Theorem: TopologicalSpace.Opens.isBasis_iff_nbhd
File: Mathlib/Topology/Sets/Opens.lean

tactic: constructor <;> intro h
state_before: ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
B : Set (Opens α)
⊢ IsBasis B ↔ ∀ {U : Opens α} {x : α}, x ∈ U → ∃ U' ∈ B, x ∈ U' ∧ U' ≤ U
state_after: case mp
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
B : Set (Opens α)
h : IsBasis B
⊢ ∀ {U : Opens α} {x : α}, x ∈ U → ∃ U' ∈ B, x ∈ U' ∧ U' ≤ U

case mpr
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
B : Set (Opens α)
h : ∀ {U : Opens α} {x : α}, x ∈ U → ∃ U' ∈ B, x ∈ U' ∧ U' ≤ U
⊢ IsBasis B

tactic: rintro ⟨sU, hU⟩ x hx
state_before: case mp
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
B : Set (Opens α)
h : IsBasis B
⊢ ∀ {U : Opens α} {x : α}, x ∈ U → ∃ U' ∈ B, x ∈ U' ∧ U' ≤ U
state_after: case mp.mk
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
B : Set (Opens α)
h : IsBasis B
sU : Set α
hU : IsOpen sU
x : α
hx : x ∈ { carrier := sU, is_open' := hU }
⊢ ∃ U' ∈ B, x ∈ U' ∧ U' ≤ { carrier := sU, is_open' := hU }

tactic: rcases h.mem_nhds_iff.mp (IsOpen.mem_nhds hU hx) with ⟨sV, ⟨⟨V, H₁, H₂⟩, hsV⟩⟩
state_before: case mp.mk
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
B : Set (Opens α)
h : IsBasis B
sU : Set α
hU : IsOpen sU
x : α
hx : x ∈ { carrier := sU, is_open' := hU }
⊢ ∃ U' ∈ B, x ∈ U' ∧ U' ≤ { carrier := sU, is_open' := hU }
state_after: case mp.mk.intro.intro.intro.intro
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
B : Set (Opens α)
h : IsBasis B
sU : Set α
hU : IsOpen sU
x : α
hx : x ∈ { carrier := sU, is_open' := hU }
sV : Set α
hsV : x ∈ sV ∧ sV ⊆ sU
V : Opens α
H₁ : V ∈ B
H₂ : ↑V = sV
⊢ ∃ U' ∈ B, x ∈ U' ∧ U' ≤ { carrier := sU, is_open' := hU }

tactic: refine ⟨V, H₁, ?_⟩
state_before: case mp.mk.intro.intro.intro.intro
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
B : Set (Opens α)
h : IsBasis B
sU : Set α
hU : IsOpen sU
x : α
hx : x ∈ { carrier := sU, is_open' := hU }
sV : Set α
hsV : x ∈ sV ∧ sV ⊆ sU
V : Opens α
H₁ : V ∈ B
H₂ : ↑V = sV
⊢ ∃ U' ∈ B, x ∈ U' ∧ U' ≤ { carrier := sU, is_open' := hU }
state_after: case mp.mk.intro.intro.intro.intro
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
B : Set (Opens α)
h : IsBasis B
sU : Set α
hU : IsOpen sU
x : α
hx : x ∈ { carrier := sU, is_open' := hU }
sV : Set α
hsV : x ∈ sV ∧ sV ⊆ sU
V : Opens α
H₁ : V ∈ B
H₂ : ↑V = sV
⊢ x ∈ V ∧ V ≤ { carrier := sU, is_open' := hU }

tactic: cases V
state_before: case mp.mk.intro.intro.intro.intro
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
B : Set (Opens α)
h : IsBasis B
sU : Set α
hU : IsOpen sU
x : α
hx : x ∈ { carrier := sU, is_open' := hU }
sV : Set α
hsV : x ∈ sV ∧ sV ⊆ sU
V : Opens α
H₁ : V ∈ B
H₂ : ↑V = sV
⊢ x ∈ V ∧ V ≤ { carrier := sU, is_open' := hU }
state_after: case mp.mk.intro.intro.intro.intro.mk
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
B : Set (Opens α)
h : IsBasis B
sU : Set α
hU : IsOpen sU
x : α
hx : x ∈ { carrier := sU, is_open' := hU }
sV : Set α
hsV : x ∈ sV ∧ sV ⊆ sU
carrier✝ : Set α
is_open'✝ : IsOpen carrier✝
H₁ : { carrier := carrier✝, is_open' := is_open'✝ } ∈ B
H₂ : ↑{ carrier := carrier✝, is_open' := is_open'✝ } = sV
⊢ x ∈ { carrier := carrier✝, is_open' := is_open'✝ } ∧
    { carrier := carrier✝, is_open' := is_open'✝ } ≤ { carrier := sU, is_open' := hU }

