mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Complex.LocallyUniformLimit

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
z : ℂ
M r : ℝ
f : ℂ → E
hr : 0 < r
hfM : ∀ w ∈ sphere z r, ‖f w‖ < M
hf : ContinuousOn f (sphere z r)
⊢ ∃ L < M, ∀ w ∈ sphere z r, ‖f w‖ ≤ L

Tactic:
have e1 : (sphere z r).Nonempty := NormedSpace.sphere_nonempty.mpr hr.le

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
z : ℂ
M r : ℝ
f : ℂ → E
hr : 0 < r
hfM : ∀ w ∈ sphere z r, ‖f w‖ < M
hf : ContinuousOn f (sphere z r)
e1 : (sphere z r).Nonempty
⊢ ∃ L < M, ∀ w ∈ sphere z r, ‖f w‖ ≤ L
Module: Mathlib.MeasureTheory.Function.Jacobian

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
ε : ℝ≥0
εpos : 0 < ε
A : E →L[ℝ] E
m : ℝ≥0 := |A.det|.toNNReal + ε
I : ENNReal.ofReal |A.det| < ↑m
δ : ℝ≥0
h : ∀ (s : Set E) (f : E → E), ApproximatesLinearOn f A s δ → μ (f '' s) ≤ ↑m * μ s
δpos : 0 < δ
⊢ ContinuousAt ContinuousLinearMap.det A

Tactic:
exact ContinuousLinearMap.continuous_det.continuousAt

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

Goal state before:
pp: case h
H : Type u_1
inst✝ : TopologicalSpace H
G G' : StructureGroupoid H
e : OpenPartialHomeomorph H H
hx : ∀ x ∈ e.source, ∃ s, IsOpen[inst✝] s ∧ x ∈ s ∧ e.restr s ∈ G.members ∩ G'.members
x : H
hex : x ∈ e.source
s : Set H
hs : IsOpen[inst✝] s ∧ x ∈ s ∧ e.restr s ∈ G.members ∩ G'.members
⊢ e.restr s ∈ G.members

Tactic:
exact hs.right.right.left

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

Goal state before:
pp: H : Type u
H' : Type u_1
M : Type u_2
M' : Type u_3
M'' : Type u_4
inst✝² : TopologicalSpace H
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
e✝ e'✝ : OpenPartialHomeomorph M H
x✝¹ : e✝ ∈ atlas H M
x✝ : e'✝ ∈ atlas H M
⊢ e✝.symm ≫ₕ e'✝ ∈ continuousGroupoid H

Tactic:
rw [continuousGroupoid, mem_groupoid_of_pregroupoid]

Goal state after:
H : Type u
H' : Type u_1
M : Type u_2
M' : Type u_3
M'' : Type u_4
inst✝² : TopologicalSpace H
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
e✝ e'✝ : OpenPartialHomeomorph M H
x✝¹ : e✝ ∈ atlas H M
x✝ : e'✝ ∈ atlas H M
⊢ (continuousPregroupoid H).property (↑(e✝.symm ≫ₕ e'✝)) (e✝.symm ≫ₕ e'✝).source ∧
    (continuousPregroupoid H).property (↑(e✝.symm ≫ₕ e'✝).symm) (e✝.symm ≫ₕ e'✝).target
Module: Mathlib.Geometry.Manifold.ChartedSpace

Goal state before:
pp: H : Type u_5
H' : Type u_6
α : Type u_7
f : H → α
⊢ (range fun p ↦ (f p.1, p.2)) = range f ×ˢ univ

Tactic:
by
  rw [prod_range_univ_eq]
  rfl

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

Goal state before:
pp: case inl
H : Type u
H' : Type u_1
M : Type u_2
M' : Type u_3
M'' : Type u_4
inst✝³ : TopologicalSpace H
inst✝² : TopologicalSpace M
inst✝¹ : TopologicalSpace M'
cm : ChartedSpace H M
cm' : ChartedSpace H M'
inst✝ : Nonempty H
x : M
⊢ Sum.inl x ∈
    (Sum.elim (fun x ↦ (ChartedSpace.chartAt x).lift_openEmbedding ⋯)
        (fun x ↦ (ChartedSpace.chartAt x).lift_openEmbedding ⋯) (Sum.inl x)).source

