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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝⁷ : TopologicalSpace H
E : Type u_3
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : IsManifold I (minSmoothness 𝕜 2) M
inst✝¹ : CompleteSpace E
n : ℕ∞ω
inst✝ : IsManifold I (n + 1) M
m : ℕ∞ω
U V : (x : M) → TangentSpace I x
s : Set M
x : M
hU : ContMDiffWithinAt I I.tangent n (fun x ↦ ⟨x, U x⟩) s x
hV : ContMDiffWithinAt I I.tangent n (fun x ↦ ⟨x, V x⟩) s x
hs : UniqueMDiff[s]
hx : x ∈ s
hmn : minSmoothness 𝕜 (m + 1) ≤ n
min2 : minSmoothness 𝕜 2 ≤ n + 1
m' : ℕ∞ω
hm' : m' ≤ m
h'm' : m' ≠ ∞
hn : 1 ≤ m' + 1
hm'n : m' + 1 ≤ n
pre_mem : ↑(extChartAt I x) ⁻¹' ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' s) ∈ 𝓝[s] x
U' : (x : E) → TangentSpace 𝓘(𝕜, E) x := mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm) U (range ↑I)
V' : (x : E) → TangentSpace 𝓘(𝕜, E) x := mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm) V (range ↑I)
A :
  ContDiffWithinAt 𝕜 m' (lieBracketWithin 𝕜 U' V' ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' s))
    ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' s) (↑(extChartAt I x) x)
B :
  ContMDiffWithinAt 𝓘(𝕜, E) 𝓘(𝕜, E).tangent m'
    (fun x₀ ↦ ⟨x₀, mlieBracketWithin 𝓘(𝕜, E) U' V' ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' s) x₀⟩)
    ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' s) (↑(extChartAt I x) x)
C :
  ContMDiffWithinAt I I.tangent m'
    (fun x_1 ↦
      ⟨x_1,
        mpullback I 𝓘(𝕜, E) (↑(extChartAt I x))
          (mlieBracketWithin 𝓘(𝕜, E) U' V' ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' s)) x_1⟩)
    s x
y : M
hy : ∀ᶠ (x_1 : M) in 𝓝[s] y, ↑(extChartAt I x) x_1 ∈ (extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' s
hyU :
  ∀ᶠ (x_1 : M) in 𝓝[s] y,
    U x_1 =
      mpullback I 𝓘(𝕜, E) (↑(extChartAt I x)) (mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm) U (range ↑I)) x_1
hyV :
  ∀ᶠ (x_1 : M) in 𝓝[s] y,
    V x_1 =
      mpullback I 𝓘(𝕜, E) (↑(extChartAt I x)) (mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm) V (range ↑I)) x_1
h'yU :
  ContMDiffWithinAt 𝓘(𝕜, E) 𝓘(𝕜, E).tangent (m' + 1)
    (fun x_1 ↦ ⟨x_1, mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm) U (range ↑I) x_1⟩)
    ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' s) (↑(extChartAt I x) y)
h'yV :
  ContMDiffWithinAt 𝓘(𝕜, E) 𝓘(𝕜, E).tangent (m' + 1)
    (fun x_1 ↦ ⟨x_1, mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm) V (range ↑I) x_1⟩)
    ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' s) (↑(extChartAt I x) y)
hy_chart : y ∈ (chartAt H x).source
hys : y ∈ s
⊢ m' + 1 ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Riemannian.Basic

