mathlib3 documentation

geometry.manifold.cont_mdiff

Smooth functions between smooth manifolds #

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We define Cⁿ functions between smooth manifolds, as functions which are Cⁿ in charts, and prove basic properties of these notions.

Main definitions and statements #

Let M and M' be two smooth manifolds, with respect to model with corners I and I'. Let f : M → M'.

We also give many basic properties of smooth functions between manifolds, following the API of smooth functions between vector spaces.

Implementation details #

Many properties follow for free from the corresponding properties of functions in vector spaces, as being Cⁿ is a local property invariant under the smooth groupoid. We take advantage of the general machinery developed in local_invariant_properties.lean to get these properties automatically. For instance, the fact that being Cⁿ does not depend on the chart one considers is given by lift_prop_within_at_indep_chart.

For this to work, the definition of cont_mdiff_within_at and friends has to follow definitionally the setup of local invariant properties. Still, we recast the definition in terms of extended charts in cont_mdiff_on_iff and cont_mdiff_iff.

Definition of smooth functions between manifolds #

def cont_diff_within_at_prop {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] (I : model_with_corners 𝕜 E H) {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] (I' : model_with_corners 𝕜 E' H') (n : ℕ∞) (f : H H') (s : set H) (x : H) :
Prop

Property in the model space of a model with corners of being C^n within at set at a point, when read in the model vector space. This property will be lifted to manifolds to define smooth functions between manifolds.

Equations
theorem cont_diff_within_at_prop_self_source {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] (I' : model_with_corners 𝕜 E' H') {n : ℕ∞} {f : E H'} {s : set E} {x : E} :
theorem cont_diff_within_at_prop_self {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {n : ℕ∞} {f : E E'} {s : set E} {x : E} :
theorem cont_diff_within_at_prop_self_target {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] (I : model_with_corners 𝕜 E H) {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {n : ℕ∞} {f : H E'} {s : set H} {x : H} :

Being Cⁿ in the model space is a local property, invariant under smooth maps. Therefore, it will lift nicely to manifolds.

theorem cont_diff_within_at_prop_mono_of_mem {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] (I : model_with_corners 𝕜 E H) {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] (I' : model_with_corners 𝕜 E' H') (n : ℕ∞) ⦃s : set H⦄ ⦃x : H⦄ ⦃t : set H⦄ ⦃f : H H'⦄ (hts : s nhds_within x t) (h : cont_diff_within_at_prop I I' n f s x) :
theorem cont_diff_within_at_prop_id {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] (I : model_with_corners 𝕜 E H) {n : ℕ∞} (x : H) :
def cont_mdiff_within_at {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] (I : model_with_corners 𝕜 E H) {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] (I' : model_with_corners 𝕜 E' H') {M' : Type u_7} [topological_space M'] [charted_space H' M'] (n : ℕ∞) (f : M M') (s : set M) (x : M) :
Prop

A function is n times continuously differentiable within a set at a point in a manifold if it is continuous and it is n times continuously differentiable in this set around this point, when read in the preferred chart at this point.

Equations
@[reducible]
def smooth_within_at {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] (I : model_with_corners 𝕜 E H) {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] (I' : model_with_corners 𝕜 E' H') {M' : Type u_7} [topological_space M'] [charted_space H' M'] (f : M M') (s : set M) (x : M) :
Prop

Abbreviation for cont_mdiff_within_at I I' ⊤ f s x. See also documentation for smooth.

Equations
def cont_mdiff_at {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] (I : model_with_corners 𝕜 E H) {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] (I' : model_with_corners 𝕜 E' H') {M' : Type u_7} [topological_space M'] [charted_space H' M'] (n : ℕ∞) (f : M M') (x : M) :
Prop

A function is n times continuously differentiable at a point in a manifold if it is continuous and it is n times continuously differentiable around this point, when read in the preferred chart at this point.

Equations
theorem cont_mdiff_at_iff {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] (I : model_with_corners 𝕜 E H) {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] (I' : model_with_corners 𝕜 E' H') {M' : Type u_7} [topological_space M'] [charted_space H' M'] {n : ℕ∞} {f : M M'} {x : M} :
@[reducible]
def smooth_at {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] (I : model_with_corners 𝕜 E H) {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] (I' : model_with_corners 𝕜 E' H') {M' : Type u_7} [topological_space M'] [charted_space H' M'] (f : M M') (x : M) :
Prop

Abbreviation for cont_mdiff_at I I' ⊤ f x. See also documentation for smooth.

Equations
def cont_mdiff_on {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] (I : model_with_corners 𝕜 E H) {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] (I' : model_with_corners 𝕜 E' H') {M' : Type u_7} [topological_space M'] [charted_space H' M'] (n : ℕ∞) (f : M M') (s : set M) :
Prop

A function is n times continuously differentiable in a set of a manifold if it is continuous and, for any pair of points, it is n times continuously differentiable on this set in the charts around these points.

Equations
@[reducible]
def smooth_on {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] (I : model_with_corners 𝕜 E H) {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] (I' : model_with_corners 𝕜 E' H') {M' : Type u_7} [topological_space M'] [charted_space H' M'] (f : M M') (s : set M) :
Prop

Abbreviation for cont_mdiff_on I I' ⊤ f s. See also documentation for smooth.

