l3lab/ntp-mathlib huggingface dataset:
/-- If a group is endowed with a topological structure coming from a group filter basis then it's a
topological group. -/
@[to_additive "If a group is endowed with a topological structure coming from a group filter basis
then it's a topological group."]
instance (priority := 100) isTopologicalGroup (B : GroupFilterBasis G) :
    @TopologicalGroup G B.topology _ := by
  letI := B.topology
  have basis := B.nhds_one_hasBasis
  have basis' := basis.prod basis
  refine' TopologicalGroup.of_nhds_one _ _ _ _
  · rw [basis'.tendsto_iff basis]
    suffices ∀ U ∈ B, ∃ V W, (V ∈ B ∧ W ∈ B) ∧ ∀ a b, a ∈ V → b ∈ W → a * b ∈ U by simpa
    intro U U_in
    rcases mul U_in with ⟨V, V_in, hV⟩
    refine' ⟨V, V, ⟨V_in, V_in⟩, _⟩
    intro a b a_in b_in
    exact hV ⟨a, b, a_in, b_in, rfl⟩
  · rw [basis.tendsto_iff basis]
    intro U U_in
    simpa using inv U_in
  · intro x₀
    rw [nhds_eq, nhds_one_eq]
    rfl
  · intro x₀
    rw [basis.tendsto_iff basis]
    intro U U_in
    

Proof state:
case refine'_4
G : Type u
inst✝ : Group G
B✝ B : GroupFilterBasis G
this : TopologicalSpace G := topology B
basis : HasBasis (𝓝 1) (fun V => V ∈ B) id
basis' : HasBasis (𝓝 1 ×ˢ 𝓝 1) (fun i => i.1 ∈ B ∧ i.2 ∈ B) fun i => id i.1 ×ˢ id i.2
x₀ : G
U : Set G
U_in : U ∈ B
⊢ ∃ ia ∈ B, ∀ x ∈ id ia, x₀ * x * x₀⁻¹ ∈ id U

Next tactic:
exact conj x₀ U_in
/-- If a ring is endowed with a topological structure coming from
a ring filter basis then it's a topological ring. -/
instance (priority := 100) isTopologicalRing {R : Type u} [Ring R] (B : RingFilterBasis R) :
    @TopologicalRing R B.topology _ := by
  

Proof state:
R✝ : Type u
inst✝¹ : Ring R✝
B✝ : RingFilterBasis R✝
R : Type u
inst✝ : Ring R
B : RingFilterBasis R
⊢ TopologicalRing R

Next tactic:
let B' := B.toAddGroupFilterBasis
/-- If a ring is endowed with a topological structure coming from
a ring filter basis then it's a topological ring. -/
instance (priority := 100) isTopologicalRing {R : Type u} [Ring R] (B : RingFilterBasis R) :
    @TopologicalRing R B.topology _ := by
  let B' := B.toAddGroupFilterBasis
  

Proof state:
R✝ : Type u
inst✝¹ : Ring R✝
B✝ : RingFilterBasis R✝
R : Type u
inst✝ : Ring R
B : RingFilterBasis R
B' : AddGroupFilterBasis R := toAddGroupFilterBasis
⊢ TopologicalRing R

Next tactic:
letI := B'.topology
/-- If a ring is endowed with a topological structure coming from
a ring filter basis then it's a topological ring. -/
instance (priority := 100) isTopologicalRing {R : Type u} [Ring R] (B : RingFilterBasis R) :
    @TopologicalRing R B.topology _ := by
  let B' := B.toAddGroupFilterBasis
  letI := B'.topology
  

Proof state:
R✝ : Type u
inst✝¹ : Ring R✝
B✝ : RingFilterBasis R✝
R : Type u
inst✝ : Ring R
B : RingFilterBasis R
B' : AddGroupFilterBasis R := toAddGroupFilterBasis
this : TopologicalSpace R := AddGroupFilterBasis.topology B'
⊢ TopologicalRing R

Next tactic:
have basis := B'.nhds_zero_hasBasis
/-- If a ring is endowed with a topological structure coming from
a ring filter basis then it's a topological ring. -/
instance (priority := 100) isTopologicalRing {R : Type u} [Ring R] (B : RingFilterBasis R) :
    @TopologicalRing R B.topology _ := by
  let B' := B.toAddGroupFilterBasis
  letI := B'.topology
  have basis := B'.nhds_zero_hasBasis
  

