Documentation

Mathlib.Geometry.Manifold.Algebra.LieGroup

Lie groups #

A Lie group is a group that is also a C^n manifold, in which the group operations of multiplication and inversion are C^n maps. Regularity of the group multiplication means that multiplication is a C^n mapping of the product manifold G × G into G.

Note that, since a manifold here is not second-countable and Hausdorff a Lie group here is not guaranteed to be second-countable (even though it can be proved it is Hausdorff). Note also that Lie groups here are not necessarily finite dimensional.

Main definitions #

Main results #

Implementation notes #

A priori, a Lie group here is a manifold with corners.

The definition of Lie group cannot require I : ModelWithCorners 𝕜 E E with the same space as the model space and as the model vector space, as one might hope, because in the product situation, the model space is ModelProd E E' and the model vector space is E × E', which are not the same, so the definition does not apply. Hence the definition should be more general, allowing I : ModelWithCorners 𝕜 E H.

structure LieAddGroup {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) (n : WithTop ENat) (G : Type u_4) [AddGroup G] [TopologicalSpace G] [ChartedSpace H G] extends ContMDiffAdd I n G :

An additive Lie group is a group and a C^n manifold at the same time in which the addition and negation operations are C^n.

Instances For
    structure LieGroup {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) (n : WithTop ENat) (G : Type u_4) [Group G] [TopologicalSpace G] [ChartedSpace H G] extends ContMDiffMul I n G :

    A (multiplicative) Lie group is a group and a C^n manifold at the same time in which the multiplication and inverse operations are C^n.

    Instances For

      Smoothness of inversion, negation, division and subtraction #

      Let f : M → G be a C^n function into a Lie group, then f is point-wise invertible with smooth inverse f. If f and g are two such functions, the quotient f / g (i.e., the point-wise product of f and the point-wise inverse of g) is also C^n.

      theorem LieGroup.of_le {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {H : Type u_2} [TopologicalSpace H] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [Group G] {m n : WithTop ENat} (hmn : LE.le m n) [h : LieGroup I n G] :
      LieGroup I m G
      theorem LieAddGroup.of_le {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {H : Type u_2} [TopologicalSpace H] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [AddGroup G] {m n : WithTop ENat} (hmn : LE.le m n) [h : LieAddGroup I n G] :
      theorem instLieGroupOfTopWithTopENat {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {H : Type u_2} [TopologicalSpace H] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [Group G] {a : WithTop ENat} [LieGroup I Top.top G] :
      LieGroup I a G
      theorem instLieGroupOfNatWithTopENat {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {H : Type u_2} [TopologicalSpace H] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [Group G] [LieGroup I 2 G] :
      LieGroup I 1 G
      theorem instLieAddGroupOfNatWithTopENat {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {H : Type u_2} [TopologicalSpace H] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [AddGroup G] [LieAddGroup I 2 G] :
      theorem contMDiff_inv {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) (n : WithTop ENat) {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [Group G] [LieGroup I n G] :
      ContMDiff I I n fun (x : G) => Inv.inv x

      In a Lie group, inversion is C^n.

      theorem contMDiff_neg {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) (n : WithTop ENat) {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [AddGroup G] [LieAddGroup I n G] :
      ContMDiff I I n fun (x : G) => Neg.neg x

      In an additive Lie group, inversion is a smooth map.

      @[deprecated contMDiff_inv (since := "2024-11-21")]
      theorem smooth_inv {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) (n : WithTop ENat) {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [Group G] [LieGroup I n G] :
      ContMDiff I I n fun (x : G) => Inv.inv x

      Alias of contMDiff_inv.


      In a Lie group, inversion is C^n.

      @[deprecated contMDiff_neg (since := "2024-11-21")]
      theorem smooth_neg {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) (n : WithTop ENat) {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [AddGroup G] [LieAddGroup I n G] :
      ContMDiff I I n fun (x : G) => Neg.neg x

      Alias of contMDiff_neg.


      In an additive Lie group, inversion is a smooth map.

      theorem topologicalGroup_of_lieGroup {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) (n : WithTop ENat) {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [Group G] [LieGroup I n G] :

      A Lie group is a topological group. This is not an instance for technical reasons, see note [Design choices about smooth algebraic structures].