Equations
def cont_mdiff {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] (I : model_with_corners 𝕜 E H) {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] (I' : model_with_corners 𝕜 E' H') {M' : Type u_7} [topological_space M'] [charted_space H' M'] (n : ℕ∞) (f : M M') :
Prop

A function is n times continuously differentiable in a manifold if it is continuous and, for any pair of points, it is n times continuously differentiable in the charts around these points.

Equations
@[reducible]
def smooth {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] (I : model_with_corners 𝕜 E H) {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] (I' : model_with_corners 𝕜 E' H') {M' : Type u_7} [topological_space M'] [charted_space H' M'] (f : M M') :
Prop

Abbreviation for cont_mdiff I I' ⊤ f. Short note to work with these abbreviations: a lemma of the form cont_mdiff_foo.bar will apply fine to an assumption smooth_foo using dot notation or normal notation. If the consequence bar of the lemma involves cont_diff, it is still better to restate the lemma replacing cont_diff with smooth both in the assumption and in the conclusion, to make it possible to use smooth consistently. This also applies to smooth_at, smooth_on and smooth_within_at.

Equations

Basic properties of smooth functions between manifolds #

theorem cont_mdiff.smooth {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M M'} (h : cont_mdiff I I' f) :
smooth I I' f
theorem smooth.cont_mdiff {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M M'} (h : smooth I I' f) :
theorem cont_mdiff_on.smooth_on {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M M'} {s : set M} (h : cont_mdiff_on I I' f s) :
smooth_on I I' f s
theorem smooth_on.cont_mdiff_on {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M M'} {s : set M} (h : smooth_on I I' f s) :
theorem cont_mdiff_at.smooth_at {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M M'} {x : M} (h : cont_mdiff_at I I' f x) :
smooth_at I I' f x
theorem smooth_at.cont_mdiff_at {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M M'} {x : M} (h : smooth_at I I' f x) :
theorem cont_mdiff_within_at.smooth_within_at {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M M'} {s : set M} {x : M} (h : cont_mdiff_within_at I I' f s x) :
smooth_within_at I I' f s x
theorem smooth_within_at.cont_mdiff_within_at {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M M'} {s : set M} {x : M} (h : smooth_within_at I I' f s x) :
theorem cont_mdiff.cont_mdiff_at {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M M'} {x : M} {n : ℕ∞} (h : cont_mdiff I I' n f) :
cont_mdiff_at I I' n f x
theorem smooth.smooth_at {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M M'} {x : M} (h : smooth I I' f) :
smooth_at I I' f x
theorem cont_mdiff_within_at_univ {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M M'} {x : M} {n : ℕ∞} :
theorem smooth_within_at_univ {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M M'} {x : M} :
theorem cont_mdiff_on_univ {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M M'} {n : ℕ∞} :
theorem smooth_on_univ {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M M'} :
theorem cont_mdiff_within_at_iff {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M M'} {s : set M} {x : M} {n : ℕ∞} :

One can reformulate smoothness within a set at a point as continuity within this set at this point, and smoothness in the corresponding extended chart.

theorem cont_mdiff_within_at_iff' {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M M'} {s : set M} {x : M} {n : ℕ∞} :

One can reformulate smoothness within a set at a point as continuity within this set at this point, and smoothness in the corresponding extended chart. This form states smoothness of f written in such a way that the set is restricted to lie within the domain/codomain of the corresponding charts. Even though this expression is more complicated than the one in cont_mdiff_within_at_iff, it is a smaller set, but their germs at ext_chart_at I x x are equal. It is sometimes useful to rewrite using this in the goal.

theorem cont_mdiff_within_at_iff_target {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M M'} {s : set M} {x : M} {n : ℕ∞} :

One can reformulate smoothness within a set at a point as continuity within this set at this point, and smoothness in the corresponding extended chart in the target.

theorem smooth_within_at_iff {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M M'} {s : set M} {x : M} :
theorem smooth_within_at_iff_target {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M M'} {s : set M} {x : M} :
theorem cont_mdiff_at_iff_target {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M M'} {n : ℕ∞} {x : M} :
theorem smooth_at_iff_target {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M M'} {x : M} :
theorem cont_mdiff_within_at_iff_of_mem_maximal_atlas {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] [Is : smooth_manifold_with_corners I M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] [I's : smooth_manifold_with_corners I' M'] {e : local_homeomorph M H} {e' : local_homeomorph M' H'} {f : M M'} {s : set M} {n : ℕ∞} {x : M} (he : e smooth_manifold_with_corners.maximal_atlas I M) (he' : e' smooth_manifold_with_corners.maximal_atlas I' M') (hx : x e.to_local_equiv.source) (hy : f x e'.to_local_equiv.source) :
theorem cont_mdiff_within_at_iff_image {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] [Is : smooth_manifold_with_corners I M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] [I's : smooth_manifold_with_corners I' M'] {e : local_homeomorph M H} {e' : local_homeomorph M' H'} {f : M M'} {s : set M} {n : ℕ∞} {x : M} (he : e smooth_manifold_with_corners.maximal_atlas I M) (he' : e' smooth_manifold_with_corners.maximal_atlas I' M') (hs : s e.to_local_equiv.source) (hx : x e.to_local_equiv.source) (hy : f x e'.to_local_equiv.source) :
cont_mdiff_within_at I I' n f s x continuous_within_at f s x cont_diff_within_at 𝕜 n ((e'.extend I') f ((e.extend I).symm)) ((e.extend I) '' s) ((e.extend I) x)

