mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Geometry.Manifold.VectorField.LieBracket

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝⁶ : TopologicalSpace H
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
s t : Set M
x : M
V W V₁ W₁ : (x : M) → TangentSpace I x
inst✝¹ : IsManifold I 2 M
inst✝ : CompleteSpace E
hV : MDiffAt[s] (T% V) x
hVs : EqOn V₁ V t
hVx : V₁ x = V x
hW : MDiffAt[s] (T% W) x
hWs : EqOn W₁ W t
hWx : W₁ x = W x
hxt : UniqueMDiffAt[t] x
h₁ : t ⊆ s
⊢ mlieBracketWithin I V₁ W₁ t x = mlieBracketWithin I V W s x

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.VectorField.LieBracket

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁴ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝¹³ : TopologicalSpace H
E : Type u_3
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
H' : Type u_5
inst✝⁸ : TopologicalSpace H'
E' : Type u_6
inst✝⁷ : NormedAddCommGroup E'
inst✝⁶ : NormedSpace 𝕜 E'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁵ : TopologicalSpace M'
inst✝⁴ : ChartedSpace H' M'
inst✝³ : IsManifold I 2 M
inst✝² : IsManifold I' 2 M'
inst✝¹ : CompleteSpace E
inst✝ : CompleteSpace E'
f : M → M'
V W : (x : M') → TangentSpace I' x
x₀ : M
s : Set M
t : Set M'
hV : MDiffAt[t] (T% V) (f x₀)
hW : MDiffAt[t] (T% W) (f x₀)
hu : UniqueMDiff[s]
hf : ContMDiffOn I I' 2 f s
hx₀ : x₀ ∈ s
ht : t ⊆ (extChartAt I' (f x₀)).source
hst : MapsTo f s t
hsymm :
  IsSymmSndFDerivWithinAt 𝕜 (↑(extChartAt I' (f x₀)) ∘ f ∘ ↑(extChartAt I x₀).symm)
    (↑(extChartAt I x₀).symm ⁻¹' s ∩ range ↑I) (↑(extChartAt I x₀) x₀)
⊢ ↑(extChartAt I x₀).symm (↑(extChartAt I x₀) x₀) = x₀

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.VectorField.LieBracket

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁴ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝¹³ : TopologicalSpace H
E : Type u_3
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
H' : Type u_5
inst✝⁸ : TopologicalSpace H'
E' : Type u_6
inst✝⁷ : NormedAddCommGroup E'
inst✝⁶ : NormedSpace 𝕜 E'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁵ : TopologicalSpace M'
inst✝⁴ : ChartedSpace H' M'
inst✝³ : IsManifold I 2 M
inst✝² : IsManifold I' 2 M'
inst✝¹ : CompleteSpace E
inst✝ : CompleteSpace E'
f : M → M'
V W : (x : M') → TangentSpace I' x
x₀ : M
s : Set M
t : Set M'
hV : MDiffAt[t] (T% V) (f x₀)
hW : MDiffAt[t] (T% W) (f x₀)
hu : UniqueMDiff[s]
hf : ContMDiffOn I I' 2 f s
hx₀ : x₀ ∈ s
ht : t ⊆ (extChartAt I' (f x₀)).source
hst : MapsTo f s t
hsymm :
  IsSymmSndFDerivWithinAt 𝕜 (↑(extChartAt I' (f x₀)) ∘ f ∘ ↑(extChartAt I x₀).symm)
    (↑(extChartAt I x₀).symm ⁻¹' s ∩ range ↑I) (↑(extChartAt I x₀) x₀)
A : ↑(extChartAt I x₀).symm (↑(extChartAt I x₀) x₀) = x₀
⊢ x₀ = ↑(extChartAt I x₀).symm (↑(extChartAt I x₀) x₀)

