File: Mathlib/Analysis/NormedSpace/HomeomorphBall.lean

full_name: PartialHomeomorph.univUnitBall
code: /-- Local homeomorphism between a real (semi)normed space and the unit ball.
See also `Homeomorph.unitBall`. -/
@[simps (config := .lemmasOnly)]
def PartialHomeomorph.univUnitBall : PartialHomeomorph E E 
goal: PartialHomeomorph E E

full_name: PartialHomeomorph.univUnitBall_symm_apply
code: /-- Local homeomorphism between a real (semi)normed space and the unit ball.
See also `Homeomorph.unitBall`. -/
@[simps (config := .lemmasOnly)]
def PartialHomeomorph.univUnitBall : PartialHomeomorph E E 
goal: ↑PartialHomeomorph.univUnitBall.symm y = (√(1 - ‖y‖ ^ 2))⁻¹ • y

full_name: PartialHomeomorph.univUnitBall_apply
code: /-- Local homeomorphism between a real (semi)normed space and the unit ball.
See also `Homeomorph.unitBall`. -/
@[simps (config := .lemmasOnly)]
def PartialHomeomorph.univUnitBall : PartialHomeomorph E E 
goal: ↑PartialHomeomorph.univUnitBall x = (√(1 + ‖x‖ ^ 2))⁻¹ • x

full_name: PartialHomeomorph.univUnitBall_target
code: /-- Local homeomorphism between a real (semi)normed space and the unit ball.
See also `Homeomorph.unitBall`. -/
@[simps (config := .lemmasOnly)]
def PartialHomeomorph.univUnitBall : PartialHomeomorph E E 
goal: PartialHomeomorph.univUnitBall.target = Metric.ball 0 1

full_name: PartialHomeomorph.univUnitBall_source
code: /-- Local homeomorphism between a real (semi)normed space and the unit ball.
See also `Homeomorph.unitBall`. -/
@[simps (config := .lemmasOnly)]
def PartialHomeomorph.univUnitBall : PartialHomeomorph E E 
goal: PartialHomeomorph.univUnitBall.source = Set.univ

full_name: PartialHomeomorph.univUnitBall_apply_zero
code: @[simp]
theorem PartialHomeomorph.univUnitBall_apply_zero : univUnitBall (0 : E) = 0 
goal: ↑PartialHomeomorph.univUnitBall 0 = 0

full_name: PartialHomeomorph.univUnitBall_symm_apply_zero
code: @[simp]
theorem PartialHomeomorph.univUnitBall_symm_apply_zero : univUnitBall.symm (0 : E) = 0 
goal: ↑PartialHomeomorph.univUnitBall.symm 0 = 0

full_name: Homeomorph.unitBall_apply_coe
code: /-- A (semi) normed real vector space is homeomorphic to the unit ball in the same space.
This homeomorphism sends `x : E` to `(1 + ‖x‖²)^(- ½) • x`.

In many cases the actual implementation is not important, so we don't mark the projection lemmas
`Homeomorph.unitBall_apply_coe` and `Homeomorph.unitBall_symm_apply` as `@[simp]`.

See also `Homeomorph.contDiff_unitBall` and `PartialHomeomorph.contDiffOn_unitBall_symm`
for smoothness properties that hold when `E` is an inner-product space. -/
@[simps! (config := .lemmasOnly)]
def Homeomorph.unitBall : E ≃ₜ ball (0 : E) 1 
goal: ∀ (a : E), ↑(Homeomorph.unitBall a) = ↑PartialHomeomorph.univUnitBall a

full_name: Homeomorph.unitBall_symm_apply
code: /-- A (semi) normed real vector space is homeomorphic to the unit ball in the same space.
This homeomorphism sends `x : E` to `(1 + ‖x‖²)^(- ½) • x`.

In many cases the actual implementation is not important, so we don't mark the projection lemmas
`Homeomorph.unitBall_apply_coe` and `Homeomorph.unitBall_symm_apply` as `@[simp]`.

See also `Homeomorph.contDiff_unitBall` and `PartialHomeomorph.contDiffOn_unitBall_symm`
for smoothness properties that hold when `E` is an inner-product space. -/
@[simps! (config := .lemmasOnly)]
def Homeomorph.unitBall : E ≃ₜ ball (0 : E) 1 
goal: ∀ (a : ↑(Metric.ball 0 1)),
  Homeomorph.unitBall.symm a = ↑(PartialHomeomorph.univUnitBall.toHomeomorphSourceTarget.symm a)

full_name: Homeomorph.unitBall
code: /-- A (semi) normed real vector space is homeomorphic to the unit ball in the same space.
This homeomorphism sends `x : E` to `(1 + ‖x‖²)^(- ½) • x`.

In many cases the actual implementation is not important, so we don't mark the projection lemmas
`Homeomorph.unitBall_apply_coe` and `Homeomorph.unitBall_symm_apply` as `@[simp]`.

