mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.OpenPartialHomeomorph.Constructions

Goal state before:
pp: X : Type u_1
Y : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
e : OpenPartialHomeomorph X Y
U : Opens X
hU : Nonempty ↥U
y : Y
hy : y ∈ (e.subtypeRestr hU).target
⊢ (Subtype.val ∘ ↑(e.subtypeRestr hU).symm) y ∈ e.source

Tactic:
·

Goal state after:
X : Type u_1
Y : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
e : OpenPartialHomeomorph X Y
U : Opens X
hU : Nonempty ↥U
y : Y
hy : y ∈ (e.subtypeRestr hU).target
⊢ (Subtype.val ∘ ↑(e.subtypeRestr hU).symm) y ∈ e.source
Module: Mathlib.Topology.OpenPartialHomeomorph.Constructions

Goal state before:
pp: case refine_1
X : Type u_1
Y : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
e : OpenPartialHomeomorph X Y
U V : Opens X
hU : Nonempty ↥U
hV : Nonempty ↥V
hUV : U ≤ V
i : ↑↑U → ↑↑V := inclusion hUV
y : Y
hy : y ∈ e.target ∩ ↑e.symm ⁻¹' (U.openPartialHomeomorphSubtypeCoe hU).target
hyV : ↑e.symm y ∈ (V.openPartialHomeomorphSubtypeCoe hV).target
⊢ ↑(V.openPartialHomeomorphSubtypeCoe hV).symm (↑e.symm y) ∈ (V.openPartialHomeomorphSubtypeCoe hV).source

pp: case refine_2
X : Type u_1
Y : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
e : OpenPartialHomeomorph X Y
U V : Opens X
hU : Nonempty ↥U
hV : Nonempty ↥V
hUV : U ≤ V
i : ↑↑U → ↑↑V := inclusion hUV
y : Y
hy : y ∈ e.target ∩ ↑e.symm ⁻¹' (U.openPartialHomeomorphSubtypeCoe hU).target
hyV : ↑e.symm y ∈ (V.openPartialHomeomorphSubtypeCoe hV).target
⊢ ↑(V.openPartialHomeomorphSubtypeCoe hV) (↑(V.openPartialHomeomorphSubtypeCoe hV).symm (↑e.symm y)) =
    ↑(V.openPartialHomeomorphSubtypeCoe hV) (i (↑(U.openPartialHomeomorphSubtypeCoe hU).symm (↑e.symm y)))

Tactic:
·

Goal state after:
case refine_1
X : Type u_1
Y : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
e : OpenPartialHomeomorph X Y
U V : Opens X
hU : Nonempty ↥U
hV : Nonempty ↥V
hUV : U ≤ V
i : ↑↑U → ↑↑V := inclusion hUV
y : Y
hy : y ∈ e.target ∩ ↑e.symm ⁻¹' (U.openPartialHomeomorphSubtypeCoe hU).target
hyV : ↑e.symm y ∈ (V.openPartialHomeomorphSubtypeCoe hV).target
⊢ ↑(V.openPartialHomeomorphSubtypeCoe hV).symm (↑e.symm y) ∈ (V.openPartialHomeomorphSubtypeCoe hV).source
Module: Mathlib.Topology.OpenPartialHomeomorph.Constructions

Goal state before:
pp: case refine_2
X : Type u_1
Y : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
e : OpenPartialHomeomorph X Y
U V : Opens X
hU : Nonempty ↥U
hV : Nonempty ↥V
hUV : U ≤ V
i : ↑↑U → ↑↑V := inclusion hUV
y : Y
hy : y ∈ e.target ∩ ↑e.symm ⁻¹' (U.openPartialHomeomorphSubtypeCoe hU).target
hyV : ↑e.symm y ∈ (V.openPartialHomeomorphSubtypeCoe hV).target
⊢ ↑(V.openPartialHomeomorphSubtypeCoe hV) (↑(V.openPartialHomeomorphSubtypeCoe hV).symm (↑e.symm y)) =
    ↑(V.openPartialHomeomorphSubtypeCoe hV) (i (↑(U.openPartialHomeomorphSubtypeCoe hU).symm (↑e.symm y)))