      An additive Lie group is an additive topological group. This is not an instance for technical reasons, see note [Design choices about smooth algebraic structures].

      theorem ContMDiffWithinAt.inv {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} {n : WithTop ENat} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [Group G] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_7} [TopologicalSpace M] [ChartedSpace H' M] [LieGroup I n G] {f : MG} {s : Set M} {x₀ : M} (hf : ContMDiffWithinAt I' I n f s x₀) :
      ContMDiffWithinAt I' I n (fun (x : M) => Inv.inv (f x)) s x₀
      theorem ContMDiffWithinAt.neg {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} {n : WithTop ENat} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [AddGroup G] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_7} [TopologicalSpace M] [ChartedSpace H' M] [LieAddGroup I n G] {f : MG} {s : Set M} {x₀ : M} (hf : ContMDiffWithinAt I' I n f s x₀) :
      ContMDiffWithinAt I' I n (fun (x : M) => Neg.neg (f x)) s x₀
      theorem ContMDiffAt.inv {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} {n : WithTop ENat} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [Group G] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_7} [TopologicalSpace M] [ChartedSpace H' M] [LieGroup I n G] {f : MG} {x₀ : M} (hf : ContMDiffAt I' I n f x₀) :
      ContMDiffAt I' I n (fun (x : M) => Inv.inv (f x)) x₀
      theorem ContMDiffAt.neg {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} {n : WithTop ENat} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [AddGroup G] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_7} [TopologicalSpace M] [ChartedSpace H' M] [LieAddGroup I n G] {f : MG} {x₀ : M} (hf : ContMDiffAt I' I n f x₀) :
      ContMDiffAt I' I n (fun (x : M) => Neg.neg (f x)) x₀
      theorem ContMDiffOn.inv {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} {n : WithTop ENat} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [Group G] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_7} [TopologicalSpace M] [ChartedSpace H' M] [LieGroup I n G] {f : MG} {s : Set M} (hf : ContMDiffOn I' I n f s) :
      ContMDiffOn I' I n (fun (x : M) => Inv.inv (f x)) s
      theorem ContMDiffOn.neg {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} {n : WithTop ENat} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [AddGroup G] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_7} [TopologicalSpace M] [ChartedSpace H' M] [LieAddGroup I n G] {f : MG} {s : Set M} (hf : ContMDiffOn I' I n f s) :
      ContMDiffOn I' I n (fun (x : M) => Neg.neg (f x)) s
      theorem ContMDiff.inv {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} {n : WithTop ENat} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [Group G] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_7} [TopologicalSpace M] [ChartedSpace H' M] [LieGroup I n G] {f : MG} (hf : ContMDiff I' I n f) :
      ContMDiff I' I n fun (x : M) => Inv.inv (f x)
      theorem ContMDiff.neg {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} {n : WithTop ENat} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [AddGroup G] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_7} [TopologicalSpace M] [ChartedSpace H' M] [LieAddGroup I n G] {f : MG} (hf : ContMDiff I' I n f) :
      ContMDiff I' I n fun (x : M) => Neg.neg (f x)
      @[deprecated ContMDiffWithinAt.inv (since := "2024-11-21")]
      theorem SmoothWithinAt.inv {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} {n : WithTop ENat} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [Group G] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_7} [TopologicalSpace M] [ChartedSpace H' M] [LieGroup I n G] {f : MG} {s : Set M} {x₀ : M} (hf : ContMDiffWithinAt I' I n f s x₀) :
      ContMDiffWithinAt I' I n (fun (x : M) => Inv.inv (f x)) s x₀

      Alias of ContMDiffWithinAt.inv.

      @[deprecated ContMDiffAt.inv (since := "2024-11-21")]
      theorem SmoothAt.inv {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} {n : WithTop ENat} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [Group G] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_7} [TopologicalSpace M] [ChartedSpace H' M] [LieGroup I n G] {f : MG} {x₀ : M} (hf : ContMDiffAt I' I n f x₀) :
      ContMDiffAt I' I n (fun (x : M) => Inv.inv (f x)) x₀

      Alias of ContMDiffAt.inv.