Proof state:
R✝ : Type u
inst✝¹ : Ring R✝
B✝ : RingFilterBasis R✝
R : Type u
inst✝ : Ring R
B : RingFilterBasis R
B' : AddGroupFilterBasis R := toAddGroupFilterBasis
this : TopologicalSpace R := AddGroupFilterBasis.topology B'
basis : HasBasis (𝓝 0) (fun V => V ∈ B') id
⊢ TopologicalRing R

Next tactic:
have basis' := basis.prod basis
/-- If a ring is endowed with a topological structure coming from
a ring filter basis then it's a topological ring. -/
instance (priority := 100) isTopologicalRing {R : Type u} [Ring R] (B : RingFilterBasis R) :
    @TopologicalRing R B.topology _ := by
  let B' := B.toAddGroupFilterBasis
  letI := B'.topology
  have basis := B'.nhds_zero_hasBasis
  have basis' := basis.prod basis
  

Proof state:
R✝ : Type u
inst✝¹ : Ring R✝
B✝ : RingFilterBasis R✝
R : Type u
inst✝ : Ring R
B : RingFilterBasis R
B' : AddGroupFilterBasis R := toAddGroupFilterBasis
this : TopologicalSpace R := AddGroupFilterBasis.topology B'
basis : HasBasis (𝓝 0) (fun V => V ∈ B') id
basis' : HasBasis (𝓝 0 ×ˢ 𝓝 0) (fun i => i.1 ∈ B' ∧ i.2 ∈ B') fun i => id i.1 ×ˢ id i.2
⊢ TopologicalRing R

Next tactic:
haveI := B'.isTopologicalAddGroup
/-- If a ring is endowed with a topological structure coming from
a ring filter basis then it's a topological ring. -/
instance (priority := 100) isTopologicalRing {R : Type u} [Ring R] (B : RingFilterBasis R) :
    @TopologicalRing R B.topology _ := by
  let B' := B.toAddGroupFilterBasis
  letI := B'.topology
  have basis := B'.nhds_zero_hasBasis
  have basis' := basis.prod basis
  haveI := B'.isTopologicalAddGroup
  

Proof state:
R✝ : Type u
inst✝¹ : Ring R✝
B✝ : RingFilterBasis R✝
R : Type u
inst✝ : Ring R
B : RingFilterBasis R
B' : AddGroupFilterBasis R := toAddGroupFilterBasis
this✝ : TopologicalSpace R := AddGroupFilterBasis.topology B'
basis : HasBasis (𝓝 0) (fun V => V ∈ B') id
basis' : HasBasis (𝓝 0 ×ˢ 𝓝 0) (fun i => i.1 ∈ B' ∧ i.2 ∈ B') fun i => id i.1 ×ˢ id i.2
this : TopologicalAddGroup R
⊢ TopologicalRing R

Next tactic:
apply TopologicalRing.of_addGroup_of_nhds_zero
/-- If a ring is endowed with a topological structure coming from
a ring filter basis then it's a topological ring. -/
instance (priority := 100) isTopologicalRing {R : Type u} [Ring R] (B : RingFilterBasis R) :
    @TopologicalRing R B.topology _ := by
  let B' := B.toAddGroupFilterBasis
  letI := B'.topology
  have basis := B'.nhds_zero_hasBasis
  have basis' := basis.prod basis
  haveI := B'.isTopologicalAddGroup
  apply TopologicalRing.of_addGroup_of_nhds_zero
  · 

Proof state:
case hmul
R✝ : Type u
inst✝¹ : Ring R✝
B✝ : RingFilterBasis R✝
R : Type u
inst✝ : Ring R
B : RingFilterBasis R
B' : AddGroupFilterBasis R := toAddGroupFilterBasis
this✝ : TopologicalSpace R := AddGroupFilterBasis.topology B'
basis : HasBasis (𝓝 0) (fun V => V ∈ B') id
basis' : HasBasis (𝓝 0 ×ˢ 𝓝 0) (fun i => i.1 ∈ B' ∧ i.2 ∈ B') fun i => id i.1 ×ˢ id i.2
this : TopologicalAddGroup R
⊢ Tendsto (uncurry fun x x_1 => x * x_1) (𝓝 0 ×ˢ 𝓝 0) (𝓝 0)

