l3lab/ntp-mathlib huggingface dataset:
/-- Every subgroups basis on a ring leads to a ring filter basis. -/
def toRingFilterBasis [Nonempty ι] {B : ι → AddSubgroup A} (hB : RingSubgroupsBasis B) :
    RingFilterBasis A where
  sets := { U | ∃ i, U = B i }
  nonempty := by
    inhabit ι
    exact ⟨B default, default, rfl⟩
  inter_sets := by
    rintro _ _ ⟨i, rfl⟩ ⟨j, rfl⟩
    cases' hB.inter i j with k hk
    use B k
    constructor
    · use k
    · exact hk
  zero' := by
    rintro _ ⟨i, rfl⟩
    exact (B i).zero_mem
  add' := by
    rintro _ ⟨i, rfl⟩
    use B i
    constructor
    · use i
    · rintro x ⟨y, z, y_in, z_in, rfl⟩
      exact (B i).add_mem y_in z_in
  neg' := by
    rintro _ ⟨i, rfl⟩
    use B i
    constructor
    · use i
    · intro x x_in
      exact (B i).neg_mem x_in
  conj' := by
    rintro x₀ _ ⟨i, rfl⟩
    use B i
    constructor
    · use i
    · simp
  mul' := by
    rintro _ ⟨i, rfl⟩
    cases' hB.mul i with k hk
    use B k
    constructor
    · use k
    · exact hk
  mul_left' := by
    rintro x₀ _ ⟨i, rfl⟩
    cases' hB.leftMul x₀ i with k hk
    use B k
    constructor
    · use k
    · exact hk
  mul_right' := by
    rintro x₀ _ ⟨i, rfl⟩
    

Proof state:
case intro
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
x₀ : A
i : ι
⊢ ∃ V ∈ AddGroupFilterBasis.toFilterBasis.sets, V ⊆ (fun x => x * x₀) ⁻¹' ↑(B i)

Next tactic:
cases' hB.rightMul x₀ i with k hk
/-- Every subgroups basis on a ring leads to a ring filter basis. -/
def toRingFilterBasis [Nonempty ι] {B : ι → AddSubgroup A} (hB : RingSubgroupsBasis B) :
    RingFilterBasis A where
  sets := { U | ∃ i, U = B i }
  nonempty := by
    inhabit ι
    exact ⟨B default, default, rfl⟩
  inter_sets := by
    rintro _ _ ⟨i, rfl⟩ ⟨j, rfl⟩
    cases' hB.inter i j with k hk
    use B k
    constructor
    · use k
    · exact hk
  zero' := by
    rintro _ ⟨i, rfl⟩
    exact (B i).zero_mem
  add' := by
    rintro _ ⟨i, rfl⟩
    use B i
    constructor
    · use i
    · rintro x ⟨y, z, y_in, z_in, rfl⟩
      exact (B i).add_mem y_in z_in
  neg' := by
    rintro _ ⟨i, rfl⟩
    use B i
    constructor
    · use i
    · intro x x_in
      exact (B i).neg_mem x_in
  conj' := by
    rintro x₀ _ ⟨i, rfl⟩
    use B i
    constructor
    · use i
    · simp
  mul' := by
    rintro _ ⟨i, rfl⟩
    cases' hB.mul i with k hk
    use B k
    constructor
    · use k
    · exact hk
  mul_left' := by
    rintro x₀ _ ⟨i, rfl⟩
    cases' hB.leftMul x₀ i with k hk
    use B k
    constructor
    · use k
    · exact hk
  mul_right' := by
    rintro x₀ _ ⟨i, rfl⟩
    cases' hB.rightMul x₀ i with k hk
    

Proof state:
case intro.intro
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
x₀ : A
i k : ι
hk : ↑(B k) ⊆ (fun x => x * x₀) ⁻¹' ↑(B i)
⊢ ∃ V ∈ AddGroupFilterBasis.toFilterBasis.sets, V ⊆ (fun x => x * x₀) ⁻¹' ↑(B i)

