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Found 106 declarations mentioning MDifferentiableOn.

• MDifferentiableOn 📋 Mathlib.Geometry.Manifold.MFDeriv.Defs
  {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') (s : Set M) : Prop
• mdifferentiableOn_empty 📋 Mathlib.Geometry.Manifold.MFDeriv.Basic
  {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'} : MDiff[∅] f
• MDifferentiableOn.continuousOn 📋 Mathlib.Geometry.Manifold.MFDeriv.Basic
  {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'} {s : Set M} (h : MDiff[s] f) : ContinuousOn f s
• mdifferentiableOn_univ 📋 Mathlib.Geometry.Manifold.MFDeriv.Basic
  {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'} : MDiff[Set.univ] f ↔ MDiff f
• MDifferentiable.mdifferentiableOn 📋 Mathlib.Geometry.Manifold.MFDeriv.Basic
  {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'} {s : Set M} (h : MDiff f) : MDiff[s] f
• MDifferentiableOn.mono 📋 Mathlib.Geometry.Manifold.MFDeriv.Basic
  {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'} {s t : Set M} (h : MDiff[t] f) (st : s ⊆ t) : MDiff[s] f
• MDifferentiableOn.mdifferentiableAt 📋 Mathlib.Geometry.Manifold.MFDeriv.Basic
  {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} {s : Set M} (h : MDiff[s] f) (hx : s ∈ nhds x) : MDiffAt f x
• MDifferentiableOn.iUnion_of_isOpen 📋 Mathlib.Geometry.Manifold.MFDeriv.Basic
  {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'} {ι : Type u_11} {s : ι → Set M} (hf : ∀ (i : ι), MDiff[s i] f) (hs : ∀ (i : ι), IsOpen (s i)) : MDiff[⋃ i, s i] f
• mdifferentiableOn_iUnion_iff_of_isOpen 📋 Mathlib.Geometry.Manifold.MFDeriv.Basic
  {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'} {ι : Type u_11} {s : ι → Set M} (hs : ∀ (i : ι), IsOpen (s i)) : MDiff[⋃ i, s i] f ↔ ∀ (i : ι), MDiff[s i] f
• ContMDiffOn.mdifferentiableOn 📋 Mathlib.Geometry.Manifold.MFDeriv.Basic
  {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'} {s : Set M} {n : WithTop ℕ∞} (hf : ContMDiffOn I I' n f s) (hn : n ≠ 0) : MDiff[s] f
• MDifferentiableOn.congr 📋 Mathlib.Geometry.Manifold.MFDeriv.Basic
  {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} (h : MDiff[s] f) (h₁ : ∀ y ∈ s, f₁ y = f y) : MDiff[s] f₁
• mdifferentiableOn_congr 📋 Mathlib.Geometry.Manifold.MFDeriv.Basic
  {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} (h₁ : ∀ y ∈ s, f₁ y = f y) : MDiff[s] f₁ ↔ MDiff[s] f
• mdifferentiable_of_mdifferentiableOn_iUnion_of_isOpen 📋 Mathlib.Geometry.Manifold.MFDeriv.Basic
  {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'} {ι : Type u_11} {s : ι → Set M} (hf : ∀ (i : ι), MDiff[s i] f) (hs : ∀ (i : ι), IsOpen (s i)) (hs' : ⋃ i, s i = Set.univ) : MDiff f
• MDifferentiableOn.congr_mono 📋 Mathlib.Geometry.Manifold.MFDeriv.Basic
  {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 t : Set M} (h : MDiff[s] f) (h' : ∀ x ∈ t, f₁ x = f x) (h₁ : t ⊆ s) : MDiff[t] f₁
• mdifferentiableOn_of_locally_mdifferentiableOn 📋 Mathlib.Geometry.Manifold.MFDeriv.Basic
  {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'} {s : Set M} (h : ∀ x ∈ s, ∃ u, IsOpen u ∧ x ∈ u ∧ MDiff[s ∩ u] f) : MDiff[s] f
• MDifferentiableOn.union_of_isOpen 📋 Mathlib.Geometry.Manifold.MFDeriv.Basic
  {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'} {s t : Set M} (hf : MDiff[s] f) (hf' : MDiff[t] f) (hs : IsOpen s) (ht : IsOpen t) : MDiff[s ∪ t] f
• mdifferentiableOn_union_iff_of_isOpen 📋 Mathlib.Geometry.Manifold.MFDeriv.Basic
  {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'} {s t : Set M} (hs : IsOpen s) (ht : IsOpen t) : MDiff[s ∪ t] f ↔ MDiff[s] f ∧ MDiff[t] f
• mdifferentiable_of_mdifferentiableOn_union_of_isOpen 📋 Mathlib.Geometry.Manifold.MFDeriv.Basic
  {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'} {s t : Set M} (hf : MDiff[s] f) (hf' : MDiff[t] f) (hst : s ∪ t = Set.univ) (hs : IsOpen s) (ht : IsOpen t) : MDiff f
• MDifferentiable.comp_mdifferentiableOn 📋 Mathlib.Geometry.Manifold.MFDeriv.Basic
  {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''} (hg : MDiff g) (hf : MDiff[s] f) : MDiff[s] (g ∘ f)
• MDifferentiableOn.comp 📋 Mathlib.Geometry.Manifold.MFDeriv.Basic
  {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'} (hg : MDiff[u] g) (hf : MDiff[s] f) (st : s ⊆ f ⁻¹' u) : MDiff[s] (g ∘ f)
• mdifferentiable_iff_target 📋 Mathlib.Geometry.Manifold.MFDeriv.Basic
  {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'} [IsManifold I 1 M] [IsManifold I' 1 M'] : MDiff f ↔ Continuous f ∧ ∀ (y : M'), MDiff[f ⁻¹' (extChartAt I' y).source] (↑(extChartAt I' y) ∘ f)
• mdifferentiableOn_iff_target 📋 Mathlib.Geometry.Manifold.MFDeriv.Basic
  {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'} {s : Set M} [IsManifold I 1 M] [IsManifold I' 1 M'] : MDiff[s] f ↔ ContinuousOn f s ∧ ∀ (y : M'), MDiff[s ∩ f ⁻¹' (extChartAt I' y).source] (↑(extChartAt I' y) ∘ f)
• mdifferentiableOn_iff_of_subset_source' 📋 Mathlib.Geometry.Manifold.MFDeriv.Basic
  {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'} {s : Set M} [IsManifold I 1 M] [IsManifold I' 1 M'] {x : M} {y : M'} (hs : s ⊆ (extChartAt I x).source) (h2s : Set.MapsTo f s (extChartAt I' y).source) : MDiff[s] f ↔ DifferentiableOn 𝕜 (↑(extChartAt I' y) ∘ f ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) '' s)
• mdifferentiableOn_iff 📋 Mathlib.Geometry.Manifold.MFDeriv.Basic
  {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'} {s : Set M} [IsManifold I 1 M] [IsManifold I' 1 M'] : MDiff[s] f ↔ ContinuousOn f s ∧ ∀ (x : M) (y : M'), DifferentiableOn 𝕜 (↑(extChartAt I' y) ∘ f ∘ ↑(extChartAt I x).symm) ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' (s ∩ f ⁻¹' (extChartAt I' y).source))
• mdifferentiableOn_iff_of_subset_source 📋 Mathlib.Geometry.Manifold.MFDeriv.Basic
  {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'} {s : Set M} [IsManifold I 1 M] [IsManifold I' 1 M'] {x : M} {y : M'} (hs : s ⊆ (chartAt H x).source) (h2s : Set.MapsTo f s (chartAt H' y).source) : MDiff[s] f ↔ ContinuousOn f s ∧ DifferentiableOn 𝕜 (↑(extChartAt I' y) ∘ f ∘ ↑(extChartAt I x).symm) (↑(extChartAt I x) '' s)
• mdifferentiableOn_iff_of_mem_maximalAtlas' 📋 Mathlib.Geometry.Manifold.MFDeriv.Basic
  {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'} {s : Set M} {e : OpenPartialHomeomorph M H} {e' : OpenPartialHomeomorph M' H'} (he : e ∈ IsManifold.maximalAtlas I 1 M) (he' : e' ∈ IsManifold.maximalAtlas I' 1 M') (hs : s ⊆ e.source) (h2s : Set.MapsTo f s e'.source) : MDiff[s] f ↔ DifferentiableOn 𝕜 (↑(e'.extend I') ∘ f ∘ ↑(e.extend I).symm) (↑(e.extend I) '' s)
• mdifferentiableOn_iff_of_mem_maximalAtlas 📋 Mathlib.Geometry.Manifold.MFDeriv.Basic
