# mathlibdocumentation

geometry.manifold.mfderiv

# The derivative of functions between smooth manifolds

Let `M` and `M'` be two smooth manifolds with corners over a field `π` (with respective models with corners `I` on `(E, H)` and `I'` on `(E', H')`), and let `f : M β M'`. We define the derivative of the function at a point, within a set or along the whole space, mimicking the API for (FrΓ©chet) derivatives. It is denoted by `mfderiv I I' f x`, where "m" stands for "manifold" and "f" for "FrΓ©chet" (as in the usual derivative `fderiv π f x`).

## Main definitions

• `unique_mdiff_on I s` : predicate saying that, at each point of the set `s`, a function can have at most one derivative. This technical condition is important when we define `mfderiv_within` below, as otherwise there is an arbitrary choice in the derivative, and many properties will fail (for instance the chain rule). This is analogous to `unique_diff_on π s` in a vector space.

Let `f` be a map between smooth manifolds. The following definitions follow the `fderiv` API.

• `mfderiv I I' f x` : the derivative of `f` at `x`, as a continuous linear map from the tangent space at `x` to the tangent space at `f x`. If the map is not differentiable, this is `0`.
• `mfderiv_within I I' f s x` : the derivative of `f` at `x` within `s`, as a continuous linear map from the tangent space at `x` to the tangent space at `f x`. If the map is not differentiable within `s`, this is `0`.
• `mdifferentiable_at I I' f x` : Prop expressing whether `f` is differentiable at `x`.
• `mdifferentiable_within_at π f s x` : Prop expressing whether `f` is differentiable within `s` at `x`.
• `has_mfderiv_at I I' f s x f'` : Prop expressing whether `f` has `f'` as a derivative at `x`.
• `has_mfderiv_within_at I I' f s x f'` : Prop expressing whether `f` has `f'` as a derivative within `s` at `x`.
• `mdifferentiable_on I I' f s` : Prop expressing that `f` is differentiable on the set `s`.
• `mdifferentiable I I' f` : Prop expressing that `f` is differentiable everywhere.
• `tangent_map I I' f` : the derivative of `f`, as a map from the tangent bundle of `M` to the tangent bundle of `M'`.

We also establish results on the differential of the identity, constant functions, charts, extended charts. For functions between vector spaces, we show that the usual notions and the manifold notions coincide.

## Implementation notes

The tangent bundle is constructed using the machinery of topological fiber bundles, for which one can define bundled morphisms and construct canonically maps from the total space of one bundle to the total space of another one. One could use this mechanism to construct directly the derivative of a smooth map. However, we want to define the derivative of any map (and let it be zero if the map is not differentiable) to avoid proof arguments everywhere. This means we have to go back to the details of the definition of the total space of a fiber bundle constructed from core, to cook up a suitable definition of the derivative. It is the following: at each point, we have a preferred chart (used to identify the fiber above the point with the model vector space in fiber bundles). Then one should read the function using these preferred charts at `x` and `f x`, and take the derivative of `f` in these charts.

Due to the fact that we are working in a model with corners, with an additional embedding `I` of the model space `H` in the model vector space `E`, the charts taking values in `E` are not the original charts of the manifold, but those ones composed with `I`, called extended charts. We define `written_in_ext_chart I I' x f` for the function `f` written in the preferred extended charts. Then the manifold derivative of `f`, at `x`, is just the usual derivative of `written_in_ext_chart I I' x f`, at the point `(ext_chart_at I x) x`.

There is a subtelty with respect to continuity: if the function is not continuous, then the image of a small open set around `x` will not be contained in the source of the preferred chart around `f x`, which means that when reading `f` in the chart one is losing some information. To avoid this, we include continuity in the definition of differentiablity (which is reasonable since with any definition, differentiability implies continuity).

Warning: the derivative (even within a subset) is a linear map on the whole tangent space. Suppose that one is given a smooth submanifold `N`, and a function which is smooth on `N` (i.e., its restriction to the subtype `N` is smooth). Then, in the whole manifold `M`, the property `mdifferentiable_on I I' f N` holds. However, `mfderiv_within I I' f N` is not uniquely defined (what values would one choose for vectors that are transverse to `N`?), which can create issues down the road. The problem here is that knowing the value of `f` along `N` does not determine the differential of `f` in all directions. This is in contrast to the case where `N` would be an open subset, or a submanifold with boundary of maximal dimension, where this issue does not appear. The predicate `unique_mdiff_on I N` indicates that the derivative along `N` is unique if it exists, and is an assumption in most statements requiring a form of uniqueness.

On a vector space, the manifold derivative and the usual derivative are equal. This means in particular that they live on the same space, i.e., the tangent space is defeq to the original vector space. To get this property is a motivation for our definition of the tangent space as a single copy of the vector space, instead of more usual definitions such as the space of derivations, or the space of equivalence classes of smooth curves in the manifold.

## Tags

Derivative, manifold

### Derivative of maps between manifolds

The derivative of a smooth map `f` between smooth manifold `M` and `M'` at `x` is a bounded linear map from the tangent space to `M` at `x`, to the tangent space to `M'` at `f x`. Since we defined the tangent space using one specific chart, the formula for the derivative is written in terms of this specific chart.

