File: Mathlib/Geometry/Manifold/MFDeriv/Atlas.lean

full_name: ModelWithCorners.hasMFDerivAt
code: protected theorem hasMFDerivAt {x} : HasMFDerivAt I 𝓘(𝕜, E) I x (ContinuousLinearMap.id _ _) 
goal: HasMFDerivAt I 𝓘(𝕜, E) (↑I) x (ContinuousLinearMap.id 𝕜 (TangentSpace I x))

full_name: ModelWithCorners.hasMFDerivWithinAt
code: protected theorem hasMFDerivWithinAt {s x} :
    HasMFDerivWithinAt I 𝓘(𝕜, E) I s x (ContinuousLinearMap.id _ _) 
goal: HasMFDerivWithinAt I 𝓘(𝕜, E) (↑I) s x (ContinuousLinearMap.id 𝕜 (TangentSpace I x))

full_name: ModelWithCorners.mdifferentiableWithinAt
code: protected theorem mdifferentiableWithinAt {s x} : MDifferentiableWithinAt I 𝓘(𝕜, E) I s x 
goal: MDifferentiableWithinAt I 𝓘(𝕜, E) (↑I) s x

full_name: ModelWithCorners.mdifferentiableAt
code: protected theorem mdifferentiableAt {x} : MDifferentiableAt I 𝓘(𝕜, E) I x 
goal: MDifferentiableAt I 𝓘(𝕜, E) (↑I) x

full_name: ModelWithCorners.mdifferentiableOn
code: protected theorem mdifferentiableOn {s} : MDifferentiableOn I 𝓘(𝕜, E) I s 
goal: MDifferentiableOn I 𝓘(𝕜, E) (↑I) s

full_name: ModelWithCorners.mdifferentiable
code: protected theorem mdifferentiable : MDifferentiable I 𝓘(𝕜, E) I 
goal: MDifferentiable I 𝓘(𝕜, E) ↑I

full_name: ModelWithCorners.hasMFDerivWithinAt_symm
code: theorem hasMFDerivWithinAt_symm {x} (hx : x ∈ range I) :
    HasMFDerivWithinAt 𝓘(𝕜, E) I I.symm (range I) x (ContinuousLinearMap.id _ _) 
goal: HasMFDerivWithinAt 𝓘(𝕜, E) I (↑I.symm) (Set.range ↑I) x (ContinuousLinearMap.id 𝕜 (TangentSpace 𝓘(𝕜, E) x))

full_name: ModelWithCorners.mdifferentiableOn_symm
code: theorem mdifferentiableOn_symm : MDifferentiableOn 𝓘(𝕜, E) I I.symm (range I) 
goal: MDifferentiableOn 𝓘(𝕜, E) I (↑I.symm) (Set.range ↑I)

full_name: mdifferentiableAt_atlas
code: theorem mdifferentiableAt_atlas (h : e ∈ atlas H M) {x : M} (hx : x ∈ e.source) :
    MDifferentiableAt I I e x 
goal: MDifferentiableAt I I (↑e) x

full_name: mdifferentiableOn_atlas
code: theorem mdifferentiableOn_atlas (h : e ∈ atlas H M) : MDifferentiableOn I I e e.source 
goal: MDifferentiableOn I I (↑e) e.source

full_name: mdifferentiableAt_atlas_symm
code: theorem mdifferentiableAt_atlas_symm (h : e ∈ atlas H M) {x : H} (hx : x ∈ e.target) :
    MDifferentiableAt I I e.symm x 
goal: MDifferentiableAt I I (↑e.symm) x

full_name: mdifferentiableOn_atlas_symm
code: theorem mdifferentiableOn_atlas_symm (h : e ∈ atlas H M) : MDifferentiableOn I I e.symm e.target 
goal: MDifferentiableOn I I (↑e.symm) e.target

full_name: mdifferentiable_of_mem_atlas
code: theorem mdifferentiable_of_mem_atlas (h : e ∈ atlas H M) : e.MDifferentiable I I 
goal: PartialHomeomorph.MDifferentiable I I e

full_name: mdifferentiable_chart
code: theorem mdifferentiable_chart (x : M) : (chartAt H x).MDifferentiable I I 
goal: PartialHomeomorph.MDifferentiable I I (chartAt H x)

full_name: tangentMap_chart
code: /-- The derivative of the chart at a base point is the chart of the tangent bundle, composed with
the identification between the tangent bundle of the model space and the product space. -/
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) 
goal: tangentMap I I (↑(chartAt H p.proj)) q = (Bundle.TotalSpace.toProd H E).symm (↑(chartAt (ModelProd H E) p) q)

full_name: tangentMap_chart_symm
code: /-- The derivative of the inverse of the chart at a base point is the inverse of the chart of the
tangent bundle, composed with the identification between the tangent bundle of the model space and
the product space. -/
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) 
goal: tangentMap I I (↑(chartAt H p.proj).symm) q = ↑(chartAt (ModelProd H E) p).symm ((Bundle.TotalSpace.toProd H E) q)

full_name: mfderiv_chartAt_eq_tangentCoordChange
code: 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 
goal: mfderiv I I (↑(chartAt H y)) x = tangentCoordChange I x y x

full_name: PartialHomeomorph.MDifferentiable.symm
code: nonrec theorem symm : e.symm.MDifferentiable I' I 
goal: PartialHomeomorph.MDifferentiable I' I e.symm

