mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Order.ScottTopology

Goal state before:
pp: α : Type u_1
inst✝ : CompleteLinearOrder α
this : TopologicalSpace α := upper α
U : Set α
⊢ IsOpen[scott α univ] U ↔ U = univ ∨ ∃ a, (Iic a)ᶜ = U

Tactic:
let := scott α univ

Goal state after:
α : Type u_1
inst✝ : CompleteLinearOrder α
this✝ : TopologicalSpace α := upper α
U : Set α
this : TopologicalSpace α := scott α univ
⊢ IsOpen[scott α univ] U ↔ U = univ ∨ ∃ a, (Iic a)ᶜ = U
Module: Mathlib.Topology.Order.NhdsSet

Goal state before:
pp: α : Type u_1
inst✝² : LinearOrder α
inst✝¹ : TopologicalSpace α
inst✝ : OrderClosedTopology α
a b c : α
h : b ≤ c
⊢ Iio c ∈ 𝓝ˢ (Iio b)

Tactic:
simpa

Goal state after:
Module: Mathlib.Topology.Order.NhdsSet

Goal state before:
pp: α : Type u_1
inst✝² : LinearOrder α
inst✝¹ : TopologicalSpace α
inst✝ : OrderClosedTopology α
a b c : α
h : a ≤ b
⊢ Ioi a ∈ 𝓝ˢ (Ioi b)

Tactic:
simpa

Goal state after:
Module: Mathlib.Topology.MetricSpace.GromovHausdorff

Goal state before:
pp: δ : ℝ
δpos : δ > 0
ε : ℝ := 2 / 5 * δ
εpos : 0 < ε
s : (p : GHSpace) → Set p.Rep
hs : ∀ (p : GHSpace), (s p).Finite ∧ univ ⊆ ⋃ x ∈ s p, ball x ε
N : GHSpace → ℕ := fun p ↦ Nat.card ↑(s p)
E : (p : GHSpace) → ↑(s p) ≃ Fin (Nat.card ↑(s p)) := fun p ↦ Finite.equivFin ↑(s p)
F : GHSpace → (n : ℕ) × (Fin n → Fin n → ℤ) := fun p ↦ ⟨N p, fun a b ↦ ⌊ε⁻¹ * dist ((E p).symm a) ((E p).symm b)⌋⟩
p q : GHSpace
hpq : F p = F q
Npq : N p = N q
Ψ : ↑(s p) → ↑(s q) := fun x ↦ (E q).symm (Fin.cast Npq ((E p) x))
Φ : ↑(s p) → q.Rep := fun x ↦ ↑(Ψ x)
x y : ↑(s p)
i : ℕ := ↑((E p) x)
hip : i < N p
hiq : i < N q
i' : i = ↑((E q) (Ψ x))
j : ℕ := ↑((E p) y)
hjp : j < N p
hjq : j < N q
j' : j = ↑((E q) (Ψ y))
this : (F p).snd ((E p) x) ((E p) y) = ⌊ε⁻¹ * dist x y⌋
⊢ (F p).snd ⟨i, hip⟩ ⟨j, hjp⟩ = ⌊ε⁻¹ * dist x y⌋

Tactic:
← this

Goal state after:
δ : ℝ
δpos : δ > 0
ε : ℝ := 2 / 5 * δ
εpos : 0 < ε
s : (p : GHSpace) → Set p.Rep
hs : ∀ (p : GHSpace), (s p).Finite ∧ univ ⊆ ⋃ x ∈ s p, ball x ε
N : GHSpace → ℕ := fun p ↦ Nat.card ↑(s p)
E : (p : GHSpace) → ↑(s p) ≃ Fin (Nat.card ↑(s p)) := fun p ↦ Finite.equivFin ↑(s p)
F : GHSpace → (n : ℕ) × (Fin n → Fin n → ℤ) := fun p ↦ ⟨N p, fun a b ↦ ⌊ε⁻¹ * dist ((E p).symm a) ((E p).symm b)⌋⟩
p q : GHSpace
hpq : F p = F q
Npq : N p = N q
Ψ : ↑(s p) → ↑(s q) := fun x ↦ (E q).symm (Fin.cast Npq ((E p) x))
Φ : ↑(s p) → q.Rep := fun x ↦ ↑(Ψ x)
x y : ↑(s p)
i : ℕ := ↑((E p) x)
hip : i < N p
hiq : i < N q
i' : i = ↑((E q) (Ψ x))
j : ℕ := ↑((E p) y)
hjp : j < N p
hjq : j < N q
j' : j = ↑((E q) (Ψ y))
this : (F p).snd ((E p) x) ((E p) y) = ⌊ε⁻¹ * dist x y⌋
⊢ (F p).snd ⟨i, hip⟩ ⟨j, hjp⟩ = (F p).snd ((E p) x) ((E p) y)
Module: Mathlib.Topology.MetricSpace.GromovHausdorffRealized