We use the names `mdifferentiable` and `mfderiv`, where the prefix letter `m` means "manifold".

def unique_mdiff_within_at {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} (I : E H) {M : Type u_4} [ M] :
set M β M β Prop

Predicate ensuring that, at a point and within a set, a function can have at most one derivative. This is expressed using the preferred chart at the considered point.

Equations
def unique_mdiff_on {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} (I : E H) {M : Type u_4} [ M] :
set M β Prop

Predicate ensuring that, at all points of a set, a function can have at most one derivative.

Equations
• = β (x : M), x β s β
@[simp]
def written_in_ext_chart_at {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} (I : E H) {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} (I' : E' H') {M' : Type u_7} [ M'] :
M β (M β M') β E β E'

Conjugating a function to write it in the preferred charts around `x`. The manifold derivative of `f` will just be the derivative of this conjugated function.

Equations
def mdifferentiable_within_at {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} (I : E H) {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} (I' : E' H') {M' : Type u_7} [ M'] :
(M β M') β set M β M β Prop

`mdifferentiable_within_at I I' f s x` indicates that the function `f` between manifolds has a derivative at the point `x` within the set `s`. This is a generalization of `differentiable_within_at` to manifolds.

We require continuity in the definition, as otherwise points close to `x` in `s` could be sent by `f` outside of the chart domain around `f x`. Then the chart could do anything to the image points, and in particular by coincidence `written_in_ext_chart_at I I' x f` could be differentiable, while this would not mean anything relevant.

Equations
def mdifferentiable_at {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} (I : E H) {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} (I' : E' H') {M' : Type u_7} [ M'] :
(M β M') β M β Prop

`mdifferentiable_at I I' f x` indicates that the function `f` between manifolds has a derivative at the point `x`. This is a generalization of `differentiable_at` to manifolds.

We require continuity in the definition, as otherwise points close to `x` could be sent by `f` outside of the chart domain around `f x`. Then the chart could do anything to the image points, and in particular by coincidence `written_in_ext_chart_at I I' x f` could be differentiable, while this would not mean anything relevant.

Equations
def mdifferentiable_on {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} (I : E H) {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} (I' : E' H') {M' : Type u_7} [ M'] :
(M β M') β set M β Prop

`mdifferentiable_on I I' f s` indicates that the function `f` between manifolds has a derivative within `s` at all points of `s`. This is a generalization of `differentiable_on` to manifolds.

Equations
• f s = β (x : M), x β s β s x
def mdifferentiable {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} (I : E H) {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} (I' : E' H') {M' : Type u_7} [ M'] :
(M β M') β Prop

`mdifferentiable I I' f` indicates that the function `f` between manifolds has a derivative everywhere. This is a generalization of `differentiable` to manifolds.

Equations
• I' f = β (x : M), f x
def local_homeomorph.mdifferentiable {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} (I : E H) {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} (I' : E' H') {M' : Type u_7} [ M'] :
M' β Prop

Prop registering if a local homeomorphism is a local diffeomorphism on its source

Equations
def has_mfderiv_within_at {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} (I : E H) {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} (I' : E' H') {M' : Type u_7} [ M'] [ M'] (f : M β M') (s : set M) (x : M) :
x βL[π] (f x)) β Prop

`has_mfderiv_within_at I I' f s x f'` indicates that the function `f` between manifolds has, at the point `x` and within the set `s`, the derivative `f'`. Here, `f'` is a continuous linear map from the tangent space at `x` to the tangent space at `f x`.

This is a generalization of `has_fderiv_within_at` to manifolds (as indicated by the prefix `m`). The order of arguments is changed as the type of the derivative `f'` depends on the choice of `x`.

We require continuity in the definition, as otherwise points close to `x` in `s` could be sent by `f` outside of the chart domain around `f x`. Then the chart could do anything to the image points, and in particular by coincidence `written_in_ext_chart_at I I' x f` could be differentiable, while this would not mean anything relevant.

Equations
def has_mfderiv_at {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} (I : E H) {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} (I' : E' H') {M' : Type u_7} [ M'] [ M'] (f : M β M') (x : M) :
x βL[π] (f x)) β Prop

`has_mfderiv_at I I' f x f'` indicates that the function `f` between manifolds has, at the point `x`, the derivative `f'`. Here, `f'` is a continuous linear map from the tangent space at `x` to the tangent space at `f x`.

We require continuity in the definition, as otherwise points close to `x` `s` could be sent by `f` outside of the chart domain around `f x`. Then the chart could do anything to the image points, and in particular by coincidence `written_in_ext_chart_at I I' x f` could be differentiable, while this would not mean anything relevant.

Equations
def mfderiv_within {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} (I : E H) {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} (I' : E' H') {M' : Type u_7} [ M'] [ M'] (f : M β M') (s : set M) (x : M) :
βL[π] (f x)

Let `f` be a function between two smooth manifolds. Then `mfderiv_within I I' f s x` is the derivative of `f` at `x` within `s`, as a continuous linear map from the tangent space at `x` to the tangent space at `f x`.