Next tactic:
rw [basis'.tendsto_iff basis]
/-- If a ring is endowed with a topological structure coming from
a ring filter basis then it's a topological ring. -/
instance (priority := 100) isTopologicalRing {R : Type u} [Ring R] (B : RingFilterBasis R) :
    @TopologicalRing R B.topology _ := by
  let B' := B.toAddGroupFilterBasis
  letI := B'.topology
  have basis := B'.nhds_zero_hasBasis
  have basis' := basis.prod basis
  haveI := B'.isTopologicalAddGroup
  apply TopologicalRing.of_addGroup_of_nhds_zero
  · rw [basis'.tendsto_iff basis]
    

Proof state:
case hmul
R✝ : Type u
inst✝¹ : Ring R✝
B✝ : RingFilterBasis R✝
R : Type u
inst✝ : Ring R
B : RingFilterBasis R
B' : AddGroupFilterBasis R := toAddGroupFilterBasis
this✝ : TopologicalSpace R := AddGroupFilterBasis.topology B'
basis : HasBasis (𝓝 0) (fun V => V ∈ B') id
basis' : HasBasis (𝓝 0 ×ˢ 𝓝 0) (fun i => i.1 ∈ B' ∧ i.2 ∈ B') fun i => id i.1 ×ˢ id i.2
this : TopologicalAddGroup R
⊢ ∀ ib ∈ B', ∃ ia, (ia.1 ∈ B' ∧ ia.2 ∈ B') ∧ ∀ x ∈ id ia.1 ×ˢ id ia.2, uncurry (fun x x_1 => x * x_1) x ∈ id ib

Next tactic:
suffices ∀ U ∈ B', ∃ V W, (V ∈ B' ∧ W ∈ B') ∧ ∀ a b, a ∈ V → b ∈ W → a * b ∈ U by simpa
/-- If a ring is endowed with a topological structure coming from
a ring filter basis then it's a topological ring. -/
instance (priority := 100) isTopologicalRing {R : Type u} [Ring R] (B : RingFilterBasis R) :
    @TopologicalRing R B.topology _ := by
  let B' := B.toAddGroupFilterBasis
  letI := B'.topology
  have basis := B'.nhds_zero_hasBasis
  have basis' := basis.prod basis
  haveI := B'.isTopologicalAddGroup
  apply TopologicalRing.of_addGroup_of_nhds_zero
  · rw [basis'.tendsto_iff basis]
    suffices ∀ U ∈ B', ∃ V W, (V ∈ B' ∧ W ∈ B') ∧ ∀ a b, a ∈ V → b ∈ W → a * b ∈ U by 

Proof state:
R✝ : Type u
inst✝¹ : Ring R✝
B✝ : RingFilterBasis R✝
R : Type u
inst✝ : Ring R
B : RingFilterBasis R
B' : AddGroupFilterBasis R := toAddGroupFilterBasis
this✝¹ : TopologicalSpace R := AddGroupFilterBasis.topology B'
basis : HasBasis (𝓝 0) (fun V => V ∈ B') id
basis' : HasBasis (𝓝 0 ×ˢ 𝓝 0) (fun i => i.1 ∈ B' ∧ i.2 ∈ B') fun i => id i.1 ×ˢ id i.2
this✝ : TopologicalAddGroup R
this : ∀ U ∈ B', ∃ V W, (V ∈ B' ∧ W ∈ B') ∧ ∀ (a b : R), a ∈ V → b ∈ W → a * b ∈ U
⊢ ∀ ib ∈ B', ∃ ia, (ia.1 ∈ B' ∧ ia.2 ∈ B') ∧ ∀ x ∈ id ia.1 ×ˢ id ia.2, uncurry (fun x x_1 => x * x_1) x ∈ id ib

Next tactic:
simpa
/-- If a ring is endowed with a topological structure coming from
a ring filter basis then it's a topological ring. -/
instance (priority := 100) isTopologicalRing {R : Type u} [Ring R] (B : RingFilterBasis R) :
    @TopologicalRing R B.topology _ := by
  let B' := B.toAddGroupFilterBasis
  letI := B'.topology
  have basis := B'.nhds_zero_hasBasis
  have basis' := basis.prod basis
  haveI := B'.isTopologicalAddGroup
  apply TopologicalRing.of_addGroup_of_nhds_zero
  · rw [basis'.tendsto_iff basis]
    suffices ∀ U ∈ B', ∃ V W, (V ∈ B' ∧ W ∈ B') ∧ ∀ a b, a ∈ V → b ∈ W → a * b ∈ U by simpa
    