Goal state before:
pp: case refine_1
X : Type u
Y : Type v
inst✝³ : MetricSpace X
inst✝² : MetricSpace Y
inst✝¹ : Nonempty X
inst✝ : Nonempty Y
f g : Cb X Y
cg : ℝ
hcg : cg ∈ lowerBounds (range ⇑g)
Hcg : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cg ≤ g x
cf : ℝ
hcf : cf ∈ lowerBounds (range ⇑f)
Hcf : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cf ≤ f x
Z : ⨆ x, ⨅ y, f (inl x, inr y) ≤ ⨆ x, ⨅ y, g (inl x, inr y) + dist f g
x✝ : X
x y : ℝ
hx : x ≤ y
⊢ (id + fun x ↦ dist f g) x ≤ (id + fun x ↦ dist f g) y

Tactic:
simpa

Goal state after:
Module: Mathlib.Topology.MetricSpace.GromovHausdorffRealized

Goal state before:
pp: case refine_1
X : Type u
Y : Type v
inst✝³ : MetricSpace X
inst✝² : MetricSpace Y
inst✝¹ : Nonempty X
inst✝ : Nonempty Y
f g : Cb X Y
cg : ℝ
hcg : cg ∈ lowerBounds (range ⇑g)
Hcg : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cg ≤ g x
cf : ℝ
hcf : cf ∈ lowerBounds (range ⇑f)
Hcf : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cf ≤ f x
Z : ⨆ x, ⨅ y, f (inl x, inr y) ≤ ⨆ x, ⨅ y, g (inl x, inr y) + dist f g
E1 : ∀ (x : X), (⨅ y, g (inl x, inr y)) + dist f g = ⨅ y, g (inl x, inr y) + dist f g
x y : ℝ
hx : x ≤ y
⊢ (id + fun x ↦ dist f g) x ≤ (id + fun x ↦ dist f g) y

Tactic:
simpa

Goal state after:
Module: Mathlib.Topology.MetricSpace.GromovHausdorffRealized

Goal state before:
pp: case refine_1
X : Type u
Y : Type v
inst✝³ : MetricSpace X
inst✝² : MetricSpace Y
inst✝¹ : Nonempty X
inst✝ : Nonempty Y
f g : Cb X Y
cg : ℝ
hcg : cg ∈ lowerBounds (range ⇑g)
Hcg : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cg ≤ g x
cf : ℝ
hcf : cf ∈ lowerBounds (range ⇑f)
Hcf : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cf ≤ f x
Z : ⨆ y, ⨅ x, f (inl x, inr y) ≤ ⨆ y, ⨅ x, g (inl x, inr y) + dist f g
y✝ : Y
x y : ℝ
hx : x ≤ y
⊢ (id + fun x ↦ dist f g) x ≤ (id + fun x ↦ dist f g) y

Tactic:
simpa

Goal state after:
Module: Mathlib.Topology.MetricSpace.GromovHausdorffRealized

Goal state before:
pp: case refine_1
X : Type u
Y : Type v
inst✝³ : MetricSpace X
inst✝² : MetricSpace Y
inst✝¹ : Nonempty X
inst✝ : Nonempty Y
f g : Cb X Y
cg : ℝ
hcg : cg ∈ lowerBounds (range ⇑g)
Hcg : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cg ≤ g x
cf : ℝ
hcf : cf ∈ lowerBounds (range ⇑f)
Hcf : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cf ≤ f x
Z : ⨆ y, ⨅ x, f (inl x, inr y) ≤ ⨆ y, ⨅ x, g (inl x, inr y) + dist f g
E1 : ∀ (y : Y), (⨅ x, g (inl x, inr y)) + dist f g = ⨅ x, g (inl x, inr y) + dist f g
x y : ℝ
hx : x ≤ y
⊢ (id + fun x ↦ dist f g) x ≤ (id + fun x ↦ dist f g) y