An alternative formulation of cont_mdiff_within_at_iff_of_mem_maximal_atlas if the set if s lies in e.source.

theorem cont_mdiff_within_at_iff_of_mem_source {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] [Is : smooth_manifold_with_corners I M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] [I's : smooth_manifold_with_corners I' M'] {f : M M'} {s : set M} {x : M} {n : ℕ∞} {x' : M} {y : M'} (hx : x' (charted_space.chart_at H x).to_local_equiv.source) (hy : f x' (charted_space.chart_at H' y).to_local_equiv.source) :

One can reformulate smoothness within a set at a point as continuity within this set at this point, and smoothness in any chart containing that point.

theorem cont_mdiff_within_at_iff_of_mem_source' {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] [Is : smooth_manifold_with_corners I M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] [I's : smooth_manifold_with_corners I' M'] {f : M M'} {s : set M} {x : M} {n : ℕ∞} {x' : M} {y : M'} (hx : x' (charted_space.chart_at H x).to_local_equiv.source) (hy : f x' (charted_space.chart_at H' y).to_local_equiv.source) :
theorem cont_mdiff_at_iff_of_mem_source {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] [Is : smooth_manifold_with_corners I M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] [I's : smooth_manifold_with_corners I' M'] {f : M M'} {x : M} {n : ℕ∞} {x' : M} {y : M'} (hx : x' (charted_space.chart_at H x).to_local_equiv.source) (hy : f x' (charted_space.chart_at H' y).to_local_equiv.source) :
theorem cont_mdiff_within_at_iff_target_of_mem_source {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] [I's : smooth_manifold_with_corners I' M'] {f : M M'} {s : set M} {n : ℕ∞} {x : M} {y : M'} (hy : f x (charted_space.chart_at H' y).to_local_equiv.source) :
theorem cont_mdiff_at_iff_target_of_mem_source {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] [I's : smooth_manifold_with_corners I' M'] {f : M M'} {n : ℕ∞} {x : M} {y : M'} (hy : f x (charted_space.chart_at H' y).to_local_equiv.source) :
theorem cont_mdiff_within_at_iff_source_of_mem_maximal_atlas {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] [Is : smooth_manifold_with_corners I M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {e : local_homeomorph M H} {f : M M'} {s : set M} {x : M} {n : ℕ∞} (he : e smooth_manifold_with_corners.maximal_atlas I M) (hx : x e.to_local_equiv.source) :
theorem cont_mdiff_within_at_iff_source_of_mem_source {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] [Is : smooth_manifold_with_corners I M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M M'} {s : set M} {x : M} {n : ℕ∞} {x' : M} (hx' : x' (charted_space.chart_at H x).to_local_equiv.source) :
theorem cont_mdiff_at_iff_source_of_mem_source {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] [Is : smooth_manifold_with_corners I M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M M'} {x : M} {n : ℕ∞} {x' : M} (hx' : x' (charted_space.chart_at H x).to_local_equiv.source) :
theorem cont_mdiff_on_iff_of_mem_maximal_atlas {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] [Is : smooth_manifold_with_corners I M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] [I's : smooth_manifold_with_corners I' M'] {e : local_homeomorph M H} {e' : local_homeomorph M' H'} {f : M M'} {s : set M} {n : ℕ∞} (he : e smooth_manifold_with_corners.maximal_atlas I M) (he' : e' smooth_manifold_with_corners.maximal_atlas I' M') (hs : s e.to_local_equiv.source) (h2s : set.maps_to f s e'.to_local_equiv.source) :
cont_mdiff_on I I' n f s continuous_on f s cont_diff_on 𝕜 n ((e'.extend I') f ((e.extend I).symm)) ((e.extend I) '' s)
theorem cont_mdiff_on_iff_of_subset_source {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] [Is : smooth_manifold_with_corners I M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] [I's : smooth_manifold_with_corners I' M'] {f : M M'} {s : set M} {n : ℕ∞} {x : M} {y : M'} (hs : s (charted_space.chart_at H x).to_local_equiv.source) (h2s : set.maps_to f s (charted_space.chart_at H' y).to_local_equiv.source) :

If the set where you want f to be smooth lies entirely in a single chart, and f maps it into a single chart, the smoothness of f on that set can be expressed by purely looking in these charts. Note: this lemma uses ext_chart_at I x '' s instead of (ext_chart_at I x).symm ⁻¹' s to ensure that this set lies in (ext_chart_at I x).target.

theorem cont_mdiff_on_iff {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] [Is : smooth_manifold_with_corners I M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] [I's : smooth_manifold_with_corners I' M'] {f : M M'} {s : set M} {n : ℕ∞} :
cont_mdiff_on I I' n f s continuous_on f s (x : M) (y : M'), cont_diff_on 𝕜 n ((ext_chart_at I' y) f ((ext_chart_at I x).symm)) ((ext_chart_at I x).target ((ext_chart_at I x).symm) ⁻¹' (s f ⁻¹' (ext_chart_at I' y).source))