Tactic:
rw [Sum.elim_inl, lift_openEmbedding_source,
        ← OpenPartialHomeomorph.lift_openEmbedding_source _ IsOpenEmbedding.inl]
      use x, cm.mem_chart_source x

Goal state after:
Module: Mathlib.MeasureTheory.Function.Jacobian

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hf : InjOn f s
ε : ℝ≥0
εpos : 0 < ε
A : E →L[ℝ] E
⊢ ContinuousAt ContinuousLinearMap.det A

Tactic:
exact ContinuousLinearMap.continuous_det.continuousAt

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

Goal state before:
pp: case mp
H : Type u_1
H' : Type u_3
inst✝¹ : TopologicalSpace H
inst✝ : TopologicalSpace H'
G : StructureGroupoid H
G' : StructureGroupoid H'
P : (H → H') → Set H → H → Prop
hG : G.LocalInvariantProp G' P
s : Set H
x : H
f : H → H'
e' : OpenPartialHomeomorph H' H'
he' : e' ∈ G'
hfs : ContinuousWithinAt f s x
hxe' : f x ∈ e'.source
h2f : f ⁻¹' e'.source ∈ 𝓝[s] x
h3f : ↑e' ∘ f ⁻¹' e'.symm.source ∈ 𝓝[s] x
h : P (↑e' ∘ f) (s ∩ ↑e' ∘ f ⁻¹' e'.symm.source) x
⊢ P f s x

Tactic:
have h2 := hG.left_invariance' (G'.symm he') inter_subset_right (e'.mapsTo hxe') h

Goal state after:
case mp
H : Type u_1
H' : Type u_3
inst✝¹ : TopologicalSpace H
inst✝ : TopologicalSpace H'
G : StructureGroupoid H
G' : StructureGroupoid H'
P : (H → H') → Set H → H → Prop
hG : G.LocalInvariantProp G' P
s : Set H
x : H
f : H → H'
e' : OpenPartialHomeomorph H' H'
he' : e' ∈ G'
hfs : ContinuousWithinAt f s x
hxe' : f x ∈ e'.source
h2f : f ⁻¹' e'.source ∈ 𝓝[s] x
h3f : ↑e' ∘ f ⁻¹' e'.symm.source ∈ 𝓝[s] x
h : P (↑e' ∘ f) (s ∩ ↑e' ∘ f ⁻¹' e'.symm.source) x
h2 : P (↑e'.symm ∘ ↑e' ∘ f) (s ∩ ↑e' ∘ f ⁻¹' e'.symm.source) x
⊢ P f s x
Module: Mathlib.Geometry.Manifold.IsManifold.Basic

Goal state before:
pp: case inr.inl.inr
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
n : ℕ∞ω
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
M' : Type u_5
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H M'
hM : IsManifold I n M
hM' : IsManifold I n M'
e e' : OpenPartialHomeomorph (M ⊕ M') H
he : e ∈ atlas H (M ⊕ M')
he' : e' ∈ atlas H (M ⊕ M')
h : Nonempty H
f : OpenPartialHomeomorph M H
hf : f ∈ atlas H M
hef : e = f.lift_openEmbedding ⋯
f' : OpenPartialHomeomorph M' H
hf' : f' ∈ atlas H M'
he'f' : e' = f'.lift_openEmbedding ⋯
x : H
⊢ x ∈ ((f.lift_openEmbedding ⋯).symm ≫ₕ f'.lift_openEmbedding ⋯).source ↔ x ∈ ∅