      @[deprecated ContMDiffOn.inv (since := "2024-11-21")]
      theorem SmoothOn.inv {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} {n : WithTop ENat} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [Group G] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_7} [TopologicalSpace M] [ChartedSpace H' M] [LieGroup I n G] {f : MG} {s : Set M} (hf : ContMDiffOn I' I n f s) :
      ContMDiffOn I' I n (fun (x : M) => Inv.inv (f x)) s

      Alias of ContMDiffOn.inv.

      @[deprecated ContMDiff.inv (since := "2024-11-21")]
      theorem Smooth.inv {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} {n : WithTop ENat} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [Group G] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_7} [TopologicalSpace M] [ChartedSpace H' M] [LieGroup I n G] {f : MG} (hf : ContMDiff I' I n f) :
      ContMDiff I' I n fun (x : M) => Inv.inv (f x)

      Alias of ContMDiff.inv.

      @[deprecated ContMDiffWithinAt.neg (since := "2024-11-21")]
      theorem SmoothWithinAt.neg {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} {n : WithTop ENat} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [AddGroup G] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_7} [TopologicalSpace M] [ChartedSpace H' M] [LieAddGroup I n G] {f : MG} {s : Set M} {x₀ : M} (hf : ContMDiffWithinAt I' I n f s x₀) :
      ContMDiffWithinAt I' I n (fun (x : M) => Neg.neg (f x)) s x₀

      Alias of ContMDiffWithinAt.neg.

      @[deprecated ContMDiffAt.neg (since := "2024-11-21")]
      theorem SmoothAt.neg {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} {n : WithTop ENat} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [AddGroup G] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_7} [TopologicalSpace M] [ChartedSpace H' M] [LieAddGroup I n G] {f : MG} {x₀ : M} (hf : ContMDiffAt I' I n f x₀) :
      ContMDiffAt I' I n (fun (x : M) => Neg.neg (f x)) x₀

      Alias of ContMDiffAt.neg.

      @[deprecated ContMDiffOn.neg (since := "2024-11-21")]
      theorem SmoothOn.neg {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} {n : WithTop ENat} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [AddGroup G] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_7} [TopologicalSpace M] [ChartedSpace H' M] [LieAddGroup I n G] {f : MG} {s : Set M} (hf : ContMDiffOn I' I n f s) :
      ContMDiffOn I' I n (fun (x : M) => Neg.neg (f x)) s

      Alias of ContMDiffOn.neg.

      @[deprecated ContMDiff.neg (since := "2024-11-21")]
      theorem Smooth.neg {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} {n : WithTop ENat} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [AddGroup G] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_7} [TopologicalSpace M] [ChartedSpace H' M] [LieAddGroup I n G] {f : MG} (hf : ContMDiff I' I n f) :
      ContMDiff I' I n fun (x : M) => Neg.neg (f x)

      Alias of ContMDiff.neg.