tactic: dsimp at H₂
state_before: case mp.mk.intro.intro.intro.intro.mk
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
B : Set (Opens α)
h : IsBasis B
sU : Set α
hU : IsOpen sU
x : α
hx : x ∈ { carrier := sU, is_open' := hU }
sV : Set α
hsV : x ∈ sV ∧ sV ⊆ sU
carrier✝ : Set α
is_open'✝ : IsOpen carrier✝
H₁ : { carrier := carrier✝, is_open' := is_open'✝ } ∈ B
H₂ : ↑{ carrier := carrier✝, is_open' := is_open'✝ } = sV
⊢ x ∈ { carrier := carrier✝, is_open' := is_open'✝ } ∧
    { carrier := carrier✝, is_open' := is_open'✝ } ≤ { carrier := sU, is_open' := hU }
state_after: case mp.mk.intro.intro.intro.intro.mk
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
B : Set (Opens α)
h : IsBasis B
sU : Set α
hU : IsOpen sU
x : α
hx : x ∈ { carrier := sU, is_open' := hU }
sV : Set α
hsV : x ∈ sV ∧ sV ⊆ sU
carrier✝ : Set α
is_open'✝ : IsOpen carrier✝
H₁ : { carrier := carrier✝, is_open' := is_open'✝ } ∈ B
H₂ : carrier✝ = sV
⊢ x ∈ { carrier := carrier✝, is_open' := is_open'✝ } ∧
    { carrier := carrier✝, is_open' := is_open'✝ } ≤ { carrier := sU, is_open' := hU }

tactic: subst H₂
state_before: case mp.mk.intro.intro.intro.intro.mk
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
B : Set (Opens α)
h : IsBasis B
sU : Set α
hU : IsOpen sU
x : α
hx : x ∈ { carrier := sU, is_open' := hU }
sV : Set α
hsV : x ∈ sV ∧ sV ⊆ sU
carrier✝ : Set α
is_open'✝ : IsOpen carrier✝
H₁ : { carrier := carrier✝, is_open' := is_open'✝ } ∈ B
H₂ : carrier✝ = sV
⊢ x ∈ { carrier := carrier✝, is_open' := is_open'✝ } ∧
    { carrier := carrier✝, is_open' := is_open'✝ } ≤ { carrier := sU, is_open' := hU }
state_after: case mp.mk.intro.intro.intro.intro.mk
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
B : Set (Opens α)
h : IsBasis B
sU : Set α
hU : IsOpen sU
x : α
hx : x ∈ { carrier := sU, is_open' := hU }
carrier✝ : Set α
is_open'✝ : IsOpen carrier✝
H₁ : { carrier := carrier✝, is_open' := is_open'✝ } ∈ B
hsV : x ∈ carrier✝ ∧ carrier✝ ⊆ sU
⊢ x ∈ { carrier := carrier✝, is_open' := is_open'✝ } ∧
    { carrier := carrier✝, is_open' := is_open'✝ } ≤ { carrier := sU, is_open' := hU }

tactic: exact hsV
state_before: case mp.mk.intro.intro.intro.intro.mk
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
B : Set (Opens α)
h : IsBasis B
sU : Set α
hU : IsOpen sU
x : α
hx : x ∈ { carrier := sU, is_open' := hU }
carrier✝ : Set α
is_open'✝ : IsOpen carrier✝
H₁ : { carrier := carrier✝, is_open' := is_open'✝ } ∈ B
hsV : x ∈ carrier✝ ∧ carrier✝ ⊆ sU
⊢ x ∈ { carrier := carrier✝, is_open' := is_open'✝ } ∧
    { carrier := carrier✝, is_open' := is_open'✝ } ≤ { carrier := sU, is_open' := hU }
state_after: no goals