See also `Homeomorph.contDiff_unitBall` and `PartialHomeomorph.contDiffOn_unitBall_symm`
for smoothness properties that hold when `E` is an inner-product space. -/
@[simps! (config := .lemmasOnly)]
def Homeomorph.unitBall : E ≃ₜ ball (0 : E) 1 
goal: E ≃ₜ ↑(Metric.ball 0 1)

full_name: Homeomorph.coe_unitBall_apply_zero
code: @[simp]
theorem Homeomorph.coe_unitBall_apply_zero :
    (Homeomorph.unitBall (0 : E) : E) = 0 
goal: ↑(Homeomorph.unitBall 0) = 0

full_name: PartialHomeomorph.unitBallBall_symm_apply
code: /-- Affine homeomorphism `(r • · +ᵥ c)` between a normed space and an add torsor over this space,
interpreted as a `PartialHomeomorph` between `Metric.ball 0 1` and `Metric.ball c r`. -/
@[simps!]
def unitBallBall (c : P) (r : ℝ) (hr : 0 < r) : PartialHomeomorph E P 
goal: ↑(PartialHomeomorph.unitBallBall c r hr).symm a = r⁻¹ • (a -ᵥ c)

full_name: PartialHomeomorph.unitBallBall_source
code: /-- Affine homeomorphism `(r • · +ᵥ c)` between a normed space and an add torsor over this space,
interpreted as a `PartialHomeomorph` between `Metric.ball 0 1` and `Metric.ball c r`. -/
@[simps!]
def unitBallBall (c : P) (r : ℝ) (hr : 0 < r) : PartialHomeomorph E P 
goal: (PartialHomeomorph.unitBallBall c r hr).source = Metric.ball 0 1

full_name: PartialHomeomorph.unitBallBall_apply
code: /-- Affine homeomorphism `(r • · +ᵥ c)` between a normed space and an add torsor over this space,
interpreted as a `PartialHomeomorph` between `Metric.ball 0 1` and `Metric.ball c r`. -/
@[simps!]
def unitBallBall (c : P) (r : ℝ) (hr : 0 < r) : PartialHomeomorph E P 
goal: ↑(PartialHomeomorph.unitBallBall c r hr) a = r • a +ᵥ c

full_name: PartialHomeomorph.unitBallBall_target
code: /-- Affine homeomorphism `(r • · +ᵥ c)` between a normed space and an add torsor over this space,
interpreted as a `PartialHomeomorph` between `Metric.ball 0 1` and `Metric.ball c r`. -/
@[simps!]
def unitBallBall (c : P) (r : ℝ) (hr : 0 < r) : PartialHomeomorph E P 
goal: (PartialHomeomorph.unitBallBall c r hr).target = Metric.ball c r

full_name: PartialHomeomorph.unitBallBall
code: /-- Affine homeomorphism `(r • · +ᵥ c)` between a normed space and an add torsor over this space,
interpreted as a `PartialHomeomorph` between `Metric.ball 0 1` and `Metric.ball c r`. -/
@[simps!]
def unitBallBall (c : P) (r : ℝ) (hr : 0 < r) : PartialHomeomorph E P 
goal: PartialHomeomorph E P

full_name: PartialHomeomorph.univBall
code: /-- If `r > 0`, then `PartialHomeomorph.univBall c r` is a smooth partial homeomorphism
with `source = Set.univ` and `target = Metric.ball c r`.
Otherwise, it is the translation by `c`.
Thus in all cases, it sends `0` to `c`, see `PartialHomeomorph.univBall_apply_zero`. -/
def univBall (c : P) (r : ℝ) : PartialHomeomorph E P 
goal: PartialHomeomorph E P

full_name: PartialHomeomorph.univBall_source
code: @[simp]
theorem univBall_source (c : P) (r : ℝ) : (univBall c r).source = univ 
goal: (PartialHomeomorph.univBall c r).source = Set.univ

full_name: PartialHomeomorph.univBall_target
code: theorem univBall_target (c : P) {r : ℝ} (hr : 0 < r) : (univBall c r).target = ball c r 
goal: (PartialHomeomorph.univBall c r).target = Metric.ball c r

full_name: PartialHomeomorph.ball_subset_univBall_target
code: theorem ball_subset_univBall_target (c : P) (r : ℝ) : ball c r ⊆ (univBall c r).target 
goal: Metric.ball c r ⊆ (PartialHomeomorph.univBall c r).target

full_name: PartialHomeomorph.univBall_apply_zero
code: @[simp]
theorem univBall_apply_zero (c : P) (r : ℝ) : univBall c r 0 = c 
goal: ↑(PartialHomeomorph.univBall c r) 0 = c

full_name: PartialHomeomorph.univBall_symm_apply_center
code: @[simp]
theorem univBall_symm_apply_center (c : P) (r : ℝ) : (univBall c r).symm c = 0 
goal: ↑(PartialHomeomorph.univBall c r).symm c = 0

full_name: PartialHomeomorph.continuous_univBall
code: @[continuity]
theorem continuous_univBall (c : P) (r : ℝ) : Continuous (univBall c r) 
goal: Continuous ↑(PartialHomeomorph.univBall c r)

full_name: PartialHomeomorph.continuousOn_univBall_symm
code: theorem continuousOn_univBall_symm (c : P) (r : ℝ) : ContinuousOn (univBall c r).symm (ball c r) 
goal: ContinuousOn (↑(PartialHomeomorph.univBall c r).symm) (Metric.ball c r)