Next tactic:
use B k
/-- Every subgroups basis on a ring leads to a ring filter basis. -/
def toRingFilterBasis [Nonempty ι] {B : ι → AddSubgroup A} (hB : RingSubgroupsBasis B) :
    RingFilterBasis A where
  sets := { U | ∃ i, U = B i }
  nonempty := by
    inhabit ι
    exact ⟨B default, default, rfl⟩
  inter_sets := by
    rintro _ _ ⟨i, rfl⟩ ⟨j, rfl⟩
    cases' hB.inter i j with k hk
    use B k
    constructor
    · use k
    · exact hk
  zero' := by
    rintro _ ⟨i, rfl⟩
    exact (B i).zero_mem
  add' := by
    rintro _ ⟨i, rfl⟩
    use B i
    constructor
    · use i
    · rintro x ⟨y, z, y_in, z_in, rfl⟩
      exact (B i).add_mem y_in z_in
  neg' := by
    rintro _ ⟨i, rfl⟩
    use B i
    constructor
    · use i
    · intro x x_in
      exact (B i).neg_mem x_in
  conj' := by
    rintro x₀ _ ⟨i, rfl⟩
    use B i
    constructor
    · use i
    · simp
  mul' := by
    rintro _ ⟨i, rfl⟩
    cases' hB.mul i with k hk
    use B k
    constructor
    · use k
    · exact hk
  mul_left' := by
    rintro x₀ _ ⟨i, rfl⟩
    cases' hB.leftMul x₀ i with k hk
    use B k
    constructor
    · use k
    · exact hk
  mul_right' := by
    rintro x₀ _ ⟨i, rfl⟩
    cases' hB.rightMul x₀ i with k hk
    use B k
    

Proof state:
case h
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
x₀ : A
i k : ι
hk : ↑(B k) ⊆ (fun x => x * x₀) ⁻¹' ↑(B i)
⊢ ↑(B k) ∈ AddGroupFilterBasis.toFilterBasis.sets ∧ ↑(B k) ⊆ (fun x => x * x₀) ⁻¹' ↑(B i)

Next tactic:
constructor
/-- Every subgroups basis on a ring leads to a ring filter basis. -/
def toRingFilterBasis [Nonempty ι] {B : ι → AddSubgroup A} (hB : RingSubgroupsBasis B) :
    RingFilterBasis A where
  sets := { U | ∃ i, U = B i }
  nonempty := by
    inhabit ι
    exact ⟨B default, default, rfl⟩
  inter_sets := by
    rintro _ _ ⟨i, rfl⟩ ⟨j, rfl⟩
    cases' hB.inter i j with k hk
    use B k
    constructor
    · use k
    · exact hk
  zero' := by
    rintro _ ⟨i, rfl⟩
    exact (B i).zero_mem
  add' := by
    rintro _ ⟨i, rfl⟩
    use B i
    constructor
    · use i
    · rintro x ⟨y, z, y_in, z_in, rfl⟩
      exact (B i).add_mem y_in z_in
  neg' := by
    rintro _ ⟨i, rfl⟩
    use B i
    constructor
    · use i
    · intro x x_in
      exact (B i).neg_mem x_in
  conj' := by
    rintro x₀ _ ⟨i, rfl⟩
    use B i
    constructor
    · use i
    · simp
  mul' := by
    rintro _ ⟨i, rfl⟩
    cases' hB.mul i with k hk
    use B k
    constructor
    · use k
    · exact hk
  mul_left' := by
    rintro x₀ _ ⟨i, rfl⟩
    cases' hB.leftMul x₀ i with k hk
    use B k
    constructor
    · use k
    · exact hk
  mul_right' := by
    rintro x₀ _ ⟨i, rfl⟩
    cases' hB.rightMul x₀ i with k hk
    use B k
    constructor
    · 

Proof state:
case h.left
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
x₀ : A
i k : ι
hk : ↑(B k) ⊆ (fun x => x * x₀) ⁻¹' ↑(B i)
⊢ ↑(B k) ∈ AddGroupFilterBasis.toFilterBasis.sets

Next tactic:
use k
/-- Every subgroups basis on a ring leads to a ring filter basis. -/
def toRingFilterBasis [Nonempty ι] {B : ι → AddSubgroup A} (hB : RingSubgroupsBasis B) :
    RingFilterBasis A where
  sets := { U | ∃ i, U = B i }
  nonempty := by
    inhabit ι
    exact ⟨B default, default, rfl⟩
  inter_sets := by
    rintro _ _ ⟨i, rfl⟩ ⟨j, rfl⟩
    cases' hB.inter i j with k hk
    use B k
    constructor
    · use k
    · exact hk
  zero' := by
    rintro _ ⟨i, rfl⟩
    exact (B i).zero_mem
  add' := by
    rintro _ ⟨i, rfl⟩
    use B i
    constructor
    · use i
    · rintro x ⟨y, z, y_in, z_in, rfl⟩
      exact (B i).add_mem y_in z_in
  neg' := by
    rintro _ ⟨i, rfl⟩
    use B i
    constructor
    · use i
    · intro x x_in
      exact (B i).neg_mem x_in
  conj' := by
    rintro x₀ _ ⟨i, rfl⟩
    use B i
    constructor
    · use i
    · simp
  mul' := by
    rintro _ ⟨i, rfl⟩
    cases' hB.mul i with k hk
    use B k
    constructor
    · use k
    · exact hk
  mul_left' := by
    rintro x₀ _ ⟨i, rfl⟩
    cases' hB.leftMul x₀ i with k hk
    use B k
    constructor
    · use k
    · exact hk
  mul_right' := by
    rintro x₀ _ ⟨i, rfl⟩
    cases' hB.rightMul x₀ i with k hk
    use B k
    constructor
    · use k
    · 