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.VectorField.LieBracket

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁴ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝¹³ : TopologicalSpace H
E : Type u_3
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
H' : Type u_5
inst✝⁸ : TopologicalSpace H'
E' : Type u_6
inst✝⁷ : NormedAddCommGroup E'
inst✝⁶ : NormedSpace 𝕜 E'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁵ : TopologicalSpace M'
inst✝⁴ : ChartedSpace H' M'
inst✝³ : IsManifold I 2 M
inst✝² : IsManifold I' 2 M'
inst✝¹ : CompleteSpace E
inst✝ : CompleteSpace E'
f : M → M'
V W : (x : M') → TangentSpace I' x
x₀ : M
s : Set M
t : Set M'
hV : MDiffAt[t] (T% V) (f x₀)
hW : MDiffAt[t] (T% W) (f x₀)
hu : UniqueMDiff[s]
hf : ContMDiffOn I I' 2 f s
hx₀ : x₀ ∈ s
ht : t ⊆ (extChartAt I' (f x₀)).source
hst : MapsTo f s t
hsymm :
  IsSymmSndFDerivWithinAt 𝕜 (↑(extChartAt I' (f x₀)) ∘ f ∘ ↑(extChartAt I x₀).symm)
    (↑(extChartAt I x₀).symm ⁻¹' s ∩ range ↑I) (↑(extChartAt I x₀) x₀)
A : ↑(extChartAt I x₀).symm (↑(extChartAt I x₀) x₀) = x₀
A' : x₀ = ↑(extChartAt I x₀).symm (↑(extChartAt I x₀) x₀)
h'f : MDiffAt[s] f x₀
⊢ ↑(extChartAt I x₀) x₀ ∈ (extChartAt I x₀).target

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.VectorField.LieBracket

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁴ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝¹³ : TopologicalSpace H
E : Type u_3
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹⁰ : TopologicalSpace M
inst✝⁹ : ChartedSpace H M
H' : Type u_5
inst✝⁸ : TopologicalSpace H'
E' : Type u_6
inst✝⁷ : NormedAddCommGroup E'
inst✝⁶ : NormedSpace 𝕜 E'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁵ : TopologicalSpace M'
inst✝⁴ : ChartedSpace H' M'
inst✝³ : IsManifold I 2 M
inst✝² : IsManifold I' 2 M'
inst✝¹ : CompleteSpace E
inst✝ : CompleteSpace E'
f : M → M'
V W : (x : M') → TangentSpace I' x
x₀ : M
s : Set M
t : Set M'
hV : MDiffAt[t] (T% V) (f x₀)
hW : MDiffAt[t] (T% W) (f x₀)
hu : UniqueMDiff[s]
hf : ContMDiffOn I I' 2 f s
hx₀ : x₀ ∈ s
ht : t ⊆ (extChartAt I' (f x₀)).source
hst : MapsTo f s t
hsymm :
  IsSymmSndFDerivWithinAt 𝕜 (↑(extChartAt I' (f x₀)) ∘ f ∘ ↑(extChartAt I x₀).symm)
    (↑(extChartAt I x₀).symm ⁻¹' s ∩ range ↑I) (↑(extChartAt I x₀) x₀)
A : ↑(extChartAt I x₀).symm (↑(extChartAt I x₀) x₀) = x₀
A' : x₀ = ↑(extChartAt I x₀).symm (↑(extChartAt I x₀) x₀)
h'f : MDiffAt[s] f x₀
⊢ (mfderiv[range ↑I] ↑(extChartAt I x₀).symm (↑(extChartAt I x₀) x₀)).inverse = mfderiv% ↑(extChartAt I x₀) x₀

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Immersion

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
E'' : Type u
F : Type u_5
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : NormedAddCommGroup E''
inst✝⁶ : NormedSpace 𝕜 E''
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
H : Type u_7
inst✝³ : TopologicalSpace H
G : Type u_9
inst✝² : TopologicalSpace G
I : ModelWithCorners 𝕜 E H
J : ModelWithCorners 𝕜 E'' G
M : Type u_11
inst✝¹ : TopologicalSpace M
N : Type u_13
inst✝ : TopologicalSpace N
f : M → N
φ : OpenPartialHomeomorph M H
ψ : OpenPartialHomeomorph N G
s : Set M
hs : IsOpen[inst✝¹] s
x✝ : ImmersionAtProp F I J M N f φ ψ
equiv : (E × F) ≃L[𝕜] E''
hf : EqOn (↑(ψ.extend J) ∘ f ∘ ↑(φ.extend I).symm) (⇑equiv ∘ fun x ↦ (x, 0)) (φ.extend I).target
⊢ ((φ.restr s).extend I).target ⊆ (φ.extend I).target