  {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'} {s : Set M} {e : OpenPartialHomeomorph M H} {e' : OpenPartialHomeomorph M' H'} (he : e ∈ IsManifold.maximalAtlas I 1 M) (he' : e' ∈ IsManifold.maximalAtlas I' 1 M') (hs : s ⊆ e.source) (h2s : Set.MapsTo f s e'.source) : MDiff[s] f ↔ ContinuousOn f s ∧ DifferentiableOn 𝕜 (↑(e'.extend I') ∘ f ∘ ↑(e.extend I).symm) (↑(e.extend I) '' s)
• DifferentiableOn.mdifferentiableOn 📋 Mathlib.Geometry.Manifold.MFDeriv.FDeriv
  {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 : E → E'} {s : Set E} : DifferentiableOn 𝕜 f s → MDiff[s] f
• MDifferentiableOn.differentiableOn 📋 Mathlib.Geometry.Manifold.MFDeriv.FDeriv
  {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 : E → E'} {s : Set E} : MDiff[s] f → DifferentiableOn 𝕜 f s
• mdifferentiableOn_iff_differentiableOn 📋 Mathlib.Geometry.Manifold.MFDeriv.FDeriv
  {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 : E → E'} {s : Set E} : MDiff[s] f ↔ DifferentiableOn 𝕜 f s
• mdifferentiableOn_id 📋 Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions
  {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] {s : Set M} : MDiff[s] id
• mdifferentiableOn_const 📋 Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions
  {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'] {s : Set M} {c : M'} : MDiff[s] fun x => c
• mdifferentiableOn_fst 📋 Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions
  {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'] {s : Set (M × M')} : MDiff[s] Prod.fst
• mdifferentiableOn_snd 📋 Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions
  {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'] {s : Set (M × M')} : MDiff[s] Prod.snd
• MDifferentiableOn.neg 📋 Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions
  {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'] {f : M → E'} {s : Set M} (hf : MDiff[s] f) : MDiff[s] (-f)
• MDifferentiableOn.sum 📋 Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions
  {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'] {s : Set M} {ι : Type} {t : Finset ι} {f : ι → M → E'} (hf : ∀ i ∈ t, MDiff[s] (f i)) : MDiff[s] (∑ i ∈ t, f i)
• MDifferentiableOn.prodMk 📋 Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions
  {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''] {s : Set M} {f : M → M'} {g : M → M''} (hf : MDiff[s] f) (hg : MDiff[s] g) : MDiff[s] fun x => (f x, g x)
• MDifferentiableOn.sub 📋 Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions
  {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'] {s : Set M} {f g : M → E'} (hf : MDiff[s] f) (hg : MDiff[s] g) : MDiff[s] (f - g)
• MDifferentiableOn.add 📋 Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions
  {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'] {f g : M → E'} {s : Set M} (hf : MDiff[s] f) (hg : MDiff[s] g) : MDiff[s] (f + g)
• mdifferentiableOn_prod_iff 📋 Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions
  {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' : Type u_14} [NormedAddCommGroup F'] [NormedSpace 𝕜 F'] {G' : Type u_15} [TopologicalSpace G'] {J' : ModelWithCorners 𝕜 F' G'} {N' : Type u_16} [TopologicalSpace N'] [ChartedSpace G' N'] {s : Set M} (f : M → M' × N') : MDiff[s] f ↔ MDiff[s] (Prod.fst ∘ f) ∧ MDiff[s] (Prod.snd ∘ f)
• MDifferentiableOn.pow 📋 Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions
  {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] {s : Set M} {F' : Type u_17} [NormedRing F'] [NormedAlgebra 𝕜 F'] {p : M → F'} (hp : MDiff[s] p) (n : ℕ) : MDiff[s] (p ^ n)
• MDifferentiableOn.const_smul 📋 Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions
  {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'] {s : Set M} {f : M → E'} (a : 𝕜) (hf : MDiff[s] f) : MDiff[s] (a • f)
• MDifferentiableOn.prod 📋 Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions
  {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] {s : Set M} {F' : Type u_17} [NormedCommRing F'] [NormedAlgebra 𝕜 F'] {ι : Type} {t : Finset ι} {f : ι → M → F'} (hf : ∀ i ∈ t, MDiff[s] (f i)) : MDiff[s] (∏ i ∈ t, f i)
• MDifferentiableOn.inv 📋 Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions
  {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] {s : Set M} {F' : Type u_17} [NormedDivisionRing F'] [NormedAlgebra 𝕜 F'] {p : M → F'} (hp : MDiff[s] p) (hp_ne : ∀ z ∈ s, p z ≠ 0) : MDiff[s] p⁻¹
• MDifferentiableOn.prodMk_space 📋 Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions
  {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'] {E'' : Type u_8} [NormedAddCommGroup E''] [NormedSpace 𝕜 E''] {s : Set M} {f : M → E'} {g : M → E''} (hf : MDiff[s] f) (hg : MDiff[s] g) : MDiff[s] fun x => (f x, g x)
• mdifferentiableOn_prod_module_iff 📋 Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions
  {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] {F₁ : Type u_17} [NormedAddCommGroup F₁] [NormedSpace 𝕜 F₁] {F₂ : Type u_18} [NormedAddCommGroup F₂] [NormedSpace 𝕜 F₂] {s : Set M} (f : M → F₁ × F₂) : MDiff[s] f ↔ MDiff[s] (Prod.fst ∘ f) ∧ MDiff[s] (Prod.snd ∘ f)
• MDifferentiableOn.prodMap 📋 Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions
  {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 : Type u_11} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_12} [TopologicalSpace G] {J : ModelWithCorners 𝕜 F G} {N : Type u_13} [TopologicalSpace N] [ChartedSpace G N] {F' : Type u_14} [NormedAddCommGroup F'] [NormedSpace 𝕜 F'] {G' : Type u_15} [TopologicalSpace G'] {J' : ModelWithCorners 𝕜 F' G'} {N' : Type u_16} [TopologicalSpace N'] [ChartedSpace G' N'] {s : Set M} {f : M → M'} {g : N → N'} {r : Set N} (hf : MDiff[s] f) (hg : MDiff[r] g) : MDiff[s ×ˢ r] (Prod.map f g)
• MDifferentiableOn.mul 📋 Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions
  {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] {s : Set M} {F' : Type u_17} [NormedRing F'] [NormedAlgebra 𝕜 F'] {p q : M → F'} (hp : MDiff[s] p) (hq : MDiff[s] q) : MDiff[s] (p * q)
• ContinuousLinearMap.mdifferentiableOn 📋 Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions
  {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] (f : E →L[𝕜] E') {s : Set E} : MDiff[s] ⇑f
• MDifferentiableOn.div 📋 Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions
  {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] {s : Set M} {F' : Type u_17} [NormedDivisionRing F'] [NormedAlgebra 𝕜 F'] {p q : M → F'} (hp : MDiff[s] p) (hq : MDiff[s] q) (hq_ne : ∀ z ∈ s, q z ≠ 0) : MDiff[s] (p / q)
• ContinuousLinearEquiv.mdifferentiableOn 📋 Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions
  {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] (f : E ≃L[𝕜] E') {s : Set E} : MDiff[s] ⇑f