      theorem ContMDiffWithinAt.div {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} {n : WithTop ENat} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [Group G] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_7} [TopologicalSpace M] [ChartedSpace H' M] [LieGroup I n G] {f g : MG} {s : Set M} {x₀ : M} (hf : ContMDiffWithinAt I' I n f s x₀) (hg : ContMDiffWithinAt I' I n g s x₀) :
      ContMDiffWithinAt I' I n (fun (x : M) => HDiv.hDiv (f x) (g x)) s x₀
      theorem ContMDiffWithinAt.sub {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} {n : WithTop ENat} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [AddGroup G] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_7} [TopologicalSpace M] [ChartedSpace H' M] [LieAddGroup I n G] {f g : MG} {s : Set M} {x₀ : M} (hf : ContMDiffWithinAt I' I n f s x₀) (hg : ContMDiffWithinAt I' I n g s x₀) :
      ContMDiffWithinAt I' I n (fun (x : M) => HSub.hSub (f x) (g x)) s x₀
      theorem ContMDiffAt.div {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} {n : WithTop ENat} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [Group G] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_7} [TopologicalSpace M] [ChartedSpace H' M] [LieGroup I n G] {f g : MG} {x₀ : M} (hf : ContMDiffAt I' I n f x₀) (hg : ContMDiffAt I' I n g x₀) :
      ContMDiffAt I' I n (fun (x : M) => HDiv.hDiv (f x) (g x)) x₀
      theorem ContMDiffAt.sub {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} {n : WithTop ENat} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [AddGroup G] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_7} [TopologicalSpace M] [ChartedSpace H' M] [LieAddGroup I n G] {f g : MG} {x₀ : M} (hf : ContMDiffAt I' I n f x₀) (hg : ContMDiffAt I' I n g x₀) :
      ContMDiffAt I' I n (fun (x : M) => HSub.hSub (f x) (g x)) x₀
      theorem ContMDiffOn.div {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} {n : WithTop ENat} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [Group G] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_7} [TopologicalSpace M] [ChartedSpace H' M] [LieGroup I n G] {f g : MG} {s : Set M} (hf : ContMDiffOn I' I n f s) (hg : ContMDiffOn I' I n g s) :
      ContMDiffOn I' I n (fun (x : M) => HDiv.hDiv (f x) (g x)) s
      theorem ContMDiffOn.sub {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} {n : WithTop ENat} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [AddGroup G] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_7} [TopologicalSpace M] [ChartedSpace H' M] [LieAddGroup I n G] {f g : MG} {s : Set M} (hf : ContMDiffOn I' I n f s) (hg : ContMDiffOn I' I n g s) :
      ContMDiffOn I' I n (fun (x : M) => HSub.hSub (f x) (g x)) s
      theorem ContMDiff.div {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} {n : WithTop ENat} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [Group G] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_7} [TopologicalSpace M] [ChartedSpace H' M] [LieGroup I n G] {f g : MG} (hf : ContMDiff I' I n f) (hg : ContMDiff I' I n g) :
      ContMDiff I' I n fun (x : M) => HDiv.hDiv (f x) (g x)
      theorem ContMDiff.sub {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} {n : WithTop ENat} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [AddGroup G] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_7} [TopologicalSpace M] [ChartedSpace H' M] [LieAddGroup I n G] {f g : MG} (hf : ContMDiff I' I n f) (hg : ContMDiff I' I n g) :
      ContMDiff I' I n fun (x : M) => HSub.hSub (f x) (g x)
      @[deprecated ContMDiffWithinAt.div (since := "2024-11-21")]
      theorem SmoothWithinAt.div {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} {n : WithTop ENat} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [Group G] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_7} [TopologicalSpace M] [ChartedSpace H' M] [LieGroup I n G] {f g : MG} {s : Set M} {x₀ : M} (hf : ContMDiffWithinAt I' I n f s x₀) (hg : ContMDiffWithinAt I' I n g s x₀) :
      ContMDiffWithinAt I' I n (fun (x : M) => HDiv.hDiv (f x) (g x)) s x₀

      Alias of ContMDiffWithinAt.div.

      @[deprecated ContMDiffAt.div (since := "2024-11-21")]
      theorem SmoothAt.div {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} {n : WithTop ENat} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [Group G] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_7} [TopologicalSpace M] [ChartedSpace H' M] [LieGroup I n G] {f g : MG} {x₀ : M} (hf : ContMDiffAt I' I n f x₀) (hg : ContMDiffAt I' I n g x₀) :
      ContMDiffAt I' I n (fun (x : M) => HDiv.hDiv (f x) (g x)) x₀

      Alias of ContMDiffAt.div.

      @[deprecated ContMDiffOn.div (since := "2024-11-21")]
      theorem SmoothOn.div {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} {n : WithTop ENat} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [Group G] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_7} [TopologicalSpace M] [ChartedSpace H' M] [LieGroup I n G] {f g : MG} {s : Set M} (hf : ContMDiffOn I' I n f s) (hg : ContMDiffOn I' I n g s) :
      ContMDiffOn I' I n (fun (x : M) => HDiv.hDiv (f x) (g x)) s

      Alias of ContMDiffOn.div.