Tactic:
·

Goal state after:
case refine_2
X : Type u_1
Y : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
e : OpenPartialHomeomorph X Y
U V : Opens X
hU : Nonempty ↥U
hV : Nonempty ↥V
hUV : U ≤ V
i : ↑↑U → ↑↑V := inclusion hUV
y : Y
hy : y ∈ e.target ∩ ↑e.symm ⁻¹' (U.openPartialHomeomorphSubtypeCoe hU).target
hyV : ↑e.symm y ∈ (V.openPartialHomeomorphSubtypeCoe hV).target
⊢ ↑(V.openPartialHomeomorphSubtypeCoe hV) (↑(V.openPartialHomeomorphSubtypeCoe hV).symm (↑e.symm y)) =
    ↑(V.openPartialHomeomorphSubtypeCoe hV) (i (↑(U.openPartialHomeomorphSubtypeCoe hU).symm (↑e.symm y)))
Module: Mathlib.Topology.OpenPartialHomeomorph.Constructions

Goal state before:
pp: case neg
X✝ : Type u_1
X'✝ : Type u_2
Y : Type u_3
Y' : Type u_4
Z✝ : Type u_5
Z' : Type u_6
inst✝⁹ : TopologicalSpace X✝
inst✝⁸ : TopologicalSpace X'✝
inst✝⁷ : TopologicalSpace Y
inst✝⁶ : TopologicalSpace Y'
inst✝⁵ : TopologicalSpace Z✝
inst✝⁴ : TopologicalSpace Z'
e✝ : OpenPartialHomeomorph X✝ Y
X : Type u_7
X' : Type u_8
Z : Type u_9
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace X'
inst✝¹ : TopologicalSpace Z
inst✝ : Nonempty Z
f : X → X'
e : OpenPartialHomeomorph X Z
hf : IsOpenEmbedding f
h✝ : ¬Nonempty X
⊢ ContinuousOn (extend f ↑e fun x ↦ Classical.arbitrary Z) (f '' e.source)

pp: case pos
X✝ : Type u_1
X'✝ : Type u_2
Y : Type u_3
Y' : Type u_4
Z✝ : Type u_5
Z' : Type u_6
inst✝⁹ : TopologicalSpace X✝
inst✝⁸ : TopologicalSpace X'✝
inst✝⁷ : TopologicalSpace Y
inst✝⁶ : TopologicalSpace Y'
inst✝⁵ : TopologicalSpace Z✝
inst✝⁴ : TopologicalSpace Z'
e✝ : OpenPartialHomeomorph X✝ Y
X : Type u_7
X' : Type u_8
Z : Type u_9
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace X'
inst✝¹ : TopologicalSpace Z
inst✝ : Nonempty Z
f : X → X'
e : OpenPartialHomeomorph X Z
hf : IsOpenEmbedding f
h✝ : Nonempty X
⊢ ContinuousOn (extend f ↑e fun x ↦ Classical.arbitrary Z) (f '' e.source)

Tactic:
·

Goal state after:
case neg
X✝ : Type u_1
X'✝ : Type u_2
Y : Type u_3
Y' : Type u_4
Z✝ : Type u_5
Z' : Type u_6
inst✝⁹ : TopologicalSpace X✝
inst✝⁸ : TopologicalSpace X'✝
inst✝⁷ : TopologicalSpace Y
inst✝⁶ : TopologicalSpace Y'
inst✝⁵ : TopologicalSpace Z✝
inst✝⁴ : TopologicalSpace Z'
e✝ : OpenPartialHomeomorph X✝ Y
X : Type u_7
X' : Type u_8
Z : Type u_9
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace X'
inst✝¹ : TopologicalSpace Z
inst✝ : Nonempty Z
f : X → X'
e : OpenPartialHomeomorph X Z
hf : IsOpenEmbedding f
h✝ : ¬Nonempty X
⊢ ContinuousOn (extend f ↑e fun x ↦ Classical.arbitrary Z) (f '' e.source)
Module: Mathlib.Topology.OpenPartialHomeomorph.Constructions

Goal state before:
pp: case h
X✝ : Type u_1
X'✝ : Type u_2
Y : Type u_3
Y' : Type u_4
Z✝ : Type u_5
Z' : Type u_6
inst✝⁹ : TopologicalSpace X✝
inst✝⁸ : TopologicalSpace X'✝
inst✝⁷ : TopologicalSpace Y
inst✝⁶ : TopologicalSpace Y'
inst✝⁵ : TopologicalSpace Z✝
inst✝⁴ : TopologicalSpace Z'
e✝ : OpenPartialHomeomorph X✝ Y
X : Type u_7
X' : Type u_8
Z : Type u_9
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace X'
inst✝¹ : TopologicalSpace Z
inst✝ : Nonempty Z
f : X → X'
e : OpenPartialHomeomorph X Z
hf : IsOpenEmbedding f
h✝ : Nonempty X
F : X' → Z := ⋯
F_eq : F = extend f ↑e fun x ↦ Classical.arbitrary Z
heq : EqOn F (↑e ∘ ↑(IsOpenEmbedding.toOpenPartialHomeomorph f hf).symm) (f '' e.source)
⊢ MapsTo (↑(IsOpenEmbedding.toOpenPartialHomeomorph f hf).symm) (f '' e.source) e.source

