ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ContinuousLinearEquiv.coe_toDiffeomorph
@[simp]
theorem coe_toDiffeomorph : ⇑e.toDiffeomorph = e 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] (e : E ≃L[𝕜] E')
Goal: ⇑e.toDiffeomorph = ⇑e
-- ContinuousLinearEquiv.symm_toDiffeomorph
@[simp]
theorem symm_toDiffeomorph : e.symm.toDiffeomorph = e.toDiffeomorph.symm 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] (e : E ≃L[𝕜] E')
Goal: e.symm.toDiffeomorph = e.toDiffeomorph.symm
-- ContinuousLinearEquiv.coe_toDiffeomorph_symm
@[simp]
theorem coe_toDiffeomorph_symm : ⇑e.toDiffeomorph.symm = e.symm 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] (e : E ≃L[𝕜] E')
Goal: ⇑e.toDiffeomorph.symm = ⇑e.symm
-- ModelWithCorners.coe_transDiffeomorph
@[simp, mfld_simps]
theorem coe_transDiffeomorph : ⇑(I.transDiffeomorph e) = e ∘ I 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H : Type u_5} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) (e : E ≃ₘ⟮𝓘(𝕜, E), 𝓘(𝕜, E')⟯ E')
Goal: ↑(I.transDiffeomorph e) = ⇑e ∘ ↑I
-- ModelWithCorners.coe_transDiffeomorph_symm
@[simp, mfld_simps]
theorem coe_transDiffeomorph_symm : ⇑(I.transDiffeomorph e).symm = I.symm ∘ e.symm 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H : Type u_5} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) (e : E ≃ₘ⟮𝓘(𝕜, E), 𝓘(𝕜, E')⟯ E')
Goal: ↑(I.transDiffeomorph e).symm = ↑I.symm ∘ ⇑e.symm
-- ModelWithCorners.transDiffeomorph_range
theorem transDiffeomorph_range : range (I.transDiffeomorph e) = e '' range I 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H : Type u_5} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) (e : E ≃ₘ⟮𝓘(𝕜, E), 𝓘(𝕜, E')⟯ E')
Goal: Set.range ↑(I.transDiffeomorph e) = ⇑e '' Set.range ↑I
-- ModelWithCorners.coe_extChartAt_transDiffeomorph
theorem coe_extChartAt_transDiffeomorph (x : M) :
    ⇑(extChartAt (I.transDiffeomorph e) x) = e ∘ extChartAt I x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H : Type u_5} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u_9} [TopologicalSpace M] [ChartedSpace H M] (e : E ≃ₘ⟮𝓘(𝕜, E), 𝓘(𝕜, E')⟯ E') (x : M)
Goal: ↑(extChartAt (I.transDiffeomorph e) x) = ⇑e ∘ ↑(extChartAt I x)
-- ModelWithCorners.coe_extChartAt_transDiffeomorph_symm
theorem coe_extChartAt_transDiffeomorph_symm (x : M) :
    ⇑(extChartAt (I.transDiffeomorph e) x).symm = (extChartAt I x).symm ∘ e.symm 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H : Type u_5} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u_9} [TopologicalSpace M] [ChartedSpace H M] (e : E ≃ₘ⟮𝓘(𝕜, E), 𝓘(𝕜, E')⟯ E') (x : M)
Goal: ↑(extChartAt (I.transDiffeomorph e) x).symm = ↑(extChartAt I x).symm ∘ ⇑e.symm
-- ModelWithCorners.extChartAt_transDiffeomorph_target
theorem extChartAt_transDiffeomorph_target (x : M) :
    (extChartAt (I.transDiffeomorph e) x).target = e.symm ⁻¹' (extChartAt I x).target 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H : Type u_5} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u_9} [TopologicalSpace M] [ChartedSpace H M] (e : E ≃ₘ⟮𝓘(𝕜, E), 𝓘(𝕜, E')⟯ E') (x : M)
Goal: (extChartAt (I.transDiffeomorph e) x).target = ⇑e.symm ⁻¹' (extChartAt I x).target
-- Diffeomorph.contMDiffWithinAt_transDiffeomorph_right
@[simp]
theorem contMDiffWithinAt_transDiffeomorph_right {f : M' → M} {x s} :