Equations
def mfderiv {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} (I : E H) {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} (I' : E' H') {M' : Type u_7} [ M'] [ M'] (f : M β M') (x : M) :
βL[π] (f x)

Let `f` be a function between two smooth manifolds. Then `mfderiv I I' f x` is the derivative of `f` at `x`, as a continuous linear map from the tangent space at `x` to the tangent space at `f x`.

Equations
def tangent_map_within {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} (I : E H) {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} (I' : E' H') {M' : Type u_7} [ M'] [ M'] :
(M β M') β set M β β M'

The derivative within a set, as a map between the tangent bundles

Equations
def tangent_map {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} (I : E H) {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} (I' : E' H') {M' : Type u_7} [ M'] [ M'] :
(M β M') β β M'

The derivative, as a map between the tangent bundles

Equations

### Unique differentiability sets in manifolds

theorem unique_mdiff_within_at_univ {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} (I : E H) {M : Type u_4} [ M] {x : M} :

theorem unique_mdiff_within_at_iff {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {s : set M} {x : M} :

theorem unique_mdiff_within_at.mono {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {x : M} {s t : set M} :
β s β t β

theorem unique_mdiff_within_at.inter' {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {x : M} {s t : set M} :
β t β π[s] x β (s β© t) x

theorem unique_mdiff_within_at.inter {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {x : M} {s t : set M} :
β t β π x β (s β© t) x

theorem is_open.unique_mdiff_within_at {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {x : M} {s : set M} :
x β s β β

theorem unique_mdiff_on.inter {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {s t : set M} :

theorem is_open.unique_mdiff_on {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {s : set M} :
β

theorem unique_mdiff_on_univ {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] :

theorem unique_mdiff_within_at.eq {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {f : M β M'} {x : M} {s : set M} [Is : M] [I's : M'] {f' fβ' : βL[π] (f x)} :
β f s x f' β f s x fβ' β f' = fβ'

`unique_mdiff_within_at` achieves its goal: it implies the uniqueness of the derivative.

theorem unique_mdiff_on.eq {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {f : M β M'} {x : M} {s : set M} [Is : M] [I's : M'] {f' fβ' : βL[π] (f x)} :
β x β s β f s x f' β f s x fβ' β f' = fβ'

### General lemmas on derivatives of functions between manifolds

We mimick the API for functions between vector spaces

theorem mdifferentiable_within_at_iff {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {f : M β M'} {s : set M} {x : M} :

theorem mfderiv_within_zero_of_not_mdifferentiable_within_at {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {f : M β M'} {x : M} {s : set M} [Is : M] [I's : M'] :
Β¬ s x β I' f s x = 0

theorem mfderiv_zero_of_not_mdifferentiable_at {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {f : M β M'} {x : M} [Is : M] [I's : M'] :
Β¬ f x β I' f x = 0

theorem has_mfderiv_within_at.mono {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {f : M β M'} {x : M} {s t : set M} [Is : M] [I's : M'] {f' : βL[π] (f x)} :
f t x f' β s β t β f s x f'

theorem has_mfderiv_at.has_mfderiv_within_at {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {f : M β M'} {x : M} {s : set M} [Is : M] [I's : M'] {f' : βL[π] (f x)} :
I' f x f' β f s x f'

theorem has_mfderiv_within_at.mdifferentiable_within_at {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {f : M β M'} {x : M} {s : set M} [Is : M] [I's : M'] {f' : βL[π] (f x)} :
f s x f' β s x

theorem has_mfderiv_at.mdifferentiable_at {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {f : M β M'} {x : M} [Is : M] [I's : M'] {f' : βL[π] (f x)} :
I' f x f' β f x

@[simp]
theorem has_mfderiv_within_at_univ {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {f : M β M'} {x : M} [Is : M] [I's : M'] {f' : βL[π] (f x)} :
x f' β I' f x f'

theorem has_mfderiv_at_unique {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {f : M β M'} {x : M} [Is : M] [I's : M'] {fβ' fβ' : βL[π] (f x)} :
I' f x fβ' β I' f x fβ' β fβ' = fβ'

theorem has_mfderiv_within_at_inter' {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {f : M β M'} {x : M} {s t : set M} [Is : M] [I's : M'] {f' : βL[π] (f x)} :
t β π[s] x β f (s β© t) x f' β f s x f')

theorem has_mfderiv_within_at_inter {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {f : M β M'} {x : M} {s t : set M} [Is : M] [I's : M'] {f' : βL[π] (f x)} :
t β π x β f (s β© t) x f' β f s x f')

theorem has_mfderiv_within_at.union {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {f : M β M'} {x : M} {s t : set M} [Is : M] [I's : M'] {f' : βL[π] (f x)} :
f s x f' β f t x f' β f (s βͺ t) x f'

theorem has_mfderiv_within_at.nhds_within {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {f : M β M'} {x : M} {s t : set M} [Is : M] [I's : M'] {f' : βL[π] (f x)} :
f s x f' β s β π[t] x β f t x f'

theorem has_mfderiv_within_at.has_mfderiv_at {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {f : M β M'} {x : M} {s : set M} [Is : M] [I's : M'] {f' : βL[π] (f x)} :
f s x f' β s β π x β I' f x f'