      @[deprecated ContMDiff.div (since := "2024-11-21")]
      theorem Smooth.div {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} {n : WithTop ENat} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [Group G] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_7} [TopologicalSpace M] [ChartedSpace H' M] [LieGroup I n G] {f g : MG} (hf : ContMDiff I' I n f) (hg : ContMDiff I' I n g) :
      ContMDiff I' I n fun (x : M) => HDiv.hDiv (f x) (g x)

      Alias of ContMDiff.div.

      @[deprecated ContMDiffWithinAt.sub (since := "2024-11-21")]
      theorem SmoothWithinAt.sub {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} {n : WithTop ENat} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [AddGroup G] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_7} [TopologicalSpace M] [ChartedSpace H' M] [LieAddGroup I n G] {f g : MG} {s : Set M} {x₀ : M} (hf : ContMDiffWithinAt I' I n f s x₀) (hg : ContMDiffWithinAt I' I n g s x₀) :
      ContMDiffWithinAt I' I n (fun (x : M) => HSub.hSub (f x) (g x)) s x₀

      Alias of ContMDiffWithinAt.sub.

      @[deprecated ContMDiffAt.sub (since := "2024-11-21")]
      theorem SmoothAt.sub {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} {n : WithTop ENat} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [AddGroup G] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_7} [TopologicalSpace M] [ChartedSpace H' M] [LieAddGroup I n G] {f g : MG} {x₀ : M} (hf : ContMDiffAt I' I n f x₀) (hg : ContMDiffAt I' I n g x₀) :
      ContMDiffAt I' I n (fun (x : M) => HSub.hSub (f x) (g x)) x₀

      Alias of ContMDiffAt.sub.

      @[deprecated ContMDiffOn.sub (since := "2024-11-21")]
      theorem SmoothOn.sub {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} {n : WithTop ENat} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [AddGroup G] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_7} [TopologicalSpace M] [ChartedSpace H' M] [LieAddGroup I n G] {f g : MG} {s : Set M} (hf : ContMDiffOn I' I n f s) (hg : ContMDiffOn I' I n g s) :
      ContMDiffOn I' I n (fun (x : M) => HSub.hSub (f x) (g x)) s

      Alias of ContMDiffOn.sub.

      @[deprecated ContMDiff.sub (since := "2024-11-21")]
      theorem Smooth.sub {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} {n : WithTop ENat} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [AddGroup G] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_7} [TopologicalSpace M] [ChartedSpace H' M] [LieAddGroup I n G] {f g : MG} (hf : ContMDiff I' I n f) (hg : ContMDiff I' I n g) :
      ContMDiff I' I n fun (x : M) => HSub.hSub (f x) (g x)

      Alias of ContMDiff.sub.

      Binary product of Lie groups

      theorem Prod.instLieGroup {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {n : WithTop ENat} {H : Type u_2} [TopologicalSpace H] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [Group G] [LieGroup I n G] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {G' : Type u_7} [TopologicalSpace G'] [ChartedSpace H' G'] [Group G'] [LieGroup I' n G'] :
      LieGroup (I.prod I') n (Prod G G')
      theorem Prod.instLieAddGroup {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {n : WithTop ENat} {H : Type u_2} [TopologicalSpace H] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [AddGroup G] [LieAddGroup I n G] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {G' : Type u_7} [TopologicalSpace G'] [ChartedSpace H' G'] [AddGroup G'] [LieAddGroup I' n G'] :
      LieAddGroup (I.prod I') n (Prod G G')

      Normed spaces are Lie groups #

      C^n manifolds with C^n inversion away from zero #

      Typeclass for C^n manifolds with 0 and Inv such that inversion is C^n at all non-zero points. (This includes multiplicative Lie groups, but also complete normed semifields.) Point-wise inversion is C^n when the function/denominator is non-zero.

      structure ContMDiffInv₀ {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) (n : WithTop ENat) (G : Type u_4) [Inv G] [Zero G] [TopologicalSpace G] [ChartedSpace H G] :

      A C^n manifold with 0 and Inv such that fun x ↦ x⁻¹ is C^n at all nonzero points. Any complete normed (semi)field has this property.