Tactic:
exact ⟨fun ⟨hx₁, hx₂⟩ ↦ by simp_all, fun hx ↦ hx.elim⟩

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

Goal state before:
pp: case inr.inr.inl
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
n : ℕ∞ω
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
M' : Type u_5
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H M'
hM : IsManifold I n M
hM' : IsManifold I n M'
e e' : OpenPartialHomeomorph (M ⊕ M') H
he : e ∈ atlas H (M ⊕ M')
he' : e' ∈ atlas H (M ⊕ M')
h : Nonempty H
f : OpenPartialHomeomorph M' H
hf : f ∈ atlas H M'
hef : e = f.lift_openEmbedding ⋯
f' : OpenPartialHomeomorph M H
hf' : f' ∈ atlas H M
he'f' : e' = f'.lift_openEmbedding ⋯
x : H
⊢ x ∈ ((f.lift_openEmbedding ⋯).symm ≫ₕ f'.lift_openEmbedding ⋯).source ↔ x ∈ ∅

Tactic:
exact ⟨fun ⟨hx₁, hx₂⟩ ↦ by simp_all, fun hx ↦ hx.elim⟩

Goal state after:
Module: Mathlib.Geometry.Manifold.ContMDiff.Constructions

Goal state before:
pp: case refine_2
𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
F : Type u_8
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
G : Type u_9
inst✝² : TopologicalSpace G
J : ModelWithCorners 𝕜 F G
N : Type u_10
inst✝¹ : TopologicalSpace N
inst✝ : ChartedSpace G N
n : WithTop ℕ∞
s : Set (M × N)
p : M × N
⊢ (↑(extChartAt I p.1) ∘ Prod.fst ∘ ↑(extChartAt (I.prod J) p).symm) (↑(extChartAt (I.prod J) p) p) =
    (↑(extChartAt (I.prod J) p) p).1

Tactic:
· exact (extChartAt I p.1).right_inv <| (extChartAt I p.1).map_source (mem_extChartAt_source _)

Goal state after:
Module: Mathlib.Geometry.Manifold.ContMDiff.Constructions

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
F₁ : Type u_14
inst✝³ : NormedAddCommGroup F₁
inst✝² : NormedSpace 𝕜 F₁
F₂ : Type u_15
inst✝¹ : NormedAddCommGroup F₂
inst✝ : NormedSpace 𝕜 F₂
x : M
n : WithTop ℕ∞
f : M → F₁ × F₂
⊢ ContMDiffAt I (𝓘(𝕜, F₁).prod 𝓘(𝕜, F₂)) n f x ↔
    ContMDiffAt I 𝓘(𝕜, F₁) n (Prod.fst ∘ f) x ∧ ContMDiffAt I 𝓘(𝕜, F₂) n (Prod.snd ∘ f) x

Tactic:
exact contMDiffAt_prod_iff f

Goal state after:
Module: Mathlib.Geometry.Manifold.ContMDiff.Defs

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝² : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H' M'
n : ℕ∞ω
f : M → M'
x : M
⊢ ContinuousAt f x ∧
      ContDiffWithinAtProp I I' n (↑(chartAt H' (f x)) ∘ f ∘ ↑(chartAt H x).symm) univ (↑(chartAt H x) x) ↔
    ContinuousAt f x ∧
      ContDiffWithinAt 𝕜 n (↑(extChartAt I' (f x)) ∘ f ∘ ↑(extChartAt I x).symm) (range ↑I) (↑(extChartAt I x) x)

Tactic:
rw [ContDiffWithinAtProp, preimage_univ, univ_inter]

Goal state after:
𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝² : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H' M'
n : ℕ∞ω
f : M → M'
x : M
⊢ ContinuousAt f x ∧
      ContDiffWithinAt 𝕜 n (↑I' ∘ (↑(chartAt H' (f x)) ∘ f ∘ ↑(chartAt H x).symm) ∘ ↑I.symm) (range ↑I)
        (↑I (↑(chartAt H x) x)) ↔
    ContinuousAt f x ∧
      ContDiffWithinAt 𝕜 n (↑(extChartAt I' (f x)) ∘ f ∘ ↑(extChartAt I x).symm) (range ↑I) (↑(extChartAt I x) x)
Module: Mathlib.Geometry.Manifold.ContMDiff.Defs