Goal state before:
pp: E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
H : Type u_2
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : RiemannianBundle fun x ↦ TangentSpace I x
inst✝¹ : IsManifold I 1 M
inst✝ : IsContinuousRiemannianBundle E fun x ↦ TangentSpace I x
x : M
C : ℝ≥0
C_pos : C > 0
hC : ∀ᶠ (y : E) in 𝓝[range ↑I] ↑(extChartAt I x) x, ‖mfderiv[range ↑I] ↑(extChartAt I x).symm y‖ₑ < ↑C
r : ℝ
r_pos : r > 0
hr :
  ball (↑(extChartAt I x) x) r ∩ range ↑I ⊆
    (extChartAt I x).target ∩ {y | ‖mfderiv[range ↑I] ↑(extChartAt I x).symm y‖ₑ < ↑C}
A : ↑(extChartAt I x) ⁻¹' (ball (↑(extChartAt I x) x) r ∩ range ↑I) ∈ 𝓝 x
y : M
hy : y ∈ ↑(extChartAt I x) ⁻¹' (ball (↑(extChartAt I x) x) r ∩ range ↑I)
h'y : y ∈ (chartAt H x).source
η : ℝ →ᴬ[ℝ] E := ContinuousAffineMap.lineMap (↑(extChartAt I x) x) (↑(extChartAt I x) y)
γ : ℝ → M := ↑(extChartAt I x).symm ∘ ⇑η
hη :
  Icc 0 1 ⊆ ⇑η ⁻¹' (extChartAt I x).target ∧ Icc 0 1 ⊆ ⇑η ⁻¹' {y | ‖mfderiv[range ↑I] ↑(extChartAt I x).symm y‖ₑ < ↑C}
η_smooth : ContMDiffOn 𝓘(ℝ, ℝ) 𝓘(ℝ, E) 1 (⇑η) (Icc 0 1)
this : riemannianEDist I x y ≤ pathELength I γ 0 1
t : ℝ
ht : t ∈ Icc 0 1
⊢ mfderiv[Icc 0 1] γ t = mfderiv[range ↑I] ↑(extChartAt I x).symm (η t) ∘SL mfderiv[Icc 0 1] ⇑η t

Tactic:
by

Goal state after:
Module: Mathlib.Topology.VectorBundle.Riemannian

Goal state before:
pp: B : Type u_1
inst✝⁷ : TopologicalSpace B
F : Type u_2
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
E : B → Type u_3
inst✝⁴ : TopologicalSpace (TotalSpace F E)
inst✝³ : (x : B) → NormedAddCommGroup (E x)
inst✝² : (x : B) → InnerProductSpace ℝ (E x)
inst✝¹ : FiberBundle F E
inst✝ : VectorBundle ℝ F E
h : IsContinuousRiemannianBundle F E
x : B
r : ℝ
hr : 1 < r
r' : ℝ
hr' : 1 < r'
r'r : r' < r
h'x : x ∈ (trivializationAt F E x).baseSet
G : E x ≃L[ℝ] F := Trivialization.continuousLinearEquivAt ℝ (trivializationAt F E x) x h'x
C : ℝ := ‖↑G‖ ^ 2
δ : ℝ
δpos : 0 < δ
hδ : (r' ^ 2)⁻¹ < 1 - δ * C
g : (x : B) → E x →L[ℝ] E x →L[ℝ] ℝ
g_cont :
  Continuous[inst✝⁷, topologicalSpaceTotalSpace (RingHom.id ℝ) F E (F →L[ℝ] ℝ) fun x ↦ E x →L[ℝ] ℝ] fun x ↦
    TotalSpace.mk' (F →L[ℝ] F →L[ℝ] ℝ) x (g x)
hg : ∀ (x : B) (v w : E x), ⟪v, w⟫ = ((g x) v) w
g' : B → F →L[ℝ] F →L[ℝ] ℝ := fun y ↦ inCoordinates F E (F →L[ℝ] ℝ) (fun x ↦ E x →L[ℝ] ℝ) x y x y (g y)
⊢ ContinuousAt g' x

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝⁷ : TopologicalSpace H
E : Type u_3
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : IsManifold I (minSmoothness 𝕜 2) M
inst✝¹ : CompleteSpace E
n : ℕ∞ω
inst✝ : IsManifold I (n + 1) M
m : ℕ∞ω
U V : (x : M) → TangentSpace I x
s : Set M
x : M
hU : ContMDiffWithinAt I I.tangent n (fun x ↦ ⟨x, U x⟩) s x
hV : ContMDiffWithinAt I I.tangent n (fun x ↦ ⟨x, V x⟩) s x
hs : UniqueMDiff[s]
hx : x ∈ s
hmn : minSmoothness 𝕜 (m + 1) ≤ n
⊢ ContMDiffWithinAt I I.tangent m (fun x₀ ↦ ⟨x₀, mlieBracketWithin I U V s x₀⟩) s x