    ContMDiffWithinAt I' (I.transDiffeomorph e) n f s x ↔ ContMDiffWithinAt I' I n f s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type u_5} [TopologicalSpace H] {H' : Type u_6} [TopologicalSpace H'] {I : ModelWithCorners 𝕜 E H} {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_9} [TopologicalSpace M] [ChartedSpace H M] {M' : Type u_10} [TopologicalSpace M'] [ChartedSpace H' M'] {n : ℕ∞} (e : E ≃ₘ⟮𝓘(𝕜, E), 𝓘(𝕜, F)⟯ F) {f : M' → M} {x : M'} {s : Set M'}
Goal: ContMDiffWithinAt I' (I.transDiffeomorph e) n f s x ↔ ContMDiffWithinAt I' I n f s x
-- Diffeomorph.contMDiffAt_transDiffeomorph_right
@[simp]
theorem contMDiffAt_transDiffeomorph_right {f : M' → M} {x} :
    ContMDiffAt I' (I.transDiffeomorph e) n f x ↔ ContMDiffAt I' I n f x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type u_5} [TopologicalSpace H] {H' : Type u_6} [TopologicalSpace H'] {I : ModelWithCorners 𝕜 E H} {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_9} [TopologicalSpace M] [ChartedSpace H M] {M' : Type u_10} [TopologicalSpace M'] [ChartedSpace H' M'] {n : ℕ∞} (e : E ≃ₘ⟮𝓘(𝕜, E), 𝓘(𝕜, F)⟯ F) {f : M' → M} {x : M'}
Goal: ContMDiffAt I' (I.transDiffeomorph e) n f x ↔ ContMDiffAt I' I n f x
-- Diffeomorph.contMDiffOn_transDiffeomorph_right
@[simp]
theorem contMDiffOn_transDiffeomorph_right {f : M' → M} {s} :
    ContMDiffOn I' (I.transDiffeomorph e) n f s ↔ ContMDiffOn I' I n f s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type u_5} [TopologicalSpace H] {H' : Type u_6} [TopologicalSpace H'] {I : ModelWithCorners 𝕜 E H} {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_9} [TopologicalSpace M] [ChartedSpace H M] {M' : Type u_10} [TopologicalSpace M'] [ChartedSpace H' M'] {n : ℕ∞} (e : E ≃ₘ⟮𝓘(𝕜, E), 𝓘(𝕜, F)⟯ F) {f : M' → M} {s : Set M'}
Goal: ContMDiffOn I' (I.transDiffeomorph e) n f s ↔ ContMDiffOn I' I n f s
-- Diffeomorph.contMDiff_transDiffeomorph_right
@[simp]
theorem contMDiff_transDiffeomorph_right {f : M' → M} :
    ContMDiff I' (I.transDiffeomorph e) n f ↔ ContMDiff I' I n f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type u_5} [TopologicalSpace H] {H' : Type u_6} [TopologicalSpace H'] {I : ModelWithCorners 𝕜 E H} {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_9} [TopologicalSpace M] [ChartedSpace H M] {M' : Type u_10} [TopologicalSpace M'] [ChartedSpace H' M'] {n : ℕ∞} (e : E ≃ₘ⟮𝓘(𝕜, E), 𝓘(𝕜, F)⟯ F) {f : M' → M}
Goal: ContMDiff I' (I.transDiffeomorph e) n f ↔ ContMDiff I' I n f
-- Diffeomorph.smooth_transDiffeomorph_right
theorem smooth_transDiffeomorph_right {f : M' → M} :
    Smooth I' (I.transDiffeomorph e) f ↔ Smooth I' I f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type u_5} [TopologicalSpace H] {H' : Type u_6} [TopologicalSpace H'] {I : ModelWithCorners 𝕜 E H} {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_9} [TopologicalSpace M] [ChartedSpace H M] {M' : Type u_10} [TopologicalSpace M'] [ChartedSpace H' M'] (e : E ≃ₘ⟮𝓘(𝕜, E), 𝓘(𝕜, F)⟯ F) {f : M' → M}
Goal: Smooth I' (I.transDiffeomorph e) f ↔ Smooth I' I f
-- Diffeomorph.contMDiffWithinAt_transDiffeomorph_left
@[simp]
theorem contMDiffWithinAt_transDiffeomorph_left {f : M → M'} {x s} :
    ContMDiffWithinAt (I.transDiffeomorph e) I' n f s x ↔ ContMDiffWithinAt I I' n f s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type u_5} [TopologicalSpace H] {H' : Type u_6} [TopologicalSpace H'] {I : ModelWithCorners 𝕜 E H} {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_9} [TopologicalSpace M] [ChartedSpace H M] {M' : Type u_10} [TopologicalSpace M'] [ChartedSpace H' M'] {n : ℕ∞} (e : E ≃ₘ⟮𝓘(𝕜, E), 𝓘(𝕜, F)⟯ F) {f : M → M'} {x : M} {s : Set M}