theorem mdifferentiable_within_at.has_mfderiv_within_at {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {f : M β M'} {x : M} {s : set M} [Is : M] [I's : M'] :
s x β f s x I' f s x)

theorem mdifferentiable_within_at.mfderiv_within {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {f : M β M'} {x : M} {s : set M} [Is : M] [I's : M'] :
s x β I' f s x = fderiv_within π x f) (β((ext_chart_at I x).symm) β»ΒΉ' s β© (β x) x)

theorem mdifferentiable_at.has_mfderiv_at {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {f : M β M'} {x : M} [Is : M] [I's : M'] :
f x β I' f x (mfderiv I I' f x)

theorem mdifferentiable_at.mfderiv {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {f : M β M'} {x : M} [Is : M] [I's : M'] :
f x β I' f x = fderiv_within π x f) (set.range βI) (β x) x)

theorem has_mfderiv_at.mfderiv {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {f : M β M'} {x : M} [Is : M] [I's : M'] {f' : βL[π] (f x)} :
I' f x f' β I' f x = f'

theorem has_mfderiv_within_at.mfderiv_within {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {f : M β M'} {x : M} {s : set M} [Is : M] [I's : M'] {f' : βL[π] (f x)} :
f s x f' β β I' f s x = f'

theorem mdifferentiable.mfderiv_within {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {f : M β M'} {x : M} {s : set M} [Is : M] [I's : M'] :
f x β β I' f s x = I' f x

theorem mfderiv_within_subset {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {f : M β M'} {x : M} {s t : set M} [Is : M] [I's : M'] :
s β t β β t x β I' f s x = I' f t x

theorem mdifferentiable_within_at.mono {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {f : M β M'} {x : M} {s t : set M} :
s β t β t x β s x

theorem mdifferentiable_within_at_univ {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {f : M β M'} {x : M} :
x β f x

theorem mdifferentiable_within_at_inter {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {f : M β M'} {x : M} {s t : set M} :
t β π x β f (s β© t) x β s x)

theorem mdifferentiable_within_at_inter' {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {f : M β M'} {x : M} {s t : set M} :
t β π[s] x β f (s β© t) x β s x)

theorem mdifferentiable_at.mdifferentiable_within_at {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {f : M β M'} {x : M} {s : set M} :
f x β s x

theorem mdifferentiable_within_at.mdifferentiable_at {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {f : M β M'} {x : M} {s : set M} :
s x β s β π x β f x

theorem mdifferentiable_on.mono {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {f : M β M'} {s t : set M} :
f t β s β t β f s

theorem mdifferentiable_on_univ {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {f : M β M'} :

theorem mdifferentiable.mdifferentiable_on {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {f : M β M'} {s : set M} :
I' f β f s

theorem mdifferentiable_on_of_locally_mdifferentiable_on {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {f : M β M'} {s : set M} :
(β (x : M), x β s β (β (u : set M), β§ x β u β§ f (s β© u))) β f s

@[simp]
theorem mfderiv_within_univ {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {f : M β M'} [Is : M] [I's : M'] :
I' f set.univ = I' f

theorem mfderiv_within_inter {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {f : M β M'} {x : M} {s t : set M} [Is : M] [I's : M'] :
t β π x β β I' f (s β© t) x = I' f s x

### Deriving continuity from differentiability on manifolds

theorem has_mfderiv_within_at.continuous_within_at {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {f : M β M'} {x : M} {s : set M} :
s x β x

theorem has_mfderiv_at.continuous_at {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {f : M β M'} {x : M} [Is : M] [I's : M'] {f' : βL[π] (f x)} :
I' f x f' β

theorem mdifferentiable_within_at.continuous_within_at {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {f : M β M'} {x : M} {s : set M} :
s x β x

theorem mdifferentiable_at.continuous_at {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {f : M β M'} {x : M} :
f x β

theorem mdifferentiable_on.continuous_on {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {f : M β M'} {s : set M} :
f s β

theorem mdifferentiable.continuous {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {f : M β M'} :
I' f β

theorem tangent_map_within_subset {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {f : M β M'} {s t : set M} [Is : M] [I's : M'] {p : M} :
s β t β β t p.fst β f s p = f t p

theorem tangent_map_within_univ {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {f : M β M'} [Is : M] [I's : M'] :
f set.univ = I' f

theorem tangent_map_within_eq_tangent_map {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {f : M β M'} {s : set M} [Is : M] [I's : M'] {p : M} :
β f p.fst β f s p = I' f p

@[simp]
theorem tangent_map_within_tangent_bundle_proj {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {f : M β M'} {s : set M} [Is : M] [I's : M'] {p : M} :
M' I' f s p) = f p)

@[simp]
theorem tangent_map_within_proj {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {f : M β M'} {s : set M} [Is : M] [I's : M'] {p : M} :
I' f s p).fst = f p.fst

@[simp]
theorem tangent_map_tangent_bundle_proj {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {f : M β M'} [Is : M] [I's : M'] {p : M} :
M' I' f p) = f p)

@[simp]
theorem tangent_map_proj {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {f : M β M'} [Is : M] [I's : M'] {p : M} :
I' f p).fst = f p.fst

