goal: (tangentBundleCore I M).baseSet i = (↑i).source
-- tangentBundleCore_coordChange_achart
theorem tangentBundleCore_coordChange_achart (x x' z : M) :
    (tangentBundleCore I M).coordChange (achart H x) (achart H x') z =
      fderivWithin 𝕜 (extChartAt I x' ∘ (extChartAt I x).symm) (range I) (extChartAt I x z) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_4} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u_6} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] (x x' z : M)
Goal: (tangentBundleCore I M).coordChange (achart H x) (achart H x') z =
  fderivWithin 𝕜 (↑(extChartAt I x') ∘ ↑(extChartAt I x).symm) (Set.range ↑I) (↑(extChartAt I x) z)
-- tangentCoordChange_def
lemma tangentCoordChange_def {x y z : M} : tangentCoordChange I x y z =
    fderivWithin 𝕜 (extChartAt I y ∘ (extChartAt I x).symm) (range I) (extChartAt I x z) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_4} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_6} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {x y z : M}
Goal: tangentCoordChange I x y z =
  fderivWithin 𝕜 (↑(extChartAt I y) ∘ ↑(extChartAt I x).symm) (Set.range ↑I) (↑(extChartAt I x) z)
-- tangentCoordChange_self
lemma tangentCoordChange_self {x z : M} {v : E} (h : z ∈ (extChartAt I x).source) :
    tangentCoordChange I x x z v = v 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_4} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_6} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {x z : M} {v : E} (h : z ∈ (extChartAt I x).source)
Goal: (tangentCoordChange I x x z) v = v
-- tangentCoordChange_comp
lemma tangentCoordChange_comp {w x y z : M} {v : E}
    (h : z ∈ (extChartAt I w).source ∩ (extChartAt I x).source ∩ (extChartAt I y).source) :
    tangentCoordChange I x y z (tangentCoordChange I w x z v) = tangentCoordChange I w y z v 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_4} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_6} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {w x y z : M} {v : E} (h : z ∈ (extChartAt I w).source ∩ (extChartAt I x).source ∩ (extChartAt I y).source)
Goal: (tangentCoordChange I x y z) ((tangentCoordChange I w x z) v) = (tangentCoordChange I w y z) v
-- hasFDerivWithinAt_tangentCoordChange
lemma hasFDerivWithinAt_tangentCoordChange {x y z : M}
    (h : z ∈ (extChartAt I x).source ∩ (extChartAt I y).source) :
    HasFDerivWithinAt ((extChartAt I y) ∘ (extChartAt I x).symm) (tangentCoordChange I x y z)
      (range I) (extChartAt I x z) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_4} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_6} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {x y z : M} (h : z ∈ (extChartAt I x).source ∩ (extChartAt I y).source)
Goal: HasFDerivWithinAt (↑(extChartAt I y) ∘ ↑(extChartAt I x).symm) (tangentCoordChange I x y z) (Set.range ↑I)
  (↑(extChartAt I x) z)
-- continuousOn_tangentCoordChange
lemma continuousOn_tangentCoordChange (x y : M) : ContinuousOn (tangentCoordChange I x y)
    ((extChartAt I x).source ∩ (extChartAt I y).source) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_4} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_6} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] (x y : M)
Goal: ContinuousOn (tangentCoordChange I x y) ((extChartAt I x).source ∩ (extChartAt I y).source)
-- TangentBundle.chartAt
protected theorem chartAt (p : TM) :
    chartAt (ModelProd H E) p =
      ((tangentBundleCore I M).toFiberBundleCore.localTriv (achart H p.1)).toPartialHomeomorph ≫ₕ
        (chartAt H p.1).prod (PartialHomeomorph.refl E) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_4} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) (M : Type u_6) [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] (p : TangentBundle I M)
Goal: chartAt (ModelProd H E) p =
  ((tangentBundleCore I M).toFiberBundleCore.localTriv (achart H p.proj)).toPartialHomeomorph ≫ₕ