One can reformulate smoothness on a set as continuity on this set, and smoothness in any extended chart.

theorem cont_mdiff_on_iff_target {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] [Is : smooth_manifold_with_corners I M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] [I's : smooth_manifold_with_corners I' M'] {f : M M'} {s : set M} {n : ℕ∞} :

One can reformulate smoothness on a set as continuity on this set, and smoothness in any extended chart in the target.

theorem smooth_on_iff {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] [Is : smooth_manifold_with_corners I M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] [I's : smooth_manifold_with_corners I' M'] {f : M M'} {s : set M} :
theorem smooth_on_iff_target {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] [Is : smooth_manifold_with_corners I M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] [I's : smooth_manifold_with_corners I' M'] {f : M M'} {s : set M} :
theorem cont_mdiff_iff {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] [Is : smooth_manifold_with_corners I M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] [I's : smooth_manifold_with_corners I' M'] {f : M M'} {n : ℕ∞} :
cont_mdiff I I' n f continuous f (x : M) (y : M'), cont_diff_on 𝕜 n ((ext_chart_at I' y) f ((ext_chart_at I x).symm)) ((ext_chart_at I x).target ((ext_chart_at I x).symm) ⁻¹' (f ⁻¹' (ext_chart_at I' y).source))

One can reformulate smoothness as continuity and smoothness in any extended chart.

theorem cont_mdiff_iff_target {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] [Is : smooth_manifold_with_corners I M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] [I's : smooth_manifold_with_corners I' M'] {f : M M'} {n : ℕ∞} :

One can reformulate smoothness as continuity and smoothness in any extended chart in the target.

theorem smooth_iff {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] [Is : smooth_manifold_with_corners I M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] [I's : smooth_manifold_with_corners I' M'] {f : M M'} :
theorem smooth_iff_target {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] [Is : smooth_manifold_with_corners I M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] [I's : smooth_manifold_with_corners I' M'] {f : M M'} :

Deducing smoothness from higher smoothness #

theorem cont_mdiff_within_at.of_le {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M M'} {s : set M} {x : M} {m n : ℕ∞} (hf : cont_mdiff_within_at I I' n f s x) (le : m n) :
theorem cont_mdiff_at.of_le {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M M'} {x : M} {m n : ℕ∞} (hf : cont_mdiff_at I I' n f x) (le : m n) :
cont_mdiff_at I I' m f x
theorem cont_mdiff_on.of_le {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M M'} {s : set M} {m n : ℕ∞} (hf : cont_mdiff_on I I' n f s) (le : m n) :
cont_mdiff_on I I' m f s
theorem cont_mdiff.of_le {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M M'} {m n : ℕ∞} (hf : cont_mdiff I I' n f) (le : m n) :
cont_mdiff I I' m f

Deducing smoothness from smoothness one step beyond #

theorem cont_mdiff_within_at.of_succ {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M M'} {s : set M} {x : M} {n : } (h : cont_mdiff_within_at I I' (n.succ) f s x) :
theorem cont_mdiff_at.of_succ {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M M'} {x : M} {n : } (h : cont_mdiff_at I I' (n.succ) f x) :
cont_mdiff_at I I' n f x
theorem cont_mdiff_on.of_succ {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M M'} {s : set M} {n : } (h : cont_mdiff_on I I' (n.succ) f s) :
cont_mdiff_on I I' n f s
theorem cont_mdiff.of_succ {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M M'} {n : } (h : cont_mdiff I I' (n.succ) f) :
cont_mdiff I I' n f

Deducing continuity from smoothness #

theorem cont_mdiff_within_at.continuous_within_at {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M M'} {s : set M} {x : M} {n : ℕ∞} (hf : cont_mdiff_within_at I I' n f s x) :
theorem cont_mdiff_at.continuous_at {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M M'} {x : M} {n : ℕ∞} (hf : cont_mdiff_at I I' n f x) :
theorem cont_mdiff_on.continuous_on {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M M'} {s : set M} {n : ℕ∞} (hf : cont_mdiff_on I I' n f s) :
theorem cont_mdiff.continuous {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M M'} {n : ℕ∞} (hf : cont_mdiff I I' n f) :

C^∞ smoothness #

theorem cont_mdiff_within_at_top {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M M'} {s : set M} {x : M} :
smooth_within_at I I' f s x (n : ), cont_mdiff_within_at I I' n f s x
theorem cont_mdiff_at_top {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M M'} {x : M} :
smooth_at I I' f x (n : ), cont_mdiff_at I I' n f x
theorem cont_mdiff_on_top {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M M'} {s : set M} :
smooth_on I I' f s (n : ), cont_mdiff_on I I' n f s
theorem cont_mdiff_top {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M M'} :
smooth I I' f (n : ), cont_mdiff I I' n f
theorem cont_mdiff_within_at_iff_nat {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M M'} {s : set M} {x : M} {n : ℕ∞} :
cont_mdiff_within_at I I' n f s x (m : ), m n cont_mdiff_within_at I I' m f s x