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Immersion

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
E'' : Type u
F : Type u_5
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : NormedAddCommGroup E''
inst✝⁶ : NormedSpace 𝕜 E''
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
H : Type u_7
inst✝³ : TopologicalSpace H
G : Type u_9
inst✝² : TopologicalSpace G
I : ModelWithCorners 𝕜 E H
J : ModelWithCorners 𝕜 E'' G
M : Type u_11
inst✝¹ : TopologicalSpace M
N : Type u_13
inst✝ : TopologicalSpace N
f g : M → N
φ : OpenPartialHomeomorph M H
ψ : OpenPartialHomeomorph N G
hfg : EqOn f g φ.source
equiv : (E × F) ≃L[𝕜] E''
hf : EqOn (↑(ψ.extend J) ∘ f ∘ ↑(φ.extend I).symm) (⇑equiv ∘ fun x ↦ (x, 0)) (φ.extend I).target
⊢ (φ.extend I).target ⊆ (φ.extend I).target

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Immersion

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
E'' : Type u
F : Type u_5
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : NormedAddCommGroup E''
inst✝⁶ : NormedSpace 𝕜 E''
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
H : Type u_7
inst✝³ : TopologicalSpace H
G : Type u_9
inst✝² : TopologicalSpace G
I : ModelWithCorners 𝕜 E H
J : ModelWithCorners 𝕜 E'' G
M : Type u_11
inst✝¹ : TopologicalSpace M
N : Type u_13
inst✝ : TopologicalSpace N
f g : M → N
φ : OpenPartialHomeomorph M H
ψ : OpenPartialHomeomorph N G
hfg : EqOn f g φ.source
equiv : (E × F) ≃L[𝕜] E''
hf : EqOn (↑(ψ.extend J) ∘ f ∘ ↑(φ.extend I).symm) (⇑equiv ∘ fun x ↦ (x, 0)) (φ.extend I).target
x : E
hx : x ∈ (φ.extend I).target
⊢ ↑(φ.extend I).symm x ∈ φ.source

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Immersion

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
E'' : Type u
F : Type u_5
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : NormedAddCommGroup E''
inst✝⁶ : NormedSpace 𝕜 E''
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
H : Type u_7
inst✝³ : TopologicalSpace H
G : Type u_9
inst✝² : TopologicalSpace G
I : ModelWithCorners 𝕜 E H
J : ModelWithCorners 𝕜 E'' G
M : Type u_11
inst✝¹ : TopologicalSpace M
N : Type u_13
inst✝ : TopologicalSpace N
f g : M → N
φ : OpenPartialHomeomorph M H
ψ : OpenPartialHomeomorph N G
hfg : EqOn f g φ.source
⊢ ImmersionAtProp F I J M N f φ ψ → ImmersionAtProp F I J M N g φ ψ

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Immersion

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹² : NontriviallyNormedField 𝕜
E : Type u_2
E'' : Type u
F : Type u_5
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : NormedAddCommGroup E''
inst✝⁸ : NormedSpace 𝕜 E''
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜 F
H : Type u_7
inst✝⁵ : TopologicalSpace H
G : Type u_9
inst✝⁴ : TopologicalSpace G
I : ModelWithCorners 𝕜 E H
J : ModelWithCorners 𝕜 E'' G
M : Type u_11
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
N : Type u_13
inst✝¹ : TopologicalSpace N
inst✝ : ChartedSpace G N
n : ℕ∞ω
f : M → N
x : M
equiv : (E × F) ≃L[𝕜] E''
domChart : OpenPartialHomeomorph M H
codChart : OpenPartialHomeomorph N G
hx : x ∈ domChart.source
hfx : f x ∈ codChart.source
hdomChart : domChart ∈ IsManifold.maximalAtlas I n M
hcodChart : codChart ∈ IsManifold.maximalAtlas J n N
hsource : domChart.source ⊆ f ⁻¹' codChart.source
hwrittenInExtend :
  EqOn (↑(codChart.extend J) ∘ f ∘ ↑(domChart.extend I).symm) (⇑equiv ∘ fun x ↦ (x, 0)) (domChart.extend I).target
⊢ IsImmersionAtOfComplement F I J n f x