Goal: ContMDiffWithinAt (I.transDiffeomorph e) I' n f s x ↔ ContMDiffWithinAt I I' n f s x
-- Diffeomorph.contMDiffAt_transDiffeomorph_left
@[simp]
theorem contMDiffAt_transDiffeomorph_left {f : M → M'} {x} :
    ContMDiffAt (I.transDiffeomorph e) I' n f x ↔ ContMDiffAt I I' n f x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type u_5} [TopologicalSpace H] {H' : Type u_6} [TopologicalSpace H'] {I : ModelWithCorners 𝕜 E H} {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_9} [TopologicalSpace M] [ChartedSpace H M] {M' : Type u_10} [TopologicalSpace M'] [ChartedSpace H' M'] {n : ℕ∞} (e : E ≃ₘ⟮𝓘(𝕜, E), 𝓘(𝕜, F)⟯ F) {f : M → M'} {x : M}
Goal: ContMDiffAt (I.transDiffeomorph e) I' n f x ↔ ContMDiffAt I I' n f x
-- Diffeomorph.contMDiffOn_transDiffeomorph_left
@[simp]
theorem contMDiffOn_transDiffeomorph_left {f : M → M'} {s} :
    ContMDiffOn (I.transDiffeomorph e) I' n f s ↔ ContMDiffOn I I' n f s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type u_5} [TopologicalSpace H] {H' : Type u_6} [TopologicalSpace H'] {I : ModelWithCorners 𝕜 E H} {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_9} [TopologicalSpace M] [ChartedSpace H M] {M' : Type u_10} [TopologicalSpace M'] [ChartedSpace H' M'] {n : ℕ∞} (e : E ≃ₘ⟮𝓘(𝕜, E), 𝓘(𝕜, F)⟯ F) {f : M → M'} {s : Set M}
Goal: ContMDiffOn (I.transDiffeomorph e) I' n f s ↔ ContMDiffOn I I' n f s
-- Diffeomorph.contMDiff_transDiffeomorph_left
@[simp]
theorem contMDiff_transDiffeomorph_left {f : M → M'} :
    ContMDiff (I.transDiffeomorph e) I' n f ↔ ContMDiff I I' n f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type u_5} [TopologicalSpace H] {H' : Type u_6} [TopologicalSpace H'] {I : ModelWithCorners 𝕜 E H} {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_9} [TopologicalSpace M] [ChartedSpace H M] {M' : Type u_10} [TopologicalSpace M'] [ChartedSpace H' M'] {n : ℕ∞} (e : E ≃ₘ⟮𝓘(𝕜, E), 𝓘(𝕜, F)⟯ F) {f : M → M'}
Goal: ContMDiff (I.transDiffeomorph e) I' n f ↔ ContMDiff I I' n f
-- Diffeomorph.smooth_transDiffeomorph_left
theorem smooth_transDiffeomorph_left {f : M → M'} :
    Smooth (I.transDiffeomorph e) I' f ↔ Smooth I I' f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type u_5} [TopologicalSpace H] {H' : Type u_6} [TopologicalSpace H'] {I : ModelWithCorners 𝕜 E H} {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_9} [TopologicalSpace M] [ChartedSpace H M] {M' : Type u_10} [TopologicalSpace M'] [ChartedSpace H' M'] (e : E ≃ₘ⟮𝓘(𝕜, E), 𝓘(𝕜, F)⟯ F) {f : M → M'}
Goal: Smooth (I.transDiffeomorph e) I' f ↔ Smooth I I' f
-- StructureGroupoid.LocalInvariantProp.section_spec
theorem StructureGroupoid.LocalInvariantProp.section_spec (hG : LocalInvariantProp G G' P)
    (U : (Opens (TopCat.of M))ᵒᵖ) (f : (hG.sheaf M M').val.obj U) : ChartedSpace.LiftProp P f 
Arguments: {H : Type u_1} [TopologicalSpace H] {H' : Type u_2} [TopologicalSpace H'] {G : StructureGroupoid H} {G' : StructureGroupoid H'} {P : (H → H') → Set H → H → Prop} (M : Type u) [TopologicalSpace M] [ChartedSpace H M] (M' : Type u) [TopologicalSpace M'] [ChartedSpace H' M'] (hG : G.LocalInvariantProp G' P) (U : (TopologicalSpace.Opens ↑(TopCat.of M))ᵒᵖ) (f : (StructureGroupoid.LocalInvariantProp.sheaf M M' hG).val.obj U)