Proof state:
case hmul
R✝ : Type u
inst✝¹ : Ring R✝
B✝ : RingFilterBasis R✝
R : Type u
inst✝ : Ring R
B : RingFilterBasis R
B' : AddGroupFilterBasis R := toAddGroupFilterBasis
this✝ : TopologicalSpace R := AddGroupFilterBasis.topology B'
basis : HasBasis (𝓝 0) (fun V => V ∈ B') id
basis' : HasBasis (𝓝 0 ×ˢ 𝓝 0) (fun i => i.1 ∈ B' ∧ i.2 ∈ B') fun i => id i.1 ×ˢ id i.2
this : TopologicalAddGroup R
⊢ ∀ U ∈ B', ∃ V W, (V ∈ B' ∧ W ∈ B') ∧ ∀ (a b : R), a ∈ V → b ∈ W → a * b ∈ U

Next tactic:
intro U U_in
/-- If a ring is endowed with a topological structure coming from
a ring filter basis then it's a topological ring. -/
instance (priority := 100) isTopologicalRing {R : Type u} [Ring R] (B : RingFilterBasis R) :
    @TopologicalRing R B.topology _ := by
  let B' := B.toAddGroupFilterBasis
  letI := B'.topology
  have basis := B'.nhds_zero_hasBasis
  have basis' := basis.prod basis
  haveI := B'.isTopologicalAddGroup
  apply TopologicalRing.of_addGroup_of_nhds_zero
  · rw [basis'.tendsto_iff basis]
    suffices ∀ U ∈ B', ∃ V W, (V ∈ B' ∧ W ∈ B') ∧ ∀ a b, a ∈ V → b ∈ W → a * b ∈ U by simpa
    intro U U_in
    

Proof state:
case hmul
R✝ : Type u
inst✝¹ : Ring R✝
B✝ : RingFilterBasis R✝
R : Type u
inst✝ : Ring R
B : RingFilterBasis R
B' : AddGroupFilterBasis R := toAddGroupFilterBasis
this✝ : TopologicalSpace R := AddGroupFilterBasis.topology B'
basis : HasBasis (𝓝 0) (fun V => V ∈ B') id
basis' : HasBasis (𝓝 0 ×ˢ 𝓝 0) (fun i => i.1 ∈ B' ∧ i.2 ∈ B') fun i => id i.1 ×ˢ id i.2
this : TopologicalAddGroup R
U : Set R
U_in : U ∈ B'
⊢ ∃ V W, (V ∈ B' ∧ W ∈ B') ∧ ∀ (a b : R), a ∈ V → b ∈ W → a * b ∈ U

Next tactic:
rcases B.mul U_in with ⟨V, V_in, hV⟩
/-- If a ring is endowed with a topological structure coming from
a ring filter basis then it's a topological ring. -/
instance (priority := 100) isTopologicalRing {R : Type u} [Ring R] (B : RingFilterBasis R) :
    @TopologicalRing R B.topology _ := by
  let B' := B.toAddGroupFilterBasis
  letI := B'.topology
  have basis := B'.nhds_zero_hasBasis
  have basis' := basis.prod basis
  haveI := B'.isTopologicalAddGroup
  apply TopologicalRing.of_addGroup_of_nhds_zero
  · rw [basis'.tendsto_iff basis]
    suffices ∀ U ∈ B', ∃ V W, (V ∈ B' ∧ W ∈ B') ∧ ∀ a b, a ∈ V → b ∈ W → a * b ∈ U by simpa
    intro U U_in
    rcases B.mul U_in with ⟨V, V_in, hV⟩
    

Proof state:
case hmul.intro.intro
R✝ : Type u
inst✝¹ : Ring R✝
B✝ : RingFilterBasis R✝
R : Type u
inst✝ : Ring R
B : RingFilterBasis R
B' : AddGroupFilterBasis R := toAddGroupFilterBasis
this✝ : TopologicalSpace R := AddGroupFilterBasis.topology B'
basis : HasBasis (𝓝 0) (fun V => V ∈ B') id
basis' : HasBasis (𝓝 0 ×ˢ 𝓝 0) (fun i => i.1 ∈ B' ∧ i.2 ∈ B') fun i => id i.1 ×ˢ id i.2
this : TopologicalAddGroup R
U : Set R
U_in : U ∈ B'
V : Set R
V_in : V ∈ B
hV : V * V ⊆ U
⊢ ∃ V W, (V ∈ B' ∧ W ∈ B') ∧ ∀ (a b : R), a ∈ V → b ∈ W → a * b ∈ U