    (chartAt H p.proj).prod (PartialHomeomorph.refl E)
-- TangentBundle.chartAt_toPartialEquiv
theorem chartAt_toPartialEquiv (p : TM) :
    (chartAt (ModelProd H E) p).toPartialEquiv =
      (tangentBundleCore I M).toFiberBundleCore.localTrivAsPartialEquiv (achart H p.1) ≫
        (chartAt H p.1).toPartialEquiv.prod (PartialEquiv.refl E) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_4} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) (M : Type u_6) [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] (p : TangentBundle I M)
Goal: (chartAt (ModelProd H E) p).toPartialEquiv =
  (tangentBundleCore I M).toFiberBundleCore.localTrivAsPartialEquiv (achart H p.proj) ≫
    (chartAt H p.proj).prod (PartialEquiv.refl E)
-- TangentBundle.trivializationAt_eq_localTriv
theorem trivializationAt_eq_localTriv (x : M) :
    trivializationAt E (TangentSpace I) x =
      (tangentBundleCore I M).toFiberBundleCore.localTriv (achart H x) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_4} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) (M : Type u_6) [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] (x : M)
Goal: trivializationAt E (TangentSpace I) x = (tangentBundleCore I M).toFiberBundleCore.localTriv (achart H x)
-- TangentBundle.trivializationAt_source
@[simp, mfld_simps]
theorem trivializationAt_source (x : M) :
    (trivializationAt E (TangentSpace I) x).source =
      π E (TangentSpace I) ⁻¹' (chartAt H x).source 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_4} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) (M : Type u_6) [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] (x : M)
Goal: (trivializationAt E (TangentSpace I) x).source = Bundle.TotalSpace.proj ⁻¹' (chartAt H x).source
-- TangentBundle.trivializationAt_target
@[simp, mfld_simps]
theorem trivializationAt_target (x : M) :
    (trivializationAt E (TangentSpace I) x).target = (chartAt H x).source ×ˢ univ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_4} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) (M : Type u_6) [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] (x : M)
Goal: (trivializationAt E (TangentSpace I) x).target = (chartAt H x).source ×ˢ Set.univ
-- TangentBundle.trivializationAt_baseSet
@[simp, mfld_simps]
theorem trivializationAt_baseSet (x : M) :
    (trivializationAt E (TangentSpace I) x).baseSet = (chartAt H x).source 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_4} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) (M : Type u_6) [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] (x : M)
Goal: (trivializationAt E (TangentSpace I) x).baseSet = (chartAt H x).source
-- TangentBundle.trivializationAt_apply
theorem trivializationAt_apply (x : M) (z : TM) :
    trivializationAt E (TangentSpace I) x z =
      (z.1, fderivWithin 𝕜 ((chartAt H x).extend I ∘ ((chartAt H z.1).extend I).symm) (range I)
        ((chartAt H z.1).extend I z.1) z.2) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_4} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) (M : Type u_6) [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] (x : M) (z : TangentBundle I M)
Goal: ↑(trivializationAt E (TangentSpace I) x) z =
  (z.proj,
    (fderivWithin 𝕜 (↑((chartAt H x).extend I) ∘ ↑((chartAt H z.proj).extend I).symm) (Set.range ↑I)
        (↑((chartAt H z.proj).extend I) z.proj))
      z.snd)
-- TangentBundle.trivializationAt_fst
@[simp, mfld_simps]
theorem trivializationAt_fst (x : M) (z : TM) : (trivializationAt E (TangentSpace I) x z).1 = z.1 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_4} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) (M : Type u_6) [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] (x : M) (z : TangentBundle I M)
Goal: (↑(trivializationAt E (TangentSpace I) x) z).1 = z.proj
-- TangentBundle.mem_chart_source_iff
@[simp, mfld_simps]
theorem mem_chart_source_iff (p q : TM) :