• ModelWithCorners.mdifferentiableOn 📋 Mathlib.Geometry.Manifold.MFDeriv.Atlas
  {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} : MDiff[s] ↑I
• ModelWithCorners.mdifferentiableOn_symm 📋 Mathlib.Geometry.Manifold.MFDeriv.Atlas
  {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) : MDiff[Set.range ↑I] ↑I.symm
• mdifferentiableOn_atlas 📋 Mathlib.Geometry.Manifold.MFDeriv.Atlas
  {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 : OpenPartialHomeomorph M H} [IsManifold I 1 M] (h : e ∈ atlas H M) : MDiff[e.source] ↑e
• mdifferentiableOn_atlas_symm 📋 Mathlib.Geometry.Manifold.MFDeriv.Atlas
  {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 : OpenPartialHomeomorph M H} [IsManifold I 1 M] (h : e ∈ atlas H M) : MDiff[e.target] ↑e.symm
• mdifferentiableOn_extChartAt_symm 📋 Mathlib.Geometry.Manifold.MFDeriv.Atlas
  {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] [IsManifold I 1 M] {x : M} : MDiff[(extChartAt I x).target] ↑(extChartAt I x).symm
• mdifferentiableOn_extChartAt 📋 Mathlib.Geometry.Manifold.MFDeriv.Atlas
  {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] [IsManifold I 1 M] {x : M} : MDiff[(chartAt H x).source] ↑(extChartAt I x)
• OpenPartialHomeomorph.mdifferentiableOn_extend 📋 Mathlib.Geometry.Manifold.MFDeriv.Atlas
  {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 : OpenPartialHomeomorph M H} (he : e ∈ IsManifold.maximalAtlas I 1 M) : MDiff[e.source] ↑(e.extend I)
• mdifferentiableOn_extend_symm 📋 Mathlib.Geometry.Manifold.MFDeriv.Atlas
  {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 : OpenPartialHomeomorph M H} (he : e ∈ IsManifold.maximalAtlas I 1 M) : MDiff[(e.extend I).target] ↑(e.extend I).symm
• UniqueMDiffOn.image_denseRange 📋 Mathlib.Geometry.Manifold.MFDeriv.UniqueDifferential
  {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'] {s : Set M} (hs : UniqueMDiff[s]) {f : M → M'} (hf : MDiff[s] f) (hd : ∀ x ∈ s, DenseRange ⇑(mfderiv[s] f x)) : UniqueMDiff[f '' s]
• Diffeomorph.mdifferentiableOn 📋 Mathlib.Geometry.Manifold.Diffeomorph
  {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] {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 : WithTop ℕ∞} (h : Diffeomorph I I' M M' n) (s : Set M) (hn : n ≠ 0) : MDiff[s] ⇑h
• IsLocalDiffeomorphOn.mdifferentiableOn 📋 Mathlib.Geometry.Manifold.LocalDiffeomorph
  {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H₁ : Type u_5} [TopologicalSpace H₁] {H₂ : Type u_6} [TopologicalSpace H₂] {I : ModelWithCorners 𝕜 E H₁} {J : ModelWithCorners 𝕜 F H₂} {M : Type u_8} [TopologicalSpace M] [ChartedSpace H₁ M] {N : Type u_9} [TopologicalSpace N] [ChartedSpace H₂ N] {n : WithTop ℕ∞} {f : M → N} {s : Set M} (hf : IsLocalDiffeomorphOn I J n f s) (hn : n ≠ 0) : MDiff[s] f
• PartialDiffeomorph.mdifferentiableOn 📋 Mathlib.Geometry.Manifold.LocalDiffeomorph
  {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H₁ : Type u_5} [TopologicalSpace H₁] {H₂ : Type u_6} [TopologicalSpace H₂] {I : ModelWithCorners 𝕜 E H₁} {J : ModelWithCorners 𝕜 F H₂} {M : Type u_8} [TopologicalSpace M] [ChartedSpace H₁ M] {N : Type u_9} [TopologicalSpace N] [ChartedSpace H₂ N] {n : WithTop ℕ∞} (Φ : PartialDiffeomorph I J M N n) (hn : n ≠ 0) : MDiff[Φ.source] ↑Φ.toPartialEquiv
• MDifferentiableOn.apply_eq_of_isPreconnected_isCompact_isOpen 📋 Mathlib.Geometry.Manifold.Complex
  {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace ℂ F] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners ℂ E H} [I.Boundaryless] {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] [IsManifold I 1 M] {f : M → F} {U : Set M} {a b : M} (hd : MDiff[U] f) (hpc : IsPreconnected U) (hc : IsCompact U) (ho : IsOpen U) (ha : a ∈ U) (hb : b ∈ U) : f a = f b
• MDifferentiableOn.eqOn_of_isPreconnected_of_isMaxOn_norm 📋 Mathlib.Geometry.Manifold.Complex
  {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace ℂ F] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners ℂ E H} [I.Boundaryless] {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] [IsManifold I 1 M] [StrictConvexSpace ℝ F] {f : M → F} {U : Set M} {c : M} (hd : MDiff[U] f) (hc : IsPreconnected U) (ho : IsOpen U) (hcU : c ∈ U) (hm : IsMaxOn (norm ∘ f) U c) : Set.EqOn f (Function.const M (f c)) U
• MDifferentiableOn.norm_eqOn_of_isPreconnected_of_isMaxOn 📋 Mathlib.Geometry.Manifold.Complex
  {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace ℂ F] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners ℂ E H} [I.Boundaryless] {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] [IsManifold I 1 M] {f : M → F} {U : Set M} {c : M} (hd : MDiff[U] f) (hc : IsPreconnected U) (ho : IsOpen U) (hcU : c ∈ U) (hm : IsMaxOn (norm ∘ f) U c) : Set.EqOn (norm ∘ f) (Function.const M ‖f c‖) U
• MDifferentiableOn.fun_smul 📋 Mathlib.Geometry.Manifold.MFDeriv.NormedSpace
  {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] {s : Set M} {V : Type u_18} [NormedAddCommGroup V] [NormedSpace 𝕜 V] {f : M → 𝕜} {g : M → V} (hf : MDiff[s] f) (hg : MDiff[s] g) : MDiff[s] fun i => f i • g i
• MDifferentiableOn.smul 📋 Mathlib.Geometry.Manifold.MFDeriv.NormedSpace
  {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] {s : Set M} {V : Type u_18} [NormedAddCommGroup V] [NormedSpace 𝕜 V] {f : M → 𝕜} {g : M → V} (hf : MDiff[s] f) (hg : MDiff[s] g) : MDiff[s] (f • g)
• MDifferentiableOn.clm_apply 📋 Mathlib.Geometry.Manifold.MFDeriv.NormedSpace
  {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] {F₁ : Type u_14} [NormedAddCommGroup F₁] [NormedSpace 𝕜 F₁] {F₂ : Type u_15} [NormedAddCommGroup F₂] [NormedSpace 𝕜 F₂] {g : M → F₁ →L[𝕜] F₂} {f : M → F₁} {s : Set M} (hg : MDiff[s] g) (hf : MDiff[s] f) : MDiff[s] fun x => (g x) (f x)
• MDifferentiableOn.clm_comp 📋 Mathlib.Geometry.Manifold.MFDeriv.NormedSpace
  {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] {F₁ : Type u_14} [NormedAddCommGroup F₁] [NormedSpace 𝕜 F₁] {F₂ : Type u_15} [NormedAddCommGroup F₂] [NormedSpace 𝕜 F₂] {F₃ : Type u_16} [NormedAddCommGroup F₃] [NormedSpace 𝕜 F₃] {g : M → F₁ →L[𝕜] F₃} {f : M → F₂ →L[𝕜] F₁} {s : Set M} (hg : MDiff[s] g) (hf : MDiff[s] f) : MDiff[s] fun x => g x ∘SL f x
• MDifferentiableOn.clm_prodMap 📋 Mathlib.Geometry.Manifold.MFDeriv.NormedSpace
  {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] {F₁ : Type u_14} [NormedAddCommGroup F₁] [NormedSpace 𝕜 F₁] {F₂ : Type u_15} [NormedAddCommGroup F₂] [NormedSpace 𝕜 F₂] {F₃ : Type u_16} [NormedAddCommGroup F₃] [NormedSpace 𝕜 F₃] {F₄ : Type u_17} [NormedAddCommGroup F₄] [NormedSpace 𝕜 F₄] {g : M → F₁ →L[𝕜] F₃} {f : M → F₂ →L[𝕜] F₄} {s : Set M} (hg : MDiff[s] g) (hf : MDiff[s] f) : MDiff[s] fun x => (g x).prodMap (f x)