Restriction to a smaller set #

theorem cont_mdiff_within_at.mono_of_mem {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M M'} {s t : set M} {x : M} {n : ℕ∞} (hf : cont_mdiff_within_at I I' n f s x) (hts : s nhds_within x t) :
theorem cont_mdiff_within_at.mono {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M M'} {s t : set M} {x : M} {n : ℕ∞} (hf : cont_mdiff_within_at I I' n f s x) (hts : t s) :
theorem cont_mdiff_within_at_congr_nhds {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M M'} {s t : set M} {x : M} {n : ℕ∞} (hst : nhds_within x s = nhds_within x t) :
theorem cont_mdiff_at.cont_mdiff_within_at {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M M'} {s : set M} {x : M} {n : ℕ∞} (hf : cont_mdiff_at I I' n f x) :
theorem smooth_at.smooth_within_at {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M M'} {s : set M} {x : M} (hf : smooth_at I I' f x) :
smooth_within_at I I' f s x
theorem cont_mdiff_on.mono {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M M'} {s t : set M} {n : ℕ∞} (hf : cont_mdiff_on I I' n f s) (hts : t s) :
cont_mdiff_on I I' n f t
theorem cont_mdiff.cont_mdiff_on {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M M'} {s : set M} {n : ℕ∞} (hf : cont_mdiff I I' n f) :
cont_mdiff_on I I' n f s
theorem smooth.smooth_on {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M M'} {s : set M} (hf : smooth I I' f) :
smooth_on I I' f s
theorem cont_mdiff_within_at_inter' {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M M'} {s t : set M} {x : M} {n : ℕ∞} (ht : t nhds_within x s) :
cont_mdiff_within_at I I' n f (s t) x cont_mdiff_within_at I I' n f s x
theorem cont_mdiff_within_at_inter {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M M'} {s t : set M} {x : M} {n : ℕ∞} (ht : t nhds x) :
cont_mdiff_within_at I I' n f (s t) x cont_mdiff_within_at I I' n f s x
theorem cont_mdiff_within_at.cont_mdiff_at {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M M'} {s : set M} {x : M} {n : ℕ∞} (h : cont_mdiff_within_at I I' n f s x) (ht : s nhds x) :
cont_mdiff_at I I' n f x
theorem smooth_within_at.smooth_at {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M M'} {s : set M} {x : M} (h : smooth_within_at I I' f s x) (ht : s nhds x) :
smooth_at I I' f x
theorem cont_mdiff_on.cont_mdiff_at {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M M'} {s : set M} {x : M} {n : ℕ∞} (h : cont_mdiff_on I I' n f s) (hx : s nhds x) :
cont_mdiff_at I I' n f x
theorem smooth_on.smooth_at {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M M'} {s : set M} {x : M} (h : smooth_on I I' f s) (hx : s nhds x) :
smooth_at I I' f x
theorem cont_mdiff_on_iff_source_of_mem_maximal_atlas {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] [Is : smooth_manifold_with_corners I M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {e : local_homeomorph M H} {f : M M'} {s : set M} {n : ℕ∞} (he : e smooth_manifold_with_corners.maximal_atlas I M) (hs : s e.to_local_equiv.source) :
cont_mdiff_on I I' n f s cont_mdiff_on (model_with_corners_self 𝕜 E) I' n (f ((e.extend I).symm)) ((e.extend I) '' s)
theorem cont_mdiff_within_at_iff_cont_mdiff_on_nhds {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] [Is : smooth_manifold_with_corners I M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] [I's : smooth_manifold_with_corners I' M'] {f : M M'} {s : set M} {x : M} {n : } :
cont_mdiff_within_at I I' n f s x (u : set M) (H_1 : u nhds_within x (has_insert.insert x s)), cont_mdiff_on I I' n f u

A function is C^n within a set at a point, for n : ℕ, if and only if it is C^n on a neighborhood of this point.

theorem cont_mdiff_at_iff_cont_mdiff_on_nhds {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] [Is : smooth_manifold_with_corners I M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] [I's : smooth_manifold_with_corners I' M'] {f : M M'} {x : M} {n : } :
cont_mdiff_at I I' n f x (u : set M) (H_1 : u nhds x), cont_mdiff_on I I' n f u

A function is C^n at a point, for n : ℕ, if and only if it is C^n on a neighborhood of this point.

theorem cont_mdiff_at_iff_cont_mdiff_at_nhds {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] [Is : smooth_manifold_with_corners I M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] [I's : smooth_manifold_with_corners I' M'] {f : M M'} {x : M} {n : } :
cont_mdiff_at I I' n f x ∀ᶠ (x' : M) in nhds x, cont_mdiff_at I I' n f x'

Note: This does not hold for n = ∞. f being C^∞ at x means that for every n, f is C^n on some neighborhood of x, but this neighborhood can depend on n.

