internlm/Lean-Github huggingface dataset:
Theorem: IsPreconnected.open_diff
Proof state before:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u✝ ∪ f v
u : Set X
x : X
m : x ∈ u
⊢ x ∈ u
Tactic:
exact m
Proof state after:
no goals
Theorem: IsPreconnected.open_diff
Proof state before:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
duv : s \ t ∩ (u ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
⊢ ∀ {x : X} {u : Set X}, IsOpen u → x ∈ f u → ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
Tactic:
intro x u o m
Proof state after:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u✝ ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
x : X
u : Set X
o : IsOpen u
m : x ∈ f u
⊢ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
Theorem: IsPreconnected.open_diff
Proof state before:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u✝ ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
x : X
u : Set X
o : IsOpen u
m : x ∈ f u
⊢ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
Tactic:
simp only [mem_union, mem_setOf, ← hf] at m
Proof state after:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u✝ ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
x : X
u : Set X
o : IsOpen u
m : x ∈ u ∨ x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
⊢ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
Theorem: IsPreconnected.open_diff
Proof state before:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u✝ ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
x : X
u : Set X
o : IsOpen u
m : x ∈ u ∨ x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
⊢ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
Tactic:
cases' m with xu m
Proof state after:
case inl
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u✝ ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
x : X
u : Set X
o : IsOpen u
xu : x ∈ u
⊢ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u

case inr
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u✝ ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
x : X
u : Set X
o : IsOpen u
m : x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
⊢ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
Theorem: IsPreconnected.open_diff
Proof state before:
case inl
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u✝ ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
x : X
u : Set X
o : IsOpen u
xu : x ∈ u
⊢ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u

case inr
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u✝ ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
x : X
u : Set X
o : IsOpen u
m : x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
⊢ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
Tactic:
exact (o.eventually_mem xu).filter_mono nhdsWithin_le_nhds
Proof state after:
case inr
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u✝ ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
x : X
u : Set X
o : IsOpen u
m : x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
⊢ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
Theorem: IsPreconnected.open_diff
Proof state before:
case inr
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u✝ v : Set X
uo : IsOpen u✝
vo : IsOpen v
suv : s \ t ⊆ u✝ ∪ v
duv : s \ t ∩ (u✝ ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u✝ ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
x : X
u : Set X
o : IsOpen u
m : x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
⊢ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
Tactic:
exact m.2.2
Proof state after:
no goals
Theorem: IsPreconnected.open_diff
Proof state before:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
duv : s \ t ∩ (u ∩ v) = ∅
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
fnon : ∀ {x : X} {u : Set X}, IsOpen u → x ∈ f u → ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
⊢ s ∩ (f u ∩ f v) = ∅
Tactic:
contrapose duv
Proof state after:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
fnon : ∀ {x : X} {u : Set X}, IsOpen u → x ∈ f u → ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
duv : ¬s ∩ (f u ∩ f v) = ∅
⊢ ¬s \ t ∩ (u ∩ v) = ∅
Theorem: IsPreconnected.open_diff
Proof state before:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
fnon : ∀ {x : X} {u : Set X}, IsOpen u → x ∈ f u → ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
duv : ¬s ∩ (f u ∩ f v) = ∅
⊢ ¬s \ t ∩ (u ∩ v) = ∅
Tactic:
simp only [← ne_eq, ← nonempty_iff_ne_empty] at duv ⊢
Proof state after:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
fnon : ∀ {x : X} {u : Set X}, IsOpen u → x ∈ f u → ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
duv : (s ∩ (f u ∩ f v)).Nonempty
⊢ (s \ t ∩ (u ∩ v)).Nonempty
Theorem: IsPreconnected.open_diff
Proof state before:
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
fnon : ∀ {x : X} {u : Set X}, IsOpen u → x ∈ f u → ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
duv : (s ∩ (f u ∩ f v)).Nonempty
⊢ (s \ t ∩ (u ∩ v)).Nonempty
Tactic:
rcases duv with ⟨x, m⟩
Proof state after:
case intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
fnon : ∀ {x : X} {u : Set X}, IsOpen u → x ∈ f u → ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
x : X
m : x ∈ s ∩ (f u ∩ f v)
⊢ (s \ t ∩ (u ∩ v)).Nonempty
Theorem: IsPreconnected.open_diff
Proof state before:
case intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
fnon : ∀ {x : X} {u : Set X}, IsOpen u → x ∈ f u → ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
x : X
m : x ∈ s ∩ (f u ∩ f v)
⊢ (s \ t ∩ (u ∩ v)).Nonempty
Tactic:
simp only [mem_inter_iff] at m
Proof state after:
case intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
fnon : ∀ {x : X} {u : Set X}, IsOpen u → x ∈ f u → ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
x : X
m : x ∈ s ∧ x ∈ f u ∧ x ∈ f v
⊢ (s \ t ∩ (u ∩ v)).Nonempty
Theorem: IsPreconnected.open_diff
Proof state before:
case intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
fnon : ∀ {x : X} {u : Set X}, IsOpen u → x ∈ f u → ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
x : X
m : x ∈ s ∧ x ∈ f u ∧ x ∈ f v
⊢ (s \ t ∩ (u ∩ v)).Nonempty
Tactic:
have b := ((so.eventually_mem m.1).filter_mono nhdsWithin_le_nhds).and
  ((fnon uo m.2.1).and (fnon vo m.2.2))
Proof state after:
case intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
fnon : ∀ {x : X} {u : Set X}, IsOpen u → x ∈ f u → ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
x : X
m : x ∈ s ∧ x ∈ f u ∧ x ∈ f v
b : ∀ᶠ (x : X) in 𝓝[tᶜ] x, x ∈ s ∧ x ∈ u ∧ x ∈ v
⊢ (s \ t ∩ (u ∩ v)).Nonempty
Theorem: IsPreconnected.open_diff
Proof state before:
case intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
fnon : ∀ {x : X} {u : Set X}, IsOpen u → x ∈ f u → ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
x : X
m : x ∈ s ∧ x ∈ f u ∧ x ∈ f v
b : ∀ᶠ (x : X) in 𝓝[tᶜ] x, x ∈ s ∧ x ∈ u ∧ x ∈ v
⊢ (s \ t ∩ (u ∩ v)).Nonempty
Tactic:
simp only [eventually_nhdsWithin_iff] at b
Proof state after:
case intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
fnon : ∀ {x : X} {u : Set X}, IsOpen u → x ∈ f u → ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
x : X
m : x ∈ s ∧ x ∈ f u ∧ x ∈ f v
b : ∀ᶠ (x : X) in 𝓝 x, x ∈ tᶜ → x ∈ s ∧ x ∈ u ∧ x ∈ v
⊢ (s \ t ∩ (u ∩ v)).Nonempty
Theorem: IsPreconnected.open_diff
Proof state before:
case intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
fnon : ∀ {x : X} {u : Set X}, IsOpen u → x ∈ f u → ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
x : X
m : x ∈ s ∧ x ∈ f u ∧ x ∈ f v
b : ∀ᶠ (x : X) in 𝓝 x, x ∈ tᶜ → x ∈ s ∧ x ∈ u ∧ x ∈ v
⊢ (s \ t ∩ (u ∩ v)).Nonempty
Tactic:
rcases eventually_nhds_iff.mp b with ⟨n, h, no, xn⟩
Proof state after:
case intro.intro.intro.intro
X : Type
inst✝⁴ : TopologicalSpace X
Y : Type
inst✝³ : TopologicalSpace Y
S : Type
inst✝² : TopologicalSpace S
inst✝¹ : ChartedSpace ℂ S
inst✝ : AnalyticManifold I S
s t : Set X
sc : ∀ (u v : Set X), IsOpen u → IsOpen v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v
so : IsOpen s
ts : Nonseparating t
u v : Set X
uo : IsOpen u
vo : IsOpen v
suv : s \ t ⊆ u ∪ v
f : Set X → Set X
hf : (fun u => u ∪ {x | x ∈ s ∧ x ∈ t ∧ ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u}) = f
mono : ∀ (u : Set X), u ⊆ f u
fopen : ∀ {u : Set X}, IsOpen u → IsOpen (f u)
mem : ∀ {x : X} {u c : Set X}, x ∈ s → x ∈ t → c ∈ 𝓝[tᶜ] x → c ⊆ u → x ∈ f u
cover : s ⊆ f u ∪ f v
fdiff : ∀ {u : Set X}, f u \ t ⊆ u
fnon : ∀ {x : X} {u : Set X}, IsOpen u → x ∈ f u → ∀ᶠ (y : X) in 𝓝[tᶜ] x, y ∈ u
x : X
m : x ∈ s ∧ x ∈ f u ∧ x ∈ f v
b : ∀ᶠ (x : X) in 𝓝 x, x ∈ tᶜ → x ∈ s ∧ x ∈ u ∧ x ∈ v
n : Set X
h : ∀ x ∈ n, x ∈ tᶜ → x ∈ s ∧ x ∈ u ∧ x ∈ v
no : IsOpen n
xn : x ∈ n
⊢ (s \ t ∩ (u ∩ v)).Nonempty