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Instances.Real

Goal state before:
pp: n✝ n : ℕ
inst✝ : NeZero n
⊢ 0 ≤ ofLp 0 0

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Instances.Real

Goal state before:
pp: n✝ n : ℕ
⊢ ∀ (i : Fin n), 0 ≤ ofLp 0 i

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Instances.Real

Goal state before:
pp: n : ℕ
inst✝ : NeZero n
x✝⁶ : EuclideanSpace ℝ (Fin n)
hx : x✝⁶ ∈ {x | 0 ≤ x.ofLp 0}
x✝⁵ : EuclideanSpace ℝ (Fin n)
hy : x✝⁵ ∈ {x | 0 ≤ x.ofLp 0}
x✝⁴ x✝³ : ℝ
x✝² : 0 ≤ x✝⁴
x✝¹ : 0 ≤ x✝³
x✝ : x✝⁴ + x✝³ = 1
⊢ x✝⁴ • x✝⁶ + x✝³ • x✝⁵ ∈ {x | 0 ≤ x.ofLp 0}

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Instances.Real

Goal state before:
pp: n : ℕ
x✝⁶ : EuclideanSpace ℝ (Fin n)
hx : x✝⁶ ∈ {x | ∀ (i : Fin n), 0 ≤ x.ofLp i}
x✝⁵ : EuclideanSpace ℝ (Fin n)
hy : x✝⁵ ∈ {x | ∀ (i : Fin n), 0 ≤ x.ofLp i}
x✝⁴ x✝³ : ℝ
x✝² : 0 ≤ x✝⁴
x✝¹ : 0 ≤ x✝³
x✝ : x✝⁴ + x✝³ = 1
i : Fin n
⊢ 0 ≤ (x✝⁴ • x✝⁶ + x✝³ • x✝⁵).ofLp i

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Instances.Real

Goal state before:
pp: n : ℕ
inst✝ : NeZero n
⊢ 0 ∈ {x | 0 ≤ x.ofLp 0}

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Instances.Real

Goal state before:
pp: n : ℕ
⊢ 0 ∈ {x | ∀ (i : Fin n), 0 ≤ x.ofLp i}

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Instances.Real

Goal state before:
pp: n : ℕ
p : ℝ≥0∞
a : ℝ
i : Fin n
⊢ interior {y | a ≤ y.ofLp i} = {y | a < y.ofLp i}

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Immersion

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹² : NontriviallyNormedField 𝕜
E : Type u_2
E'' : Type u
F : Type u_5
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : NormedAddCommGroup E''
inst✝⁸ : NormedSpace 𝕜 E''
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜 F
H : Type u_7
inst✝⁵ : TopologicalSpace H
G : Type u_9
inst✝⁴ : TopologicalSpace G
I : ModelWithCorners 𝕜 E H
J : ModelWithCorners 𝕜 E'' G
M : Type u_11
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
N : Type u_13
inst✝¹ : TopologicalSpace N
inst✝ : ChartedSpace G N
n : ℕ∞ω
f : M → N
x : M
h : IsImmersionAtOfComplement F I J n f x
⊢ f '' h.domChart.source ⊆ h.codChart.source

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Instances.Real

Goal state before:
pp: n : ℕ
p : ℝ≥0∞
a : ℝ
i : Fin n
⊢ closure {y | a ≤ y.ofLp i} = {y | a ≤ y.ofLp i}

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Instances.Real

Goal state before:
pp: n : ℕ
p : ℝ≥0∞
a : ℝ
i : Fin n
⊢ closure {y | a ≤ y.ofLp i} = {y | a ≤ y.ofLp i}

Tactic:
by
  let f : PiLp p (fun _ : Fin n ↦ ℝ) → ℝ := fun x ↦ x i
  change closure (f ⁻¹' Ici a) = f ⁻¹' Ici a
  rw [← (PiLp.isOpenMap_apply p _ i).preimage_closure_eq_closure_preimage, closure_Ici]
  fun_prop