• MDifferentiableOn.clm_precomp 📋 Mathlib.Geometry.Manifold.MFDeriv.NormedSpace
  {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] {F₁ : Type u_14} [NormedAddCommGroup F₁] [NormedSpace 𝕜 F₁] {F₂ : Type u_15} [NormedAddCommGroup F₂] [NormedSpace 𝕜 F₂] {F₃ : Type u_16} [NormedAddCommGroup F₃] [NormedSpace 𝕜 F₃] {f : M → F₁ →L[𝕜] F₂} {s : Set M} (hf : MDiff[s] f) : MDiff[s] fun y => ContinuousLinearMap.precomp F₃ (f y)
• MDifferentiableOn.clm_postcomp 📋 Mathlib.Geometry.Manifold.MFDeriv.NormedSpace
  {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] {F₁ : Type u_14} [NormedAddCommGroup F₁] [NormedSpace 𝕜 F₁] {F₂ : Type u_15} [NormedAddCommGroup F₂] [NormedSpace 𝕜 F₂] {F₃ : Type u_16} [NormedAddCommGroup F₃] [NormedSpace 𝕜 F₃] {f : M → F₂ →L[𝕜] F₃} {s : Set M} (hf : MDiff[s] f) : MDiff[s] fun y => ContinuousLinearMap.postcomp F₁ (f y)
• MDifferentiableOn.cle_arrowCongr 📋 Mathlib.Geometry.Manifold.MFDeriv.NormedSpace
  {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] {F₁ : Type u_14} [NormedAddCommGroup F₁] [NormedSpace 𝕜 F₁] {F₂ : Type u_15} [NormedAddCommGroup F₂] [NormedSpace 𝕜 F₂] {F₃ : Type u_16} [NormedAddCommGroup F₃] [NormedSpace 𝕜 F₃] {F₄ : Type u_17} [NormedAddCommGroup F₄] [NormedSpace 𝕜 F₄] {f : M → F₁ ≃L[𝕜] F₂} {g : M → F₃ ≃L[𝕜] F₄} {s : Set M} (hf : MDiff[s] fun x => ↑(f x).symm) (hg : MDiff[s] fun x => ↑(g x)) : MDiff[s] fun y => ↑((f y).arrowCongr (g y))
• Bundle.mdifferentiableOn_proj 📋 Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable
  {𝕜 : Type u_1} {B : Type u_2} {F : Type u_3} (E : B → Type u_5) [NontriviallyNormedField 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] {EB : Type u_6} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {HB : Type u_7} [TopologicalSpace HB] {IB : ModelWithCorners 𝕜 EB HB} [TopologicalSpace B] [ChartedSpace HB B] [FiberBundle F E] {s : Set (Bundle.TotalSpace F E)} : MDiff[s] Bundle.TotalSpace.proj
• Bundle.mdifferentiableOn_section_of_subsingleton 📋 Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable
  {𝕜 : Type u_1} {B : Type u_2} {F : Type u_3} (E : B → Type u_5) [NontriviallyNormedField 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] {EB : Type u_6} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {HB : Type u_7} [TopologicalSpace HB] {IB : ModelWithCorners 𝕜 EB HB} [TopologicalSpace B] [ChartedSpace HB B] [FiberBundle F E] {s : (x : B) → E x} {u : Set B} [Subsingleton F] : MDiff[u] fun x => ⟨x, s x⟩
• Bundle.mdifferentiableOn_zeroSection 📋 Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable
  (𝕜 : Type u_1) {B : Type u_2} {F : Type u_3} (E : B → Type u_5) [NontriviallyNormedField 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] {EB : Type u_6} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {HB : Type u_7} [TopologicalSpace HB] {IB : ModelWithCorners 𝕜 EB HB} [TopologicalSpace B] [ChartedSpace HB B] [FiberBundle F E] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module 𝕜 (E x)] [VectorBundle 𝕜 F E] {t : Set B} : MDiff[t] (Bundle.zeroSection F E)
• FiberBundle.exists_mdifferentiableOn_extend 📋 Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable
  {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] (F : Type u_5) [NormedAddCommGroup F] {V : M → Type u_6} [TopologicalSpace (Bundle.TotalSpace F V)] [(x : M) → AddCommGroup (V x)] [(x : M) → TopologicalSpace (V x)] [FiberBundle F V] [NormedSpace 𝕜 F] [(x : M) → Module 𝕜 (V x)] [VectorBundle 𝕜 F V] [ContMDiffVectorBundle 1 F V I] {x₀ : M} (σ₀ : V x₀) : ∃ s ∈ nhds x₀, MDiff[s] fun x' => ⟨x', FiberBundle.extend F σ₀ x'⟩
• mdifferentiableOn_neg_section 📋 Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable
  {𝕜 : Type u_1} {B : Type u_2} {F : Type u_3} {E : B → Type u_5} [TopologicalSpace B] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] [NormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 F] [FiberBundle F E] [(x : B) → AddCommGroup (E x)] [(x : B) → Module 𝕜 (E x)] [VectorBundle 𝕜 F E] {HB : Type u_6} [TopologicalSpace HB] [ChartedSpace HB B] {EB : Type u_7} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {I : ModelWithCorners 𝕜 EB HB} {s : (x : B) → E x} {u : Set B} (hs : MDiff[u] fun x => ⟨x, s x⟩) : MDiff[u] fun x => ⟨x, (-s) x⟩
• MDifferentiableOn.sum_section 📋 Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable
  {𝕜 : Type u_1} {B : Type u_2} {F : Type u_3} {E : B → Type u_5} [TopologicalSpace B] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] [NormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 F] [FiberBundle F E] [(x : B) → AddCommGroup (E x)] [(x : B) → Module 𝕜 (E x)] [VectorBundle 𝕜 F E] {HB : Type u_6} [TopologicalSpace HB] [ChartedSpace HB B] {EB : Type u_7} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {I : ModelWithCorners 𝕜 EB HB} {u : Set B} {ι : Type u_8} {s : Finset ι} {t : ι → (x : B) → E x} (hs : ∀ i ∈ s, MDiff[u] fun x => ⟨x, t i x⟩) : MDiff[u] fun x => ⟨x, ∑ i ∈ s, t i x⟩
• MDifferentiableOn.finsum_section_of_locallyFinite 📋 Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable
  {𝕜 : Type u_1} {B : Type u_2} {F : Type u_3} {E : B → Type u_5} [TopologicalSpace B] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] [NormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 F] [FiberBundle F E] [(x : B) → AddCommGroup (E x)] [(x : B) → Module 𝕜 (E x)] [VectorBundle 𝕜 F E] {HB : Type u_6} [TopologicalSpace HB] [ChartedSpace HB B] {EB : Type u_7} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {I : ModelWithCorners 𝕜 EB HB} {u : Set B} {ι : Type u_8} {t : ι → (x : B) → E x} (ht : LocallyFinite fun i => {x | t i x ≠ 0}) (ht' : ∀ (i : ι), MDiff[u] fun x => ⟨x, t i x⟩) : MDiff[u] fun x => ⟨x, ∑ᶠ (i : ι), t i x⟩
• MDifferentiableOn.sum_section_of_locallyFinite 📋 Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable
  {𝕜 : Type u_1} {B : Type u_2} {F : Type u_3} {E : B → Type u_5} [TopologicalSpace B] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] [NormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 F] [FiberBundle F E] [(x : B) → AddCommGroup (E x)] [(x : B) → Module 𝕜 (E x)] [VectorBundle 𝕜 F E] {HB : Type u_6} [TopologicalSpace HB] [ChartedSpace HB B] {EB : Type u_7} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {I : ModelWithCorners 𝕜 EB HB} {u : Set B} {ι : Type u_8} {t : ι → (x : B) → E x} (ht : LocallyFinite fun i => {x | t i x ≠ 0}) (ht' : ∀ (i : ι), MDiff[u] fun x => ⟨x, t i x⟩) : MDiff[u] fun x => ⟨x, ∑' (i : ι), t i x⟩
• Bundle.Trivialization.mdifferentiableOn_section_iff 📋 Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable
  {𝕜 : Type u_1} {B : Type u_2} {F : Type u_3} {E : B → Type u_5} [NontriviallyNormedField 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] {EB : Type u_6} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {HB : Type u_7} [TopologicalSpace HB] {IB : ModelWithCorners 𝕜 EB HB} [TopologicalSpace B] [ChartedSpace HB B] [FiberBundle F E] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module 𝕜 (E x)] [VectorBundle 𝕜 F E] [ContMDiffVectorBundle 1 F E IB] {s : (x : B) → E x} {a : Set B} (e : Bundle.Trivialization F Bundle.TotalSpace.proj) [MemTrivializationAtlas e] (ha : IsOpen a) (ha' : a ⊆ e.baseSet) : (MDiff[a] fun x => ⟨x, s x⟩) ↔ MDiff[a] fun x => (↑e ⟨x, s x⟩).2
• Bundle.Trivialization.mdifferentiableOn_section_baseSet_iff 📋 Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable
  {𝕜 : Type u_1} {B : Type u_2} {F : Type u_3} {E : B → Type u_5} [NontriviallyNormedField 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] {EB : Type u_6} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {HB : Type u_7} [TopologicalSpace HB] {IB : ModelWithCorners 𝕜 EB HB} [TopologicalSpace B] [ChartedSpace HB B] [FiberBundle F E] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module 𝕜 (E x)] [VectorBundle 𝕜 F E] [ContMDiffVectorBundle 1 F E IB] {s : (x : B) → E x} (e : Bundle.Trivialization F Bundle.TotalSpace.proj) [MemTrivializationAtlas e] : (MDiff[e.baseSet] fun x => ⟨x, s x⟩) ↔ MDiff[e.baseSet] fun x => (↑e ⟨x, s x⟩).2
• MDifferentiableOn.smul_const_section 📋 Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable
  {𝕜 : Type u_1} {B : Type u_2} {F : Type u_3} {E : B → Type u_5} [TopologicalSpace B] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] [NormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 F] [FiberBundle F E] [(x : B) → AddCommGroup (E x)] [(x : B) → Module 𝕜 (E x)] [VectorBundle 𝕜 F E] {HB : Type u_6} [TopologicalSpace HB] [ChartedSpace HB B] {EB : Type u_7} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {I : ModelWithCorners 𝕜 EB HB} {a : 𝕜} {s : (x : B) → E x} {u : Set B} (hs : MDiff[u] fun x => ⟨x, s x⟩) : MDiff[u] fun x => ⟨x, (a • s) x⟩
• MDifferentiableOn.coordChange 📋 Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable
  {𝕜 : Type u_1} {B : Type u_2} {F : Type u_3} {M : Type u_4} {E : B → Type u_5} [NontriviallyNormedField 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] {EB : Type u_6} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {HB : Type u_7} [TopologicalSpace HB] {IB : ModelWithCorners 𝕜 EB HB} {EM : Type u_9} [NormedAddCommGroup EM] [NormedSpace 𝕜 EM] {HM : Type u_10} [TopologicalSpace HM] {IM : ModelWithCorners 𝕜 EM HM} [TopologicalSpace M] [ChartedSpace HM M] [TopologicalSpace B] [ChartedSpace HB B] [FiberBundle F E] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module 𝕜 (E x)] {e e' : Bundle.Trivialization F Bundle.TotalSpace.proj} [MemTrivializationAtlas e] [MemTrivializationAtlas e'] [VectorBundle 𝕜 F E] [ContMDiffVectorBundle 1 F E IB] {s : Set M} {f : M → B} {g : M → F} (hf : MDiff[s] f) (hg : MDiff[s] g) (he : Set.MapsTo f s e.baseSet) (he' : Set.MapsTo f s e'.baseSet) : MDiff[s] fun y => e.coordChange e' (f y) (g y)
• mDifferentiableOn_sub_section 📋 Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable
  {𝕜 : Type u_1} {B : Type u_2} {F : Type u_3} {E : B → Type u_5} [TopologicalSpace B] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] [NormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 F] [FiberBundle F E] [(x : B) → AddCommGroup (E x)] [(x : B) → Module 𝕜 (E x)] [VectorBundle 𝕜 F E] {HB : Type u_6} [TopologicalSpace HB] [ChartedSpace HB B] {EB : Type u_7} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {I : ModelWithCorners 𝕜 EB HB} {s t : (x : B) → E x} {u : Set B} (hs : MDiff[u] fun x => ⟨x, s x⟩) (ht : MDiff[u] fun x => ⟨x, t x⟩) : MDiff[u] fun x => ⟨x, (s - t) x⟩
• mdifferentiableOn_add_section 📋 Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable
  {𝕜 : Type u_1} {B : Type u_2} {F : Type u_3} {E : B → Type u_5} [TopologicalSpace B] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] [NormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 F] [FiberBundle F E] [(x : B) → AddCommGroup (E x)] [(x : B) → Module 𝕜 (E x)] [VectorBundle 𝕜 F E] {HB : Type u_6} [TopologicalSpace HB] [ChartedSpace HB B] {EB : Type u_7} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {I : ModelWithCorners 𝕜 EB HB} {s t : (x : B) → E x} {u : Set B} (hs : MDiff[u] fun x => ⟨x, s x⟩) (ht : MDiff[u] fun x => ⟨x, t x⟩) : MDiff[u] fun x => ⟨x, (s + t) x⟩
• MDifferentiableOn.smul_section 📋 Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable
  {𝕜 : Type u_1} {B : Type u_2} {F : Type u_3} {E : B → Type u_5} [TopologicalSpace B] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] [NormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 F] [FiberBundle F E] [(x : B) → AddCommGroup (E x)] [(x : B) → Module 𝕜 (E x)] [VectorBundle 𝕜 F E] {HB : Type u_6} [TopologicalSpace HB] [ChartedSpace HB B] {EB : Type u_7} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {I : ModelWithCorners 𝕜 EB HB} {f : B → 𝕜} {s : (x : B) → E x} {u : Set B} (hf : MDiff[u] f) (hs : MDiff[u] fun x => ⟨x, s x⟩) : MDiff[u] fun x => ⟨x, (f • s) x⟩
• MDifferentiableOn.smul_section_of_tsupport 📋 Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable
  {𝕜 : Type u_1} {B : Type u_2} {F : Type u_3} {E : B → Type u_5} [TopologicalSpace B] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] [NormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 F] [FiberBundle F E] [(x : B) → AddCommGroup (E x)] [(x : B) → Module 𝕜 (E x)] [VectorBundle 𝕜 F E] {HB : Type u_6} [TopologicalSpace HB] [ChartedSpace HB B] {EB : Type u_7} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {I : ModelWithCorners 𝕜 EB HB} {u : Set B} {s : (x : B) → E x} {ψ : B → 𝕜} (hψ : MDiff[u] ψ) (ht : IsOpen u) (ht' : tsupport ψ ⊆ u) (hs : MDiff[u] fun x => ⟨x, s x⟩) : MDiff fun x => ⟨x, (ψ • s) x⟩
• mdifferentiableOn_coordChangeL 📋 Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable
  {𝕜 : Type u_1} {B : Type u_2} {F : Type u_3} {E : B → Type u_5} [NontriviallyNormedField 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] {EB : Type u_6} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {HB : Type u_7} [TopologicalSpace HB] {IB : ModelWithCorners 𝕜 EB HB} [TopologicalSpace B] [ChartedSpace HB B] [FiberBundle F E] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module 𝕜 (E x)] (e e' : Bundle.Trivialization F Bundle.TotalSpace.proj) [MemTrivializationAtlas e] [MemTrivializationAtlas e'] [VectorBundle 𝕜 F E] [ContMDiffVectorBundle 1 F E IB] : MDiff[e.baseSet ∩ e'.baseSet] fun b => ↑(Bundle.Trivialization.coordChangeL 𝕜 e e' b)