Next tactic:
refine' ⟨V, V, ⟨V_in, V_in⟩, _⟩
/-- If a ring is endowed with a topological structure coming from
a ring filter basis then it's a topological ring. -/
instance (priority := 100) isTopologicalRing {R : Type u} [Ring R] (B : RingFilterBasis R) :
    @TopologicalRing R B.topology _ := by
  let B' := B.toAddGroupFilterBasis
  letI := B'.topology
  have basis := B'.nhds_zero_hasBasis
  have basis' := basis.prod basis
  haveI := B'.isTopologicalAddGroup
  apply TopologicalRing.of_addGroup_of_nhds_zero
  · rw [basis'.tendsto_iff basis]
    suffices ∀ U ∈ B', ∃ V W, (V ∈ B' ∧ W ∈ B') ∧ ∀ a b, a ∈ V → b ∈ W → a * b ∈ U by simpa
    intro U U_in
    rcases B.mul U_in with ⟨V, V_in, hV⟩
    refine' ⟨V, V, ⟨V_in, V_in⟩, _⟩
    

Proof state:
case hmul.intro.intro
R✝ : Type u
inst✝¹ : Ring R✝
B✝ : RingFilterBasis R✝
R : Type u
inst✝ : Ring R
B : RingFilterBasis R
B' : AddGroupFilterBasis R := toAddGroupFilterBasis
this✝ : TopologicalSpace R := AddGroupFilterBasis.topology B'
basis : HasBasis (𝓝 0) (fun V => V ∈ B') id
basis' : HasBasis (𝓝 0 ×ˢ 𝓝 0) (fun i => i.1 ∈ B' ∧ i.2 ∈ B') fun i => id i.1 ×ˢ id i.2
this : TopologicalAddGroup R
U : Set R
U_in : U ∈ B'
V : Set R
V_in : V ∈ B
hV : V * V ⊆ U
⊢ ∀ (a b : R), a ∈ V → b ∈ V → a * b ∈ U

Next tactic:
intro a b a_in b_in
/-- If a ring is endowed with a topological structure coming from
a ring filter basis then it's a topological ring. -/
instance (priority := 100) isTopologicalRing {R : Type u} [Ring R] (B : RingFilterBasis R) :
    @TopologicalRing R B.topology _ := by
  let B' := B.toAddGroupFilterBasis
  letI := B'.topology
  have basis := B'.nhds_zero_hasBasis
  have basis' := basis.prod basis
  haveI := B'.isTopologicalAddGroup
  apply TopologicalRing.of_addGroup_of_nhds_zero
  · rw [basis'.tendsto_iff basis]
    suffices ∀ U ∈ B', ∃ V W, (V ∈ B' ∧ W ∈ B') ∧ ∀ a b, a ∈ V → b ∈ W → a * b ∈ U by simpa
    intro U U_in
    rcases B.mul U_in with ⟨V, V_in, hV⟩
    refine' ⟨V, V, ⟨V_in, V_in⟩, _⟩
    intro a b a_in b_in
    

Proof state:
case hmul.intro.intro
R✝ : Type u
inst✝¹ : Ring R✝
B✝ : RingFilterBasis R✝
R : Type u
inst✝ : Ring R
B : RingFilterBasis R
B' : AddGroupFilterBasis R := toAddGroupFilterBasis
this✝ : TopologicalSpace R := AddGroupFilterBasis.topology B'
basis : HasBasis (𝓝 0) (fun V => V ∈ B') id
basis' : HasBasis (𝓝 0 ×ˢ 𝓝 0) (fun i => i.1 ∈ B' ∧ i.2 ∈ B') fun i => id i.1 ×ˢ id i.2
this : TopologicalAddGroup R
U : Set R
U_in : U ∈ B'
V : Set R
V_in : V ∈ B
hV : V * V ⊆ U
a b : R
a_in : a ∈ V
b_in : b ∈ V
⊢ a * b ∈ U

Next tactic:
exact hV ⟨a, b, a_in, b_in, rfl⟩
/-- If a ring is endowed with a topological structure coming from
a ring filter basis then it's a topological ring. -/
instance (priority := 100) isTopologicalRing {R : Type u} [Ring R] (B : RingFilterBasis R) :
    @TopologicalRing R B.topology _ := by
  let B' := B.toAddGroupFilterBasis
  letI := B'.topology
  have basis := B'.nhds_zero_hasBasis
  have basis' := basis.prod basis
  haveI := B'.isTopologicalAddGroup
  apply TopologicalRing.of_addGroup_of_nhds_zero
  · rw [basis'.tendsto_iff basis]
    suffices ∀ U ∈ B', ∃ V W, (V ∈ B' ∧ W ∈ B') ∧ ∀ a b, a ∈ V → b ∈ W → a * b ∈ U by simpa
    intro U U_in
    rcases B.mul U_in with ⟨V, V_in, hV⟩
    refine' ⟨V, V, ⟨V_in, V_in⟩, _⟩
    intro a b a_in b_in
    exact hV ⟨a, b, a_in, b_in, rfl⟩
  · 

