internlm/Lean-Github huggingface dataset:
Theorem: exists_image_ne_top
Proof state before:
case left.left.h
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
M : Set ↑(D' φ f) := ⋂₀ Ch
⊢ IsClosed M
Tactic:
apply isClosed_sInter
Proof state after:
case left.left.h.a
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
M : Set ↑(D' φ f) := ⋂₀ Ch
⊢ ∀ (t : Set ↑(D' φ f)), t ∈ Ch → IsClosed t
Theorem: exists_image_ne_top
Proof state before:
case left.left.h.a
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
M : Set ↑(D' φ f) := ⋂₀ Ch
⊢ ∀ (t : Set ↑(D' φ f)), t ∈ Ch → IsClosed t
Tactic:
intro N hN
Proof state after:
case left.left.h.a
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
M : Set ↑(D' φ f) := ⋂₀ Ch
N : Set ↑(D' φ f)
hN : N ∈ Ch
⊢ IsClosed N
Theorem: exists_image_ne_top
Proof state before:
case left.left.h.a
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
M : Set ↑(D' φ f) := ⋂₀ Ch
N : Set ↑(D' φ f)
hN : N ∈ Ch
⊢ IsClosed N
Tactic:
have N_comp := (h hN).1
Proof state after:
case left.left.h.a
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
M : Set ↑(D' φ f) := ⋂₀ Ch
N : Set ↑(D' φ f)
hN : N ∈ Ch
N_comp : CompactSpace ↑N
⊢ IsClosed N
Theorem: exists_image_ne_top
Proof state before:
case left.left.h.a
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
M : Set ↑(D' φ f) := ⋂₀ Ch
N : Set ↑(D' φ f)
hN : N ∈ Ch
N_comp : CompactSpace ↑N
⊢ IsClosed N
Tactic:
rw [←isCompact_iff_compactSpace] at N_comp
Proof state after:
case left.left.h.a
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
M : Set ↑(D' φ f) := ⋂₀ Ch
N : Set ↑(D' φ f)
hN : N ∈ Ch
N_comp : IsCompact N
⊢ IsClosed N
Theorem: exists_image_ne_top
Proof state before:
case left.left.h.a
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
M : Set ↑(D' φ f) := ⋂₀ Ch
N : Set ↑(D' φ f)
hN : N ∈ Ch
N_comp : IsCompact N
⊢ IsClosed N
Tactic:
exact IsCompact.isClosed N_comp
Proof state after:
no goals
Theorem: exists_image_ne_top
Proof state before:
case left.right
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
M : Set ↑(D' φ f) := ⋂₀ Ch
⊢ π₁ φ f '' M = univ
Tactic:
by_cases h₂ : Nonempty Ch
Proof state after:
case pos
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
M : Set ↑(D' φ f) := ⋂₀ Ch
h₂ : Nonempty ↑Ch
⊢ π₁ φ f '' M = univ

case neg
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
M : Set ↑(D' φ f) := ⋂₀ Ch
h₂ : ¬Nonempty ↑Ch
⊢ π₁ φ f '' M = univ
Theorem: exists_image_ne_top
Proof state before:
case neg
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
M : Set ↑(D' φ f) := ⋂₀ Ch
h₂ : ¬Nonempty ↑Ch
⊢ π₁ φ f '' M = univ
Tactic:
simp at h₂
Proof state after:
case neg
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
M : Set ↑(D' φ f) := ⋂₀ Ch
h₂ : ∀ (x : Set ↑(D' φ f)), ¬x ∈ Ch
⊢ π₁ φ f '' M = univ
Theorem: exists_image_ne_top
Proof state before:
case neg
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
M : Set ↑(D' φ f) := ⋂₀ Ch
h₂ : ∀ (x : Set ↑(D' φ f)), ¬x ∈ Ch
⊢ π₁ φ f '' M = univ
Tactic:
rw [←eq_empty_iff_forall_not_mem] at h₂
Proof state after:
case neg
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
M : Set ↑(D' φ f) := ⋂₀ Ch
h₂ : Ch = ∅
⊢ π₁ φ f '' M = univ
Theorem: exists_image_ne_top
Proof state before:
case neg
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
M : Set ↑(D' φ f) := ⋂₀ Ch
h₂ : Ch = ∅
⊢ π₁ φ f '' M = univ
Tactic:
revert M
Proof state after:
case neg
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
h₂ : Ch = ∅
⊢ let M := ⋂₀ Ch;
  π₁ φ f '' M = univ
Theorem: exists_image_ne_top
Proof state before:
case neg
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
h₂ : Ch = ∅
⊢ let M := ⋂₀ Ch;
  π₁ φ f '' M = univ
Tactic:
rw [h₂, sInter_empty]
Proof state after:
case neg
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
h₂ : Ch = ∅
⊢ let M := univ;
  π₁ φ f '' M = univ
Theorem: exists_image_ne_top
Proof state before:
case neg
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
h₂ : Ch = ∅
⊢ let M := univ;
  π₁ φ f '' M = univ
Tactic:
simp
Proof state after:
case neg
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
h₂ : Ch = ∅
⊢ range (π₁ φ f) = univ
Theorem: exists_image_ne_top
Proof state before:
case neg
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
h₂ : Ch = ∅
⊢ range (π₁ φ f) = univ
Tactic:
exact range_iff_surjective.mpr <| fun a => ⟨⟨⟨_, _⟩, (hf' <| φ a).choose_spec.symm⟩, rfl⟩
Proof state after:
no goals
Theorem: exists_image_ne_top
Proof state before:
case pos
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
M : Set ↑(D' φ f) := ⋂₀ Ch
h₂ : Nonempty ↑Ch
⊢ π₁ φ f '' M = univ
Tactic:
ext x
Proof state after:
case pos.h
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
M : Set ↑(D' φ f) := ⋂₀ Ch
h₂ : Nonempty ↑Ch
x : A
⊢ x ∈ π₁ φ f '' M ↔ x ∈ univ
Theorem: exists_image_ne_top
Proof state before:
case pos.h
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
M : Set ↑(D' φ f) := ⋂₀ Ch
h₂ : Nonempty ↑Ch
x : A
⊢ x ∈ π₁ φ f '' M ↔ x ∈ univ
Tactic:
refine' ⟨ fun _ => trivial , fun _ => _ ⟩
Proof state after:
case pos.h
A : Type u_1
B : Type u_2
C : Type u_3
D : Type ?u.12569
E : Set D
inst✝⁸ : TopologicalSpace A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : TopologicalSpace C
inst✝⁵ : TopologicalSpace D
ρ : ↑E → A
ρ_cont : Continuous ρ
ρ_surj : Function.Surjective ρ
π : D → A
π_cont : Continuous π
π_surj : Function.Surjective π
image_ne_top : ∀ (E₀ : Set ↑E), E₀ ≠ ⊤ → IsClosed E₀ → ρ '' E₀ ≠ ⊤
inst✝⁴ : T2Space A
inst✝³ : CompactSpace A
inst✝² : T2Space B
inst✝¹ : CompactSpace B
inst✝ : T2Space C
φ : A → C
f : B → C
hφ : Continuous φ
hf : Continuous f
hf' : Function.Surjective f
this : CompactSpace ↑(D' φ f)
S : Set (Set ↑(D' φ f)) := {E | CompactSpace ↑E ∧ π₁ φ f '' E = univ}
Ch : Set (Set ↑(D' φ f))
h : Ch ⊆ S
hCh : IsChain (fun x x_1 => x ⊆ x_1) Ch
M : Set ↑(D' φ f) := ⋂₀ Ch
h₂ : Nonempty ↑Ch
x : A
x✝ : x ∈ univ
⊢ x ∈ π₁ φ f '' M