• MDifferentiableOn.coordChangeL 📋 Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable
  {𝕜 : Type u_1} {B : Type u_2} {F : Type u_3} {M : Type u_4} {E : B → Type u_5} [NontriviallyNormedField 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] {EB : Type u_6} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {HB : Type u_7} [TopologicalSpace HB] {IB : ModelWithCorners 𝕜 EB HB} {EM : Type u_9} [NormedAddCommGroup EM] [NormedSpace 𝕜 EM] {HM : Type u_10} [TopologicalSpace HM] {IM : ModelWithCorners 𝕜 EM HM} [TopologicalSpace M] [ChartedSpace HM M] [TopologicalSpace B] [ChartedSpace HB B] [FiberBundle F E] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module 𝕜 (E x)] {e e' : Bundle.Trivialization F Bundle.TotalSpace.proj} [MemTrivializationAtlas e] [MemTrivializationAtlas e'] [VectorBundle 𝕜 F E] [ContMDiffVectorBundle 1 F E IB] {s : Set M} {f : M → B} (hf : MDiff[s] f) (he : Set.MapsTo f s e.baseSet) (he' : Set.MapsTo f s e'.baseSet) : MDiff[s] fun y => ↑(Bundle.Trivialization.coordChangeL 𝕜 e e' (f y))
• mdifferentiableOn_symm_coordChangeL 📋 Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable
  {𝕜 : Type u_1} {B : Type u_2} {F : Type u_3} {E : B → Type u_5} [NontriviallyNormedField 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] {EB : Type u_6} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {HB : Type u_7} [TopologicalSpace HB] {IB : ModelWithCorners 𝕜 EB HB} [TopologicalSpace B] [ChartedSpace HB B] [FiberBundle F E] [(x : B) → AddCommMonoid (E x)] [(x : B) → Module 𝕜 (E x)] (e e' : Bundle.Trivialization F Bundle.TotalSpace.proj) [MemTrivializationAtlas e] [MemTrivializationAtlas e'] [VectorBundle 𝕜 F E] [ContMDiffVectorBundle 1 F E IB] : MDiff[e.baseSet ∩ e'.baseSet] fun b => ↑(Bundle.Trivialization.coordChangeL 𝕜 e e' b).symm
• MDifferentiableOn.clm_bundle_apply 📋 Mathlib.Geometry.Manifold.VectorBundle.Hom
  {𝕜 : Type u_1} {B : Type u_2} {F₁ : Type u_3} {F₂ : Type u_4} {M : Type u_6} [NontriviallyNormedField 𝕜] {E₁ : B → Type u_7} [(x : B) → AddCommGroup (E₁ x)] [(x : B) → Module 𝕜 (E₁ x)] [NormedAddCommGroup F₁] [NormedSpace 𝕜 F₁] [TopologicalSpace (Bundle.TotalSpace F₁ E₁)] [(x : B) → TopologicalSpace (E₁ x)] {E₂ : B → Type u_8} [(x : B) → AddCommGroup (E₂ x)] [(x : B) → Module 𝕜 (E₂ x)] [NormedAddCommGroup F₂] [NormedSpace 𝕜 F₂] [TopologicalSpace (Bundle.TotalSpace F₂ E₂)] [(x : B) → TopologicalSpace (E₂ x)] {EB : Type u_10} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {HB : Type u_11} [TopologicalSpace HB] {IB : ModelWithCorners 𝕜 EB HB} [TopologicalSpace B] [ChartedSpace HB B] {EM : Type u_12} [NormedAddCommGroup EM] [NormedSpace 𝕜 EM] {HM : Type u_13} [TopologicalSpace HM] {IM : ModelWithCorners 𝕜 EM HM} [TopologicalSpace M] [ChartedSpace HM M] [FiberBundle F₁ E₁] [VectorBundle 𝕜 F₁ E₁] [FiberBundle F₂ E₂] [VectorBundle 𝕜 F₂ E₂] {b : M → B} {v : (x : M) → E₁ (b x)} {s : Set M} [∀ (x : B), IsTopologicalAddGroup (E₂ x)] [∀ (x : B), ContinuousSMul 𝕜 (E₂ x)] {ϕ : (x : M) → E₁ (b x) →L[𝕜] E₂ (b x)} (hϕ : MDiff[s] fun m => ⟨b m, ϕ m⟩) (hv : MDiff[s] fun m => ⟨b m, v m⟩) : MDiff[s] fun m => ⟨b m, (ϕ m) (v m)⟩
• mdifferentiableOn_continuousLinearMapCoordChange 📋 Mathlib.Geometry.Manifold.VectorBundle.Hom
  {𝕜 : Type u_1} {B : Type u_2} {F₁ : Type u_3} {F₂ : Type u_4} {E₁ : B → Type u_6} {E₂ : B → Type u_7} [NontriviallyNormedField 𝕜] [(x : B) → AddCommGroup (E₁ x)] [(x : B) → Module 𝕜 (E₁ x)] [NormedAddCommGroup F₁] [NormedSpace 𝕜 F₁] [TopologicalSpace (Bundle.TotalSpace F₁ E₁)] [(x : B) → TopologicalSpace (E₁ x)] [(x : B) → AddCommGroup (E₂ x)] [(x : B) → Module 𝕜 (E₂ x)] [NormedAddCommGroup F₂] [NormedSpace 𝕜 F₂] [TopologicalSpace (Bundle.TotalSpace F₂ E₂)] [(x : B) → TopologicalSpace (E₂ x)] {EB : Type u_8} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {HB : Type u_9} [TopologicalSpace HB] {IB : ModelWithCorners 𝕜 EB HB} [TopologicalSpace B] [ChartedSpace HB B] [FiberBundle F₁ E₁] [VectorBundle 𝕜 F₁ E₁] [FiberBundle F₂ E₂] [VectorBundle 𝕜 F₂ E₂] {e₁ e₁' : Bundle.Trivialization F₁ Bundle.TotalSpace.proj} {e₂ e₂' : Bundle.Trivialization F₂ Bundle.TotalSpace.proj} [ContMDiffVectorBundle 1 F₁ E₁ IB] [ContMDiffVectorBundle 1 F₂ E₂ IB] [MemTrivializationAtlas e₁] [MemTrivializationAtlas e₁'] [MemTrivializationAtlas e₂] [MemTrivializationAtlas e₂'] : MDiff[e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet)] (Bundle.Pretrivialization.continuousLinearMapCoordChange (RingHom.id 𝕜) e₁ e₁' e₂ e₂')
• MDifferentiableOn.clm_bundle_apply₂ 📋 Mathlib.Geometry.Manifold.VectorBundle.Hom
  {𝕜 : Type u_1} {B : Type u_2} {F₁ : Type u_3} {F₂ : Type u_4} {F₃ : Type u_5} {M : Type u_6} [NontriviallyNormedField 𝕜] {E₁ : B → Type u_7} [(x : B) → AddCommGroup (E₁ x)] [(x : B) → Module 𝕜 (E₁ x)] [NormedAddCommGroup F₁] [NormedSpace 𝕜 F₁] [TopologicalSpace (Bundle.TotalSpace F₁ E₁)] [(x : B) → TopologicalSpace (E₁ x)] {E₂ : B → Type u_8} [(x : B) → AddCommGroup (E₂ x)] [(x : B) → Module 𝕜 (E₂ x)] [NormedAddCommGroup F₂] [NormedSpace 𝕜 F₂] [TopologicalSpace (Bundle.TotalSpace F₂ E₂)] [(x : B) → TopologicalSpace (E₂ x)] {E₃ : B → Type u_9} [(x : B) → AddCommGroup (E₃ x)] [(x : B) → Module 𝕜 (E₃ x)] [NormedAddCommGroup F₃] [NormedSpace 𝕜 F₃] [TopologicalSpace (Bundle.TotalSpace F₃ E₃)] [(x : B) → TopologicalSpace (E₃ x)] {EB : Type u_10} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {HB : Type u_11} [TopologicalSpace HB] {IB : ModelWithCorners 𝕜 EB HB} [TopologicalSpace B] [ChartedSpace HB B] {EM : Type u_12} [NormedAddCommGroup EM] [NormedSpace 𝕜 EM] {HM : Type u_13} [TopologicalSpace HM] {IM : ModelWithCorners 𝕜 EM HM} [TopologicalSpace M] [ChartedSpace HM M] [FiberBundle F₁ E₁] [VectorBundle 𝕜 F₁ E₁] [FiberBundle F₂ E₂] [VectorBundle 𝕜 F₂ E₂] [FiberBundle F₃ E₃] [VectorBundle 𝕜 F₃ E₃] {b : M → B} {v : (x : M) → E₁ (b x)} {s : Set M} [∀ (x : B), IsTopologicalAddGroup (E₃ x)] [∀ (x : B), ContinuousSMul 𝕜 (E₃ x)] {ψ : (x : M) → E₁ (b x) →L[𝕜] E₂ (b x) →L[𝕜] E₃ (b x)} {w : (x : M) → E₂ (b x)} (hψ : MDiff[s] fun m => ⟨b m, ψ m⟩) (hv : MDiff[s] fun m => ⟨b m, v m⟩) (hw : MDiff[s] fun m => ⟨b m, w m⟩) : MDiff[s] fun m => ⟨b m, ((ψ m) (v m)) (w m)⟩
• MDifferentiableOn.mpullback_vectorField_preimage 📋 Mathlib.Geometry.Manifold.VectorField.Pullback
  {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} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {H' : Type u_5} [TopologicalSpace H'] {E' : Type u_6} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} {V : (x : M') → TangentSpace I' x} {n : WithTop ℕ∞} {t : Set M'} [IsManifold I 2 M] [IsManifold I' 2 M'] [CompleteSpace E] (hV : MDiff[t] fun x => ⟨x, V x⟩) (hf : ContMDiff I I' n f) (hf' : ∀ x ∈ f ⁻¹' t, (mfderiv% f x).IsInvertible) (hmn : 2 ≤ n) : MDiff[f ⁻¹' t] fun x => ⟨x, VectorField.mpullback I I' f V x⟩