Proof state:
case h.right
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
x₀ : A
i k : ι
hk : ↑(B k) ⊆ (fun x => x * x₀) ⁻¹' ↑(B i)
⊢ ↑(B k) ⊆ (fun x => x * x₀) ⁻¹' ↑(B i)

Next tactic:
exact hk
theorem hasBasis_nhds_zero : HasBasis (@nhds A hB.topology 0) (fun _ => True) fun i => B i :=
  ⟨by
    

Proof state:
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
⊢ ∀ (t : Set A), t ∈ 𝓝 0 ↔ ∃ i, True ∧ ↑(B i) ⊆ t

Next tactic:
intro s
theorem hasBasis_nhds_zero : HasBasis (@nhds A hB.topology 0) (fun _ => True) fun i => B i :=
  ⟨by
    intro s
    

Proof state:
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
s : Set A
⊢ s ∈ 𝓝 0 ↔ ∃ i, True ∧ ↑(B i) ⊆ s

Next tactic:
rw [hB.toRingFilterBasis.toAddGroupFilterBasis.nhds_zero_hasBasis.mem_iff]
theorem hasBasis_nhds_zero : HasBasis (@nhds A hB.topology 0) (fun _ => True) fun i => B i :=
  ⟨by
    intro s
    rw [hB.toRingFilterBasis.toAddGroupFilterBasis.nhds_zero_hasBasis.mem_iff]
    

Proof state:
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
s : Set A
⊢ (∃ i ∈ RingFilterBasis.toAddGroupFilterBasis, id i ⊆ s) ↔ ∃ i, True ∧ ↑(B i) ⊆ s

Next tactic:
constructor
theorem hasBasis_nhds_zero : HasBasis (@nhds A hB.topology 0) (fun _ => True) fun i => B i :=
  ⟨by
    intro s
    rw [hB.toRingFilterBasis.toAddGroupFilterBasis.nhds_zero_hasBasis.mem_iff]
    constructor
    · 

Proof state:
case mp
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
s : Set A
⊢ (∃ i ∈ RingFilterBasis.toAddGroupFilterBasis, id i ⊆ s) → ∃ i, True ∧ ↑(B i) ⊆ s

Next tactic:
rintro ⟨-, ⟨i, rfl⟩, hi⟩
theorem hasBasis_nhds_zero : HasBasis (@nhds A hB.topology 0) (fun _ => True) fun i => B i :=
  ⟨by
    intro s
    rw [hB.toRingFilterBasis.toAddGroupFilterBasis.nhds_zero_hasBasis.mem_iff]
    constructor
    · rintro ⟨-, ⟨i, rfl⟩, hi⟩
      

Proof state:
case mp.intro.intro.intro
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
s : Set A
i : ι
hi : id ↑(B i) ⊆ s
⊢ ∃ i, True ∧ ↑(B i) ⊆ s

Next tactic:
exact ⟨i, trivial, hi⟩
theorem hasBasis_nhds_zero : HasBasis (@nhds A hB.topology 0) (fun _ => True) fun i => B i :=
  ⟨by
    intro s
    rw [hB.toRingFilterBasis.toAddGroupFilterBasis.nhds_zero_hasBasis.mem_iff]
    constructor
    · rintro ⟨-, ⟨i, rfl⟩, hi⟩
      exact ⟨i, trivial, hi⟩
    · 

Proof state:
case mpr
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
s : Set A
⊢ (∃ i, True ∧ ↑(B i) ⊆ s) → ∃ i ∈ RingFilterBasis.toAddGroupFilterBasis, id i ⊆ s