      • contMDiffAt_inv₀ x : G : Ne x 0ContMDiffAt I I n (fun (y : G) => Inv.inv y) x

        Inversion is C^n away from 0.

      Instances For
        @[deprecated ContMDiffInv₀ (since := "2025-01-09")]
        def SmoothInv₀ {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) (n : WithTop ENat) (G : Type u_4) [Inv G] [Zero G] [TopologicalSpace G] [ChartedSpace H G] :

        Alias of ContMDiffInv₀.


        A C^n manifold with 0 and Inv such that fun x ↦ x⁻¹ is C^n at all nonzero points. Any complete normed (semi)field has this property.

        Equations
        Instances For
          theorem ContMDiffInv₀.of_le {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {H : Type u_2} [TopologicalSpace H] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [Inv G] [Zero G] {m n : WithTop ENat} (hmn : LE.le m n) [h : ContMDiffInv₀ I n G] :
          theorem instContMDiffInv₀OfNatWithTopENat {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {H : Type u_2} [TopologicalSpace H] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [Inv G] [Zero G] [ContMDiffInv₀ I 2 G] :
          theorem contMDiffAt_inv₀ {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {n : WithTop ENat} {H : Type u_2} [TopologicalSpace H] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [Inv G] [Zero G] [ContMDiffInv₀ I n G] {x : G} (hx : Ne x 0) :
          ContMDiffAt I I n (fun (y : G) => Inv.inv y) x
          @[deprecated contMDiffAt_inv₀ (since := "2024-11-21")]
          theorem smoothAt_inv₀ {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {n : WithTop ENat} {H : Type u_2} [TopologicalSpace H] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [Inv G] [Zero G] [ContMDiffInv₀ I n G] {x : G} (hx : Ne x 0) :
          ContMDiffAt I I n (fun (y : G) => Inv.inv y) x

          Alias of contMDiffAt_inv₀.

          In a manifold with C^n inverse away from 0, the inverse is continuous away from 0. This is not an instance for technical reasons, see note [Design choices about smooth algebraic structures].

          @[deprecated hasContinuousInv₀_of_hasContMDiffInv₀ (since := "2025-01-09")]

          Alias of hasContinuousInv₀_of_hasContMDiffInv₀.


          In a manifold with C^n inverse away from 0, the inverse is continuous away from 0. This is not an instance for technical reasons, see note [Design choices about smooth algebraic structures].

          @[deprecated contMDiffOn_inv₀ (since := "2024-11-21")]
          theorem smoothOn_inv₀ {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {n : WithTop ENat} {H : Type u_2} [TopologicalSpace H] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [Inv G] [Zero G] [ContMDiffInv₀ I n G] :

          Alias of contMDiffOn_inv₀.

          @[deprecated contMDiffOn_inv₀ (since := "2024-11-21")]
          theorem SmoothOn_inv₀ {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {n : WithTop ENat} {H : Type u_2} [TopologicalSpace H] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [Inv G] [Zero G] [ContMDiffInv₀ I n G] :

          Alias of contMDiffOn_inv₀.