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝² : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H' M'
e : OpenPartialHomeomorph M H
f : M → M'
s : Set M
n : ℕ∞ω
he : e ∈ maximalAtlas I n M
hs : s ⊆ e.source
x : M
hx : x ∈ s
⊢ ContMDiffWithinAt I I' n f s x ↔
    ContMDiffWithinAt 𝓘(𝕜, E) I' n (f ∘ ↑(e.extend I).symm) (↑(e.extend I) '' s) (↑(e.extend I) x)

Tactic:
contMDiffWithinAt_iff_source_of_mem_maximalAtlas he (hs hx)

Goal state after:
𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝² : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H' M'
e : OpenPartialHomeomorph M H
f : M → M'
s : Set M
n : ℕ∞ω
he : e ∈ maximalAtlas I n M
hs : s ⊆ e.source
x : M
hx : x ∈ s
⊢ ContMDiffWithinAt 𝓘(𝕜, E) I' n (f ∘ ↑(e.extend I).symm) (↑(e.extend I).symm ⁻¹' s ∩ range ↑I) (↑(e.extend I) x) ↔
    ContMDiffWithinAt 𝓘(𝕜, E) I' n (f ∘ ↑(e.extend I).symm) (↑(e.extend I) '' s) (↑(e.extend I) x)
Module: Mathlib.Geometry.Manifold.ContMDiff.Atlas

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
H : Type u_3
inst✝ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
n : ℕ∞ω
x : E
hx : x ∈ range ↑I
⊢ ContDiffWithinAt 𝕜 n (↑(extChartAt I (↑I.symm x)) ∘ ↑I.symm ∘ ↑(extChartAt 𝓘(𝕜, E) x).symm)
    (↑(extChartAt 𝓘(𝕜, E) x).symm ⁻¹' range ↑I ∩ range ↑𝓘(𝕜, E)) (↑(extChartAt 𝓘(𝕜, E) x) x)

Tactic:
simpa using contDiffWithinAt_id.congr (fun y hy ↦ by simp [hy]) (by simp [hx])

Goal state after:
Module: Mathlib.Geometry.Manifold.ContMDiff.Atlas

Goal state before:
pp: case left
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
H : Type u_3
inst✝² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
n : ℕ∞ω
x : M
⊢ ContinuousWithinAt (↑(extChartAt I x).symm) (extChartAt I x).target (↑(extChartAt I x) x)

Tactic:
apply ContinuousAt.continuousWithinAt

Goal state after:
case left
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
H : Type u_3
inst✝² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H M
n : ℕ∞ω
x : M
⊢ ContinuousAt (↑(extChartAt I x).symm) (↑(extChartAt I x) x)
Module: Mathlib.Geometry.Manifold.ContMDiff.Atlas

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
n : ℕ∞ω
M' : Type u_5
inst✝² : TopologicalSpace M'
inst✝¹ : IsManifold I n M
inst✝ : ChartedSpace H M'
IsM' : IsManifold I n M'
f : OpenPartialHomeomorph M M'
h : LiftPropOn (contDiffGroupoid n I).IsLocalStructomorphWithinAt (↑f) f.source
X : M'
hX : X ∈ f.symm.source
x : M := ↑f.symm X
hx : x ∈ f.source
c : OpenPartialHomeomorph M H := chartAt H x
c' : OpenPartialHomeomorph M' H := chartAt H X
hxf :
  (contDiffGroupoid n I).IsLocalStructomorphWithinAt (↑(chartAt H (↑f x)) ∘ ↑f ∘ ↑(chartAt H x).symm)
    (↑(chartAt H x).symm ⁻¹' f.source) (↑(chartAt H x) x)
h2x : ↑(chartAt H X) X ∈ ↑(chartAt H X).symm ⁻¹' f.symm.source
e : OpenPartialHomeomorph H H
he : e ∈ contDiffGroupoid n I
h2e : e.source ⊆ (c.symm ≫ₕ f ≫ₕ c').source
hef : EqOn (↑c' ∘ ↑f ∘ ↑c.symm) (↑e) e.source
hex : ↑c x ∈ e.source
hfX : ↑f.symm X ∈ c.source
⊢ ↑c' X = ↑c' (↑f (↑c.symm (↑c x)))