Next tactic:
rintro ⟨i, -, hi⟩
theorem hasBasis_nhds_zero : HasBasis (@nhds A hB.topology 0) (fun _ => True) fun i => B i :=
  ⟨by
    intro s
    rw [hB.toRingFilterBasis.toAddGroupFilterBasis.nhds_zero_hasBasis.mem_iff]
    constructor
    · rintro ⟨-, ⟨i, rfl⟩, hi⟩
      exact ⟨i, trivial, hi⟩
    · rintro ⟨i, -, hi⟩
      

Proof state:
case mpr.intro.intro
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
s : Set A
i : ι
hi : ↑(B i) ⊆ s
⊢ ∃ i ∈ RingFilterBasis.toAddGroupFilterBasis, id i ⊆ s

Next tactic:
exact ⟨B i, ⟨i, rfl⟩, hi⟩
theorem hasBasis_nhds (a : A) :
    HasBasis (@nhds A hB.topology a) (fun _ => True) fun i => { b | b - a ∈ B i } :=
  ⟨by
    

Proof state:
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
a : A
⊢ ∀ (t : Set A), t ∈ 𝓝 a ↔ ∃ i, True ∧ {b | b - a ∈ B i} ⊆ t

Next tactic:
intro s
theorem hasBasis_nhds (a : A) :
    HasBasis (@nhds A hB.topology a) (fun _ => True) fun i => { b | b - a ∈ B i } :=
  ⟨by
    intro s
    

Proof state:
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
a : A
s : Set A
⊢ s ∈ 𝓝 a ↔ ∃ i, True ∧ {b | b - a ∈ B i} ⊆ s

Next tactic:
rw [(hB.toRingFilterBasis.toAddGroupFilterBasis.nhds_hasBasis a).mem_iff]
theorem hasBasis_nhds (a : A) :
    HasBasis (@nhds A hB.topology a) (fun _ => True) fun i => { b | b - a ∈ B i } :=
  ⟨by
    intro s
    rw [(hB.toRingFilterBasis.toAddGroupFilterBasis.nhds_hasBasis a).mem_iff]
    

Proof state:
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
a : A
s : Set A
⊢ (∃ i ∈ RingFilterBasis.toAddGroupFilterBasis, (fun y => a + y) '' i ⊆ s) ↔ ∃ i, True ∧ {b | b - a ∈ B i} ⊆ s

Next tactic:
simp only [true_and]
theorem hasBasis_nhds (a : A) :
    HasBasis (@nhds A hB.topology a) (fun _ => True) fun i => { b | b - a ∈ B i } :=
  ⟨by
    intro s
    rw [(hB.toRingFilterBasis.toAddGroupFilterBasis.nhds_hasBasis a).mem_iff]
    simp only [true_and]
    

Proof state:
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
a : A
s : Set A
⊢ (∃ i ∈ RingFilterBasis.toAddGroupFilterBasis, (fun y => a + y) '' i ⊆ s) ↔ ∃ i, {b | b - a ∈ B i} ⊆ s

Next tactic:
constructor
theorem hasBasis_nhds (a : A) :
    HasBasis (@nhds A hB.topology a) (fun _ => True) fun i => { b | b - a ∈ B i } :=
  ⟨by
    intro s
    rw [(hB.toRingFilterBasis.toAddGroupFilterBasis.nhds_hasBasis a).mem_iff]
    simp only [true_and]
    constructor
    · 

Proof state:
case mp
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
a : A
s : Set A
⊢ (∃ i ∈ RingFilterBasis.toAddGroupFilterBasis, (fun y => a + y) '' i ⊆ s) → ∃ i, {b | b - a ∈ B i} ⊆ s

Next tactic:
rintro ⟨-, ⟨i, rfl⟩, hi⟩
theorem hasBasis_nhds (a : A) :
    HasBasis (@nhds A hB.topology a) (fun _ => True) fun i => { b | b - a ∈ B i } :=
  ⟨by
    intro s
    rw [(hB.toRingFilterBasis.toAddGroupFilterBasis.nhds_hasBasis a).mem_iff]
    simp only [true_and]
    constructor
    · rintro ⟨-, ⟨i, rfl⟩, hi⟩
      

Proof state:
case mp.intro.intro.intro
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
a : A
s : Set A
i : ι
hi : (fun y => a + y) '' ↑(B i) ⊆ s
⊢ ∃ i, {b | b - a ∈ B i} ⊆ s