          theorem ContMDiffWithinAt.inv₀ {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {n : WithTop ENat} {H : Type u_2} [TopologicalSpace H] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [Inv G] [Zero G] {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 : MG} [ContMDiffInv₀ I n G] {s : Set M} {a : M} (hf : ContMDiffWithinAt I' I n f s a) (ha : Ne (f a) 0) :
          ContMDiffWithinAt I' I n (fun (x : M) => Inv.inv (f x)) s a
          theorem ContMDiffAt.inv₀ {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {n : WithTop ENat} {H : Type u_2} [TopologicalSpace H] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [Inv G] [Zero G] {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 : MG} [ContMDiffInv₀ I n G] {a : M} (hf : ContMDiffAt I' I n f a) (ha : Ne (f a) 0) :
          ContMDiffAt I' I n (fun (x : M) => Inv.inv (f x)) a
          theorem ContMDiff.inv₀ {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {n : WithTop ENat} {H : Type u_2} [TopologicalSpace H] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [Inv G] [Zero G] {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 : MG} [ContMDiffInv₀ I n G] (hf : ContMDiff I' I n f) (h0 : ∀ (x : M), Ne (f x) 0) :
          ContMDiff I' I n fun (x : M) => Inv.inv (f x)
          theorem ContMDiffOn.inv₀ {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {n : WithTop ENat} {H : Type u_2} [TopologicalSpace H] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [Inv G] [Zero G] {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 : MG} [ContMDiffInv₀ I n G] {s : Set M} (hf : ContMDiffOn I' I n f s) (h0 : ∀ (x : M), Membership.mem s xNe (f x) 0) :
          ContMDiffOn I' I n (fun (x : M) => Inv.inv (f x)) s
          @[deprecated ContMDiffWithinAt.inv₀ (since := "2024-11-21")]
          theorem SmoothWithinAt.inv₀ {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {n : WithTop ENat} {H : Type u_2} [TopologicalSpace H] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [Inv G] [Zero G] {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 : MG} [ContMDiffInv₀ I n G] {s : Set M} {a : M} (hf : ContMDiffWithinAt I' I n f s a) (ha : Ne (f a) 0) :
          ContMDiffWithinAt I' I n (fun (x : M) => Inv.inv (f x)) s a

          Alias of ContMDiffWithinAt.inv₀.

          @[deprecated ContMDiffAt.inv₀ (since := "2024-11-21")]
          theorem SmoothAt.inv₀ {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {n : WithTop ENat} {H : Type u_2} [TopologicalSpace H] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [Inv G] [Zero G] {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 : MG} [ContMDiffInv₀ I n G] {a : M} (hf : ContMDiffAt I' I n f a) (ha : Ne (f a) 0) :
          ContMDiffAt I' I n (fun (x : M) => Inv.inv (f x)) a

          Alias of ContMDiffAt.inv₀.

          @[deprecated ContMDiffOn.inv₀ (since := "2024-11-21")]
          theorem SmoothOn.inv₀ {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {n : WithTop ENat} {H : Type u_2} [TopologicalSpace H] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [Inv G] [Zero G] {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 : MG} [ContMDiffInv₀ I n G] {s : Set M} (hf : ContMDiffOn I' I n f s) (h0 : ∀ (x : M), Membership.mem s xNe (f x) 0) :
          ContMDiffOn I' I n (fun (x : M) => Inv.inv (f x)) s

          Alias of ContMDiffOn.inv₀.

          @[deprecated ContMDiff.inv₀ (since := "2024-11-21")]
          theorem Smooth.inv₀ {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {n : WithTop ENat} {H : Type u_2} [TopologicalSpace H] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [Inv G] [Zero G] {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 : MG} [ContMDiffInv₀ I n G] (hf : ContMDiff I' I n f) (h0 : ∀ (x : M), Ne (f x) 0) :
          ContMDiff I' I n fun (x : M) => Inv.inv (f x)

          Alias of ContMDiff.inv₀.

          Point-wise division of C^n functions #

          If [ContMDiffMul I n N] and [ContMDiffInv₀ I n N], point-wise division of C^n functions f : M → N is C^n whenever the denominator is non-zero. (This includes N being a completely normed field.)