pp: case hf
X✝ : Type u_1
X'✝ : Type u_2
Y : Type u_3
Y' : Type u_4
Z✝ : Type u_5
Z' : Type u_6
inst✝⁹ : TopologicalSpace X✝
inst✝⁸ : TopologicalSpace X'✝
inst✝⁷ : TopologicalSpace Y
inst✝⁶ : TopologicalSpace Y'
inst✝⁵ : TopologicalSpace Z✝
inst✝⁴ : TopologicalSpace Z'
e✝ : OpenPartialHomeomorph X✝ Y
X : Type u_7
X' : Type u_8
Z : Type u_9
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace X'
inst✝¹ : TopologicalSpace Z
inst✝ : Nonempty Z
f : X → X'
e : OpenPartialHomeomorph X Z
hf : IsOpenEmbedding f
h✝ : Nonempty X
F : X' → Z := ⋯
F_eq : F = extend f ↑e fun x ↦ Classical.arbitrary Z
heq : EqOn F (↑e ∘ ↑(IsOpenEmbedding.toOpenPartialHomeomorph f hf).symm) (f '' e.source)
⊢ ContinuousOn (↑(IsOpenEmbedding.toOpenPartialHomeomorph f hf).symm) (f '' e.source)

Tactic:
·

Goal state after:
case h
X✝ : Type u_1
X'✝ : Type u_2
Y : Type u_3
Y' : Type u_4
Z✝ : Type u_5
Z' : Type u_6
inst✝⁹ : TopologicalSpace X✝
inst✝⁸ : TopologicalSpace X'✝
inst✝⁷ : TopologicalSpace Y
inst✝⁶ : TopologicalSpace Y'
inst✝⁵ : TopologicalSpace Z✝
inst✝⁴ : TopologicalSpace Z'
e✝ : OpenPartialHomeomorph X✝ Y
X : Type u_7
X' : Type u_8
Z : Type u_9
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace X'
inst✝¹ : TopologicalSpace Z
inst✝ : Nonempty Z
f : X → X'
e : OpenPartialHomeomorph X Z
hf : IsOpenEmbedding f
h✝ : Nonempty X
F : X' → Z := ⋯
F_eq : F = extend f ↑e fun x ↦ Classical.arbitrary Z
heq : EqOn F (↑e ∘ ↑(IsOpenEmbedding.toOpenPartialHomeomorph f hf).symm) (f '' e.source)
⊢ MapsTo (↑(IsOpenEmbedding.toOpenPartialHomeomorph f hf).symm) (f '' e.source) e.source
Module: Mathlib.Topology.OpenPartialHomeomorph.Constructions

Goal state before:
pp: case h
X : Type u_7
X' : Type u_8
Z : Type u_9
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace X'
inst✝¹ : TopologicalSpace Z
inst✝ : Nonempty Z
f : X → X'
e e' : OpenPartialHomeomorph X Z
hf : IsOpenEmbedding f
z : Z
⊢ ↑((e.lift_openEmbedding hf).symm.trans (e'.lift_openEmbedding hf)) z = ↑(e.symm.trans e') z

pp: case hinv
X : Type u_7
X' : Type u_8
Z : Type u_9
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace X'
inst✝¹ : TopologicalSpace Z
inst✝ : Nonempty Z
f : X → X'
e e' : OpenPartialHomeomorph X Z
hf : IsOpenEmbedding f
z : Z
⊢ ↑((e.lift_openEmbedding hf).symm.trans (e'.lift_openEmbedding hf)).symm z = ↑(e.symm.trans e').symm z

pp: case hs
X : Type u_7
X' : Type u_8
Z : Type u_9
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace X'
inst✝¹ : TopologicalSpace Z
inst✝ : Nonempty Z
f : X → X'
e e' : OpenPartialHomeomorph X Z
hf : IsOpenEmbedding f
z : Z
⊢ z ∈ ((e.lift_openEmbedding hf).symm.trans (e'.lift_openEmbedding hf)).source ↔ z ∈ (e.symm.trans e').source