Tactic:
rw [c.left_inv hfX, f.right_inv hX]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁴ : NormedAddCommGroup E
inst✝¹³ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹¹ : TopologicalSpace M
inst✝¹⁰ : ChartedSpace H M
E' : Type u_5
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁷ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁶ : TopologicalSpace M'
inst✝⁵ : ChartedSpace H' M'
E'' : Type u_8
inst✝⁴ : NormedAddCommGroup E''
inst✝³ : NormedSpace 𝕜 E''
H'' : Type u_9
inst✝² : TopologicalSpace H''
I'' : ModelWithCorners 𝕜 E'' H''
M'' : Type u_10
inst✝¹ : TopologicalSpace M''
inst✝ : ChartedSpace H'' M''
f : M → M'
s : Set M
g : M' → M''
hg : MDiff g
hf : MDiff[s] f
⊢ MDiff[s] (g ∘ f)

Tactic:
rw [← mdifferentiableOn_univ] at hg
  exact hg.comp hf (by simp)

Goal state after:
Module: Mathlib.Geometry.Manifold.MFDeriv.Atlas

Goal state before:
pp: case h
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : IsManifold I 1 M
x : M
y : E
hy : y ∈ (extChartAt I x).target
h'y : ↑(extChartAt I x).symm y ∈ (extChartAt I x).source
h''y : ↑(extChartAt I x).symm y ∈ (chartAt H x).source
U' : UniqueMDiffAt[(extChartAt I x).source] (↑(extChartAt I x).symm y)
this :
  mfderiv% ↑(extChartAt I x) (↑(extChartAt I x).symm y) =
    mfderiv[(extChartAt I x).source] ↑(extChartAt I x) (↑(extChartAt I x).symm y)
z : M
hz : z ∈ (extChartAt I x).source
⊢ z ∈ ↑(extChartAt I x) ⁻¹' range ↑I

Tactic:
apply extChartAt_target_subset_range x

Goal state after:
case h
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : IsManifold I 1 M
x : M
y : E
hy : y ∈ (extChartAt I x).target
h'y : ↑(extChartAt I x).symm y ∈ (extChartAt I x).source
h''y : ↑(extChartAt I x).symm y ∈ (chartAt H x).source
U' : UniqueMDiffAt[(extChartAt I x).source] (↑(extChartAt I x).symm y)
this :
  mfderiv% ↑(extChartAt I x) (↑(extChartAt I x).symm y) =
    mfderiv[(extChartAt I x).source] ↑(extChartAt I x) (↑(extChartAt I x).symm y)
z : M
hz : z ∈ (extChartAt I x).source
⊢ ↑(extChartAt I x) z ∈ (extChartAt I x).target
Module: Mathlib.Topology.Compactness.Paracompact

Goal state before:
pp: X : Type v
Y : Type w
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : ParacompactSpace Y
e : X → Y
he : IsClosedEmbedding e
α : Type v
s : α → Set X
ho : ∀ (a : α), IsOpen[inst✝²] (s a)
hu : ⋃ a, s a = univ
⊢ ∃ β t, (∀ (b : β), IsOpen[inst✝²] (t b)) ∧ ⋃ b, t b = univ ∧ LocallyFinite t ∧ ∀ (b : β), ∃ a, t b ⊆ s a

Tactic:
choose U hUo hU using fun a ↦ he.isOpen_iff.1 (ho a)

Goal state after:
X : Type v
Y : Type w
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : ParacompactSpace Y
e : X → Y
he : IsClosedEmbedding e
α : Type v
s : α → Set X
ho : ∀ (a : α), IsOpen[inst✝²] (s a)
hu : ⋃ a, s a = univ
U : α → Set Y
hUo : ∀ (a : α), IsOpen[inst✝¹] (U a)
hU : ∀ (a : α), e ⁻¹' U a = s a
⊢ ∃ β t, (∀ (b : β), IsOpen[inst✝²] (t b)) ∧ ⋃ b, t b = univ ∧ LocallyFinite t ∧ ∀ (b : β), ∃ a, t b ⊆ s a
Module: Mathlib.Topology.Compactness.Paracompact