Proof state:
case hmul_left
R✝ : Type u
inst✝¹ : Ring R✝
B✝ : RingFilterBasis R✝
R : Type u
inst✝ : Ring R
B : RingFilterBasis R
B' : AddGroupFilterBasis R := toAddGroupFilterBasis
this✝ : TopologicalSpace R := AddGroupFilterBasis.topology B'
basis : HasBasis (𝓝 0) (fun V => V ∈ B') id
basis' : HasBasis (𝓝 0 ×ˢ 𝓝 0) (fun i => i.1 ∈ B' ∧ i.2 ∈ B') fun i => id i.1 ×ˢ id i.2
this : TopologicalAddGroup R
⊢ ∀ (x₀ : R), Tendsto (fun x => x₀ * x) (𝓝 0) (𝓝 0)

Next tactic:
intro x₀
/-- If a ring is endowed with a topological structure coming from
a ring filter basis then it's a topological ring. -/
instance (priority := 100) isTopologicalRing {R : Type u} [Ring R] (B : RingFilterBasis R) :
    @TopologicalRing R B.topology _ := by
  let B' := B.toAddGroupFilterBasis
  letI := B'.topology
  have basis := B'.nhds_zero_hasBasis
  have basis' := basis.prod basis
  haveI := B'.isTopologicalAddGroup
  apply TopologicalRing.of_addGroup_of_nhds_zero
  · rw [basis'.tendsto_iff basis]
    suffices ∀ U ∈ B', ∃ V W, (V ∈ B' ∧ W ∈ B') ∧ ∀ a b, a ∈ V → b ∈ W → a * b ∈ U by simpa
    intro U U_in
    rcases B.mul U_in with ⟨V, V_in, hV⟩
    refine' ⟨V, V, ⟨V_in, V_in⟩, _⟩
    intro a b a_in b_in
    exact hV ⟨a, b, a_in, b_in, rfl⟩
  · intro x₀
    

Proof state:
case hmul_left
R✝ : Type u
inst✝¹ : Ring R✝
B✝ : RingFilterBasis R✝
R : Type u
inst✝ : Ring R
B : RingFilterBasis R
B' : AddGroupFilterBasis R := toAddGroupFilterBasis
this✝ : TopologicalSpace R := AddGroupFilterBasis.topology B'
basis : HasBasis (𝓝 0) (fun V => V ∈ B') id
basis' : HasBasis (𝓝 0 ×ˢ 𝓝 0) (fun i => i.1 ∈ B' ∧ i.2 ∈ B') fun i => id i.1 ×ˢ id i.2
this : TopologicalAddGroup R
x₀ : R
⊢ Tendsto (fun x => x₀ * x) (𝓝 0) (𝓝 0)

Next tactic:
rw [basis.tendsto_iff basis]
/-- If a ring is endowed with a topological structure coming from
a ring filter basis then it's a topological ring. -/
instance (priority := 100) isTopologicalRing {R : Type u} [Ring R] (B : RingFilterBasis R) :
    @TopologicalRing R B.topology _ := by
  let B' := B.toAddGroupFilterBasis
  letI := B'.topology
  have basis := B'.nhds_zero_hasBasis
  have basis' := basis.prod basis
  haveI := B'.isTopologicalAddGroup
  apply TopologicalRing.of_addGroup_of_nhds_zero
  · rw [basis'.tendsto_iff basis]
    suffices ∀ U ∈ B', ∃ V W, (V ∈ B' ∧ W ∈ B') ∧ ∀ a b, a ∈ V → b ∈ W → a * b ∈ U by simpa
    intro U U_in
    rcases B.mul U_in with ⟨V, V_in, hV⟩
    refine' ⟨V, V, ⟨V_in, V_in⟩, _⟩
    intro a b a_in b_in
    exact hV ⟨a, b, a_in, b_in, rfl⟩
  · intro x₀
    rw [basis.tendsto_iff basis]
    