### Congruence lemmas for derivatives on manifolds

theorem has_mfderiv_within_at.congr_of_eventually_eq {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {f fβ : M β M'} {x : M} {s : set M} [Is : M] [I's : M'] {f' : βL[π] (f x)} :
f s x f' β fβ =αΆ [π[s] x] f β fβ x = f x β fβ s x f'

theorem has_mfderiv_within_at.congr_mono {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {f fβ : M β M'} {x : M} {s t : set M} [Is : M] [I's : M'] {f' : βL[π] (f x)} :
f s x f' β (β (x : M), x β t β fβ x = f x) β fβ x = f x β t β s β fβ t x f'

theorem has_mfderiv_at.congr_of_eventually_eq {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {f fβ : M β M'} {x : M} [Is : M] [I's : M'] {f' : βL[π] (f x)} :
I' f x f' β fβ =αΆ [π x] f β I' fβ x f'

theorem mdifferentiable_within_at.congr_of_eventually_eq {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {f fβ : M β M'} {x : M} {s : set M} [Is : M] [I's : M'] :
s x β fβ =αΆ [π[s] x] f β fβ x = f x β fβ s x

theorem filter.eventually_eq.mdifferentiable_within_at_iff {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} (I : E H) {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} (I' : E' H') {M' : Type u_7} [ M'] {f fβ : M β M'} {x : M} {s : set M} [Is : M] [I's : M'] :
fβ =αΆ [π[s] x] f β fβ x = f x β f s x β fβ s x)

theorem mdifferentiable_within_at.congr_mono {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {f fβ : M β M'} {x : M} {s t : set M} [Is : M] [I's : M'] :
s x β (β (x : M), x β t β fβ x = f x) β fβ x = f x β t β s β fβ t x

theorem mdifferentiable_within_at.congr {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {f fβ : M β M'} {x : M} {s : set M} [Is : M] [I's : M'] :
s x β (β (x : M), x β s β fβ x = f x) β fβ x = f x β fβ s x

theorem mdifferentiable_on.congr_mono {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {f fβ : M β M'} {s t : set M} [Is : M] [I's : M'] :
f s β (β (x : M), x β t β fβ x = f x) β t β s β fβ t

theorem mdifferentiable_at.congr_of_eventually_eq {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {f fβ : M β M'} {x : M} [Is : M] [I's : M'] :
f x β fβ =αΆ [π x] f β fβ x

theorem mdifferentiable_within_at.mfderiv_within_congr_mono {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {f fβ : M β M'} {x : M} {s t : set M} [Is : M] [I's : M'] :
s x β (β (x : M), x β t β fβ x = f x) β fβ x = f x β β t β s β I' fβ t x = I' f s x

theorem filter.eventually_eq.mfderiv_within_eq {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {f fβ : M β M'} {x : M} {s : set M} [Is : M] [I's : M'] :
β fβ =αΆ [π[s] x] f β fβ x = f x β I' fβ s x = I' f s x

theorem mfderiv_within_congr {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {f fβ : M β M'} {x : M} {s : set M} [Is : M] [I's : M'] :
β (β (x : M), x β s β fβ x = f x) β fβ x = f x β I' fβ s x = I' f s x

theorem tangent_map_within_congr {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {f fβ : M β M'} {s : set M} [Is : M] [I's : M'] (h : β (x : M), x β s β f x = fβ x) (p : M) :
p.fst β s β β f s p = fβ s p

theorem filter.eventually_eq.mfderiv_eq {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {f fβ : M β M'} {x : M} [Is : M] [I's : M'] :
fβ =αΆ [π x] f β I' fβ x = I' f x

### Composition lemmas

theorem written_in_ext_chart_comp {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {E'' : Type u_8} [normed_group E''] [normed_space π E''] {H'' : Type u_9} [topological_space H''] {I'' : E'' H''} {M'' : Type u_10} [topological_space M''] [ M''] {f : M β M'} {x : M} {s : set M} {g : M' β M''} :
x β {y : E | x (g β f) y = I'' (f x) g β x f) y} β π[β((ext_chart_at I x).symm) β»ΒΉ' s β© ] β x) x

theorem has_mfderiv_within_at.comp {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {E'' : Type u_8} [normed_group E''] [normed_space π E''] {H'' : Type u_9} [topological_space H''] {I'' : E'' H''} {M'' : Type u_10} [topological_space M''] [ M''] {f : M β M'} (x : M) {s : set M} {g : M' β M''} {u : set M'} [Is : M] [I's : M'] [I''s : M''] {f' : βL[π] (f x)} {g' : (f x) βL[π] (g (f x))} :
I'' g u (f x) g' β f s x f' β s β f β»ΒΉ' u β (g β f) s x (g'.comp f')

theorem has_mfderiv_at.comp {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {E'' : Type u_8} [normed_group E''] [normed_space π E''] {H'' : Type u_9} [topological_space H''] {I'' : E'' H''} {M'' : Type u_10} [topological_space M''] [ M''] {f : M β M'} (x : M) {g : M' β M''} [Is : M] [I's : M'] [I''s : M''] {f' : βL[π] (f x)} {g' : (f x) βL[π] (g (f x))} :
I'' g (f x) g' β I' f x f' β I'' (g β f) x (g'.comp f')