Congruence lemmas #

theorem cont_mdiff_within_at.congr {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f f₁ : M M'} {s : set M} {x : M} {n : ℕ∞} (h : cont_mdiff_within_at I I' n f s x) (h₁ : (y : M), y s f₁ y = f y) (hx : f₁ x = f x) :
cont_mdiff_within_at I I' n f₁ s x
theorem cont_mdiff_within_at_congr {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f f₁ : M M'} {s : set M} {x : M} {n : ℕ∞} (h₁ : (y : M), y s f₁ y = f y) (hx : f₁ x = f x) :
cont_mdiff_within_at I I' n f₁ s x cont_mdiff_within_at I I' n f s x
theorem cont_mdiff_within_at.congr_of_eventually_eq {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f f₁ : M M'} {s : set M} {x : M} {n : ℕ∞} (h : cont_mdiff_within_at I I' n f s x) (h₁ : f₁ =ᶠ[nhds_within x s] f) (hx : f₁ x = f x) :
cont_mdiff_within_at I I' n f₁ s x
theorem filter.eventually_eq.cont_mdiff_within_at_iff {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f f₁ : M M'} {s : set M} {x : M} {n : ℕ∞} (h₁ : f₁ =ᶠ[nhds_within x s] f) (hx : f₁ x = f x) :
cont_mdiff_within_at I I' n f₁ s x cont_mdiff_within_at I I' n f s x
theorem cont_mdiff_at.congr_of_eventually_eq {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f f₁ : M M'} {x : M} {n : ℕ∞} (h : cont_mdiff_at I I' n f x) (h₁ : f₁ =ᶠ[nhds x] f) :
cont_mdiff_at I I' n f₁ x
theorem filter.eventually_eq.cont_mdiff_at_iff {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f f₁ : M M'} {x : M} {n : ℕ∞} (h₁ : f₁ =ᶠ[nhds x] f) :
cont_mdiff_at I I' n f₁ x cont_mdiff_at I I' n f x
theorem cont_mdiff_on.congr {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f f₁ : M M'} {s : set M} {n : ℕ∞} (h : cont_mdiff_on I I' n f s) (h₁ : (y : M), y s f₁ y = f y) :
cont_mdiff_on I I' n f₁ s
theorem cont_mdiff_on_congr {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f f₁ : M M'} {s : set M} {n : ℕ∞} (h₁ : (y : M), y s f₁ y = f y) :
cont_mdiff_on I I' n f₁ s cont_mdiff_on I I' n f s

Locality #

theorem cont_mdiff_on_of_locally_cont_mdiff_on {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M M'} {s : set M} {n : ℕ∞} (h : (x : M), x s ( (u : set M), is_open u x u cont_mdiff_on I I' n f (s u))) :
cont_mdiff_on I I' n f s

Being C^n is a local property.

theorem cont_mdiff_of_locally_cont_mdiff_on {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {f : M M'} {n : ℕ∞} (h : (x : M), (u : set M), is_open u x u cont_mdiff_on I I' n f u) :
cont_mdiff I I' n f

Smoothness of the composition of smooth functions between manifolds #

theorem cont_mdiff_within_at.comp {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {E'' : Type u_8} [normed_add_comm_group E''] [normed_space 𝕜 E''] {H'' : Type u_9} [topological_space H''] {I'' : model_with_corners 𝕜 E'' H''} {M'' : Type u_10} [topological_space M''] [charted_space H'' M''] {f : M M'} {s : set M} {n : ℕ∞} {t : set M'} {g : M' M''} (x : M) (hg : cont_mdiff_within_at I' I'' n g t (f x)) (hf : cont_mdiff_within_at I I' n f s x) (st : set.maps_to f s t) :
cont_mdiff_within_at I I'' n (g f) s x

The composition of C^n functions within domains at points is C^n.

theorem cont_mdiff_within_at.comp_of_eq {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {E'' : Type u_8} [normed_add_comm_group E''] [normed_space 𝕜 E''] {H'' : Type u_9} [topological_space H''] {I'' : model_with_corners 𝕜 E'' H''} {M'' : Type u_10} [topological_space M''] [charted_space H'' M''] {f : M M'} {s : set M} {n : ℕ∞} {t : set M'} {g : M' M''} {x : M} {y : M'} (hg : cont_mdiff_within_at I' I'' n g t y) (hf : cont_mdiff_within_at I I' n f s x) (st : set.maps_to f s t) (hx : f x = y) :
cont_mdiff_within_at I I'' n (g f) s x

See note [comp_of_eq lemmas]

theorem smooth_within_at.comp {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {E'' : Type u_8} [normed_add_comm_group E''] [normed_space 𝕜 E''] {H'' : Type u_9} [topological_space H''] {I'' : model_with_corners 𝕜 E'' H''} {M'' : Type u_10} [topological_space M''] [charted_space H'' M''] {f : M M'} {s : set M} {t : set M'} {g : M' M''} (x : M) (hg : smooth_within_at I' I'' g t (f x)) (hf : smooth_within_at I I' f s x) (st : set.maps_to f s t) :
smooth_within_at I I'' (g f) s x

The composition of C^∞ functions within domains at points is C^∞.

theorem cont_mdiff_on.comp {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {E'' : Type u_8} [normed_add_comm_group E''] [normed_space 𝕜 E''] {H'' : Type u_9} [topological_space H''] {I'' : model_with_corners 𝕜 E'' H''} {M'' : Type u_10} [topological_space M''] [charted_space H'' M''] {f : M M'} {s : set M} {n : ℕ∞} {t : set M'} {g : M' M''} (hg : cont_mdiff_on I' I'' n g t) (hf : cont_mdiff_on I I' n f s) (st : s f ⁻¹' t) :
cont_mdiff_on I I'' n (g f) s