Tactic:
·

Goal state after:
case h
X : Type u_7
X' : Type u_8
Z : Type u_9
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace X'
inst✝¹ : TopologicalSpace Z
inst✝ : Nonempty Z
f : X → X'
e e' : OpenPartialHomeomorph X Z
hf : IsOpenEmbedding f
z : Z
⊢ ↑((e.lift_openEmbedding hf).symm.trans (e'.lift_openEmbedding hf)) z = ↑(e.symm.trans e') z
Module: Mathlib.Topology.OpenPartialHomeomorph.Constructions

Goal state before:
pp: case hinv
X : Type u_7
X' : Type u_8
Z : Type u_9
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace X'
inst✝¹ : TopologicalSpace Z
inst✝ : Nonempty Z
f : X → X'
e e' : OpenPartialHomeomorph X Z
hf : IsOpenEmbedding f
z : Z
⊢ ↑((e.lift_openEmbedding hf).symm.trans (e'.lift_openEmbedding hf)).symm z = ↑(e.symm.trans e').symm z

pp: case hs
X : Type u_7
X' : Type u_8
Z : Type u_9
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace X'
inst✝¹ : TopologicalSpace Z
inst✝ : Nonempty Z
f : X → X'
e e' : OpenPartialHomeomorph X Z
hf : IsOpenEmbedding f
z : Z
⊢ z ∈ ((e.lift_openEmbedding hf).symm.trans (e'.lift_openEmbedding hf)).source ↔ z ∈ (e.symm.trans e').source

Tactic:
·

Goal state after:
case hinv
X : Type u_7
X' : Type u_8
Z : Type u_9
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace X'
inst✝¹ : TopologicalSpace Z
inst✝ : Nonempty Z
f : X → X'
e e' : OpenPartialHomeomorph X Z
hf : IsOpenEmbedding f
z : Z
⊢ ↑((e.lift_openEmbedding hf).symm.trans (e'.lift_openEmbedding hf)).symm z = ↑(e.symm.trans e').symm z
Module: Mathlib.Topology.OpenPartialHomeomorph.Constructions

Goal state before:
pp: case hs
X : Type u_7
X' : Type u_8
Z : Type u_9
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace X'
inst✝¹ : TopologicalSpace Z
inst✝ : Nonempty Z
f : X → X'
e e' : OpenPartialHomeomorph X Z
hf : IsOpenEmbedding f
z : Z
⊢ z ∈ ((e.lift_openEmbedding hf).symm.trans (e'.lift_openEmbedding hf)).source ↔ z ∈ (e.symm.trans e').source

Tactic:
·

Goal state after:
case hs
X : Type u_7
X' : Type u_8
Z : Type u_9
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace X'
inst✝¹ : TopologicalSpace Z
inst✝ : Nonempty Z
f : X → X'
e e' : OpenPartialHomeomorph X Z
hf : IsOpenEmbedding f
z : Z
⊢ z ∈ ((e.lift_openEmbedding hf).symm.trans (e'.lift_openEmbedding hf)).source ↔ z ∈ (e.symm.trans e').source
Module: Mathlib.Topology.FiberBundle.Basic

Goal state before:
pp: B : Type u_2
F : Type u_3
inst✝⁴ : TopologicalSpace B
inst✝³ : TopologicalSpace F
E : B → Type u_5
inst✝² : TopologicalSpace (TotalSpace F E)
inst✝¹ : (b : B) → TopologicalSpace (E b)
inst✝ : FiberBundle F E
s : (x : B) → E x
a : Set B
x₀ : B
⊢ ContinuousWithinAt (fun x ↦ TotalSpace.mk' F x (s x)) a x₀ ↔
    ContinuousWithinAt (fun x ↦ (↑(trivializationAt F E x₀) { proj := x, snd := s x }).2) a x₀

Tactic:
by
  simp_rw [continuousWithinAt_totalSpace, and_iff_right_iff_imp]
  intro; exact continuousWithinAt_id

Goal state after:
Module: Mathlib.Topology.FiberBundle.Basic

Goal state before:
pp: case inl
B : Type u_2
F : Type u_3
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace F
E : B → Type u_5
inst✝⁴ : TopologicalSpace (TotalSpace F E)
inst✝³ : (b : B) → TopologicalSpace (E b)
inst✝² : ConditionallyCompleteLinearOrder B
inst✝¹ : OrderTopology B
inst✝ : FiberBundle F E
a b : B
ea : Trivialization F TotalSpace.proj
hea : a ∈ ea.baseSet
h✝ : b < a
⊢ ∃ e, Icc a b ⊆ e.baseSet

pp: case inr
B : Type u_2
F : Type u_3
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace F
E : B → Type u_5
inst✝⁴ : TopologicalSpace (TotalSpace F E)
inst✝³ : (b : B) → TopologicalSpace (E b)
inst✝² : ConditionallyCompleteLinearOrder B
inst✝¹ : OrderTopology B
inst✝ : FiberBundle F E
a b : B
ea : Trivialization F TotalSpace.proj
hea : a ∈ ea.baseSet
hab : a ≤ b
⊢ ∃ e, Icc a b ⊆ e.baseSet