The chain rule.

theorem has_mfderiv_at.comp_has_mfderiv_within_at {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {E'' : Type u_8} [normed_group E''] [normed_space π E''] {H'' : Type u_9} [topological_space H''] {I'' : E'' H''} {M'' : Type u_10} [topological_space M''] [ M''] {f : M β M'} (x : M) {s : set M} {g : M' β M''} [Is : M] [I's : M'] [I''s : M''] {f' : βL[π] (f x)} {g' : (f x) βL[π] (g (f x))} :
I'' g (f x) g' β f s x f' β (g β f) s x (g'.comp f')

theorem mdifferentiable_within_at.comp {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {E'' : Type u_8} [normed_group E''] [normed_space π E''] {H'' : Type u_9} [topological_space H''] {I'' : E'' H''} {M'' : Type u_10} [topological_space M''] [ M''] {f : M β M'} (x : M) {s : set M} {g : M' β M''} {u : set M'} [Is : M] [I's : M'] [I''s : M''] :
g u (f x) β s x β s β f β»ΒΉ' u β (g β f) s x

theorem mdifferentiable_at.comp {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {E'' : Type u_8} [normed_group E''] [normed_space π E''] {H'' : Type u_9} [topological_space H''] {I'' : E'' H''} {M'' : Type u_10} [topological_space M''] [ M''] {f : M β M'} (x : M) {g : M' β M''} [Is : M] [I's : M'] [I''s : M''] :
I'' g (f x) β f x β I'' (g β f) x

theorem mfderiv_within_comp {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {E'' : Type u_8} [normed_group E''] [normed_space π E''] {H'' : Type u_9} [topological_space H''] {I'' : E'' H''} {M'' : Type u_10} [topological_space M''] [ M''] {f : M β M'} (x : M) {s : set M} {g : M' β M''} {u : set M'} [Is : M] [I's : M'] [I''s : M''] :
g u (f x) β s x β s β f β»ΒΉ' u β β I'' (g β f) s x = I'' g u (f x)).comp I' f s x)

theorem mfderiv_comp {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {E'' : Type u_8} [normed_group E''] [normed_space π E''] {H'' : Type u_9} [topological_space H''] {I'' : E'' H''} {M'' : Type u_10} [topological_space M''] [ M''] {f : M β M'} (x : M) {g : M' β M''} [Is : M] [I's : M'] [I''s : M''] :
I'' g (f x) β f x β I'' (g β f) x = (mfderiv I' I'' g (f x)).comp (mfderiv I I' f x)

theorem mdifferentiable_on.comp {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {E'' : Type u_8} [normed_group E''] [normed_space π E''] {H'' : Type u_9} [topological_space H''] {I'' : E'' H''} {M'' : Type u_10} [topological_space M''] [ M''] {f : M β M'} {s : set M} {g : M' β M''} {u : set M'} [Is : M] [I's : M'] [I''s : M''] :
I'' g u β f s β s β f β»ΒΉ' u β I'' (g β f) s

theorem mdifferentiable.comp {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {E'' : Type u_8} [normed_group E''] [normed_space π E''] {H'' : Type u_9} [topological_space H''] {I'' : E'' H''} {M'' : Type u_10} [topological_space M''] [ M''] {f : M β M'} {g : M' β M''} [Is : M] [I's : M'] [I''s : M''] :
I'' g β I' f β I'' (g β f)

theorem tangent_map_within_comp_at {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {E'' : Type u_8} [normed_group E''] [normed_space π E''] {H'' : Type u_9} [topological_space H''] {I'' : E'' H''} {M'' : Type u_10} [topological_space M''] [ M''] {f : M β M'} {s : set M} {g : M' β M''} {u : set M'} [Is : M] [I's : M'] [I''s : M''] (p : M) :
g u (f p.fst) β s p.fst β s β f β»ΒΉ' u β β I'' (g β f) s p = I'' g u I' f s p)

theorem tangent_map_comp_at {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {E'' : Type u_8} [normed_group E''] [normed_space π E''] {H'' : Type u_9} [topological_space H''] {I'' : E'' H''} {M'' : Type u_10} [topological_space M''] [ M''] {f : M β M'} {g : M' β M''} [Is : M] [I's : M'] [I''s : M''] (p : M) :
I'' g (f p.fst) β f p.fst β I'' (g β f) p = I'' g I' f p)

theorem tangent_map_comp {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {E'' : Type u_8} [normed_group E''] [normed_space π E''] {H'' : Type u_9} [topological_space H''] {I'' : E'' H''} {M'' : Type u_10} [topological_space M''] [ M''] {f : M β M'} {g : M' β M''} [Is : M] [I's : M'] [I''s : M''] :
I'' g β I' f β I'' (g β f) = I'' g β I' f