Proof state:
case hmul_left
R✝ : Type u
inst✝¹ : Ring R✝
B✝ : RingFilterBasis R✝
R : Type u
inst✝ : Ring R
B : RingFilterBasis R
B' : AddGroupFilterBasis R := toAddGroupFilterBasis
this✝ : TopologicalSpace R := AddGroupFilterBasis.topology B'
basis : HasBasis (𝓝 0) (fun V => V ∈ B') id
basis' : HasBasis (𝓝 0 ×ˢ 𝓝 0) (fun i => i.1 ∈ B' ∧ i.2 ∈ B') fun i => id i.1 ×ˢ id i.2
this : TopologicalAddGroup R
x₀ : R
⊢ ∀ ib ∈ B', ∃ ia ∈ B', ∀ x ∈ id ia, x₀ * x ∈ id ib

Next tactic:
intro U
/-- If a ring is endowed with a topological structure coming from
a ring filter basis then it's a topological ring. -/
instance (priority := 100) isTopologicalRing {R : Type u} [Ring R] (B : RingFilterBasis R) :
    @TopologicalRing R B.topology _ := by
  let B' := B.toAddGroupFilterBasis
  letI := B'.topology
  have basis := B'.nhds_zero_hasBasis
  have basis' := basis.prod basis
  haveI := B'.isTopologicalAddGroup
  apply TopologicalRing.of_addGroup_of_nhds_zero
  · rw [basis'.tendsto_iff basis]
    suffices ∀ U ∈ B', ∃ V W, (V ∈ B' ∧ W ∈ B') ∧ ∀ a b, a ∈ V → b ∈ W → a * b ∈ U by simpa
    intro U U_in
    rcases B.mul U_in with ⟨V, V_in, hV⟩
    refine' ⟨V, V, ⟨V_in, V_in⟩, _⟩
    intro a b a_in b_in
    exact hV ⟨a, b, a_in, b_in, rfl⟩
  · intro x₀
    rw [basis.tendsto_iff basis]
    intro U
    

Proof state:
case hmul_left
R✝ : Type u
inst✝¹ : Ring R✝
B✝ : RingFilterBasis R✝
R : Type u
inst✝ : Ring R
B : RingFilterBasis R
B' : AddGroupFilterBasis R := toAddGroupFilterBasis
this✝ : TopologicalSpace R := AddGroupFilterBasis.topology B'
basis : HasBasis (𝓝 0) (fun V => V ∈ B') id
basis' : HasBasis (𝓝 0 ×ˢ 𝓝 0) (fun i => i.1 ∈ B' ∧ i.2 ∈ B') fun i => id i.1 ×ˢ id i.2
this : TopologicalAddGroup R
x₀ : R
U : Set R
⊢ U ∈ B' → ∃ ia ∈ B', ∀ x ∈ id ia, x₀ * x ∈ id U

Next tactic:
simpa using B.mul_left x₀
/-- If a ring is endowed with a topological structure coming from
a ring filter basis then it's a topological ring. -/
instance (priority := 100) isTopologicalRing {R : Type u} [Ring R] (B : RingFilterBasis R) :
    @TopologicalRing R B.topology _ := by
  let B' := B.toAddGroupFilterBasis
  letI := B'.topology
  have basis := B'.nhds_zero_hasBasis
  have basis' := basis.prod basis
  haveI := B'.isTopologicalAddGroup
  apply TopologicalRing.of_addGroup_of_nhds_zero
  · rw [basis'.tendsto_iff basis]
    suffices ∀ U ∈ B', ∃ V W, (V ∈ B' ∧ W ∈ B') ∧ ∀ a b, a ∈ V → b ∈ W → a * b ∈ U by simpa
    intro U U_in
    rcases B.mul U_in with ⟨V, V_in, hV⟩
    refine' ⟨V, V, ⟨V_in, V_in⟩, _⟩
    intro a b a_in b_in
    exact hV ⟨a, b, a_in, b_in, rfl⟩
  · intro x₀
    rw [basis.tendsto_iff basis]
    intro U
    simpa using B.mul_left x₀
  · 

Proof state:
case hmul_right
R✝ : Type u
inst✝¹ : Ring R✝
B✝ : RingFilterBasis R✝
R : Type u
inst✝ : Ring R
B : RingFilterBasis R
B' : AddGroupFilterBasis R := toAddGroupFilterBasis
this✝ : TopologicalSpace R := AddGroupFilterBasis.topology B'
basis : HasBasis (𝓝 0) (fun V => V ∈ B') id
basis' : HasBasis (𝓝 0 ×ˢ 𝓝 0) (fun i => i.1 ∈ B' ∧ i.2 ∈ B') fun i => id i.1 ×ˢ id i.2
this : TopologicalAddGroup R
⊢ ∀ (x₀ : R), Tendsto (fun x => x * x₀) (𝓝 0) (𝓝 0)