• MDifferentiableOn.mpullbackWithin_vectorField_inter 📋 Mathlib.Geometry.Manifold.VectorField.Pullback
  {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} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {H' : Type u_5} [TopologicalSpace H'] {E' : Type u_6} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} {s : Set M} {V : (x : M') → TangentSpace I' x} {n : WithTop ℕ∞} {t : Set M'} [IsManifold I 2 M] [IsManifold I' 2 M'] [CompleteSpace E] (hV : MDiff[t] fun x => ⟨x, V x⟩) (hf : ContMDiffOn I I' n f s) (hf' : ∀ x ∈ s ∩ f ⁻¹' t, (mfderiv[s] f x).IsInvertible) (hs : UniqueMDiff[s]) (hmn : 2 ≤ n) : MDiff[s ∩ f ⁻¹' t] fun x => ⟨x, VectorField.mpullbackWithin I I' f V s x⟩
• Manifold.pathELength_comp_of_antitoneOn 📋 Mathlib.Geometry.Manifold.Riemannian.PathELength
  {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type u_2} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type u_3} [TopologicalSpace M] [ChartedSpace H M] [(x : M) → ENorm (TangentSpace I x)] {a b : ℝ} {γ : ℝ → M} [∀ (x : M), ENormSMulClass ℝ (TangentSpace I x)] {f : ℝ → ℝ} (h : a ≤ b) (hf : AntitoneOn f (Set.Icc a b)) (h'f : DifferentiableOn ℝ f (Set.Icc a b)) (hγ : MDiff[Set.Icc (f b) (f a)] γ) : Manifold.pathELength I (γ ∘ f) a b = Manifold.pathELength I γ (f b) (f a)
• Manifold.pathELength_comp_of_monotoneOn 📋 Mathlib.Geometry.Manifold.Riemannian.PathELength
  {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type u_2} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type u_3} [TopologicalSpace M] [ChartedSpace H M] [(x : M) → ENorm (TangentSpace I x)] {a b : ℝ} {γ : ℝ → M} [∀ (x : M), ENormSMulClass ℝ (TangentSpace I x)] {f : ℝ → ℝ} (h : a ≤ b) (hf : MonotoneOn f (Set.Icc a b)) (h'f : DifferentiableOn ℝ f (Set.Icc a b)) (hγ : MDiff[Set.Icc (f a) (f b)] γ) : Manifold.pathELength I (γ ∘ f) a b = Manifold.pathELength I γ (f a) (f b)
• MDifferentiableOn.inner_bundle 📋 Mathlib.Geometry.Manifold.VectorBundle.Riemannian
  {EB : Type u_1} [NormedAddCommGroup EB] [NormedSpace ℝ EB] {HB : Type u_2} [TopologicalSpace HB] {IB : ModelWithCorners ℝ EB HB} {B : Type u_3} [TopologicalSpace B] [ChartedSpace HB B] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace ℝ F] {E : B → Type u_5} [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → NormedAddCommGroup (E x)] [(x : B) → InnerProductSpace ℝ (E x)] [FiberBundle F E] [VectorBundle ℝ F E] {EM : Type u_6} [NormedAddCommGroup EM] [NormedSpace ℝ EM] {HM : Type u_7} [TopologicalSpace HM] {IM : ModelWithCorners ℝ EM HM} {M : Type u_8} [TopologicalSpace M] [ChartedSpace HM M] [h : IsContMDiffRiemannianBundle IB 1 F E] {b : M → B} {v w : (x : M) → E (b x)} {s : Set M} (hv : MDiff[s] fun m => ⟨b m, v m⟩) (hw : MDiff[s] fun m => ⟨b m, w m⟩) : MDiff[s] fun b_1 => inner ℝ (v b_1) (w b_1)
• IsLocalFrameOn.mdifferentiableOn_of_coeff 📋 Mathlib.Geometry.Manifold.VectorBundle.LocalFrame
  {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] {F : Type u_5} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {V : M → Type u_6} [TopologicalSpace (Bundle.TotalSpace F V)] [(x : M) → AddCommGroup (V x)] [(x : M) → Module 𝕜 (V x)] [(x : M) → TopologicalSpace (V x)] [FiberBundle F V] {ι : Type u_7} {s : ι → (x : M) → V x} {u : Set M} {t : (x : M) → V x} [VectorBundle 𝕜 F V] (hs : IsLocalFrameOn I F 1 s u) [FiniteDimensional 𝕜 F] (h : ∀ (i : ι), MDiff[u] ((LinearMap.piApply (hs.coeff i)) t)) : MDiff[u] fun x => ⟨x, t x⟩
• mdifferentiableOn_localFrameCoeff 📋 Mathlib.Geometry.Manifold.VectorBundle.LocalFrame
  {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] {F : Type u_5} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {V : M → Type u_6} [TopologicalSpace (Bundle.TotalSpace F V)] [(x : M) → AddCommGroup (V x)] [(x : M) → Module 𝕜 (V x)] [(x : M) → TopologicalSpace (V x)] [FiberBundle F V] [VectorBundle 𝕜 F V] [ContMDiffVectorBundle 1 F V I] {e : Bundle.Trivialization F Bundle.TotalSpace.proj} [MemTrivializationAtlas e] {ι : Type u_7} (b : Module.Basis ι 𝕜 F) {s : (x : M) → V x} {t : Set M} [FiniteDimensional 𝕜 F] [CompleteSpace 𝕜] (ht : IsOpen t) (ht' : t ⊆ e.baseSet) (hs : MDiff[t] fun x => ⟨x, s x⟩) (i : ι) : MDiff[t] ((LinearMap.piApply (Bundle.Trivialization.localFrameCoeff I e b i)) s)
• mdifferentiableOn_localFrame_coeff 📋 Mathlib.Geometry.Manifold.VectorBundle.LocalFrame
  {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] {F : Type u_5} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {V : M → Type u_6} [TopologicalSpace (Bundle.TotalSpace F V)] [(x : M) → AddCommGroup (V x)] [(x : M) → Module 𝕜 (V x)] [(x : M) → TopologicalSpace (V x)] [FiberBundle F V] [VectorBundle 𝕜 F V] [ContMDiffVectorBundle 1 F V I] {e : Bundle.Trivialization F Bundle.TotalSpace.proj} [MemTrivializationAtlas e] {ι : Type u_7} (b : Module.Basis ι 𝕜 F) {s : (x : M) → V x} {t : Set M} [FiniteDimensional 𝕜 F] [CompleteSpace 𝕜] (ht : IsOpen t) (ht' : t ⊆ e.baseSet) (hs : MDiff[t] fun x => ⟨x, s x⟩) (i : ι) : MDiff[t] ((LinearMap.piApply (Bundle.Trivialization.localFrameCoeff I e b i)) s)
• mdifferentiableOn_baseSet_localFrameCoeff 📋 Mathlib.Geometry.Manifold.VectorBundle.LocalFrame
  {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] {F : Type u_5} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {V : M → Type u_6} [TopologicalSpace (Bundle.TotalSpace F V)] [(x : M) → AddCommGroup (V x)] [(x : M) → Module 𝕜 (V x)] [(x : M) → TopologicalSpace (V x)] [FiberBundle F V] [VectorBundle 𝕜 F V] [ContMDiffVectorBundle 1 F V I] {e : Bundle.Trivialization F Bundle.TotalSpace.proj} [MemTrivializationAtlas e] {ι : Type u_7} (b : Module.Basis ι 𝕜 F) {s : (x : M) → V x} [FiniteDimensional 𝕜 F] [CompleteSpace 𝕜] (hs : MDiff[e.baseSet] fun x => ⟨x, s x⟩) (i : ι) : MDiff[e.baseSet] ((LinearMap.piApply (Bundle.Trivialization.localFrameCoeff I e b i)) s)
• mdifferentiableOn_baseSet_localFrame_coeff 📋 Mathlib.Geometry.Manifold.VectorBundle.LocalFrame
  {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] {F : Type u_5} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {V : M → Type u_6} [TopologicalSpace (Bundle.TotalSpace F V)] [(x : M) → AddCommGroup (V x)] [(x : M) → Module 𝕜 (V x)] [(x : M) → TopologicalSpace (V x)] [FiberBundle F V] [VectorBundle 𝕜 F V] [ContMDiffVectorBundle 1 F V I] {e : Bundle.Trivialization F Bundle.TotalSpace.proj} [MemTrivializationAtlas e] {ι : Type u_7} (b : Module.Basis ι 𝕜 F) {s : (x : M) → V x} [FiniteDimensional 𝕜 F] [CompleteSpace 𝕜] (hs : MDiff[e.baseSet] fun x => ⟨x, s x⟩) (i : ι) : MDiff[e.baseSet] ((LinearMap.piApply (Bundle.Trivialization.localFrameCoeff I e b i)) s)