The composition of C^n functions on domains is C^n.

theorem smooth_on.comp {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {E'' : Type u_8} [normed_add_comm_group E''] [normed_space 𝕜 E''] {H'' : Type u_9} [topological_space H''] {I'' : model_with_corners 𝕜 E'' H''} {M'' : Type u_10} [topological_space M''] [charted_space H'' M''] {f : M M'} {s : set M} {t : set M'} {g : M' M''} (hg : smooth_on I' I'' g t) (hf : smooth_on I I' f s) (st : s f ⁻¹' t) :
smooth_on I I'' (g f) s

The composition of C^∞ functions on domains is C^∞.

theorem cont_mdiff_on.comp' {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {E'' : Type u_8} [normed_add_comm_group E''] [normed_space 𝕜 E''] {H'' : Type u_9} [topological_space H''] {I'' : model_with_corners 𝕜 E'' H''} {M'' : Type u_10} [topological_space M''] [charted_space H'' M''] {f : M M'} {s : set M} {n : ℕ∞} {t : set M'} {g : M' M''} (hg : cont_mdiff_on I' I'' n g t) (hf : cont_mdiff_on I I' n f s) :
cont_mdiff_on I I'' n (g f) (s f ⁻¹' t)

The composition of C^n functions on domains is C^n.

theorem smooth_on.comp' {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {E'' : Type u_8} [normed_add_comm_group E''] [normed_space 𝕜 E''] {H'' : Type u_9} [topological_space H''] {I'' : model_with_corners 𝕜 E'' H''} {M'' : Type u_10} [topological_space M''] [charted_space H'' M''] {f : M M'} {s : set M} {t : set M'} {g : M' M''} (hg : smooth_on I' I'' g t) (hf : smooth_on I I' f s) :
smooth_on I I'' (g f) (s f ⁻¹' t)

The composition of C^∞ functions is C^∞.

theorem cont_mdiff.comp {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {E'' : Type u_8} [normed_add_comm_group E''] [normed_space 𝕜 E''] {H'' : Type u_9} [topological_space H''] {I'' : model_with_corners 𝕜 E'' H''} {M'' : Type u_10} [topological_space M''] [charted_space H'' M''] {f : M M'} {n : ℕ∞} {g : M' M''} (hg : cont_mdiff I' I'' n g) (hf : cont_mdiff I I' n f) :
cont_mdiff I I'' n (g f)

The composition of C^n functions is C^n.

theorem smooth.comp {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {E'' : Type u_8} [normed_add_comm_group E''] [normed_space 𝕜 E''] {H'' : Type u_9} [topological_space H''] {I'' : model_with_corners 𝕜 E'' H''} {M'' : Type u_10} [topological_space M''] [charted_space H'' M''] {f : M M'} {g : M' M''} (hg : smooth I' I'' g) (hf : smooth I I' f) :
smooth I I'' (g f)

The composition of C^∞ functions is C^∞.

theorem cont_mdiff_within_at.comp' {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {E'' : Type u_8} [normed_add_comm_group E''] [normed_space 𝕜 E''] {H'' : Type u_9} [topological_space H''] {I'' : model_with_corners 𝕜 E'' H''} {M'' : Type u_10} [topological_space M''] [charted_space H'' M''] {f : M M'} {s : set M} {n : ℕ∞} {t : set M'} {g : M' M''} (x : M) (hg : cont_mdiff_within_at I' I'' n g t (f x)) (hf : cont_mdiff_within_at I I' n f s x) :
cont_mdiff_within_at I I'' n (g f) (s f ⁻¹' t) x

The composition of C^n functions within domains at points is C^n.

theorem smooth_within_at.comp' {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {E'' : Type u_8} [normed_add_comm_group E''] [normed_space 𝕜 E''] {H'' : Type u_9} [topological_space H''] {I'' : model_with_corners 𝕜 E'' H''} {M'' : Type u_10} [topological_space M''] [charted_space H'' M''] {f : M M'} {s : set M} {t : set M'} {g : M' M''} (x : M) (hg : smooth_within_at I' I'' g t (f x)) (hf : smooth_within_at I I' f s x) :
smooth_within_at I I'' (g f) (s f ⁻¹' t) x

The composition of C^∞ functions within domains at points is C^∞.

theorem cont_mdiff_at.comp_cont_mdiff_within_at {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {E'' : Type u_8} [normed_add_comm_group E''] [normed_space 𝕜 E''] {H'' : Type u_9} [topological_space H''] {I'' : model_with_corners 𝕜 E'' H''} {M'' : Type u_10} [topological_space M''] [charted_space H'' M''] {f : M M'} {s : set M} {n : ℕ∞} {g : M' M''} (x : M) (hg : cont_mdiff_at I' I'' n g (f x)) (hf : cont_mdiff_within_at I I' n f s x) :
cont_mdiff_within_at I I'' n (g f) s x

g ∘ f is C^n within s at x if g is C^n at f x and f is C^n within s at x.