Goal: ChartedSpace.LiftProp P ↑f
-- ModelWithCorners.hasMFDerivAt
protected theorem hasMFDerivAt {x} : HasMFDerivAt I 𝓘(𝕜, E) I x (ContinuousLinearMap.id _ _) 
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) {x : H}
Goal: HasMFDerivAt I 𝓘(𝕜, E) (↑I) x (ContinuousLinearMap.id 𝕜 (TangentSpace I x))
-- ModelWithCorners.hasMFDerivWithinAt
protected theorem hasMFDerivWithinAt {s x} :
    HasMFDerivWithinAt I 𝓘(𝕜, E) I s x (ContinuousLinearMap.id _ _) 
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) {s : Set H} {x : H}
Goal: HasMFDerivWithinAt I 𝓘(𝕜, E) (↑I) s x (ContinuousLinearMap.id 𝕜 (TangentSpace I x))
-- ModelWithCorners.mdifferentiableWithinAt
protected theorem mdifferentiableWithinAt {s x} : MDifferentiableWithinAt I 𝓘(𝕜, E) I s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {s : Set H} {x : H}
Goal: MDifferentiableWithinAt I 𝓘(𝕜, E) (↑I) s x
-- ModelWithCorners.mdifferentiableAt
protected theorem mdifferentiableAt {x} : MDifferentiableAt I 𝓘(𝕜, E) I x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {x : H}
Goal: MDifferentiableAt I 𝓘(𝕜, E) (↑I) x
-- ModelWithCorners.mdifferentiableOn
protected theorem mdifferentiableOn {s} : MDifferentiableOn I 𝓘(𝕜, E) I s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {s : Set H}
Goal: MDifferentiableOn I 𝓘(𝕜, E) (↑I) s
-- ModelWithCorners.mdifferentiable
protected theorem mdifferentiable : MDifferentiable I 𝓘(𝕜, E) 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)
Goal: MDifferentiable I 𝓘(𝕜, E) ↑I
-- ModelWithCorners.hasMFDerivWithinAt_symm
theorem hasMFDerivWithinAt_symm {x} (hx : x ∈ range I) :
    HasMFDerivWithinAt 𝓘(𝕜, E) I I.symm (range I) x (ContinuousLinearMap.id _ _) 
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) {x : E} (hx : x ∈ Set.range ↑I)
Goal: HasMFDerivWithinAt 𝓘(𝕜, E) I (↑I.symm) (Set.range ↑I) x (ContinuousLinearMap.id 𝕜 (TangentSpace 𝓘(𝕜, E) x))
-- ModelWithCorners.mdifferentiableOn_symm
theorem mdifferentiableOn_symm : MDifferentiableOn 𝓘(𝕜, E) I 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)
Goal: MDifferentiableOn 𝓘(𝕜, E) I (↑I.symm) (Set.range ↑I)
-- mdifferentiableAt_atlas
theorem mdifferentiableAt_atlas (h : e ∈ atlas H M) {x : M} (hx : x ∈ e.source) :
    MDifferentiableAt I I 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} (h : e ∈ atlas H M) {x : M} (hx : x ∈ e.source)
Goal: MDifferentiableAt I I (↑e) x
-- mdifferentiableOn_atlas
theorem mdifferentiableOn_atlas (h : e ∈ atlas H M) : MDifferentiableOn I I 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} (h : e ∈ atlas H M)
Goal: MDifferentiableOn I I (↑e) e.source
-- mdifferentiableAt_atlas_symm
theorem mdifferentiableAt_atlas_symm (h : e ∈ atlas H M) {x : H} (hx : x ∈ e.target) :
    MDifferentiableAt I I 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} (h : e ∈ atlas H M) {x : H} (hx : x ∈ e.target)
Goal: MDifferentiableAt I I (↑e.symm) x
-- mdifferentiableOn_atlas_symm
theorem mdifferentiableOn_atlas_symm (h : e ∈ atlas H M) : MDifferentiableOn I I 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} (h : e ∈ atlas H M)
Goal: MDifferentiableOn I I (↑e.symm) e.target
-- mdifferentiable_of_mem_atlas
theorem mdifferentiable_of_mem_atlas (h : e ∈ atlas H M) : e.MDifferentiable I I 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] {e : PartialHomeomorph M H} (h : e ∈ atlas H M)