full_name: PartialHomeomorph.MDifferentiable.mdifferentiableAt
code: protected theorem mdifferentiableAt {x : M} (hx : x ∈ e.source) : MDifferentiableAt I I' e x 
goal: MDifferentiableAt I I' (↑e) x

full_name: PartialHomeomorph.MDifferentiable.mdifferentiableAt_symm
code: theorem mdifferentiableAt_symm {x : M'} (hx : x ∈ e.target) : MDifferentiableAt I' I e.symm x 
goal: MDifferentiableAt I' I (↑e.symm) x

full_name: PartialHomeomorph.MDifferentiable.symm_comp_deriv
code: 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) 
goal: (mfderiv I' I (↑e.symm) (↑e x)).comp (mfderiv I I' (↑e) x) = ContinuousLinearMap.id 𝕜 (TangentSpace I x)

full_name: PartialHomeomorph.MDifferentiable.comp_symm_deriv
code: theorem comp_symm_deriv {x : M'} (hx : x ∈ e.target) :
    (mfderiv I I' e (e.symm x)).comp (mfderiv I' I e.symm x) =
      ContinuousLinearMap.id 𝕜 (TangentSpace I' x) 
goal: (mfderiv I I' (↑e) (↑e.symm x)).comp (mfderiv I' I (↑e.symm) x) = ContinuousLinearMap.id 𝕜 (TangentSpace I' x)

full_name: PartialHomeomorph.MDifferentiable.mfderiv
code: /-- The derivative of a differentiable partial homeomorphism, as a continuous linear equivalence
between the tangent spaces at `x` and `e x`. -/
protected def mfderiv {x : M} (hx : x ∈ e.source) : TangentSpace I x ≃L[𝕜] TangentSpace I' (e x) 
goal: TangentSpace I x ≃L[𝕜] TangentSpace I' (↑e x)

full_name: PartialHomeomorph.MDifferentiable.mfderiv_bijective
code: theorem mfderiv_bijective {x : M} (hx : x ∈ e.source) : Function.Bijective (mfderiv I I' e x) 
goal: Function.Bijective ⇑(mfderiv I I' (↑e) x)

full_name: PartialHomeomorph.MDifferentiable.mfderiv_injective
code: theorem mfderiv_injective {x : M} (hx : x ∈ e.source) : Function.Injective (mfderiv I I' e x) 
goal: Function.Injective ⇑(mfderiv I I' (↑e) x)

full_name: PartialHomeomorph.MDifferentiable.mfderiv_surjective
code: theorem mfderiv_surjective {x : M} (hx : x ∈ e.source) : Function.Surjective (mfderiv I I' e x) 
goal: Function.Surjective ⇑(mfderiv I I' (↑e) x)

full_name: PartialHomeomorph.MDifferentiable.ker_mfderiv_eq_bot
code: theorem ker_mfderiv_eq_bot {x : M} (hx : x ∈ e.source) : LinearMap.ker (mfderiv I I' e x) = ⊥ 
goal: LinearMap.ker (mfderiv I I' (↑e) x) = ⊥

full_name: PartialHomeomorph.MDifferentiable.range_mfderiv_eq_top
code: theorem range_mfderiv_eq_top {x : M} (hx : x ∈ e.source) : LinearMap.range (mfderiv I I' e x) = ⊤ 
goal: LinearMap.range (mfderiv I I' (↑e) x) = ⊤

full_name: PartialHomeomorph.MDifferentiable.range_mfderiv_eq_univ
code: theorem range_mfderiv_eq_univ {x : M} (hx : x ∈ e.source) : range (mfderiv I I' e x) = univ 
goal: Set.range ⇑(mfderiv I I' (↑e) x) = Set.univ

full_name: PartialHomeomorph.MDifferentiable.trans
code: theorem trans (he' : e'.MDifferentiable I' I'') : (e.trans e').MDifferentiable I I'' 
goal: PartialHomeomorph.MDifferentiable I I'' (e ≫ₕ e')

full_name: hasMFDerivAt_extChartAt
code: theorem hasMFDerivAt_extChartAt (h : y ∈ (chartAt H x).source) :
    HasMFDerivAt I 𝓘(𝕜, E) (extChartAt I x) y (mfderiv I I (chartAt H x) y : _) 
goal: HasMFDerivAt I 𝓘(𝕜, E) (↑(extChartAt I x)) y (mfderiv I I (↑(chartAt H x)) y)

full_name: hasMFDerivWithinAt_extChartAt
code: theorem hasMFDerivWithinAt_extChartAt (h : y ∈ (chartAt H x).source) :
    HasMFDerivWithinAt I 𝓘(𝕜, E) (extChartAt I x) s y (mfderiv I I (chartAt H x) y : _) 
goal: HasMFDerivWithinAt I 𝓘(𝕜, E) (↑(extChartAt I x)) s y (mfderiv I I (↑(chartAt H x)) y)

full_name: mdifferentiableAt_extChartAt
code: theorem mdifferentiableAt_extChartAt (h : y ∈ (chartAt H x).source) :
    MDifferentiableAt I 𝓘(𝕜, E) (extChartAt I x) y 
goal: MDifferentiableAt I 𝓘(𝕜, E) (↑(extChartAt I x)) y

full_name: mdifferentiableOn_extChartAt
code: theorem mdifferentiableOn_extChartAt :
    MDifferentiableOn I 𝓘(𝕜, E) (extChartAt I x) (chartAt H x).source 
goal: MDifferentiableOn I 𝓘(𝕜, E) (↑(extChartAt I x)) (chartAt H x).source