Goal state after:
Module: Mathlib.Geometry.Manifold.Instances.Real

Goal state before:
pp: n : ℕ
p : ℝ≥0∞
a : ℝ
i : Fin n
⊢ closure {y | a < y.ofLp i} = {y | a ≤ y.ofLp i}

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Immersion

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹² : NontriviallyNormedField 𝕜
E : Type u_2
E'' : Type u
F : Type u_5
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : NormedAddCommGroup E''
inst✝⁸ : NormedSpace 𝕜 E''
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜 F
H : Type u_7
inst✝⁵ : TopologicalSpace H
G : Type u_9
inst✝⁴ : TopologicalSpace G
I : ModelWithCorners 𝕜 E H
J : ModelWithCorners 𝕜 E'' G
M : Type u_11
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
N : Type u_13
inst✝¹ : TopologicalSpace N
inst✝ : ChartedSpace G N
n : ℕ∞ω
f : M → N
x : M
h : IsImmersionAtOfComplement F I J n f x
⊢ (⇑h.equiv ∘ fun x ↦ (x, 0)) '' (h.domChart.extend I).target ⊆ (h.codChart.extend J).target

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Instances.Real

Goal state before:
pp: n : ℕ
p : ℝ≥0∞
a : ℝ
i : Fin n
⊢ frontier {y | a ≤ y.ofLp i} = {y | a = y.ofLp i}

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Instances.Real

Goal state before:
pp: n : ℕ
p : ℝ≥0∞
a : ℝ
f : Fin n → (PiLp p fun x ↦ ℝ) → ℝ := fun i x ↦ x.ofLp i
⊢ {y | ∀ (i : Fin n), a ≤ y.ofLp i} = ⋂ i, f i ⁻¹' Ici a

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Instances.Real

Goal state before:
pp: n : ℕ
p : ℝ≥0∞
a : ℝ
f : Fin n → (PiLp p fun x ↦ ℝ) → ℝ := fun i x ↦ x.ofLp i
h : {y | ∀ (i : Fin n), a ≤ y.ofLp i} = ⋂ i, f i ⁻¹' Ici a
⊢ {y | ∀ (i : Fin n), a < y.ofLp i} = ⋂ i, f i ⁻¹' Ioi a

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Immersion

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁴ : NontriviallyNormedField 𝕜
E : Type u_2
E'' : Type u
F : Type u_5
F' : Type u_6
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace 𝕜 E
inst✝¹¹ : NormedAddCommGroup E''
inst✝¹⁰ : NormedSpace 𝕜 E''
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
inst✝⁷ : NormedAddCommGroup F'
inst✝⁶ : NormedSpace 𝕜 F'
H : Type u_7
inst✝⁵ : TopologicalSpace H
G : Type u_9
inst✝⁴ : TopologicalSpace G
I : ModelWithCorners 𝕜 E H
J : ModelWithCorners 𝕜 E'' G
M : Type u_11
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
N : Type u_13
inst✝¹ : TopologicalSpace N
inst✝ : ChartedSpace G N
n : ℕ∞ω
f : M → N
x : M
h : IsImmersionAtOfComplement F I J n f x
e : F ≃L[𝕜] F'
⊢ IsImmersionAtOfComplement F' I J n f x

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Instances.Real

Goal state before:
pp: n : ℕ
p : ℝ≥0∞
a : ℝ
⊢ interior {y | ∀ (i : Fin n), a ≤ y.ofLp i} = {y | ∀ (i : Fin n), a < y.ofLp i}

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Instances.Real

Goal state before:
pp: n : ℕ
inst✝ : NeZero n
x : EuclideanSpace ℝ (Fin n)
⊢ 0 ≤ (toLp 2 (update x.ofLp 0 (max (x.ofLp 0) 0))).ofLp 0

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Instances.Real

Goal state before:
pp: n : ℕ
inst✝ : NeZero n
x✝¹ : EuclideanHalfSpace n
xval : EuclideanSpace ℝ (Fin n)
xprop : 0 ≤ xval.ofLp 0
x✝ : ⟨xval, xprop⟩ ∈ univ
⊢ ⟨toLp 2 (update (↑⟨xval, xprop⟩).ofLp 0 (max ((↑⟨xval, xprop⟩).ofLp 0) 0)), ⋯⟩ = ⟨xval, xprop⟩

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Instances.Real

Goal state before:
pp: n : ℕ
inst✝ : NeZero n
x✝ : EuclideanSpace ℝ (Fin n)
hx : x✝ ∈ {x | 0 ≤ x.ofLp 0}
⊢ ↑⟨toLp 2 (update x✝.ofLp 0 (max (x✝.ofLp 0) 0)), ⋯⟩ = x✝

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Instances.Real

Goal state before:
pp: n : ℕ
inst✝ : NeZero n
⊢ if h : IsRCLikeNormedField ℝ then Convex ℝ (range Subtype.val) else range Subtype.val = univ

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Instances.Real

Goal state before:
pp: n : ℕ
inst✝ : NeZero n
⊢ (toLp 2 fun i ↦ 1) ∈ {y | 0 < y.ofLp 0}

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Instances.Real

Goal state before:
pp: n : ℕ
inst✝ : NeZero n
⊢ (interior (range Subtype.val)).Nonempty

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Instances.Real

Goal state before:
pp: n : ℕ
inst✝ : NeZero n
⊢ Continuous fun x ↦ ⟨toLp 2 (update x.ofLp 0 (max (x.ofLp 0) 0)), ⋯⟩

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Instances.Real

Goal state before:
pp: n : ℕ
x : EuclideanSpace ℝ (Fin n)
i : Fin n
⊢ 0 ≤ (toLp 2 fun i ↦ max (x.ofLp i) 0).ofLp i

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Instances.Real

Goal state before:
pp: n : ℕ
x : EuclideanQuadrant n
x✝ : x ∈ univ
⊢ ⟨toLp 2 fun i ↦ max ((↑x).ofLp i) 0, ⋯⟩ = x

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Instances.Real

Goal state before:
pp: n : ℕ
x : EuclideanSpace ℝ (Fin n)
hx : x ∈ {x | ∀ (i : Fin n), 0 ≤ x.ofLp i}
⊢ ↑⟨toLp 2 fun i ↦ max (x.ofLp i) 0, ⋯⟩ = x

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Instances.Real

Goal state before:
pp: n : ℕ
⊢ if h : IsRCLikeNormedField ℝ then Convex ℝ (range Subtype.val) else range Subtype.val = univ

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Instances.Real

Goal state before:
pp: n : ℕ
⊢ (toLp 2 fun i ↦ 1) ∈ {y | ∀ (i : Fin n), 0 < y.ofLp i}