    p ∈ (chartAt (ModelProd H E) q).source ↔ p.1 ∈ (chartAt H q.1).source 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_4} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) (M : Type u_6) [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] (p q : TangentBundle I M)
Goal: p ∈ (chartAt (ModelProd H E) q).source ↔ p.proj ∈ (chartAt H q.proj).source
-- TangentBundle.mem_chart_target_iff
@[simp, mfld_simps]
theorem mem_chart_target_iff (p : H × E) (q : TM) :
    p ∈ (chartAt (ModelProd H E) q).target ↔ p.1 ∈ (chartAt H q.1).target 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_4} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) (M : Type u_6) [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] (p : H × E) (q : TangentBundle I M)
Goal: p ∈ (chartAt (ModelProd H E) q).target ↔ p.1 ∈ (chartAt H q.proj).target
-- TangentBundle.coe_chartAt_fst
@[simp, mfld_simps]
theorem coe_chartAt_fst (p q : TM) : ((chartAt (ModelProd H E) q) p).1 = chartAt H q.1 p.1 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_4} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) (M : Type u_6) [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] (p q : TangentBundle I M)
Goal: (↑(chartAt (ModelProd H E) q) p).1 = ↑(chartAt H q.proj) p.proj
-- TangentBundle.coe_chartAt_symm_fst
@[simp, mfld_simps]
theorem coe_chartAt_symm_fst (p : H × E) (q : TM) :
    ((chartAt (ModelProd H E) q).symm p).1 = ((chartAt H q.1).symm : H → M) p.1 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_4} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) (M : Type u_6) [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] (p : H × E) (q : TangentBundle I M)
Goal: (↑(chartAt (ModelProd H E) q).symm p).proj = ↑(chartAt H q.proj).symm p.1
-- TangentBundle.trivializationAt_continuousLinearMapAt
@[simp, mfld_simps]
theorem trivializationAt_continuousLinearMapAt {b₀ b : M}
    (hb : b ∈ (trivializationAt E (TangentSpace I) b₀).baseSet) :
    (trivializationAt E (TangentSpace I) b₀).continuousLinearMapAt 𝕜 b =
      (tangentBundleCore I M).coordChange (achart H b) (achart H b₀) b 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_4} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) (M : Type u_6) [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {b₀ b : M} (hb : b ∈ (trivializationAt E (TangentSpace I) b₀).baseSet)
Goal: Trivialization.continuousLinearMapAt 𝕜 (trivializationAt E (TangentSpace I) b₀) b =
  (tangentBundleCore I M).coordChange (achart H b) (achart H b₀) b
-- TangentBundle.trivializationAt_symmL
@[simp, mfld_simps]
theorem trivializationAt_symmL {b₀ b : M}
    (hb : b ∈ (trivializationAt E (TangentSpace I) b₀).baseSet) :
    (trivializationAt E (TangentSpace I) b₀).symmL 𝕜 b =
      (tangentBundleCore I M).coordChange (achart H b₀) (achart H b) b 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_4} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) (M : Type u_6) [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {b₀ b : M} (hb : b ∈ (trivializationAt E (TangentSpace I) b₀).baseSet)
Goal: Trivialization.symmL 𝕜 (trivializationAt E (TangentSpace I) b₀) b =
  (tangentBundleCore I M).coordChange (achart H b₀) (achart H b) b
-- TangentBundle.coordChange_model_space
@[simp high, mfld_simps]
theorem coordChange_model_space (b b' x : F) :
    (tangentBundleCore 𝓘(𝕜, F) F).coordChange (achart F b) (achart F b') x = 1 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_8} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (b b' x : F)
Goal: (tangentBundleCore 𝓘(𝕜, F) F).coordChange (achart F b) (achart F b') x = 1
-- TangentBundle.symmL_model_space
@[simp high, mfld_simps]
theorem symmL_model_space (b b' : F) :
    (trivializationAt F (TangentSpace 𝓘(𝕜, F)) b).symmL 𝕜 b' = (1 : F →L[𝕜] F) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_8} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (b b' : F)
Goal: Trivialization.symmL 𝕜 (trivializationAt F (TangentSpace 𝓘(𝕜, F)) b) b' = 1