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝⁷ : TopologicalSpace H
E : Type u_3
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : IsManifold I (minSmoothness 𝕜 2) M
inst✝¹ : CompleteSpace E
m n : ℕ∞
inst✝ : IsManifold I (↑n + 1) M
U V : (x : M) → TangentSpace I x
x : M
hU : ContMDiffAt I I.tangent (↑n) (fun x ↦ ⟨x, U x⟩) x
hV : ContMDiffAt I I.tangent (↑n) (fun x ↦ ⟨x, V x⟩) x
hmn : minSmoothness 𝕜 (↑m + 1) ≤ ↑n
⊢ ContMDiffAt I I.tangent (↑m) (fun x₀ ↦ ⟨x₀, mlieBracket I U V x₀⟩) x

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝⁷ : TopologicalSpace H
E : Type u_3
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : IsManifold I (minSmoothness 𝕜 2) M
inst✝¹ : CompleteSpace E
m n : ℕ∞
inst✝ : IsManifold I (↑n + 1) M
U V : (x : M) → TangentSpace I x
hU : ContMDiff I I.tangent ↑n fun x ↦ ⟨x, U x⟩
hV : ContMDiff I I.tangent ↑n fun x ↦ ⟨x, V x⟩
hmn : minSmoothness 𝕜 (↑m + 1) ≤ ↑n
⊢ ContMDiff I I.tangent ↑m fun x₀ ↦ ⟨x₀, mlieBracket I U V x₀⟩

Tactic:
by

Goal state after:
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
inst✝¹ : IsManifold I (minSmoothness 𝕜 3) M
inst✝ : CompleteSpace E
U V W : (x : M) → TangentSpace I x
s : Set M
x : M
hs : UniqueMDiff[s]
h's : x ∈ closure[inst✝³] (interior s)
hx : x ∈ s
hU : ContMDiffWithinAt I I.tangent (minSmoothness 𝕜 2) (fun x ↦ ⟨x, U x⟩) s x
hV : ContMDiffWithinAt I I.tangent (minSmoothness 𝕜 2) (fun x ↦ ⟨x, V x⟩) s x
hW : ContMDiffWithinAt I I.tangent (minSmoothness 𝕜 2) (fun x ↦ ⟨x, W x⟩) s x
⊢ minSmoothness 𝕜 2 + 1 ≤ minSmoothness 𝕜 3

Tactic:
by

Goal state after:
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
inst✝¹ : IsManifold I (minSmoothness 𝕜 3) M
inst✝ : CompleteSpace E
U V W : (x : M) → TangentSpace I x
s : Set M
x : M
hs : UniqueMDiff[s]
h's : x ∈ closure[inst✝³] (interior s)
hx : x ∈ s
hU : ContMDiffWithinAt I I.tangent (minSmoothness 𝕜 2) (fun x ↦ ⟨x, U x⟩) s x
hV : ContMDiffWithinAt I I.tangent (minSmoothness 𝕜 2) (fun x ↦ ⟨x, V x⟩) s x
hW : ContMDiffWithinAt I I.tangent (minSmoothness 𝕜 2) (fun x ↦ ⟨x, W x⟩) s x
A : minSmoothness 𝕜 2 + 1 ≤ minSmoothness 𝕜 3
s_inter_mem : s ∩ (extChartAt I x).source ∈ 𝓝[s] x
y : M
hy : y ∈ s ∩ (extChartAt I x).source
⊢ ↑(extChartAt I x) y ∈ ↑(extChartAt I x).symm ⁻¹' s

Tactic:
by

Goal state after:
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
inst✝¹ : IsManifold I (minSmoothness 𝕜 3) M
inst✝ : CompleteSpace E
U V W : (x : M) → TangentSpace I x
s : Set M
x : M
hs : UniqueMDiff[s]
h's : x ∈ closure[inst✝³] (interior s)
hx : x ∈ s
hU : ContMDiffWithinAt I I.tangent (minSmoothness 𝕜 2) (fun x ↦ ⟨x, U x⟩) s x
hV : ContMDiffWithinAt I I.tangent (minSmoothness 𝕜 2) (fun x ↦ ⟨x, V x⟩) s x
hW : ContMDiffWithinAt I I.tangent (minSmoothness 𝕜 2) (fun x ↦ ⟨x, W x⟩) s x
A : minSmoothness 𝕜 2 + 1 ≤ minSmoothness 𝕜 3
s_inter_mem : s ∩ (extChartAt I x).source ∈ 𝓝[s] x
⊢ ↑(extChartAt I x) ⁻¹' ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' s) ∈ 𝓝[s] x