Goal state before:
pp: X : Type v
Y : Type w
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : ParacompactSpace Y
e : X → Y
he : IsClosedEmbedding e
α : Type v
s : α → Set X
ho : ∀ (a : α), IsOpen[inst✝²] (s a)
U : α → Set Y
hUo : ∀ (a : α), IsOpen[inst✝¹] (U a)
hU : ∀ (a : α), e ⁻¹' U a = s a
hu : ⋃ a, e ⁻¹' U a = univ
⊢ range e ⊆ ⋃ i, U i

Tactic:
simpa only [range_subset_iff, mem_iUnion, iUnion_eq_univ_iff] using! hu

Goal state after:
Module: Mathlib.Geometry.Manifold.VectorBundle.ContMDiffSection

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
F : Type u_5
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜 F
n : ℕ∞ω
V : M → Type u_6
inst✝⁵ : TopologicalSpace (TotalSpace F V)
inst✝⁴ : (x : M) → TopologicalSpace (V x)
inst✝³ : FiberBundle F V
inst✝² : (x : M) → AddCommGroup (V x)
inst✝¹ : (x : M) → Module 𝕜 (V x)
inst✝ : VectorBundle 𝕜 F V
s t : (x : M) → V x
x₀ : M
hs : ContMDiffAt I (I.prod 𝓘(𝕜, F)) n (fun x ↦ ⟨x, s x⟩) x₀
ht : ContMDiffAt I (I.prod 𝓘(𝕜, F)) n (fun x ↦ ⟨x, t x⟩) x₀
⊢ ContMDiffAt I (I.prod 𝓘(𝕜, F)) n (fun x ↦ ⟨x, (s + t) x⟩) x₀

Tactic:
rw [← contMDiffWithinAt_univ] at hs ⊢

Goal state after:
𝕜 : Type u_1
inst✝¹³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
F : Type u_5
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜 F
n : ℕ∞ω
V : M → Type u_6
inst✝⁵ : TopologicalSpace (TotalSpace F V)
inst✝⁴ : (x : M) → TopologicalSpace (V x)
inst✝³ : FiberBundle F V
inst✝² : (x : M) → AddCommGroup (V x)
inst✝¹ : (x : M) → Module 𝕜 (V x)
inst✝ : VectorBundle 𝕜 F V
s t : (x : M) → V x
x₀ : M
hs : ContMDiffWithinAt I (I.prod 𝓘(𝕜, F)) n (fun x ↦ ⟨x, s x⟩) Set.univ x₀
ht : ContMDiffAt I (I.prod 𝓘(𝕜, F)) n (fun x ↦ ⟨x, t x⟩) x₀
⊢ ContMDiffWithinAt I (I.prod 𝓘(𝕜, F)) n (fun x ↦ ⟨x, (s + t) x⟩) Set.univ x₀
Module: Mathlib.Geometry.Manifold.VectorBundle.ContMDiffSection

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
F : Type u_5
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜 F
n : ℕ∞ω
V : M → Type u_6
inst✝⁵ : TopologicalSpace (TotalSpace F V)
inst✝⁴ : (x : M) → TopologicalSpace (V x)
inst✝³ : FiberBundle F V
inst✝² : (x : M) → AddCommGroup (V x)
inst✝¹ : (x : M) → Module 𝕜 (V x)
inst✝ : VectorBundle 𝕜 F V
s : (x : M) → V x
x₀ : M
hs : ContMDiffAt I (I.prod 𝓘(𝕜, F)) n (fun x ↦ ⟨x, s x⟩) x₀
⊢ ContMDiffAt I (I.prod 𝓘(𝕜, F)) n (fun x ↦ ⟨x, (-s) x⟩) x₀