### Differentiability of specific functions

#### Identity

theorem has_mfderiv_at_id {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} (I : E H) {M : Type u_4} [ M] (x : M) :
id x x))

theorem has_mfderiv_within_at_id {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} (I : E H) {M : Type u_4} [ M] (s : set M) (x : M) :
s x x))

theorem mdifferentiable_at_id {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} (I : E H) {M : Type u_4} [ M] {x : M} :
x

theorem mdifferentiable_within_at_id {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} (I : E H) {M : Type u_4} [ M] {s : set M} {x : M} :
x

theorem mdifferentiable_id {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} (I : E H) {M : Type u_4} [ M]  :

theorem mdifferentiable_on_id {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} (I : E H) {M : Type u_4} [ M] {s : set M} :
s

@[simp]
theorem mfderiv_id {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} (I : E H) {M : Type u_4} [ M] {x : M} :
I id x = x)

theorem mfderiv_within_id {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} (I : E H) {M : Type u_4} [ M] {s : set M} {x : M} :
β id s x = x)

@[simp]
theorem tangent_map_id {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} (I : E H) {M : Type u_4} [ M]  :
I id = id

theorem tangent_map_within_id {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} (I : E H) {M : Type u_4} [ M] {s : set M} {p : M} :
p) β s p = p

#### Constants

theorem has_mfderiv_at_const {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} (I : E H) {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} (I' : E' H') {M' : Type u_7} [ M'] [ M'] (c : M') (x : M) :
I' (Ξ» (y : M), c) x 0

theorem has_mfderiv_within_at_const {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} (I : E H) {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} (I' : E' H') {M' : Type u_7} [ M'] [ M'] (c : M') (s : set M) (x : M) :
(Ξ» (y : M), c) s x 0

theorem mdifferentiable_at_const {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} (I : E H) {M : Type u_4} [ M] {x : M} {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} (I' : E' H') {M' : Type u_7} [ M'] [ M'] {c : M'} :
(Ξ» (y : M), c) x

theorem mdifferentiable_within_at_const {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} (I : E H) {M : Type u_4} [ M] {s : set M} {x : M} {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} (I' : E' H') {M' : Type u_7} [ M'] [ M'] {c : M'} :
(Ξ» (y : M), c) s x

theorem mdifferentiable_const {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} (I : E H) {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} (I' : E' H') {M' : Type u_7} [ M'] [ M'] {c : M'} :
I' (Ξ» (y : M), c)

theorem mdifferentiable_on_const {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} (I : E H) {M : Type u_4} [ M] {s : set M} {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} (I' : E' H') {M' : Type u_7} [ M'] [ M'] {c : M'} :
(Ξ» (y : M), c) s

@[simp]
theorem mfderiv_const {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} (I : E H) {M : Type u_4} [ M] {x : M} {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} (I' : E' H') {M' : Type u_7} [ M'] [ M'] {c : M'} :
I' (Ξ» (y : M), c) x = 0

theorem mfderiv_within_const {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} (I : E H) {M : Type u_4} [ M] {s : set M} {x : M} {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} (I' : E' H') {M' : Type u_7} [ M'] [ M'] {c : M'} :
β I' (Ξ» (y : M), c) s x = 0

#### Model with corners

theorem model_with_corners.mdifferentiable {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} (I : E H) :
π(π, E) βI

theorem model_with_corners.mdifferentiable_on_symm {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} (I : E H) :

theorem mdifferentiable_at_atlas {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} (I : E H) {M : Type u_4} [ M] {e : H} (h : e β M) {x : M} :
β x

theorem mdifferentiable_on_atlas {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} (I : E H) {M : Type u_4} [ M] {e : H} :
e β M β

theorem mdifferentiable_at_atlas_symm {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} (I : E H) {M : Type u_4} [ M] {e : H} (h : e β M) {x : H} :
β β(e.symm) x

theorem mdifferentiable_on_atlas_symm {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} (I : E H) {M : Type u_4} [ M] {e : H} :

theorem mdifferentiable_of_mem_atlas {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} (I : E H) {M : Type u_4} [ M] {e : H} :
e β M β

theorem mdifferentiable_chart {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} (I : E H) {M : Type u_4} [ M] (x : M) :
(chart_at H x)

theorem tangent_map_chart {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} (I : E H) {M : Type u_4} [ M] {p q : M} :

The derivative of the chart at a base point is the chart of the tangent bundle, composed with the identification between the tangent bundle of the model space and the product space.

theorem tangent_map_chart_symm {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} (I : E H) {M : Type u_4} [ M] {p : M} {q : H} :

The derivative of the inverse of the chart at a base point is the inverse of the chart of the tangent bundle, composed with the identification between the tangent bundle of the model space and the product space.

### Relations between vector space derivative and manifold derivative

The manifold derivative `mfderiv`, when considered on the model vector space with its trivial manifold structure, coincides with the usual Frechet derivative `fderiv`. In this section, we prove this and related statements.

theorem unique_mdiff_within_at_iff_unique_diff_within_at {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {s : set E} {x : E} :
s x β s x

theorem unique_mdiff_on_iff_unique_diff_on {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {s : set E} :

@[simp]
theorem written_in_ext_chart_model_space {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {E' : Type u_3} [normed_group E'] [normed_space π E'] {f : E β E'} {x : E} :
π(π, E') x f = f

theorem mdifferentiable_within_at_iff_differentiable_within_at {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {E' : Type u_3} [normed_group E'] [normed_space π E'] {f : E β E'} {s : set E} {x : E} :
π(π, E') f s x β f s x