Tactic:
simpa

Goal state after:
Module: Mathlib.Topology.MetricSpace.GromovHausdorffRealized

Goal state before:
pp: case refine_1
X : Type u
Y : Type v
inst✝³ : MetricSpace X
inst✝² : MetricSpace Y
inst✝¹ : CompactSpace X
inst✝ : CompactSpace Y
⊢ ∀ (f : (X ⊕ Y) × (X ⊕ Y) →ᵇ ℝ) (x : (X ⊕ Y) × (X ⊕ Y)), f ∈ candidatesB X Y → f x ∈ Icc 0 ↑(maxVar X Y)

Tactic:
rintro f ⟨x1, x2⟩ hf
    simp only [Set.mem_Icc]
    exact ⟨candidates_nonneg hf, candidates_le_maxVar hf⟩

Goal state after:
Module: Mathlib.Topology.Sion

Goal state before:
pp: E : Type u_1
F : Type u_2
β : Type u_3
inst✝¹ : LinearOrder β
X : Set E
Y : Set F
f : E → F → β
inst✝ : TopologicalSpace E
ne_X : X.Nonempty
kX : IsCompact X
hfy : ∀ y ∈ Y, LowerSemicontinuousOn (fun x ↦ f x y) X
y : ↑Y
b : β
h : ∀ y ∈ Y, ∃ x ∈ X, f x y ≤ b
⊢ (sublevelLeft X f b ↑y).Nonempty

Tactic:
by
  rcases y with ⟨y, hy⟩
  obtain ⟨x, hx, hx_le⟩ := LowerSemicontinuousOn.exists_isMinOn ne_X kX (hfy y hy)
  obtain ⟨x', hx', hx'b⟩ := h y hy
  exact ⟨⟨x, hx⟩, le_trans (hx_le hx') hx'b⟩

Goal state after:
Module: Mathlib.Topology.Sheaves.Sheafify

Goal state before:
pp: case h.refine_3
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥↑U) → F.stalk ↑y
hU : (Sheafify.isLocallyGerm F).pred fU
fV : (y : ↥↑V) → F.stalk ↑y
hV : (Sheafify.isLocallyGerm F).pred fV
U' : Opens ↑X
mU : ↑⟨x, ⋯⟩ ∈ U'
iU : U' ⟶ ↑U
gU : F.obj (op U')
V' : Opens ↑X
mV : ↑⟨x, ⋯⟩ ∈ V'
iV : V' ⟶ ↑V
gV : F.obj (op V')
e : (ConcreteCategory.hom (F.germ U' x ⋯)) gU = (ConcreteCategory.hom (F.germ V' x ⋯)) gV
W : Opens ↑X
mW : x ∈ W
iU' : W ⟶ U'
iV' : W ⟶ V'
e' : (ConcreteCategory.hom (F.map iU'.op)) gU = (ConcreteCategory.hom (F.map iV'.op)) gV
w : ↥↑⟨W ⊓ (U' ⊓ V'), ⋯⟩
wV : (fun x_1 ↦ fV (iV x_1)) ⟨↑w, ⋯⟩ = (ConcreteCategory.hom (F.germ V' ↑⟨↑w, ⋯⟩ ⋯)) gV
wU : (fun x_1 ↦ fU (iU x_1)) ⟨↑w, ⋯⟩ = (ConcreteCategory.hom (F.germ U' ↑⟨↑w, ⋯⟩ ⋯)) gU
⊢ (ConcreteCategory.hom (F.germ U' ↑⟨↑w, ⋯⟩ ⋯)) gU = (ConcreteCategory.hom (F.germ V' ↑⟨↑w, ⋯⟩ ⋯)) gV

Tactic:
rw [← F.germ_res iU' w w.2.1, ← F.germ_res iV' w w.2.1,
      CategoryTheory.types_comp_apply, CategoryTheory.types_comp_apply, e']

Goal state after:
Module: Mathlib.Topology.SmallInductiveDimension

Goal state before:
pp: m : ℕ
IH :
  ∀ {X : Type u_1} [inst : TopologicalSpace X] {m_1 : ℕ},
    m_1 ≤ m → HasSmallInductiveDimensionLT X m_1 → HasSmallInductiveDimensionLT X m
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : IsEmpty X
hmn : 0 ≤ m + 1
⊢ IsTopologicalBasis ∅