Goal: PartialHomeomorph.MDifferentiable I I e
-- mdifferentiable_chart
theorem mdifferentiable_chart (x : M) : (chartAt H x).MDifferentiable I I 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] (x : M)
Goal: PartialHomeomorph.MDifferentiable I I (chartAt H x)
-- tangentMap_chart
theorem tangentMap_chart {p q : TangentBundle I M} (h : q.1 ∈ (chartAt H p.1).source) :
    tangentMap I I (chartAt H p.1) q =
      (TotalSpace.toProd _ _).symm
        ((chartAt (ModelProd H E) p : TangentBundle I M → ModelProd H E) q) 
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] {p q : TangentBundle I M} (h : q.proj ∈ (chartAt H p.proj).source)
Goal: tangentMap I I (↑(chartAt H p.proj)) q = (Bundle.TotalSpace.toProd H E).symm (↑(chartAt (ModelProd H E) p) q)
-- tangentMap_chart_symm
theorem tangentMap_chart_symm {p : TangentBundle I M} {q : TangentBundle I H}
    (h : q.1 ∈ (chartAt H p.1).target) :
    tangentMap I I (chartAt H p.1).symm q =
      (chartAt (ModelProd H E) p).symm (TotalSpace.toProd H E q) 
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] {p : TangentBundle I M} {q : TangentBundle I H} (h : q.proj ∈ (chartAt H p.proj).target)
Goal: tangentMap I I (↑(chartAt H p.proj).symm) q = ↑(chartAt (ModelProd H E) p).symm ((Bundle.TotalSpace.toProd H E) q)
-- mfderiv_chartAt_eq_tangentCoordChange
lemma mfderiv_chartAt_eq_tangentCoordChange {x y : M} (hsrc : x ∈ (chartAt H y).source) :
    mfderiv I I (chartAt H y) x = tangentCoordChange I x y 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] {x y : M} (hsrc : x ∈ (chartAt H y).source)
Goal: mfderiv I I (↑(chartAt H y)) x = tangentCoordChange I x y x
-- PartialHomeomorph.MDifferentiable.symm
nonrec theorem symm : e.symm.MDifferentiable I' I 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {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 : PartialHomeomorph M M'} (he : PartialHomeomorph.MDifferentiable I I' e)
Goal: PartialHomeomorph.MDifferentiable I' I e.symm
-- PartialHomeomorph.MDifferentiable.mdifferentiableAt
protected theorem mdifferentiableAt {x : M} (hx : x ∈ e.source) : MDifferentiableAt I I' 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] {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 : PartialHomeomorph M M'} (he : PartialHomeomorph.MDifferentiable I I' e) {x : M} (hx : x ∈ e.source)
Goal: MDifferentiableAt I I' (↑e) x
-- PartialHomeomorph.MDifferentiable.mdifferentiableAt_symm
theorem mdifferentiableAt_symm {x : M'} (hx : x ∈ e.target) : MDifferentiableAt I' I 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] {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 : PartialHomeomorph M M'} (he : PartialHomeomorph.MDifferentiable I I' e) {x : M'} (hx : x ∈ e.target)
Goal: MDifferentiableAt I' I (↑e.symm) x
-- PartialHomeomorph.MDifferentiable.symm_comp_deriv
theorem symm_comp_deriv {x : M} (hx : x ∈ e.source) :
    (mfderiv I' I e.symm (e x)).comp (mfderiv I I' e x) =
      ContinuousLinearMap.id 𝕜 (TangentSpace I x) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {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 : PartialHomeomorph M M'} (he : PartialHomeomorph.MDifferentiable I I' e) [SmoothManifoldWithCorners I M] [SmoothManifoldWithCorners I' M'] {x : M} (hx : x ∈ e.source)
Goal: (mfderiv I' I (↑e.symm) (↑e x)).comp (mfderiv I I' (↑e) x) = ContinuousLinearMap.id 𝕜 (TangentSpace I x)