Tactic:
·

Goal state after:
case inl
B : Type u_2
F : Type u_3
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace F
E : B → Type u_5
inst✝⁴ : TopologicalSpace (TotalSpace F E)
inst✝³ : (b : B) → TopologicalSpace (E b)
inst✝² : ConditionallyCompleteLinearOrder B
inst✝¹ : OrderTopology B
inst✝ : FiberBundle F E
a b : B
ea : Trivialization F TotalSpace.proj
hea : a ∈ ea.baseSet
h✝ : b < a
⊢ ∃ e, Icc a b ⊆ e.baseSet
Module: Mathlib.Topology.FiberBundle.Basic

Goal state before:
pp: case inl
B : Type u_2
F : Type u_3
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace F
E : B → Type u_5
inst✝⁴ : TopologicalSpace (TotalSpace F E)
inst✝³ : (b : B) → TopologicalSpace (E b)
inst✝² : ConditionallyCompleteLinearOrder B
inst✝¹ : OrderTopology B
inst✝ : FiberBundle F E
a b : B
ea : Trivialization F TotalSpace.proj
hea : a ∈ ea.baseSet
hab : a ≤ b
s : Set B := {x | x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet}
ha : a ∈ s
sne : s.Nonempty
hsb : b ∈ upperBounds s
sbd : BddAbove s
c : B := sSup s
hsc : IsLUB s c
hc : c ∈ Icc a b
heq : a = c
⊢ c ∈ s

pp: case inr
B : Type u_2
F : Type u_3
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace F
E : B → Type u_5
inst✝⁴ : TopologicalSpace (TotalSpace F E)
inst✝³ : (b : B) → TopologicalSpace (E b)
inst✝² : ConditionallyCompleteLinearOrder B
inst✝¹ : OrderTopology B
inst✝ : FiberBundle F E
a b : B
ea : Trivialization F TotalSpace.proj
hea : a ∈ ea.baseSet
hab : a ≤ b
s : Set B := {x | x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet}
ha : a ∈ s
sne : s.Nonempty
hsb : b ∈ upperBounds s
sbd : BddAbove s
c : B := sSup s
hsc : IsLUB s c
hc : c ∈ Icc a b
hlt : a < c
⊢ c ∈ s

Tactic:
·

Goal state after:
case inl
B : Type u_2
F : Type u_3
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace F
E : B → Type u_5
inst✝⁴ : TopologicalSpace (TotalSpace F E)
inst✝³ : (b : B) → TopologicalSpace (E b)
inst✝² : ConditionallyCompleteLinearOrder B
inst✝¹ : OrderTopology B
inst✝ : FiberBundle F E
a b : B
ea : Trivialization F TotalSpace.proj
hea : a ∈ ea.baseSet
hab : a ≤ b
s : Set B := {x | x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet}
ha : a ∈ s
sne : s.Nonempty
hsb : b ∈ upperBounds s
sbd : BddAbove s
c : B := sSup s
hsc : IsLUB s c
hc : c ∈ Icc a b
heq : a = c
⊢ c ∈ s
Module: Mathlib.Topology.FiberBundle.Basic

Goal state before:
pp: case inr.inl
B : Type u_2
F : Type u_3
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace F
E : B → Type u_5
inst✝⁴ : TopologicalSpace (TotalSpace F E)
inst✝³ : (b : B) → TopologicalSpace (E b)
inst✝² : ConditionallyCompleteLinearOrder B
inst✝¹ : OrderTopology B
inst✝ : FiberBundle F E
a b : B
ea : Trivialization F TotalSpace.proj
hea : a ∈ ea.baseSet
hab : a ≤ b
s : Set B := {x | x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet}
ha : a ∈ s
sne : s.Nonempty
hsb : b ∈ upperBounds s
sbd : BddAbove s
c : B := sSup s
hsc : IsLUB s c
hc : c ∈ Icc a b
ec : Trivialization F TotalSpace.proj
hec : Icc a c ⊆ ec.baseSet
heq : c = b
⊢ ∃ e, Icc a b ⊆ e.baseSet