theorem smooth_at.comp_smooth_within_at {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {E'' : Type u_8} [normed_add_comm_group E''] [normed_space 𝕜 E''] {H'' : Type u_9} [topological_space H''] {I'' : model_with_corners 𝕜 E'' H''} {M'' : Type u_10} [topological_space M''] [charted_space H'' M''] {f : M M'} {s : set M} {g : M' M''} (x : M) (hg : smooth_at I' I'' g (f x)) (hf : smooth_within_at I I' f s x) :
smooth_within_at I I'' (g f) s x

g ∘ f is C^∞ within s at x if g is C^∞ at f x and f is C^∞ within s at x.

theorem cont_mdiff_at.comp {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {E'' : Type u_8} [normed_add_comm_group E''] [normed_space 𝕜 E''] {H'' : Type u_9} [topological_space H''] {I'' : model_with_corners 𝕜 E'' H''} {M'' : Type u_10} [topological_space M''] [charted_space H'' M''] {f : M M'} {n : ℕ∞} {g : M' M''} (x : M) (hg : cont_mdiff_at I' I'' n g (f x)) (hf : cont_mdiff_at I I' n f x) :
cont_mdiff_at I I'' n (g f) x

The composition of C^n functions at points is C^n.

theorem cont_mdiff_at.comp_of_eq {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {E'' : Type u_8} [normed_add_comm_group E''] [normed_space 𝕜 E''] {H'' : Type u_9} [topological_space H''] {I'' : model_with_corners 𝕜 E'' H''} {M'' : Type u_10} [topological_space M''] [charted_space H'' M''] {f : M M'} {n : ℕ∞} {g : M' M''} {x : M} {y : M'} (hg : cont_mdiff_at I' I'' n g y) (hf : cont_mdiff_at I I' n f x) (hx : f x = y) :
cont_mdiff_at I I'' n (g f) x

See note [comp_of_eq lemmas]

theorem smooth_at.comp {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {E'' : Type u_8} [normed_add_comm_group E''] [normed_space 𝕜 E''] {H'' : Type u_9} [topological_space H''] {I'' : model_with_corners 𝕜 E'' H''} {M'' : Type u_10} [topological_space M''] [charted_space H'' M''] {f : M M'} {g : M' M''} (x : M) (hg : smooth_at I' I'' g (f x)) (hf : smooth_at I I' f x) :
smooth_at I I'' (g f) x

The composition of C^∞ functions at points is C^∞.

theorem cont_mdiff.comp_cont_mdiff_on {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {E'' : Type u_8} [normed_add_comm_group E''] [normed_space 𝕜 E''] {H'' : Type u_9} [topological_space H''] {I'' : model_with_corners 𝕜 E'' H''} {M'' : Type u_10} [topological_space M''] [charted_space H'' M''] {n : ℕ∞} {f : M M'} {g : M' M''} {s : set M} (hg : cont_mdiff I' I'' n g) (hf : cont_mdiff_on I I' n f s) :
cont_mdiff_on I I'' n (g f) s
theorem smooth.comp_smooth_on {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {E'' : Type u_8} [normed_add_comm_group E''] [normed_space 𝕜 E''] {H'' : Type u_9} [topological_space H''] {I'' : model_with_corners 𝕜 E'' H''} {M'' : Type u_10} [topological_space M''] [charted_space H'' M''] {f : M M'} {g : M' M''} {s : set M} (hg : smooth I' I'' g) (hf : smooth_on I I' f s) :
smooth_on I I'' (g f) s
theorem cont_mdiff_on.comp_cont_mdiff {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {E'' : Type u_8} [normed_add_comm_group E''] [normed_space 𝕜 E''] {H'' : Type u_9} [topological_space H''] {I'' : model_with_corners 𝕜 E'' H''} {M'' : Type u_10} [topological_space M''] [charted_space H'' M''] {f : M M'} {n : ℕ∞} {t : set M'} {g : M' M''} (hg : cont_mdiff_on I' I'' n g t) (hf : cont_mdiff I I' n f) (ht : (x : M), f x t) :
cont_mdiff I I'' n (g f)
theorem smooth_on.comp_smooth {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_4} [topological_space M] [charted_space H M] {E' : Type u_5} [normed_add_comm_group E'] [normed_space 𝕜 E'] {H' : Type u_6} [topological_space H'] {I' : model_with_corners 𝕜 E' H'} {M' : Type u_7} [topological_space M'] [charted_space H' M'] {E'' : Type u_8} [normed_add_comm_group E''] [normed_space 𝕜 E''] {H'' : Type u_9} [topological_space H''] {I'' : model_with_corners 𝕜 E'' H''} {M'' : Type u_10} [topological_space M''] [charted_space H'' M''] {f : M M'} {t : set M'} {g : M' M''} (hg : smooth_on I' I'' g t) (hf : smooth I I' f) (ht : (x : M), f x t) :
smooth I I'' (g f)

Atlas members are smooth #

theorem cont_mdiff_model {𝕜 : Type u_1} [nontrivially_normed_field 𝕜] {E : Type u_2} [normed_add_comm_group E] [normed_space 𝕜 E] {H : Type u_3} [topological_space H] {I : model_with_corners 𝕜 E H} {n : ℕ∞} :

An atlas member is C^n for any n.

The inverse of an atlas member is C^n for any n.