Tactic:
simpa

Goal state after:
Module: Mathlib.Topology.Spectral.ConstructibleTopology

Goal state before:
pp: case inr
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
s : Set X
hs : s ∈ {s | IsClosed[inst✝¹] s ∧ IsCompact sᶜ}
⊢ IsCompact s

Tactic:
exact hs.1.isCompact

Goal state after:
Module: Mathlib.Topology.Spectral.ConstructibleTopology

Goal state before:
pp: case pos
X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : CompactSpace X
inst✝² : QuasiSober X
inst✝¹ : PrespectralSpace X
inst✝ : QuasiSeparatedSpace X
𝔅 : Set (Set X) := constructibleTopologySubbasis X
𝒮 : Set (Set (Set X)) := {P | P ⊆ 𝔅 ∧ (∀ Q ⊆ P, Q.Finite → (⋂₀ Q).Nonempty) ∧ ⋂₀ P = ∅}
s : Set (Set X)
hs : s ∈ 𝒮
B : Set (Set X)
hsB : s ⊆ B
hB : Maximal (fun x ↦ x ∈ 𝒮) B
B' : Set (Set X) := {s | s ∈ B ∧ IsClosed[inst✝⁴] s}
Z : Set X := ⋂₀ B'
hZ_closed : IsClosed[inst✝⁴] Z
hZ_nonempty : Z.Nonempty
hZ_irred : IsIrreducible Z
hη : IsGenericPoint hZ_irred.genericPoint Z
⊢ ∀ t ∈ B, hZ_irred.genericPoint ∈ t

Tactic:
intro i hi

Goal state after:
case pos
X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : CompactSpace X
inst✝² : QuasiSober X
inst✝¹ : PrespectralSpace X
inst✝ : QuasiSeparatedSpace X
𝔅 : Set (Set X) := constructibleTopologySubbasis X
𝒮 : Set (Set (Set X)) := {P | P ⊆ 𝔅 ∧ (∀ Q ⊆ P, Q.Finite → (⋂₀ Q).Nonempty) ∧ ⋂₀ P = ∅}
s : Set (Set X)
hs : s ∈ 𝒮
B : Set (Set X)
hsB : s ⊆ B
hB : Maximal (fun x ↦ x ∈ 𝒮) B
B' : Set (Set X) := {s | s ∈ B ∧ IsClosed[inst✝⁴] s}
Z : Set X := ⋂₀ B'
hZ_closed : IsClosed[inst✝⁴] Z
hZ_nonempty : Z.Nonempty
hZ_irred : IsIrreducible Z
hη : IsGenericPoint hZ_irred.genericPoint Z
i : Set X
hi : i ∈ B
⊢ hZ_irred.genericPoint ∈ i
Module: Mathlib.Topology.Spectral.ConstructibleTopology

Goal state before:
pp: X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : CompactSpace X
inst✝² : QuasiSober X
inst✝¹ : PrespectralSpace X
inst✝ : QuasiSeparatedSpace X
𝔅 : Set (Set X) := constructibleTopologySubbasis X
𝒮 : Set (Set (Set X)) := {P | P ⊆ 𝔅 ∧ (∀ Q ⊆ P, Q.Finite → (⋂₀ Q).Nonempty) ∧ ⋂₀ P = ∅}
s : Set (Set X)
hs : s ∈ 𝒮
B : Set (Set X)
hsB : s ⊆ B
hB : Maximal (fun x ↦ x ∈ 𝒮) B
B' : Set (Set X) := {s | s ∈ B ∧ IsClosed[inst✝⁴] s}
Z : Set X := ⋂₀ B'
hZ_closed : IsClosed[inst✝⁴] Z
hZ_nonempty : Z.Nonempty
hZ_irred : IsIrreducible Z
hη : IsGenericPoint hZ_irred.genericPoint Z
i : Set X
hi : i ∈ B
hiB' : i ∉ B'
hi_cmpt : IsCompact i
a : Set (Set X)
ha : a ⊆ B'
hfa : a.Finite
⊢ (insert i a).Finite

Tactic:
simpa

Goal state after:
Module: Mathlib.Topology.Subpath

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
a b : X
γ : Path a b
t₀ t₁ t₂ : ↑I
γ₁ : (t : ↑I) → Path (γ t₀) (γ (Icc.convexComb t₁ t₂ t)) := fun t ↦ γ.subpath t₀ (Icc.convexComb t₁ t₂ t)
⊢ Continuous fun x ↦ ((γ.subpath t₀ (Icc.convexComb t₁ t₂ x.1)).trans (γ.subpath (Icc.convexComb t₁ t₂ x.1) t₂)) x.2