Next tactic:
use i
theorem hasBasis_nhds (a : A) :
    HasBasis (@nhds A hB.topology a) (fun _ => True) fun i => { b | b - a ∈ B i } :=
  ⟨by
    intro s
    rw [(hB.toRingFilterBasis.toAddGroupFilterBasis.nhds_hasBasis a).mem_iff]
    simp only [true_and]
    constructor
    · rintro ⟨-, ⟨i, rfl⟩, hi⟩
      use i
      

Proof state:
case h
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
a : A
s : Set A
i : ι
hi : (fun y => a + y) '' ↑(B i) ⊆ s
⊢ {b | b - a ∈ B i} ⊆ s

Next tactic:
suffices h : { b : A | b - a ∈ B i } = (fun y => a + y) '' ↑(B i)
theorem hasBasis_nhds (a : A) :
    HasBasis (@nhds A hB.topology a) (fun _ => True) fun i => { b | b - a ∈ B i } :=
  ⟨by
    intro s
    rw [(hB.toRingFilterBasis.toAddGroupFilterBasis.nhds_hasBasis a).mem_iff]
    simp only [true_and]
    constructor
    · rintro ⟨-, ⟨i, rfl⟩, hi⟩
      use i
      suffices h : { b : A | b - a ∈ B i } = (fun y => a + y) '' ↑(B i)
      · 

Proof state:
case h
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
a : A
s : Set A
i : ι
hi : (fun y => a + y) '' ↑(B i) ⊆ s
h : {b | b - a ∈ B i} = (fun y => a + y) '' ↑(B i)
⊢ {b | b - a ∈ B i} ⊆ s

Next tactic:
rw [h]
theorem hasBasis_nhds (a : A) :
    HasBasis (@nhds A hB.topology a) (fun _ => True) fun i => { b | b - a ∈ B i } :=
  ⟨by
    intro s
    rw [(hB.toRingFilterBasis.toAddGroupFilterBasis.nhds_hasBasis a).mem_iff]
    simp only [true_and]
    constructor
    · rintro ⟨-, ⟨i, rfl⟩, hi⟩
      use i
      suffices h : { b : A | b - a ∈ B i } = (fun y => a + y) '' ↑(B i)
      · rw [h]
        

Proof state:
case h
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
a : A
s : Set A
i : ι
hi : (fun y => a + y) '' ↑(B i) ⊆ s
h : {b | b - a ∈ B i} = (fun y => a + y) '' ↑(B i)
⊢ (fun y => a + y) '' ↑(B i) ⊆ s

Next tactic:
assumption
theorem hasBasis_nhds (a : A) :
    HasBasis (@nhds A hB.topology a) (fun _ => True) fun i => { b | b - a ∈ B i } :=
  ⟨by
    intro s
    rw [(hB.toRingFilterBasis.toAddGroupFilterBasis.nhds_hasBasis a).mem_iff]
    simp only [true_and]
    constructor
    · rintro ⟨-, ⟨i, rfl⟩, hi⟩
      use i
      suffices h : { b : A | b - a ∈ B i } = (fun y => a + y) '' ↑(B i)
      · rw [h]
        assumption
      

Proof state:
case h
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
a : A
s : Set A
i : ι
hi : (fun y => a + y) '' ↑(B i) ⊆ s
⊢ {b | b - a ∈ B i} = (fun y => a + y) '' ↑(B i)

Next tactic:
simp only [image_add_left, neg_add_eq_sub]
theorem hasBasis_nhds (a : A) :
    HasBasis (@nhds A hB.topology a) (fun _ => True) fun i => { b | b - a ∈ B i } :=
  ⟨by
    intro s
    rw [(hB.toRingFilterBasis.toAddGroupFilterBasis.nhds_hasBasis a).mem_iff]
    simp only [true_and]
    constructor
    · rintro ⟨-, ⟨i, rfl⟩, hi⟩
      use i
      suffices h : { b : A | b - a ∈ B i } = (fun y => a + y) '' ↑(B i)
      · rw [h]
        assumption
      simp only [image_add_left, neg_add_eq_sub]
      

Proof state:
case h
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
a : A
s : Set A
i : ι
hi : (fun y => a + y) '' ↑(B i) ⊆ s
⊢ {b | b - a ∈ B i} = (fun x => x - a) ⁻¹' ↑(B i)

Next tactic:
ext b
theorem hasBasis_nhds (a : A) :
    HasBasis (@nhds A hB.topology a) (fun _ => True) fun i => { b | b - a ∈ B i } :=
  ⟨by
    intro s
    rw [(hB.toRingFilterBasis.toAddGroupFilterBasis.nhds_hasBasis a).mem_iff]
    simp only [true_and]
    constructor
    · rintro ⟨-, ⟨i, rfl⟩, hi⟩
      use i
      suffices h : { b : A | b - a ∈ B i } = (fun y => a + y) '' ↑(B i)
      · rw [h]
        assumption
      simp only [image_add_left, neg_add_eq_sub]
      ext b
      