tactic: refine isTopologicalBasis_of_isOpen_of_nhds ?_ ?_
state_before: case mpr
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
B : Set (Opens α)
h : ∀ {U : Opens α} {x : α}, x ∈ U → ∃ U' ∈ B, x ∈ U' ∧ U' ≤ U
⊢ IsBasis B
state_after: case mpr.refine_1
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
B : Set (Opens α)
h : ∀ {U : Opens α} {x : α}, x ∈ U → ∃ U' ∈ B, x ∈ U' ∧ U' ≤ U
⊢ ∀ u ∈ SetLike.coe '' B, IsOpen u

case mpr.refine_2
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
B : Set (Opens α)
h : ∀ {U : Opens α} {x : α}, x ∈ U → ∃ U' ∈ B, x ∈ U' ∧ U' ≤ U
⊢ ∀ (a : α) (u : Set α), a ∈ u → IsOpen u → ∃ v ∈ SetLike.coe '' B, a ∈ v ∧ v ⊆ u

tactic: rintro sU ⟨U, -, rfl⟩
state_before: case mpr.refine_1
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
B : Set (Opens α)
h : ∀ {U : Opens α} {x : α}, x ∈ U → ∃ U' ∈ B, x ∈ U' ∧ U' ≤ U
⊢ ∀ u ∈ SetLike.coe '' B, IsOpen u
state_after: case mpr.refine_1.intro.intro
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
B : Set (Opens α)
h : ∀ {U : Opens α} {x : α}, x ∈ U → ∃ U' ∈ B, x ∈ U' ∧ U' ≤ U
U : Opens α
⊢ IsOpen ↑U

tactic: exact U.2
state_before: case mpr.refine_1.intro.intro
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
B : Set (Opens α)
h : ∀ {U : Opens α} {x : α}, x ∈ U → ∃ U' ∈ B, x ∈ U' ∧ U' ≤ U
U : Opens α
⊢ IsOpen ↑U
state_after: no goals

tactic: intro x sU hx hsU
state_before: case mpr.refine_2
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
B : Set (Opens α)
h : ∀ {U : Opens α} {x : α}, x ∈ U → ∃ U' ∈ B, x ∈ U' ∧ U' ≤ U
⊢ ∀ (a : α) (u : Set α), a ∈ u → IsOpen u → ∃ v ∈ SetLike.coe '' B, a ∈ v ∧ v ⊆ u
state_after: case mpr.refine_2
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
B : Set (Opens α)
h : ∀ {U : Opens α} {x : α}, x ∈ U → ∃ U' ∈ B, x ∈ U' ∧ U' ≤ U
x : α
sU : Set α
hx : x ∈ sU
hsU : IsOpen sU
⊢ ∃ v ∈ SetLike.coe '' B, x ∈ v ∧ v ⊆ sU

tactic: rcases @h ⟨sU, hsU⟩ x hx with ⟨V, hV, H⟩
state_before: case mpr.refine_2
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
B : Set (Opens α)
h : ∀ {U : Opens α} {x : α}, x ∈ U → ∃ U' ∈ B, x ∈ U' ∧ U' ≤ U
x : α
sU : Set α
hx : x ∈ sU
hsU : IsOpen sU
⊢ ∃ v ∈ SetLike.coe '' B, x ∈ v ∧ v ⊆ sU
state_after: case mpr.refine_2.intro.intro
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
B : Set (Opens α)
h : ∀ {U : Opens α} {x : α}, x ∈ U → ∃ U' ∈ B, x ∈ U' ∧ U' ≤ U
x : α
sU : Set α
hx : x ∈ sU
hsU : IsOpen sU
V : Opens α
hV : V ∈ B
H : x ∈ V ∧ V ≤ { carrier := sU, is_open' := hsU }
⊢ ∃ v ∈ SetLike.coe '' B, x ∈ v ∧ v ⊆ sU

tactic: exact ⟨V, ⟨V, hV, rfl⟩, H⟩
state_before: case mpr.refine_2.intro.intro
ι : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
B : Set (Opens α)
h : ∀ {U : Opens α} {x : α}, x ∈ U → ∃ U' ∈ B, x ∈ U' ∧ U' ≤ U
x : α
sU : Set α
hx : x ∈ sU
hsU : IsOpen sU
V : Opens α
hV : V ∈ B
H : x ∈ V ∧ V ≤ { carrier := sU, is_open' := hsU }
⊢ ∃ v ∈ SetLike.coe '' B, x ∈ v ∧ v ⊆ sU
state_after: no goals