For maps between vector spaces, `mdifferentiable_within_at` and `fdifferentiable_within_at` coincide

theorem mdifferentiable_at_iff_differentiable_at {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {E' : Type u_3} [normed_group E'] [normed_space π E'] {f : E β E'} {x : E} :
π(π, E') f x β f x

For maps between vector spaces, `mdifferentiable_at` and `differentiable_at` coincide

theorem mdifferentiable_on_iff_differentiable_on {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {E' : Type u_3} [normed_group E'] [normed_space π E'] {f : E β E'} {s : set E} :
π(π, E') f s β f s

For maps between vector spaces, `mdifferentiable_on` and `differentiable_on` coincide

theorem mdifferentiable_iff_differentiable {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {E' : Type u_3} [normed_group E'] [normed_space π E'] {f : E β E'} :
mdifferentiable π(π, E) π(π, E') f β differentiable π f

For maps between vector spaces, `mdifferentiable` and `differentiable` coincide

theorem mfderiv_within_eq_fderiv_within {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {E' : Type u_3} [normed_group E'] [normed_space π E'] {f : E β E'} {s : set E} {x : E} :
mfderiv_within π(π, E) π(π, E') f s x = fderiv_within π f s x

For maps between vector spaces, `mfderiv_within` and `fderiv_within` coincide

theorem mfderiv_eq_fderiv {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {E' : Type u_3} [normed_group E'] [normed_space π E'] {f : E β E'} {x : E} :
mfderiv π(π, E) π(π, E') f x = fderiv π f x

For maps between vector spaces, `mfderiv` and `fderiv` coincide

### Differentiable local homeomorphisms

theorem local_homeomorph.mdifferentiable.symm {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {e : M'} :
β

theorem local_homeomorph.mdifferentiable.mdifferentiable_at {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {e : M'} (he : e) {x : M} :
β βe x

theorem local_homeomorph.mdifferentiable.mdifferentiable_at_symm {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {e : M'} (he : e) {x : M'} :
β β(e.symm) x

theorem local_homeomorph.mdifferentiable.symm_comp_deriv {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {e : M'} (he : e) [ M'] {x : M} :
β (mfderiv I' I β(e.symm) (βe x)).comp (mfderiv I I' βe x) = x)

theorem local_homeomorph.mdifferentiable.comp_symm_deriv {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {e : M'} (he : e) [ M'] {x : M'} :
β (mfderiv I I' βe (β(e.symm) x)).comp (mfderiv I' I β(e.symm) x) = x)

def local_homeomorph.mdifferentiable.mfderiv {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {e : M'} (he : e) [ M'] {x : M} :
β x βL[π] (βe x))

The derivative of a differentiable local homeomorphism, as a continuous linear equivalence between the tangent spaces at `x` and `e x`.

Equations
theorem local_homeomorph.mdifferentiable.mfderiv_bijective {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {e : M'} (he : e) [ M'] {x : M} :

theorem local_homeomorph.mdifferentiable.mfderiv_surjective {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {e : M'} (he : e) [ M'] {x : M} :

theorem local_homeomorph.mdifferentiable.range_mfderiv_eq_univ {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {e : M'} (he : e) [ M'] {x : M} :

theorem local_homeomorph.mdifferentiable.trans {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {E'' : Type u_8} [normed_group E''] [normed_space π E''] {H'' : Type u_9} [topological_space H''] {I'' : E'' H''} {M'' : Type u_10} [topological_space M''] [ M''] {e : M'} (he : e) {e' : M''} [ M'] [ M''] :
e' β (e β«β e')

### Unique derivative sets in manifolds

theorem unique_mdiff_on.unique_mdiff_on_preimage {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {s : set M} [ M'] (hs : s) {e : M'} :

If a set has the unique differential property, then its image under a local diffeomorphism also has the unique differential property.

theorem unique_mdiff_on.unique_diff_on {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {s : set M} (hs : s) (x : M) :

If a set in a manifold has the unique derivative property, then its pullback by any extended chart, in the vector space, also has the unique derivative property.

theorem unique_mdiff_on.unique_diff_on_inter_preimage {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {E' : Type u_5} [normed_group E'] [normed_space π E'] {H' : Type u_6} {I' : E' H'} {M' : Type u_7} [ M'] {s : set M} (hs : s) (x : M) (y : M') {f : M β M'} :

When considering functions between manifolds, this statement shows up often. It entails the unique differential of the pullback in extended charts of the set where the function can be read in the charts.

theorem unique_mdiff_on.smooth_bundle_preimage {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {s : set M} {F : Type u_8} [normed_group F] [normed_space π F] (Z : F) :
β unique_mdiff_on (I.prod π(π, F))

In a smooth fiber bundle constructed from core, the preimage under the projection of a set with unique differential in the basis also has unique differential.

theorem unique_mdiff_on.tangent_bundle_proj_preimage {π : Type u_1} [nondiscrete_normed_field π] {E : Type u_2} [normed_group E] [normed_space π E] {H : Type u_3} {I : E H} {M : Type u_4} [ M] {s : set M} :
β β»ΒΉ' s)