Tactic:
rw [← contMDiffWithinAt_univ] at hs ⊢

Goal state after:
𝕜 : Type u_1
inst✝¹³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
F : Type u_5
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜 F
n : ℕ∞ω
V : M → Type u_6
inst✝⁵ : TopologicalSpace (TotalSpace F V)
inst✝⁴ : (x : M) → TopologicalSpace (V x)
inst✝³ : FiberBundle F V
inst✝² : (x : M) → AddCommGroup (V x)
inst✝¹ : (x : M) → Module 𝕜 (V x)
inst✝ : VectorBundle 𝕜 F V
s : (x : M) → V x
x₀ : M
hs : ContMDiffWithinAt I (I.prod 𝓘(𝕜, F)) n (fun x ↦ ⟨x, s x⟩) Set.univ x₀
⊢ ContMDiffWithinAt I (I.prod 𝓘(𝕜, F)) n (fun x ↦ ⟨x, (-s) x⟩) Set.univ x₀
Module: Mathlib.Topology.PartitionOfUnity

Goal state before:
pp: ι : Type u
X : Type v
inst✝² : TopologicalSpace X
s : Set X
ρ : PartitionOfUnity ι X s
x₀ : X
M : Type u_2
inst✝¹ : AddCommMonoid M
inst✝ : Module ℝ M
φ : ι → X → M
⊢ ∑ i ∈ ρ.finsupport x₀, (ρ i) x₀ • φ i x₀ = ∑ᶠ (i : ι), (ρ i) x₀ • φ i x₀

Tactic:
apply (finsum_eq_sum_of_support_subset _ _).symm

Goal state after:
ι : Type u
X : Type v
inst✝² : TopologicalSpace X
s : Set X
ρ : PartitionOfUnity ι X s
x₀ : X
M : Type u_2
inst✝¹ : AddCommMonoid M
inst✝ : Module ℝ M
φ : ι → X → M
⊢ (support fun i ↦ (ρ i) x₀ • φ i x₀) ⊆ ↑(ρ.finsupport x₀)
Module: Mathlib.Geometry.Manifold.VectorBundle.ContMDiffSection

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
F : Type u_5
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜 F
n : ℕ∞ω
V : M → Type u_6
inst✝⁵ : TopologicalSpace (TotalSpace F V)
inst✝⁴ : (x : M) → TopologicalSpace (V x)
inst✝³ : FiberBundle F V
inst✝² : (x : M) → AddCommGroup (V x)
inst✝¹ : (x : M) → Module 𝕜 (V x)
inst✝ : VectorBundle 𝕜 F V
f : M → 𝕜
s : (x : M) → V x
x₀ : M
hf : ContMDiffAt I 𝓘(𝕜, 𝕜) n f x₀
hs : ContMDiffAt I (I.prod 𝓘(𝕜, F)) n (fun x ↦ ⟨x, s x⟩) x₀
⊢ ContMDiffAt I (I.prod 𝓘(𝕜, F)) n (fun x ↦ ⟨x, (f • s) x⟩) x₀

Tactic:
rw [← contMDiffWithinAt_univ] at hs ⊢

Goal state after:
𝕜 : Type u_1
inst✝¹³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹⁰ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁹ : TopologicalSpace M
inst✝⁸ : ChartedSpace H M
F : Type u_5
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜 F
n : ℕ∞ω
V : M → Type u_6
inst✝⁵ : TopologicalSpace (TotalSpace F V)
inst✝⁴ : (x : M) → TopologicalSpace (V x)
inst✝³ : FiberBundle F V
inst✝² : (x : M) → AddCommGroup (V x)
inst✝¹ : (x : M) → Module 𝕜 (V x)
inst✝ : VectorBundle 𝕜 F V
f : M → 𝕜
s : (x : M) → V x
x₀ : M
hf : ContMDiffAt I 𝓘(𝕜, 𝕜) n f x₀
hs : ContMDiffWithinAt I (I.prod 𝓘(𝕜, F)) n (fun x ↦ ⟨x, s x⟩) Set.univ x₀
⊢ ContMDiffWithinAt I (I.prod 𝓘(𝕜, F)) n (fun x ↦ ⟨x, (f • s) x⟩) Set.univ x₀