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Sheaf.LocallyRingedSpace

Goal state before:
pp: 𝕜 : Type u
inst✝¹⁵ : NontriviallyNormedField 𝕜
EM : Type u_1
inst✝¹⁴ : NormedAddCommGroup EM
inst✝¹³ : NormedSpace 𝕜 EM
HM : Type u_2
inst✝¹² : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
M : Type u
inst✝¹¹ : TopologicalSpace M
inst✝¹⁰ : ChartedSpace HM M
EN : Type u_3
inst✝⁹ : NormedAddCommGroup EN
inst✝⁸ : NormedSpace 𝕜 EN
HN : Type u_4
inst✝⁷ : TopologicalSpace HN
IN : ModelWithCorners 𝕜 EN HN
N : Type u
inst✝⁶ : TopologicalSpace N
inst✝⁵ : ChartedSpace HN N
EP : Type u_5
inst✝⁴ : NormedAddCommGroup EP
inst✝³ : NormedSpace 𝕜 EP
HP : Type u_6
inst✝² : TopologicalSpace HP
IP : ModelWithCorners 𝕜 EP HP
P : Type u
inst✝¹ : TopologicalSpace P
inst✝ : ChartedSpace HP P
U : Opens M
V : Opens ↑↑(ChartedSpace.locallyRingedSpace IM ↥U).toPresheafedSpace
⊢ IsIso
    ((LocallyRingedSpace.Hom.toShHom (ChartedSpace.locallyRingedSpaceMap Subtype.val ⋯)).hom.c.app
      (Opposite.op (⋯.functor.obj V)))

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Riemannian.Basic

Goal state before:
pp: E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
H : Type u_2
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : RiemannianBundle fun x ↦ TangentSpace I x
inst✝¹ : IsManifold I 1 M
inst✝ : IsContinuousRiemannianBundle E fun x ↦ TangentSpace I x
x : M
⊢ ∃ C > 0, ∀ᶠ (y : M) in 𝓝 x, riemannianEDist I x y ≤ ↑C * edist (↑(extChartAt I x) x) (↑(extChartAt I x) y)

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Sheaf.LocallyRingedSpace

Goal state before:
pp: 𝕜 : Type u
inst✝¹⁵ : NontriviallyNormedField 𝕜
EM : Type u_1
inst✝¹⁴ : NormedAddCommGroup EM
inst✝¹³ : NormedSpace 𝕜 EM
HM : Type u_2
inst✝¹² : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
M : Type u
inst✝¹¹ : TopologicalSpace M
inst✝¹⁰ : ChartedSpace HM M
EN : Type u_3
inst✝⁹ : NormedAddCommGroup EN
inst✝⁸ : NormedSpace 𝕜 EN
HN : Type u_4
inst✝⁷ : TopologicalSpace HN
IN : ModelWithCorners 𝕜 EN HN
N : Type u
inst✝⁶ : TopologicalSpace N
inst✝⁵ : ChartedSpace HN N
EP : Type u_5
inst✝⁴ : NormedAddCommGroup EP
inst✝³ : NormedSpace 𝕜 EP
HP : Type u_6
inst✝² : TopologicalSpace HP
IP : ModelWithCorners 𝕜 EP HP
P : Type u
inst✝¹ : TopologicalSpace P
inst✝ : ChartedSpace HP P
U : Opens M
⊢ Set.range ⇑(ConcreteCategory.hom ((locallyRingedSpace IM M).ofRestrict ⋯).base) ⊆
    Set.range ⇑(ConcreteCategory.hom (locallyRingedSpaceMap Subtype.val ⋯).base)