pp: case inr.inr
B : Type u_2
F : Type u_3
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace F
E : B → Type u_5
inst✝⁴ : TopologicalSpace (TotalSpace F E)
inst✝³ : (b : B) → TopologicalSpace (E b)
inst✝² : ConditionallyCompleteLinearOrder B
inst✝¹ : OrderTopology B
inst✝ : FiberBundle F E
a b : B
ea : Trivialization F TotalSpace.proj
hea : a ∈ ea.baseSet
hab : a ≤ b
s : Set B := {x | x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet}
ha : a ∈ s
sne : s.Nonempty
hsb : b ∈ upperBounds s
sbd : BddAbove s
c : B := sSup s
hsc : IsLUB s c
hc : c ∈ Icc a b
ec : Trivialization F TotalSpace.proj
hec : Icc a c ⊆ ec.baseSet
hlt : c < b
⊢ ∃ e, Icc a b ⊆ e.baseSet

Tactic:
·

Goal state after:
case inr.inl
B : Type u_2
F : Type u_3
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace F
E : B → Type u_5
inst✝⁴ : TopologicalSpace (TotalSpace F E)
inst✝³ : (b : B) → TopologicalSpace (E b)
inst✝² : ConditionallyCompleteLinearOrder B
inst✝¹ : OrderTopology B
inst✝ : FiberBundle F E
a b : B
ea : Trivialization F TotalSpace.proj
hea : a ∈ ea.baseSet
hab : a ≤ b
s : Set B := {x | x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet}
ha : a ∈ s
sne : s.Nonempty
hsb : b ∈ upperBounds s
sbd : BddAbove s
c : B := sSup s
hsc : IsLUB s c
hc : c ∈ Icc a b
ec : Trivialization F TotalSpace.proj
hec : Icc a c ⊆ ec.baseSet
heq : c = b
⊢ ∃ e, Icc a b ⊆ e.baseSet
Module: Mathlib.Topology.FiberBundle.Basic

Goal state before:
pp: case inr.inr
B : Type u_2
F : Type u_3
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace F
E : B → Type u_5
inst✝⁴ : TopologicalSpace (TotalSpace F E)
inst✝³ : (b : B) → TopologicalSpace (E b)
inst✝² : ConditionallyCompleteLinearOrder B
inst✝¹ : OrderTopology B
inst✝ : FiberBundle F E
a b : B
ea : Trivialization F TotalSpace.proj
hea : a ∈ ea.baseSet
hab : a ≤ b
s : Set B := {x | x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet}
ha : a ∈ s
sne : s.Nonempty
hsb : b ∈ upperBounds s
sbd : BddAbove s
c : B := sSup s
hsc : IsLUB s c
hc : c ∈ Icc a b
ec : Trivialization F TotalSpace.proj
hec : Icc a c ⊆ ec.baseSet
hlt : c < b
d : B
hdcb : d ∈ Ioc c b
hd : ∃ e, Icc a d ⊆ e.baseSet
⊢ ∃ e, Icc a b ⊆ e.baseSet

pp: B : Type u_2
F : Type u_3
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace F
E : B → Type u_5
inst✝⁴ : TopologicalSpace (TotalSpace F E)
inst✝³ : (b : B) → TopologicalSpace (E b)
inst✝² : ConditionallyCompleteLinearOrder B
inst✝¹ : OrderTopology B
inst✝ : FiberBundle F E
a b : B
ea : Trivialization F TotalSpace.proj
hea : a ∈ ea.baseSet
hab : a ≤ b
s : Set B := ⋯
ha : a ∈ s
sne : s.Nonempty
hsb : b ∈ upperBounds s
sbd : BddAbove s
c : B := ⋯
hsc : IsLUB s c
hc : c ∈ Icc a b
ec : Trivialization F TotalSpace.proj
hec : Icc a c ⊆ ec.baseSet
hlt : c < b
⊢ ∃ d ∈ Ioc c b, ∃ e, Icc a d ⊆ e.baseSet

Tactic:
·

Goal state after:
case inr.inr
B : Type u_2
F : Type u_3
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace F
E : B → Type u_5
inst✝⁴ : TopologicalSpace (TotalSpace F E)
inst✝³ : (b : B) → TopologicalSpace (E b)
inst✝² : ConditionallyCompleteLinearOrder B
inst✝¹ : OrderTopology B
inst✝ : FiberBundle F E
a b : B
ea : Trivialization F TotalSpace.proj
hea : a ∈ ea.baseSet
hab : a ≤ b
s : Set B := {x | x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet}
ha : a ∈ s
sne : s.Nonempty
hsb : b ∈ upperBounds s
sbd : BddAbove s
c : B := sSup s
hsc : IsLUB s c
hc : c ∈ Icc a b
ec : Trivialization F TotalSpace.proj
hec : Icc a c ⊆ ec.baseSet
hlt : c < b
d : B
hdcb : d ∈ Ioc c b
hd : ∃ e, Icc a d ⊆ e.baseSet
⊢ ∃ e, Icc a b ⊆ e.baseSet
Module: Mathlib.Topology.FiberBundle.Basic