Next tactic:
intro x₀
/-- If a ring is endowed with a topological structure coming from
a ring filter basis then it's a topological ring. -/
instance (priority := 100) isTopologicalRing {R : Type u} [Ring R] (B : RingFilterBasis R) :
    @TopologicalRing R B.topology _ := by
  let B' := B.toAddGroupFilterBasis
  letI := B'.topology
  have basis := B'.nhds_zero_hasBasis
  have basis' := basis.prod basis
  haveI := B'.isTopologicalAddGroup
  apply TopologicalRing.of_addGroup_of_nhds_zero
  · rw [basis'.tendsto_iff basis]
    suffices ∀ U ∈ B', ∃ V W, (V ∈ B' ∧ W ∈ B') ∧ ∀ a b, a ∈ V → b ∈ W → a * b ∈ U by simpa
    intro U U_in
    rcases B.mul U_in with ⟨V, V_in, hV⟩
    refine' ⟨V, V, ⟨V_in, V_in⟩, _⟩
    intro a b a_in b_in
    exact hV ⟨a, b, a_in, b_in, rfl⟩
  · intro x₀
    rw [basis.tendsto_iff basis]
    intro U
    simpa using B.mul_left x₀
  · intro x₀
    

Proof state:
case hmul_right
R✝ : Type u
inst✝¹ : Ring R✝
B✝ : RingFilterBasis R✝
R : Type u
inst✝ : Ring R
B : RingFilterBasis R
B' : AddGroupFilterBasis R := toAddGroupFilterBasis
this✝ : TopologicalSpace R := AddGroupFilterBasis.topology B'
basis : HasBasis (𝓝 0) (fun V => V ∈ B') id
basis' : HasBasis (𝓝 0 ×ˢ 𝓝 0) (fun i => i.1 ∈ B' ∧ i.2 ∈ B') fun i => id i.1 ×ˢ id i.2
this : TopologicalAddGroup R
x₀ : R
⊢ Tendsto (fun x => x * x₀) (𝓝 0) (𝓝 0)

Next tactic:
rw [basis.tendsto_iff basis]
/-- If a ring is endowed with a topological structure coming from
a ring filter basis then it's a topological ring. -/
instance (priority := 100) isTopologicalRing {R : Type u} [Ring R] (B : RingFilterBasis R) :
    @TopologicalRing R B.topology _ := by
  let B' := B.toAddGroupFilterBasis
  letI := B'.topology
  have basis := B'.nhds_zero_hasBasis
  have basis' := basis.prod basis
  haveI := B'.isTopologicalAddGroup
  apply TopologicalRing.of_addGroup_of_nhds_zero
  · rw [basis'.tendsto_iff basis]
    suffices ∀ U ∈ B', ∃ V W, (V ∈ B' ∧ W ∈ B') ∧ ∀ a b, a ∈ V → b ∈ W → a * b ∈ U by simpa
    intro U U_in
    rcases B.mul U_in with ⟨V, V_in, hV⟩
    refine' ⟨V, V, ⟨V_in, V_in⟩, _⟩
    intro a b a_in b_in
    exact hV ⟨a, b, a_in, b_in, rfl⟩
  · intro x₀
    rw [basis.tendsto_iff basis]
    intro U
    simpa using B.mul_left x₀
  · intro x₀
    rw [basis.tendsto_iff basis]
    

Proof state:
case hmul_right
R✝ : Type u
inst✝¹ : Ring R✝
B✝ : RingFilterBasis R✝
R : Type u
inst✝ : Ring R
B : RingFilterBasis R
B' : AddGroupFilterBasis R := toAddGroupFilterBasis
this✝ : TopologicalSpace R := AddGroupFilterBasis.topology B'
basis : HasBasis (𝓝 0) (fun V => V ∈ B') id
basis' : HasBasis (𝓝 0 ×ˢ 𝓝 0) (fun i => i.1 ∈ B' ∧ i.2 ∈ B') fun i => id i.1 ×ˢ id i.2
this : TopologicalAddGroup R
x₀ : R
⊢ ∀ ib ∈ B', ∃ ia ∈ B', ∀ x ∈ id ia, x * x₀ ∈ id ib

Next tactic:
intro U