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Sheaf.LocallyRingedSpace

Goal state before:
pp: 𝕜 : Type u
inst✝¹⁵ : NontriviallyNormedField 𝕜
EM : Type u_1
inst✝¹⁴ : NormedAddCommGroup EM
inst✝¹³ : NormedSpace 𝕜 EM
HM : Type u_2
inst✝¹² : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
M : Type u
inst✝¹¹ : TopologicalSpace M
inst✝¹⁰ : ChartedSpace HM M
EN : Type u_3
inst✝⁹ : NormedAddCommGroup EN
inst✝⁸ : NormedSpace 𝕜 EN
HN : Type u_4
inst✝⁷ : TopologicalSpace HN
IN : ModelWithCorners 𝕜 EN HN
N : Type u
inst✝⁶ : TopologicalSpace N
inst✝⁵ : ChartedSpace HN N
EP : Type u_5
inst✝⁴ : NormedAddCommGroup EP
inst✝³ : NormedSpace 𝕜 EP
HP : Type u_6
inst✝² : TopologicalSpace HP
IP : ModelWithCorners 𝕜 EP HP
P : Type u
inst✝¹ : TopologicalSpace P
inst✝ : ChartedSpace HP P
U : Opens M
⊢ Set.range ⇑(ConcreteCategory.hom (locallyRingedSpaceMap Subtype.val ⋯).base) ⊆
    Set.range ⇑(ConcreteCategory.hom ((locallyRingedSpace IM M).ofRestrict ⋯).base)

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Riemannian.Basic

Goal state before:
pp: E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
H : Type u_2
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : RiemannianBundle fun x ↦ TangentSpace I x
inst✝¹ : IsManifold I 1 M
inst✝ : IsContinuousRiemannianBundle E fun x ↦ TangentSpace I x
x : M
c : ℝ≥0∞
hc : 0 < c
C : ℝ≥0
C_pos : C > 0
hC : ∀ᶠ (y : M) in 𝓝 x, riemannianEDist I x y ≤ ↑C * edist (↑(extChartAt I x) x) (↑(extChartAt I x) y)
⊢ ↑C ≠ ∞

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Riemannian.Basic

Goal state before:
pp: E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
H : Type u_2
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : RiemannianBundle fun x ↦ TangentSpace I x
inst✝¹ : IsManifold I 1 M
inst✝ : IsContinuousRiemannianBundle E fun x ↦ TangentSpace I x
x : M
c : ℝ≥0∞
hc : 0 < c
C : ℝ≥0
C_pos : C > 0
hC : ∀ᶠ (y : M) in 𝓝 x, riemannianEDist I x y ≤ ↑C * edist (↑(extChartAt I x) x) (↑(extChartAt I x) y)
⊢ ↑(extChartAt I x) ⁻¹' eball (↑(extChartAt I x) x) (c / ↑C) ∈ 𝓝 x

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Sheaf.LocallyRingedSpace

Goal state before:
pp: 𝕜 : Type u
inst✝¹⁵ : NontriviallyNormedField 𝕜
EM : Type u_1
inst✝¹⁴ : NormedAddCommGroup EM
inst✝¹³ : NormedSpace 𝕜 EM
HM : Type u_2
inst✝¹² : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
M : Type u
inst✝¹¹ : TopologicalSpace M
inst✝¹⁰ : ChartedSpace HM M
EN : Type u_3
inst✝⁹ : NormedAddCommGroup EN
inst✝⁸ : NormedSpace 𝕜 EN
HN : Type u_4
inst✝⁷ : TopologicalSpace HN
IN : ModelWithCorners 𝕜 EN HN
N : Type u
inst✝⁶ : TopologicalSpace N
inst✝⁵ : ChartedSpace HN N
EP : Type u_5
inst✝⁴ : NormedAddCommGroup EP
inst✝³ : NormedSpace 𝕜 EP
HP : Type u_6
inst✝² : TopologicalSpace HP
IP : ModelWithCorners 𝕜 EP HP
P : Type u
inst✝¹ : TopologicalSpace P
inst✝ : ChartedSpace HP P
U : Opens M
⊢ LocallyRingedSpace.IsOpenImmersion.lift (locallyRingedSpaceMap Subtype.val ⋯) ((locallyRingedSpace IM M).ofRestrict ⋯)
        ⋯ ≫
      LocallyRingedSpace.IsOpenImmersion.lift ((locallyRingedSpace IM M).ofRestrict ⋯)
        (locallyRingedSpaceMap Subtype.val ⋯) ⋯ =
    𝟙 ((locallyRingedSpace IM M).restrict ⋯)