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Instances.Real

Goal state before:
pp: n : ℕ
⊢ (interior (range Subtype.val)).Nonempty

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Immersion

Goal state before:
pp: 𝕜 : Type u_1
inst✝²⁸ : NontriviallyNormedField 𝕜
E : Type u_2
E' : Type u_3
E''' : Type u_4
E'' : Type u
F : Type u_5
F' : Type u_6
inst✝²⁷ : NormedAddCommGroup E
inst✝²⁶ : NormedSpace 𝕜 E
inst✝²⁵ : NormedAddCommGroup E'
inst✝²⁴ : NormedSpace 𝕜 E'
inst✝²³ : NormedAddCommGroup E''
inst✝²² : NormedSpace 𝕜 E''
inst✝²¹ : NormedAddCommGroup E'''
inst✝²⁰ : NormedSpace 𝕜 E'''
inst✝¹⁹ : NormedAddCommGroup F
inst✝¹⁸ : NormedSpace 𝕜 F
inst✝¹⁷ : NormedAddCommGroup F'
inst✝¹⁶ : NormedSpace 𝕜 F'
H : Type u_7
inst✝¹⁵ : TopologicalSpace H
H' : Type u_8
inst✝¹⁴ : TopologicalSpace H'
G : Type u_9
inst✝¹³ : TopologicalSpace G
G' : Type u_10
inst✝¹² : TopologicalSpace G'
I : ModelWithCorners 𝕜 E H
I' : ModelWithCorners 𝕜 E' H'
J : ModelWithCorners 𝕜 E'' G
J' : ModelWithCorners 𝕜 E''' G'
M : Type u_11
inst✝¹¹ : TopologicalSpace M
inst✝¹⁰ : ChartedSpace H M
M' : Type u_12
inst✝⁹ : TopologicalSpace M'
inst✝⁸ : ChartedSpace H' M'
N : Type u_13
inst✝⁷ : TopologicalSpace N
inst✝⁶ : ChartedSpace G N
N' : Type u_14
inst✝⁵ : TopologicalSpace N'
inst✝⁴ : ChartedSpace G' N'
n : ℕ∞ω
x✝¹ : M
f✝ : M → N
g✝ : M' → N'
x' : M'
inst✝³ : IsManifold I n M
inst✝² : IsManifold I' n M'
inst✝¹ : IsManifold J n N
inst✝ : IsManifold J' n N'
hf : IsImmersionAtOfComplement F I J n f✝ x✝¹
hg : IsImmersionAtOfComplement F' I' J' n g✝ x'
f : M → N
φ₁ : OpenPartialHomeomorph M H
ψ₁ : OpenPartialHomeomorph N G
g : M' → N'
φ₂ : OpenPartialHomeomorph M' H'
ψ₂ : OpenPartialHomeomorph N' G'
equiv₁ : (E × F) ≃L[𝕜] E''
hfprop : EqOn (↑(ψ₁.extend J) ∘ f ∘ ↑(φ₁.extend I).symm) (⇑equiv₁ ∘ fun x ↦ (x, 0)) (φ₁.extend I).target
equiv₂ : (E' × F') ≃L[𝕜] E'''
hgprop : EqOn (↑(ψ₂.extend J') ∘ g ∘ ↑(φ₂.extend I').symm) (⇑equiv₂ ∘ fun x ↦ (x, 0)) (φ₂.extend I').target
x : E × E'
x✝ : x ∈ (φ₁.extend I).target ×ˢ (φ₂.extend I').target
hx : x.1 ∈ (φ₁.extend I).target
hx' : x.2 ∈ (φ₂.extend I').target
⊢ (Prod.fst ∘ ↑((ψ₁.extend J).prod (ψ₂.extend J')) ∘ Prod.map f g ∘ ↑((φ₁.extend I).prod (φ₂.extend I')).symm) x =
    (Prod.fst ∘
        ⇑((ContinuousLinearEquiv.prodProdProdComm 𝕜 E E' F F').trans (equiv₁.prodCongr equiv₂)) ∘ fun x ↦ (x, 0))
      x

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Instances.Real

Goal state before:
pp: n : ℕ
inst✝ : NeZero n
⊢ interior (range ↑(𝓡∂ n)) = interior {y | 0 ≤ y.ofLp 0}

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Instances.Real

Goal state before:
pp: n : ℕ
inst✝ : NeZero n
⊢ interior (range ↑(𝓡∂ n)) = {y | 0 < y.ofLp 0}

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Instances.Real

Goal state before:
pp: n : ℕ
inst✝ : NeZero n
⊢ frontier (range ↑(𝓡∂ n)) = frontier {y | 0 ≤ y.ofLp 0}

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Instances.Real

Goal state before:
pp: n : ℕ
inst✝ : NeZero n
⊢ frontier (range ↑(𝓡∂ n)) = {y | 0 = y.ofLp 0}

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Instances.Real

Goal state before:
pp: x y : ℝ
h : Fact (x < y)
z : EuclideanHalfSpace 1
⊢ min ((↑z).ofLp 0 + x) y ∈ Icc x y

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Instances.Real

Goal state before:
pp: x y : ℝ
h : Fact (x < y)
⊢ ∀ ⦃x_1 : ↑(Icc x y)⦄, x_1 ∈ {z | ↑z < y} → ⟨toLp 2 fun x_2 ↦ ↑x_1 - x, ⋯⟩ ∈ {z | (↑z).ofLp 0 < y - x}

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Instances.Real

Goal state before:
pp: x y : ℝ
h : Fact (x < y)
⊢ ∀ ⦃x_1 : EuclideanHalfSpace 1⦄, x_1 ∈ {z | (↑z).ofLp 0 < y - x} → ⟨min ((↑x_1).ofLp 0 + x) y, ⋯⟩ ∈ {z | ↑z < y}