          theorem ContMDiffWithinAt.div₀ {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {n : WithTop ENat} {H : Type u_2} [TopologicalSpace H] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [GroupWithZero G] [ContMDiffInv₀ I n G] [ContMDiffMul I n G] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_7} [TopologicalSpace M] [ChartedSpace H' M] {f g : MG} {s : Set M} {a : M} (hf : ContMDiffWithinAt I' I n f s a) (hg : ContMDiffWithinAt I' I n g s a) (h₀ : Ne (g a) 0) :
          ContMDiffWithinAt I' I n (HDiv.hDiv f g) s a
          theorem ContMDiffOn.div₀ {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {n : WithTop ENat} {H : Type u_2} [TopologicalSpace H] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [GroupWithZero G] [ContMDiffInv₀ I n G] [ContMDiffMul I n G] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_7} [TopologicalSpace M] [ChartedSpace H' M] {f g : MG} {s : Set M} (hf : ContMDiffOn I' I n f s) (hg : ContMDiffOn I' I n g s) (h₀ : ∀ (x : M), Membership.mem s xNe (g x) 0) :
          ContMDiffOn I' I n (HDiv.hDiv f g) s
          theorem ContMDiffAt.div₀ {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {n : WithTop ENat} {H : Type u_2} [TopologicalSpace H] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [GroupWithZero G] [ContMDiffInv₀ I n G] [ContMDiffMul I n G] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_7} [TopologicalSpace M] [ChartedSpace H' M] {f g : MG} {a : M} (hf : ContMDiffAt I' I n f a) (hg : ContMDiffAt I' I n g a) (h₀ : Ne (g a) 0) :
          ContMDiffAt I' I n (HDiv.hDiv f g) a
          theorem ContMDiff.div₀ {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {n : WithTop ENat} {H : Type u_2} [TopologicalSpace H] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [GroupWithZero G] [ContMDiffInv₀ I n G] [ContMDiffMul I n G] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_7} [TopologicalSpace M] [ChartedSpace H' M] {f g : MG} (hf : ContMDiff I' I n f) (hg : ContMDiff I' I n g) (h₀ : ∀ (x : M), Ne (g x) 0) :
          ContMDiff I' I n (HDiv.hDiv f g)
          @[deprecated ContMDiffWithinAt.div₀ (since := "2024-11-21")]
          theorem SmoothWithinAt.div₀ {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {n : WithTop ENat} {H : Type u_2} [TopologicalSpace H] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [GroupWithZero G] [ContMDiffInv₀ I n G] [ContMDiffMul I n G] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_7} [TopologicalSpace M] [ChartedSpace H' M] {f g : MG} {s : Set M} {a : M} (hf : ContMDiffWithinAt I' I n f s a) (hg : ContMDiffWithinAt I' I n g s a) (h₀ : Ne (g a) 0) :
          ContMDiffWithinAt I' I n (HDiv.hDiv f g) s a

          Alias of ContMDiffWithinAt.div₀.

          @[deprecated ContMDiffAt.div₀ (since := "2024-11-21")]
          theorem SmoothAt.div₀ {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {n : WithTop ENat} {H : Type u_2} [TopologicalSpace H] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [GroupWithZero G] [ContMDiffInv₀ I n G] [ContMDiffMul I n G] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_7} [TopologicalSpace M] [ChartedSpace H' M] {f g : MG} {a : M} (hf : ContMDiffAt I' I n f a) (hg : ContMDiffAt I' I n g a) (h₀ : Ne (g a) 0) :
          ContMDiffAt I' I n (HDiv.hDiv f g) a

          Alias of ContMDiffAt.div₀.

          @[deprecated ContMDiffOn.div₀ (since := "2024-11-21")]
          theorem SmoothOn.div₀ {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {n : WithTop ENat} {H : Type u_2} [TopologicalSpace H] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [GroupWithZero G] [ContMDiffInv₀ I n G] [ContMDiffMul I n G] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_7} [TopologicalSpace M] [ChartedSpace H' M] {f g : MG} {s : Set M} (hf : ContMDiffOn I' I n f s) (hg : ContMDiffOn I' I n g s) (h₀ : ∀ (x : M), Membership.mem s xNe (g x) 0) :
          ContMDiffOn I' I n (HDiv.hDiv f g) s

          Alias of ContMDiffOn.div₀.

          @[deprecated ContMDiff.div₀ (since := "2024-11-21")]
          theorem Smooth.div₀ {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {n : WithTop ENat} {H : Type u_2} [TopologicalSpace H] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [GroupWithZero G] [ContMDiffInv₀ I n G] [ContMDiffMul I n G] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_7} [TopologicalSpace M] [ChartedSpace H' M] {f g : MG} (hf : ContMDiff I' I n f) (hg : ContMDiff I' I n g) (h₀ : ∀ (x : M), Ne (g x) 0) :
          ContMDiff I' I n (HDiv.hDiv f g)

          Alias of ContMDiff.div₀.