Tactic:
by

Goal state after:
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
inst✝¹ : IsManifold I (minSmoothness 𝕜 3) M
inst✝ : CompleteSpace E
U V W : (x : M) → TangentSpace I x
s : Set M
x : M
hs : UniqueMDiff[s]
h's : x ∈ closure[inst✝³] (interior s)
hx : x ∈ s
hU : ContMDiffWithinAt I I.tangent (minSmoothness 𝕜 2) (fun x ↦ ⟨x, U x⟩) s x
hV : ContMDiffWithinAt I I.tangent (minSmoothness 𝕜 2) (fun x ↦ ⟨x, V x⟩) s x
hW : ContMDiffWithinAt I I.tangent (minSmoothness 𝕜 2) (fun x ↦ ⟨x, W x⟩) s x
A : minSmoothness 𝕜 2 + 1 ≤ minSmoothness 𝕜 3
s_inter_mem : s ∩ (extChartAt I x).source ∈ 𝓝[s] x
pre_mem : ↑(extChartAt I x) ⁻¹' ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' s) ∈ 𝓝[s] x
U' : (x : E) → TangentSpace 𝓘(𝕜, E) x := mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm) U (range ↑I)
V' : (x : E) → TangentSpace 𝓘(𝕜, E) x := mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm) V (range ↑I)
W' : (x : E) → TangentSpace 𝓘(𝕜, E) x := mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm) W (range ↑I)
J0U :
  ContMDiffWithinAt 𝓘(𝕜, E) 𝓘(𝕜, E).tangent (minSmoothness 𝕜 2) (fun x ↦ ⟨x, U' x⟩)
    ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' s) (↑(extChartAt I x) x)
J0V :
  ContMDiffWithinAt 𝓘(𝕜, E) 𝓘(𝕜, E).tangent (minSmoothness 𝕜 2) (fun x ↦ ⟨x, V' x⟩)
    ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' s) (↑(extChartAt I x) x)
J0W :
  ContMDiffWithinAt 𝓘(𝕜, E) 𝓘(𝕜, E).tangent (minSmoothness 𝕜 2) (fun x ↦ ⟨x, W' x⟩)
    ((extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' s) (↑(extChartAt I x) x)
⊢ minSmoothness 𝕜 2 ≠ ∞

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Sheaf.LocallyRingedSpace

Goal state before:
pp: 𝕜 : Type u
inst✝¹⁵ : NontriviallyNormedField 𝕜
EM : Type u_1
inst✝¹⁴ : NormedAddCommGroup EM
inst✝¹³ : NormedSpace 𝕜 EM
HM : Type u_2
inst✝¹² : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
M : Type u
inst✝¹¹ : TopologicalSpace M
inst✝¹⁰ : ChartedSpace HM M
EN : Type u_3
inst✝⁹ : NormedAddCommGroup EN
inst✝⁸ : NormedSpace 𝕜 EN
HN : Type u_4
inst✝⁷ : TopologicalSpace HN
IN : ModelWithCorners 𝕜 EN HN
N : Type u
inst✝⁶ : TopologicalSpace N
inst✝⁵ : ChartedSpace HN N
EP : Type u_5
inst✝⁴ : NormedAddCommGroup EP
inst✝³ : NormedSpace 𝕜 EP
HP : Type u_6
inst✝² : TopologicalSpace HP
IP : ModelWithCorners 𝕜 EP HP
P : Type u
inst✝¹ : TopologicalSpace P
inst✝ : ChartedSpace HP P
U : Opens M
⊢ LocallyRingedSpace.IsOpenImmersion.lift ((locallyRingedSpace IM M).ofRestrict ⋯) (locallyRingedSpaceMap Subtype.val ⋯)
        ⋯ ≫
      LocallyRingedSpace.IsOpenImmersion.lift (locallyRingedSpaceMap Subtype.val ⋯)
        ((locallyRingedSpace IM M).ofRestrict ⋯) ⋯ =
    𝟙 (locallyRingedSpace IM ↥U)