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Immersion

Goal state before:
pp: 𝕜 : Type u_1
inst✝²⁸ : NontriviallyNormedField 𝕜
E : Type u_2
E' : Type u_3
E''' : Type u_4
E'' : Type u
F : Type u_5
F' : Type u_6
inst✝²⁷ : NormedAddCommGroup E
inst✝²⁶ : NormedSpace 𝕜 E
inst✝²⁵ : NormedAddCommGroup E'
inst✝²⁴ : NormedSpace 𝕜 E'
inst✝²³ : NormedAddCommGroup E''
inst✝²² : NormedSpace 𝕜 E''
inst✝²¹ : NormedAddCommGroup E'''
inst✝²⁰ : NormedSpace 𝕜 E'''
inst✝¹⁹ : NormedAddCommGroup F
inst✝¹⁸ : NormedSpace 𝕜 F
inst✝¹⁷ : NormedAddCommGroup F'
inst✝¹⁶ : NormedSpace 𝕜 F'
H : Type u_7
inst✝¹⁵ : TopologicalSpace H
H' : Type u_8
inst✝¹⁴ : TopologicalSpace H'
G : Type u_9
inst✝¹³ : TopologicalSpace G
G' : Type u_10
inst✝¹² : TopologicalSpace G'
I : ModelWithCorners 𝕜 E H
I' : ModelWithCorners 𝕜 E' H'
J : ModelWithCorners 𝕜 E'' G
J' : ModelWithCorners 𝕜 E''' G'
M : Type u_11
inst✝¹¹ : TopologicalSpace M
inst✝¹⁰ : ChartedSpace H M
M' : Type u_12
inst✝⁹ : TopologicalSpace M'
inst✝⁸ : ChartedSpace H' M'
N : Type u_13
inst✝⁷ : TopologicalSpace N
inst✝⁶ : ChartedSpace G N
N' : Type u_14
inst✝⁵ : TopologicalSpace N'
inst✝⁴ : ChartedSpace G' N'
n : ℕ∞ω
x✝¹ : M
f✝ : M → N
g✝ : M' → N'
x' : M'
inst✝³ : IsManifold I n M
inst✝² : IsManifold I' n M'
inst✝¹ : IsManifold J n N
inst✝ : IsManifold J' n N'
hf : IsImmersionAtOfComplement F I J n f✝ x✝¹
hg : IsImmersionAtOfComplement F' I' J' n g✝ x'
f : M → N
φ₁ : OpenPartialHomeomorph M H
ψ₁ : OpenPartialHomeomorph N G
g : M' → N'
φ₂ : OpenPartialHomeomorph M' H'
ψ₂ : OpenPartialHomeomorph N' G'
equiv₁ : (E × F) ≃L[𝕜] E''
hfprop : EqOn (↑(ψ₁.extend J) ∘ f ∘ ↑(φ₁.extend I).symm) (⇑equiv₁ ∘ fun x ↦ (x, 0)) (φ₁.extend I).target
equiv₂ : (E' × F') ≃L[𝕜] E'''
hgprop : EqOn (↑(ψ₂.extend J') ∘ g ∘ ↑(φ₂.extend I').symm) (⇑equiv₂ ∘ fun x ↦ (x, 0)) (φ₂.extend I').target
x : E × E'
x✝ : x ∈ (φ₁.extend I).target ×ˢ (φ₂.extend I').target
hx : x.1 ∈ (φ₁.extend I).target
hx' : x.2 ∈ (φ₂.extend I').target
⊢ (Prod.snd ∘ ↑((ψ₁.extend J).prod (ψ₂.extend J')) ∘ Prod.map f g ∘ ↑((φ₁.extend I).prod (φ₂.extend I')).symm) x =
    (Prod.snd ∘
        ⇑((ContinuousLinearEquiv.prodProdProdComm 𝕜 E E' F F').trans (equiv₁.prodCongr equiv₂)) ∘ fun x ↦ (x, 0))
      x

Tactic:
by

Goal state after: