Theorem: RestrictGenTopology.of_seq
File: Mathlib/Topology/RestrictGenTopology.lean

Proof steps:
state_before: X : Type u_1
inst✝¹ : TopologicalSpace X
S : Set (Set X)
t : Set X
x : X
inst✝ : SequentialSpace X
h : ∀ ⦃u : ℕ → X⦄ ⦃x : X⦄, Tendsto u atTop (𝓝 x) → insert x (range u) ∈ S
⊢ RestrictGenTopology S
state_after: X : Type u_1
inst✝¹ : TopologicalSpace X
S : Set (Set X)
t✝ : Set X
x✝ : X
inst✝ : SequentialSpace X
h : ∀ ⦃u : ℕ → X⦄ ⦃x : X⦄, Tendsto u atTop (𝓝 x) → insert x (range u) ∈ S
t : Set X
ht : ∀ s ∈ S, IsClosed (Subtype.val ⁻¹' t)
u : ℕ → X
x : X
hut : ∀ (n : ℕ), u n ∈ t
hux : Tendsto u atTop (𝓝 x)
⊢ x ∈ t
tactic: refine of_isClosed fun t ht ↦ IsSeqClosed.isClosed fun u x hut hux ↦ ?_
state_before: X : Type u_1
inst✝¹ : TopologicalSpace X
S : Set (Set X)
t✝ : Set X
x✝ : X
inst✝ : SequentialSpace X
h : ∀ ⦃u : ℕ → X⦄ ⦃x : X⦄, Tendsto u atTop (𝓝 x) → insert x (range u) ∈ S
t : Set X
ht : ∀ s ∈ S, IsClosed (Subtype.val ⁻¹' t)
u : ℕ → X
x : X
hut : ∀ (n : ℕ), u n ∈ t
hux : Tendsto u atTop (𝓝 x)
⊢ x ∈ t
state_after: case intro.intro
X : Type u_1
inst✝¹ : TopologicalSpace X
S : Set (Set X)
t✝ : Set X
x✝ : X
inst✝ : SequentialSpace X
h : ∀ ⦃u : ℕ → X⦄ ⦃x : X⦄, Tendsto u atTop (𝓝 x) → insert x (range u) ∈ S
t : Set X
ht : ∀ s ∈ S, IsClosed (Subtype.val ⁻¹' t)
u : ℕ → X
x : X
hut : ∀ (n : ℕ), u n ∈ t
hux : Tendsto u atTop (𝓝 x)
s : Set X
hsc : IsClosed s
hst : Subtype.val ⁻¹' s = Subtype.val ⁻¹' t
⊢ x ∈ t
tactic: rcases isClosed_induced_iff.1 (ht _ (h hux)) with ⟨s, hsc, hst⟩
state_before: case intro.intro
X : Type u_1
inst✝¹ : TopologicalSpace X
S : Set (Set X)
t✝ : Set X
x✝ : X
inst✝ : SequentialSpace X
h : ∀ ⦃u : ℕ → X⦄ ⦃x : X⦄, Tendsto u atTop (𝓝 x) → insert x (range u) ∈ S
t : Set X
ht : ∀ s ∈ S, IsClosed (Subtype.val ⁻¹' t)
u : ℕ → X
x : X
hut : ∀ (n : ℕ), u n ∈ t
hux : Tendsto u atTop (𝓝 x)
s : Set X
hsc : IsClosed s
hst : Subtype.val ⁻¹' s = Subtype.val ⁻¹' t
⊢ x ∈ t
state_after: case intro.intro
X : Type u_1
inst✝¹ : TopologicalSpace X
S : Set (Set X)
t✝ : Set X
x✝ : X
inst✝ : SequentialSpace X
h : ∀ ⦃u : ℕ → X⦄ ⦃x : X⦄, Tendsto u atTop (𝓝 x) → insert x (range u) ∈ S
t : Set X
ht : ∀ s ∈ S, IsClosed (Subtype.val ⁻¹' t)
u : ℕ → X
x : X
hut : ∀ (n : ℕ), u n ∈ t
hux : Tendsto u atTop (𝓝 x)
s : Set X
hsc : IsClosed s
hst : ∀ (x_1 : X), x_1 ∈ insert x (range u) ∩ s ↔ x_1 ∈ insert x (range u) ∩ t
⊢ x ∈ t
tactic: rw [Subtype.preimage_val_eq_preimage_val_iff, Set.ext_iff] at hst
state_before: case intro.intro
X : Type u_1
inst✝¹ : TopologicalSpace X
S : Set (Set X)
t✝ : Set X
x✝ : X
inst✝ : SequentialSpace X
h : ∀ ⦃u : ℕ → X⦄ ⦃x : X⦄, Tendsto u atTop (𝓝 x) → insert x (range u) ∈ S
t : Set X
ht : ∀ s ∈ S, IsClosed (Subtype.val ⁻¹' t)
u : ℕ → X
x : X
hut : ∀ (n : ℕ), u n ∈ t
hux : Tendsto u atTop (𝓝 x)
s : Set X
hsc : IsClosed s
hst : ∀ (x_1 : X), x_1 ∈ insert x (range u) ∩ s ↔ x_1 ∈ insert x (range u) ∩ t
⊢ x ∈ t
state_after: case intro.intro
X : Type u_1
inst✝¹ : TopologicalSpace X
S : Set (Set X)
t✝ : Set X
x✝ : X
inst✝ : SequentialSpace X
h : ∀ ⦃u : ℕ → X⦄ ⦃x : X⦄, Tendsto u atTop (𝓝 x) → insert x (range u) ∈ S
t : Set X
ht : ∀ s ∈ S, IsClosed (Subtype.val ⁻¹' t)
u : ℕ → X
x : X
hut : ∀ (n : ℕ), u n ∈ t
hux : Tendsto u atTop (𝓝 x)
s : Set X
hsc : IsClosed s
hst : ∀ (x_1 : X), x_1 ∈ insert x (range u) ∩ s ↔ x_1 ∈ insert x (range u) ∩ t
⊢ x ∈ s
tactic: suffices x ∈ s by specialize hst x; simp_all
state_before: case intro.intro
X : Type u_1
inst✝¹ : TopologicalSpace X
S : Set (Set X)
t✝ : Set X
x✝ : X
inst✝ : SequentialSpace X
h : ∀ ⦃u : ℕ → X⦄ ⦃x : X⦄, Tendsto u atTop (𝓝 x) → insert x (range u) ∈ S
t : Set X
ht : ∀ s ∈ S, IsClosed (Subtype.val ⁻¹' t)
u : ℕ → X
x : X
hut : ∀ (n : ℕ), u n ∈ t
hux : Tendsto u atTop (𝓝 x)
s : Set X
hsc : IsClosed s
hst : ∀ (x_1 : X), x_1 ∈ insert x (range u) ∩ s ↔ x_1 ∈ insert x (range u) ∩ t
⊢ x ∈ s
state_after: case intro.intro
X : Type u_1
inst✝¹ : TopologicalSpace X
S : Set (Set X)
t✝ : Set X
x✝ : X
inst✝ : SequentialSpace X
h : ∀ ⦃u : ℕ → X⦄ ⦃x : X⦄, Tendsto u atTop (𝓝 x) → insert x (range u) ∈ S
t : Set X
ht : ∀ s ∈ S, IsClosed (Subtype.val ⁻¹' t)
u : ℕ → X
x : X
hut : ∀ (n : ℕ), u n ∈ t
hux : Tendsto u atTop (𝓝 x)
s : Set X
hsc : IsClosed s
hst : ∀ (x_1 : X), x_1 ∈ insert x (range u) ∩ s ↔ x_1 ∈ insert x (range u) ∩ t
k : ℕ
⊢ u k ∈ s
tactic: refine hsc.mem_of_tendsto hux <| eventually_of_forall fun k ↦ ?_
state_before: case intro.intro
X : Type u_1
inst✝¹ : TopologicalSpace X
S : Set (Set X)
t✝ : Set X
x✝ : X
inst✝ : SequentialSpace X
h : ∀ ⦃u : ℕ → X⦄ ⦃x : X⦄, Tendsto u atTop (𝓝 x) → insert x (range u) ∈ S
t : Set X
ht : ∀ s ∈ S, IsClosed (Subtype.val ⁻¹' t)
u : ℕ → X
x : X
hut : ∀ (n : ℕ), u n ∈ t
hux : Tendsto u atTop (𝓝 x)
s : Set X
hsc : IsClosed s
hst : ∀ (x_1 : X), x_1 ∈ insert x (range u) ∩ s ↔ x_1 ∈ insert x (range u) ∩ t
k : ℕ
⊢ u k ∈ s
state_after: case intro.intro
X : Type u_1
inst✝¹ : TopologicalSpace X
S : Set (Set X)
t✝ : Set X
x✝ : X
inst✝ : SequentialSpace X
h : ∀ ⦃u : ℕ → X⦄ ⦃x : X⦄, Tendsto u atTop (𝓝 x) → insert x (range u) ∈ S
t : Set X
ht : ∀ s ∈ S, IsClosed (Subtype.val ⁻¹' t)
u : ℕ → X
x : X
hut : ∀ (n : ℕ), u n ∈ t
hux : Tendsto u atTop (𝓝 x)
s : Set X
hsc : IsClosed s
k : ℕ
hst : u k ∈ insert x (range u) ∩ s ↔ u k ∈ insert x (range u) ∩ t
⊢ u k ∈ s
tactic: specialize hst (u k)
state_before: case intro.intro
X : Type u_1
inst✝¹ : TopologicalSpace X
S : Set (Set X)
t✝ : Set X
x✝ : X
inst✝ : SequentialSpace X
h : ∀ ⦃u : ℕ → X⦄ ⦃x : X⦄, Tendsto u atTop (𝓝 x) → insert x (range u) ∈ S
t : Set X
ht : ∀ s ∈ S, IsClosed (Subtype.val ⁻¹' t)
u : ℕ → X
x : X
hut : ∀ (n : ℕ), u n ∈ t
hux : Tendsto u atTop (𝓝 x)
s : Set X
hsc : IsClosed s
k : ℕ
hst : u k ∈ insert x (range u) ∩ s ↔ u k ∈ insert x (range u) ∩ t
⊢ u k ∈ s
state_after: no goals
tactic: simp_all