Goal state before:
pp: case pos
B : Type u_2
F : Type u_3
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace F
E : B → Type u_5
inst✝⁴ : TopologicalSpace (TotalSpace F E)
inst✝³ : (b : B) → TopologicalSpace (E b)
inst✝² : ConditionallyCompleteLinearOrder B
inst✝¹ : OrderTopology B
inst✝ : FiberBundle F E
a b : B
ea : Trivialization F TotalSpace.proj
hea : a ∈ ea.baseSet
hab : a ≤ b
s : Set B := {x | x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet}
ha : a ∈ s
sne : s.Nonempty
hsb : b ∈ upperBounds s
sbd : BddAbove s
c : B := sSup s
hsc : IsLUB s c
hc : c ∈ Icc a b
ec : Trivialization F TotalSpace.proj
hec : Icc a c ⊆ ec.baseSet
hlt : c < b
d : B
hdcb : d ∈ Ioc c b
hd : Ico c d ⊆ ec.baseSet
had : Ico a d ⊆ ec.baseSet
he : Disjoint (Iio d) (Ioi c)
⊢ ∃ d ∈ Ioc c b, ∃ e, Icc a d ⊆ e.baseSet

pp: case neg
B : Type u_2
F : Type u_3
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace F
E : B → Type u_5
inst✝⁴ : TopologicalSpace (TotalSpace F E)
inst✝³ : (b : B) → TopologicalSpace (E b)
inst✝² : ConditionallyCompleteLinearOrder B
inst✝¹ : OrderTopology B
inst✝ : FiberBundle F E
a b : B
ea : Trivialization F TotalSpace.proj
hea : a ∈ ea.baseSet
hab : a ≤ b
s : Set B := {x | x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet}
ha : a ∈ s
sne : s.Nonempty
hsb : b ∈ upperBounds s
sbd : BddAbove s
c : B := sSup s
hsc : IsLUB s c
hc : c ∈ Icc a b
ec : Trivialization F TotalSpace.proj
hec : Icc a c ⊆ ec.baseSet
hlt : c < b
d : B
hdcb : d ∈ Ioc c b
hd : Ico c d ⊆ ec.baseSet
had : Ico a d ⊆ ec.baseSet
he : ¬Disjoint (Iio d) (Ioi c)
⊢ ∃ d ∈ Ioc c b, ∃ e, Icc a d ⊆ e.baseSet

Tactic:
·

Goal state after:
case pos
B : Type u_2
F : Type u_3
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace F
E : B → Type u_5
inst✝⁴ : TopologicalSpace (TotalSpace F E)
inst✝³ : (b : B) → TopologicalSpace (E b)
inst✝² : ConditionallyCompleteLinearOrder B
inst✝¹ : OrderTopology B
inst✝ : FiberBundle F E
a b : B
ea : Trivialization F TotalSpace.proj
hea : a ∈ ea.baseSet
hab : a ≤ b
s : Set B := {x | x ∈ Icc a b ∧ ∃ e, Icc a x ⊆ e.baseSet}
ha : a ∈ s
sne : s.Nonempty
hsb : b ∈ upperBounds s
sbd : BddAbove s
c : B := sSup s
hsc : IsLUB s c
hc : c ∈ Icc a b
ec : Trivialization F TotalSpace.proj
hec : Icc a c ⊆ ec.baseSet
hlt : c < b
d : B
hdcb : d ∈ Ioc c b
hd : Ico c d ⊆ ec.baseSet
had : Ico a d ⊆ ec.baseSet
he : Disjoint (Iio d) (Ioi c)
⊢ ∃ d ∈ Ioc c b, ∃ e, Icc a d ⊆ e.baseSet
Module: Mathlib.Topology.FiberBundle.Trivialization

Goal state before:
pp: case h₁.h
B : Type u_1
F : Type u_2
Z : Type u_4
inst✝¹ : TopologicalSpace B
inst✝ : TopologicalSpace F
proj : Z → B
e e' : Pretrivialization F proj
h₁ : ∀ (x : Z), ↑e x = ↑e' x
h₂ : ∀ (x : B × F), ↑e.symm x = ↑e'.symm x
h₃ : e.baseSet = e'.baseSet
x✝ : Z
⊢ ↑e.toPartialEquiv x✝ = ↑e'.toPartialEquiv x✝

pp: case h₁.hsymm
B : Type u_1
F : Type u_2
Z : Type u_4
inst✝¹ : TopologicalSpace B
inst✝ : TopologicalSpace F
proj : Z → B
e e' : Pretrivialization F proj
h₁ : ∀ (x : Z), ↑e x = ↑e' x
h₂ : ∀ (x : B × F), ↑e.symm x = ↑e'.symm x
h₃ : e.baseSet = e'.baseSet
x✝ : B × F
⊢ ↑e.symm x✝ = ↑e'.symm x✝

pp: case h₁.hs
B : Type u_1
F : Type u_2
Z : Type u_4
inst✝¹ : TopologicalSpace B
inst✝ : TopologicalSpace F
proj : Z → B
e e' : Pretrivialization F proj
h₁ : ∀ (x : Z), ↑e x = ↑e' x
h₂ : ∀ (x : B × F), ↑e.symm x = ↑e'.symm x
h₃ : e.baseSet = e'.baseSet
⊢ e.source = e'.source

Tactic:
·

Goal state after:
case h₁.h
B : Type u_1
F : Type u_2
Z : Type u_4
inst✝¹ : TopologicalSpace B
inst✝ : TopologicalSpace F
proj : Z → B
e e' : Pretrivialization F proj
h₁ : ∀ (x : Z), ↑e x = ↑e' x
h₂ : ∀ (x : B × F), ↑e.symm x = ↑e'.symm x
h₃ : e.baseSet = e'.baseSet
x✝ : Z
⊢ ↑e.toPartialEquiv x✝ = ↑e'.toPartialEquiv x✝
Module: Mathlib.Topology.FiberBundle.Trivialization

Goal state before:
pp: case h₁.hsymm
B : Type u_1
F : Type u_2
Z : Type u_4
inst✝¹ : TopologicalSpace B
inst✝ : TopologicalSpace F
proj : Z → B
e e' : Pretrivialization F proj
h₁ : ∀ (x : Z), ↑e x = ↑e' x
h₂ : ∀ (x : B × F), ↑e.symm x = ↑e'.symm x
h₃ : e.baseSet = e'.baseSet
x✝ : B × F
⊢ ↑e.symm x✝ = ↑e'.symm x✝

pp: case h₁.hs
B : Type u_1
F : Type u_2
Z : Type u_4
inst✝¹ : TopologicalSpace B
inst✝ : TopologicalSpace F
proj : Z → B
e e' : Pretrivialization F proj
h₁ : ∀ (x : Z), ↑e x = ↑e' x
h₂ : ∀ (x : B × F), ↑e.symm x = ↑e'.symm x
h₃ : e.baseSet = e'.baseSet
⊢ e.source = e'.source

Tactic:
·

Goal state after:
case h₁.hsymm
B : Type u_1
F : Type u_2
Z : Type u_4
inst✝¹ : TopologicalSpace B
inst✝ : TopologicalSpace F
proj : Z → B
e e' : Pretrivialization F proj
h₁ : ∀ (x : Z), ↑e x = ↑e' x
h₂ : ∀ (x : B × F), ↑e.symm x = ↑e'.symm x
h₃ : e.baseSet = e'.baseSet
x✝ : B × F
⊢ ↑e.symm x✝ = ↑e'.symm x✝
Module: Mathlib.Topology.FiberBundle.Trivialization

Goal state before:
pp: case h₁.hs
B : Type u_1
F : Type u_2
Z : Type u_4
inst✝¹ : TopologicalSpace B
inst✝ : TopologicalSpace F
proj : Z → B
e e' : Pretrivialization F proj
h₁ : ∀ (x : Z), ↑e x = ↑e' x
h₂ : ∀ (x : B × F), ↑e.symm x = ↑e'.symm x
h₃ : e.baseSet = e'.baseSet
⊢ e.source = e'.source

Tactic:
·

Goal state after:
case h₁.hs
B : Type u_1
F : Type u_2
Z : Type u_4
inst✝¹ : TopologicalSpace B
inst✝ : TopologicalSpace F
proj : Z → B
e e' : Pretrivialization F proj
h₁ : ∀ (x : Z), ↑e x = ↑e' x
h₂ : ∀ (x : B × F), ↑e.symm x = ↑e'.symm x
h₃ : e.baseSet = e'.baseSet
⊢ e.source = e'.source