Tactic:
let γ₂ (t : I) := γ.subpath (Icc.convexComb t₁ t₂ t) t₂

Goal state after:
X : Type u_1
inst✝ : TopologicalSpace X
a b : X
γ : Path a b
t₀ t₁ t₂ : ↑I
γ₁ : (t : ↑I) → Path (γ t₀) (γ (Icc.convexComb t₁ t₂ t)) := fun t ↦ γ.subpath t₀ (Icc.convexComb t₁ t₂ t)
γ₂ : (t : ↑I) → Path (γ (Icc.convexComb t₁ t₂ t)) (γ t₂) := fun t ↦ γ.subpath (Icc.convexComb t₁ t₂ t) t₂
⊢ Continuous fun x ↦ ((γ.subpath t₀ (Icc.convexComb t₁ t₂ x.1)).trans (γ.subpath (Icc.convexComb t₁ t₂ x.1) t₂)) x.2
Module: Mathlib.Topology.Subpath

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
a b : X
γ : Path a b
t₀ t₁ t₂ : ↑I
⊢ Continuous fun x ↦ ((γ.subpath t₀ (Icc.convexComb t₁ t₂ x.1)).trans (γ.subpath (Icc.convexComb t₁ t₂ x.1) t₂)) x.2

Tactic:
let γ₁ (t : I) := γ.subpath t₀ (Icc.convexComb t₁ t₂ t)
    let γ₂ (t : I) := γ.subpath (Icc.convexComb t₁ t₂ t) t₂
    refine Path.trans_continuous_family γ₁ ?_ γ₂ ?_ <;>
    refine γ.subpath_continuous_family.comp (.prodMk ?_ <| .prodMk ?_ ?_) <;>
    fun_prop

Goal state after:
Module: Mathlib.Topology.Subpath

Goal state before:
pp: case zero
X : Type u_1
inst✝ : TopologicalSpace X
a b : X
n : ℕ
γ : Path a b
t : Fin (0 + 1) → ↑I
⊢ (Path.refl ((⇑γ ∘ t) 0)).Homotopy (Path.refl (γ (t 0)))

Tactic:
exact refl _

Goal state after:
Module: Mathlib.Topology.Sion

Goal state before:
pp: case right
E : Type u_1
F : Type u_2
β : Type u_3
inst✝¹¹ : CompleteLinearOrder β
inst✝¹⁰ : DenselyOrdered β
X : Set E
Y : Set F
f : E → F → β
inst✝⁹ : TopologicalSpace E
inst✝⁸ : AddCommGroup E
inst✝⁷ : Module ℝ E
inst✝⁶ : IsTopologicalAddGroup E
inst✝⁵ : ContinuousSMul ℝ E
ne_X : X.Nonempty
cX : Convex ℝ X
kX : IsCompact X
hfy : ∀ y ∈ Y, LowerSemicontinuousOn (fun x ↦ f x y) X
hfy' : ∀ y ∈ Y, QuasiconvexOn ℝ X fun x ↦ f x y
inst✝⁴ : TopologicalSpace F
inst✝³ : AddCommGroup F
inst✝² : Module ℝ F
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousSMul ℝ F
cY : Convex ℝ Y
ne_Y : Y.Nonempty
kY : IsCompact Y
hfx : ∀ x ∈ X, UpperSemicontinuousOn (fun y ↦ f x y) Y
hfx' : ∀ x ∈ X, QuasiconcaveOn ℝ Y fun y ↦ f x y
hlsc : LowerSemicontinuousOn (fun x ↦ ⨆ y ∈ Y, f x y) X
husc : UpperSemicontinuousOn (fun y ↦ ⨅ x ∈ X, f x y) Y
a : E
ha : a ∈ X
ha' : IsMinOn (fun x ↦ ⨆ y ∈ Y, f x y) X a
b : F
hb : b ∈ Y
hb' : IsMaxOn (fun y ↦ ⨅ x ∈ X, f x y) Y b
⊢ ⨅ i ∈ X, (fun x ↦ ⨆ y ∈ Y, f x y) i ≤ ⨆ i ∈ Y, (fun y ↦ ⨅ x ∈ X, f x y) i