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Manifold.Riemannian.Basic

Goal state before:
pp: E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
H : Type u_2
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_3
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : RiemannianBundle fun x ↦ TangentSpace I x
inst✝¹ : IsManifold I 1 M
inst✝ : IsContinuousRiemannianBundle E fun x ↦ TangentSpace I x
x : M
c : ℝ≥0∞
hc : 0 < c
C : ℝ≥0
C_pos : C > 0
hC : ∀ᶠ (y : M) in 𝓝 x, riemannianEDist I x y ≤ ↑C * edist (↑(extChartAt I x) x) (↑(extChartAt I x) y)
this : ↑(extChartAt I x) ⁻¹' eball (↑(extChartAt I x) x) (c / ↑C) ∈ 𝓝 x
y : M
hy : y ∈ ↑(extChartAt I x) ⁻¹' eball (↑(extChartAt I x) x) (c / ↑C)
h'y : riemannianEDist I x y ≤ ↑C * edist (↑(extChartAt I x) x) (↑(extChartAt I x) y)
⊢ edist (↑(extChartAt I x) x) (↑(extChartAt I x) y) < c / ↑C

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E'' : Type u_3
F : Type u_5
H : Type u_7
G : Type u_9
E : Type u
inst✝¹⁰ : NontriviallyNormedField 𝕜
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
inst✝³ : TopologicalSpace H
inst✝² : TopologicalSpace G
I : ModelWithCorners 𝕜 E H
J : ModelWithCorners 𝕜 E'' G
M : Type u_11
N : Type u_13
inst✝¹ : TopologicalSpace M
inst✝ : TopologicalSpace N
f : M → N
φ : OpenPartialHomeomorph M H
ψ : OpenPartialHomeomorph N G
s : Set M
hs : IsOpen[inst✝¹] s
x✝ : SubmersionAtProp F I J M N f φ ψ
equiv : E ≃L[𝕜] E'' × F
hf : EqOn (↑(ψ.extend J) ∘ f ∘ ↑(φ.extend I).symm) (Prod.fst ∘ ⇑equiv) (φ.extend I).target
⊢ ((φ.restr s).extend I).target ⊆ (φ.extend I).target

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E'' : Type u_3
F : Type u_5
H : Type u_7
G : Type u_9
E : Type u
inst✝¹⁰ : NontriviallyNormedField 𝕜
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
inst✝³ : TopologicalSpace H
inst✝² : TopologicalSpace G
I : ModelWithCorners 𝕜 E H
J : ModelWithCorners 𝕜 E'' G
M : Type u_11
N : Type u_13
inst✝¹ : TopologicalSpace M
inst✝ : TopologicalSpace N
f g : M → N
φ : OpenPartialHomeomorph M H
ψ : OpenPartialHomeomorph N G
hfg : EqOn f g φ.source
equiv : E ≃L[𝕜] E'' × F
hf : EqOn (↑(ψ.extend J) ∘ f ∘ ↑(φ.extend I).symm) (Prod.fst ∘ ⇑equiv) (φ.extend I).target
⊢ (φ.extend I).target ⊆ (φ.extend I).target

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E'' : Type u_3
F : Type u_5
H : Type u_7
G : Type u_9
E : Type u
inst✝¹⁰ : NontriviallyNormedField 𝕜
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
inst✝³ : TopologicalSpace H
inst✝² : TopologicalSpace G
I : ModelWithCorners 𝕜 E H
J : ModelWithCorners 𝕜 E'' G
M : Type u_11
N : Type u_13
inst✝¹ : TopologicalSpace M
inst✝ : TopologicalSpace N
f g : M → N
φ : OpenPartialHomeomorph M H
ψ : OpenPartialHomeomorph N G
hfg : EqOn f g φ.source
equiv : E ≃L[𝕜] E'' × F
hf : EqOn (↑(ψ.extend J) ∘ f ∘ ↑(φ.extend I).symm) (Prod.fst ∘ ⇑equiv) (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.Submersion

Goal state before:
pp: 𝕜 : Type u_1
E'' : Type u_3
F : Type u_5
H : Type u_7
G : Type u_9
E : Type u
inst✝¹⁰ : NontriviallyNormedField 𝕜
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
inst✝³ : TopologicalSpace H
inst✝² : TopologicalSpace G
I : ModelWithCorners 𝕜 E H
J : ModelWithCorners 𝕜 E'' G
M : Type u_11
N : Type u_13
inst✝¹ : TopologicalSpace M
inst✝ : TopologicalSpace N
f g : M → N
φ : OpenPartialHomeomorph M H
ψ : OpenPartialHomeomorph N G
hfg : EqOn f g φ.source
⊢ SubmersionAtProp F I J M N f φ ψ → SubmersionAtProp F I J M N g φ ψ

Tactic:
by

Goal state after: