Theorem: FiberBundle.exists_trivialization_Icc_subset
File: Mathlib/Topology/FiberBundle/Basic.lean

Proof steps:
state_before: ι : Type u_1
B : Type u_2
F : Type u_3
X : Type u_4
inst✝⁷ : TopologicalSpace X
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
⊢ ∃ e, Icc a b ⊆ e.baseSet
state_after: case intro
ι : Type u_1
B : Type u_2
F : Type u_3
X : Type u_4
inst✝⁷ : TopologicalSpace X
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
⊢ ∃ e, Icc a b ⊆ e.baseSet
tactic: obtain ⟨ea, hea⟩ : ∃ ea : Trivialization F (π F E), a ∈ ea.baseSet :=
    ⟨trivializationAt F E a, mem_baseSet_trivializationAt F E a⟩
state_before: case intro
ι : Type u_1
B : Type u_2
F : Type u_3
X : Type u_4
inst✝⁷ : TopologicalSpace X
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
⊢ ∃ e, Icc a b ⊆ e.baseSet
state_after: case intro.inl
ι : Type u_1
B : Type u_2
F : Type u_3
X : Type u_4
inst✝⁷ : TopologicalSpace X
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 : b < a
⊢ ∃ e, Icc a b ⊆ e.baseSet

case intro.inr
ι : Type u_1
B : Type u_2
F : Type u_3
X : Type u_4
inst✝⁷ : TopologicalSpace X
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: cases' lt_or_le b a with hab hab
state_before: case intro.inr
ι : Type u_1
B : Type u_2
F : Type u_3
X : Type u_4
inst✝⁷ : TopologicalSpace X
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
state_after: case intro.inr
ι : Type u_1
B : Type u_2
F : Type u_3
X : Type u_4
inst✝⁷ : TopologicalSpace X
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}
⊢ ∃ e, Icc a b ⊆ e.baseSet
tactic: set s : Set B := { x ∈ Icc a b | ∃ e : Trivialization F (π F E), Icc a x ⊆ e.baseSet }
state_before: case intro.inr
ι : Type u_1
B : Type u_2
F : Type u_3
X : Type u_4
inst✝⁷ : TopologicalSpace X
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}
⊢ ∃ e, Icc a b ⊆ e.baseSet
state_after: case intro.inr
ι : Type u_1
B : Type u_2
F : Type u_3
X : Type u_4
inst✝⁷ : TopologicalSpace X
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
⊢ ∃ e, Icc a b ⊆ e.baseSet
tactic: have ha : a ∈ s := ⟨left_mem_Icc.2 hab, ea, by simp [hea]⟩
state_before: case intro.inr
ι : Type u_1
B : Type u_2
F : Type u_3
X : Type u_4
inst✝⁷ : TopologicalSpace X
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
⊢ ∃ e, Icc a b ⊆ e.baseSet
state_after: case intro.inr
ι : Type u_1
B : Type u_2
F : Type u_3
X : Type u_4
inst✝⁷ : TopologicalSpace X
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
⊢ ∃ e, Icc a b ⊆ e.baseSet
tactic: have sne : s.Nonempty := ⟨a, ha⟩
state_before: case intro.inr
ι : Type u_1
B : Type u_2
F : Type u_3
X : Type u_4
inst✝⁷ : TopologicalSpace X
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
⊢ ∃ e, Icc a b ⊆ e.baseSet
state_after: case intro.inr
ι : Type u_1
B : Type u_2
F : Type u_3
X : Type u_4
inst✝⁷ : TopologicalSpace X
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
⊢ ∃ e, Icc a b ⊆ e.baseSet
tactic: have hsb : b ∈ upperBounds s := fun x hx => hx.1.2
state_before: case intro.inr
ι : Type u_1
B : Type u_2
F : Type u_3
X : Type u_4
inst✝⁷ : TopologicalSpace X
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
⊢ ∃ e, Icc a b ⊆ e.baseSet
state_after: case intro.inr
ι : Type u_1
B : Type u_2
F : Type u_3
X : Type u_4
inst✝⁷ : TopologicalSpace X
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
⊢ ∃ e, Icc a b ⊆ e.baseSet
tactic: have sbd : BddAbove s := ⟨b, hsb⟩
state_before: case intro.inr
ι : Type u_1
B : Type u_2
F : Type u_3
X : Type u_4
inst✝⁷ : TopologicalSpace X
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
⊢ ∃ e, Icc a b ⊆ e.baseSet
state_after: case intro.inr
ι : Type u_1
B : Type u_2
F : Type u_3
X : Type u_4
inst✝⁷ : TopologicalSpace X
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
⊢ ∃ e, Icc a b ⊆ e.baseSet
tactic: set c := sSup s
state_before: case intro.inr
ι : Type u_1
B : Type u_2
F : Type u_3
X : Type u_4
inst✝⁷ : TopologicalSpace X
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
⊢ ∃ e, Icc a b ⊆ e.baseSet
state_after: case intro.inr
ι : Type u_1
B : Type u_2
F : Type u_3
X : Type u_4
inst✝⁷ : TopologicalSpace X
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
⊢ ∃ e, Icc a b ⊆ e.baseSet
tactic: have hsc : IsLUB s c := isLUB_csSup sne sbd
state_before: case intro.inr
ι : Type u_1
B : Type u_2
F : Type u_3
X : Type u_4
inst✝⁷ : TopologicalSpace X
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
⊢ ∃ e, Icc a b ⊆ e.baseSet
state_after: case intro.inr
ι : Type u_1
B : Type u_2
F : Type u_3
X : Type u_4
inst✝⁷ : TopologicalSpace X
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
⊢ ∃ e, Icc a b ⊆ e.baseSet
tactic: have hc : c ∈ Icc a b := ⟨hsc.1 ha, hsc.2 hsb⟩
state_before: case intro.inr.intro.intro
ι : Type u_1
B : Type u_2
F : Type u_3
X : Type u_4
inst✝⁷ : TopologicalSpace X
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
⊢ ∃ e, Icc a b ⊆ e.baseSet
state_after: case intro.inr.intro.intro.inl
ι : Type u_1
B : Type u_2
F : Type u_3
X : Type u_4
inst✝⁷ : TopologicalSpace X
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

case intro.inr.intro.intro.inr
ι : Type u_1
B : Type u_2
F : Type u_3
X : Type u_4
inst✝⁷ : TopologicalSpace X
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: rcases hc.2.eq_or_lt with heq | hlt
state_before: case intro.inr.intro.intro.inr
ι : Type u_1
B : Type u_2
F : Type u_3
X : Type u_4
inst✝⁷ : TopologicalSpace X
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
state_after: case intro.inr.intro.intro.inr.intro.intro
ι : Type u_1
B : Type u_2
F : Type u_3
X : Type u_4
inst✝⁷ : TopologicalSpace X
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

ι : Type u_1
B : Type u_2
F : Type u_3
X : Type u_4
inst✝⁷ : TopologicalSpace X
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 ∈ Ioc c b, ∃ e, Icc a d ⊆ e.baseSet
tactic: rsuffices ⟨d, hdcb, hd⟩ : ∃ d ∈ Ioc c b, ∃ e : Trivialization F (π F E), Icc a d ⊆ e.baseSet
state_before: ι : Type u_1
B : Type u_2
F : Type u_3
X : Type u_4
inst✝⁷ : TopologicalSpace X
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 ∈ Ioc c b, ∃ e, Icc a d ⊆ e.baseSet
state_after: case intro.intro
ι : Type u_1
B : Type u_2
F : Type u_3
X : Type u_4
inst✝⁷ : TopologicalSpace X
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
⊢ ∃ d ∈ Ioc c b, ∃ e, Icc a d ⊆ e.baseSet
tactic: obtain ⟨d, hdcb, hd⟩ : ∃ d ∈ Ioc c b, Ico c d ⊆ ec.baseSet :=
    (mem_nhdsWithin_Ici_iff_exists_mem_Ioc_Ico_subset hlt).1
      (mem_nhdsWithin_of_mem_nhds <| IsOpen.mem_nhds ec.open_baseSet (hec ⟨hc.1, le_rfl⟩))
state_before: case intro.intro
ι : Type u_1
B : Type u_2
F : Type u_3
X : Type u_4
inst✝⁷ : TopologicalSpace X
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
⊢ ∃ d ∈ Ioc c b, ∃ e, Icc a d ⊆ e.baseSet
state_after: case intro.intro
ι : Type u_1
B : Type u_2
F : Type u_3
X : Type u_4
inst✝⁷ : TopologicalSpace X
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
⊢ ∃ d ∈ Ioc c b, ∃ e, Icc a d ⊆ e.baseSet
tactic: have had : Ico a d ⊆ ec.baseSet := Ico_subset_Icc_union_Ico.trans (union_subset hec hd)
state_before: case intro.intro
ι : Type u_1
B : Type u_2
F : Type u_3
X : Type u_4
inst✝⁷ : TopologicalSpace X
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
⊢ ∃ d ∈ Ioc c b, ∃ e, Icc a d ⊆ e.baseSet
state_after: case pos
ι : Type u_1
B : Type u_2
F : Type u_3
X : Type u_4
inst✝⁷ : TopologicalSpace X
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

case neg
ι : Type u_1
B : Type u_2
F : Type u_3
X : Type u_4
inst✝⁷ : TopologicalSpace X
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: by_cases he : Disjoint (Iio d) (Ioi c)