About

Loogle searches Lean and Mathlib definitions and theorems.

You can use Loogle from within the Lean4 VSCode language extension using the Loogle command from the command palette. You can also try the #loogle command from LeanSearchClient, the CLI version, the Loogle VS Code extension, the lean.nvim integration or the Zulip bot.

Usage

Loogle finds definitions and lemmas in various ways:

1. By constant:
   🔍 Real.sin
   finds all lemmas whose statement somehow mentions the sine function.

2. By lemma name substring:
   🔍 "differ"
   finds all lemmas that have "differ" somewhere in their lemma name.

3. By subexpression:
   🔍 _ * (_ ^ _)
   finds all lemmas whose statements somewhere include a product where the second argument is raised to some power.

   The pattern can also be non-linear, as in
   🔍 Real.sqrt ?a * Real.sqrt ?a

   If the pattern has parameters, they are matched in any order. Both of these will find List.map:
   🔍 (?a -> ?b) -> List ?a -> List ?b
   🔍 List ?a -> (?a -> ?b) -> List ?b

4. By main conclusion:
   🔍 |- tsum _ = _ * tsum _
   finds all lemmas where the conclusion (the subexpression to the right of all → and ∀) has the given shape.

   As before, if the pattern has parameters, they are matched against the hypotheses of the lemma in any order; for example,
   🔍 |- _ < _ → tsum _ < tsum _
   will find tsum_lt_tsum even though the hypothesis f i < g i is not the last.

5. You can filter for definitions vs theorems: Using ⊢ (_ : Type _) finds all definitions which provide data while ⊢ (_ : Prop) finds all theorems (and definitions of proofs).

If you pass more than one such search filter, separated by commas Loogle will return lemmas which match all of them. The search
🔍 Real.sin, "two", tsum, _ * _, _ ^ _, |- _ < _ → _
would find all lemmas which mention the constants Real.sin and tsum, have "two" as a substring of the lemma name, include a product and a power somewhere in the type, and have a hypothesis of the form _ < _ (if there were any such lemmas). Metavariables (?a) are assigned independently in each filter.

The #lucky button will directly send you to the documentation of the first hit.

Source code

You can find the source code for this service at https://github.com/nomeata/loogle. The https://loogle.lean-lang.org/ service is provided by the Lean FRO. Please review the Lean FRO Terms of Use and Privacy Policy.

This is Loogle revision 9f11169 serving mathlib revision 69fae59