Tactic:
rw [minimax' ne_X cX kX hfy hfy' cY hfx hfx']

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

Goal state before:
pp: case intro.refine_1
𝕜 : Type u_1
ι : Type u_2
inst✝²² : NontriviallyNormedField 𝕜
B : Type u_3
inst✝²¹ : TopologicalSpace B
F₁ : Type u_4
inst✝²⁰ : NormedAddCommGroup F₁
inst✝¹⁹ : NormedSpace 𝕜 F₁
E₁ : B → Type u_5
inst✝¹⁸ : (x : B) → AddCommGroup (E₁ x)
inst✝¹⁷ : (x : B) → Module 𝕜 (E₁ x)
inst✝¹⁶ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_6
inst✝¹⁵ : NormedAddCommGroup F₂
inst✝¹⁴ : NormedSpace 𝕜 F₂
E₂ : B → Type u_7
inst✝¹³ : (x : B) → AddCommGroup (E₂ x)
inst✝¹² : (x : B) → Module 𝕜 (E₂ x)
inst✝¹¹ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝¹⁰ : (x : B) → TopologicalSpace (E₁ x)
inst✝⁹ : FiberBundle F₁ E₁
inst✝⁸ : (x : B) → TopologicalSpace (E₂ x)
inst✝⁷ : FiberBundle F₂ E₂
e₁ e₁' : Trivialization F₁ TotalSpace.proj
e₂ e₂' : Trivialization F₂ TotalSpace.proj
inst✝⁶ : Finite ι
inst✝⁵ : VectorBundle 𝕜 F₁ E₁
inst✝⁴ : VectorBundle 𝕜 F₂ E₂
inst✝³ : MemTrivializationAtlas e₁
inst✝² : MemTrivializationAtlas e₁'
inst✝¹ : MemTrivializationAtlas e₂
inst✝ : MemTrivializationAtlas e₂'
val✝ : Fintype ι
⊢ ContinuousOn
    (fun b ↦
      (ContinuousLinearMap.compContinuousAlternatingMapCLM 𝕜 F₁ F₂ F₂ ι) ↑(Trivialization.coordChangeL 𝕜 e₂ e₂' b))
    (e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet))

Tactic:
refine map_continuous (ContinuousLinearMap.compContinuousAlternatingMapCLM (ι := ι) 𝕜 F₁ F₂ F₂)
      |>.comp_continuousOn ((continuousOn_coordChange 𝕜 e₂ e₂').mono ?_)

Goal state after:
case intro.refine_1
𝕜 : Type u_1
ι : Type u_2
inst✝²² : NontriviallyNormedField 𝕜
B : Type u_3
inst✝²¹ : TopologicalSpace B
F₁ : Type u_4
inst✝²⁰ : NormedAddCommGroup F₁
inst✝¹⁹ : NormedSpace 𝕜 F₁
E₁ : B → Type u_5
inst✝¹⁸ : (x : B) → AddCommGroup (E₁ x)
inst✝¹⁷ : (x : B) → Module 𝕜 (E₁ x)
inst✝¹⁶ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_6
inst✝¹⁵ : NormedAddCommGroup F₂
inst✝¹⁴ : NormedSpace 𝕜 F₂
E₂ : B → Type u_7
inst✝¹³ : (x : B) → AddCommGroup (E₂ x)
inst✝¹² : (x : B) → Module 𝕜 (E₂ x)
inst✝¹¹ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝¹⁰ : (x : B) → TopologicalSpace (E₁ x)
inst✝⁹ : FiberBundle F₁ E₁
inst✝⁸ : (x : B) → TopologicalSpace (E₂ x)
inst✝⁷ : FiberBundle F₂ E₂
e₁ e₁' : Trivialization F₁ TotalSpace.proj
e₂ e₂' : Trivialization F₂ TotalSpace.proj
inst✝⁶ : Finite ι
inst✝⁵ : VectorBundle 𝕜 F₁ E₁
inst✝⁴ : VectorBundle 𝕜 F₂ E₂
inst✝³ : MemTrivializationAtlas e₁
inst✝² : MemTrivializationAtlas e₁'
inst✝¹ : MemTrivializationAtlas e₂
inst✝ : MemTrivializationAtlas e₂'
val✝ : Fintype ι
⊢ e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet) ⊆ e₂.baseSet ∩ e₂'.baseSet