Proof state:
case h.h
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
a : A
s : Set A
i : ι
hi : (fun y => a + y) '' ↑(B i) ⊆ s
b : A
⊢ b ∈ {b | b - a ∈ B i} ↔ b ∈ (fun x => x - a) ⁻¹' ↑(B i)

Next tactic:
simp
theorem hasBasis_nhds (a : A) :
    HasBasis (@nhds A hB.topology a) (fun _ => True) fun i => { b | b - a ∈ B i } :=
  ⟨by
    intro s
    rw [(hB.toRingFilterBasis.toAddGroupFilterBasis.nhds_hasBasis a).mem_iff]
    simp only [true_and]
    constructor
    · rintro ⟨-, ⟨i, rfl⟩, hi⟩
      use i
      suffices h : { b : A | b - a ∈ B i } = (fun y => a + y) '' ↑(B i)
      · rw [h]
        assumption
      simp only [image_add_left, neg_add_eq_sub]
      ext b
      simp
    · 

Proof state:
case mpr
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
a : A
s : Set A
⊢ (∃ i, {b | b - a ∈ B i} ⊆ s) → ∃ i ∈ RingFilterBasis.toAddGroupFilterBasis, (fun y => a + y) '' i ⊆ s

Next tactic:
rintro ⟨i, hi⟩
theorem hasBasis_nhds (a : A) :
    HasBasis (@nhds A hB.topology a) (fun _ => True) fun i => { b | b - a ∈ B i } :=
  ⟨by
    intro s
    rw [(hB.toRingFilterBasis.toAddGroupFilterBasis.nhds_hasBasis a).mem_iff]
    simp only [true_and]
    constructor
    · rintro ⟨-, ⟨i, rfl⟩, hi⟩
      use i
      suffices h : { b : A | b - a ∈ B i } = (fun y => a + y) '' ↑(B i)
      · rw [h]
        assumption
      simp only [image_add_left, neg_add_eq_sub]
      ext b
      simp
    · rintro ⟨i, hi⟩
      

Proof state:
case mpr.intro
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
a : A
s : Set A
i : ι
hi : {b | b - a ∈ B i} ⊆ s
⊢ ∃ i ∈ RingFilterBasis.toAddGroupFilterBasis, (fun y => a + y) '' i ⊆ s

Next tactic:
use B i
theorem hasBasis_nhds (a : A) :
    HasBasis (@nhds A hB.topology a) (fun _ => True) fun i => { b | b - a ∈ B i } :=
  ⟨by
    intro s
    rw [(hB.toRingFilterBasis.toAddGroupFilterBasis.nhds_hasBasis a).mem_iff]
    simp only [true_and]
    constructor
    · rintro ⟨-, ⟨i, rfl⟩, hi⟩
      use i
      suffices h : { b : A | b - a ∈ B i } = (fun y => a + y) '' ↑(B i)
      · rw [h]
        assumption
      simp only [image_add_left, neg_add_eq_sub]
      ext b
      simp
    · rintro ⟨i, hi⟩
      use B i
      

Proof state:
case h
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
a : A
s : Set A
i : ι
hi : {b | b - a ∈ B i} ⊆ s
⊢ ↑(B i) ∈ RingFilterBasis.toAddGroupFilterBasis ∧ (fun y => a + y) '' ↑(B i) ⊆ s

Next tactic:
constructor
theorem hasBasis_nhds (a : A) :
    HasBasis (@nhds A hB.topology a) (fun _ => True) fun i => { b | b - a ∈ B i } :=
  ⟨by
    intro s
    rw [(hB.toRingFilterBasis.toAddGroupFilterBasis.nhds_hasBasis a).mem_iff]
    simp only [true_and]
    constructor
    · rintro ⟨-, ⟨i, rfl⟩, hi⟩
      use i
      suffices h : { b : A | b - a ∈ B i } = (fun y => a + y) '' ↑(B i)
      · rw [h]
        assumption
      simp only [image_add_left, neg_add_eq_sub]
      ext b
      simp
    · rintro ⟨i, hi⟩
      use B i
      constructor
      · 

Proof state:
case h.left
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
a : A
s : Set A
i : ι
hi : {b | b - a ∈ B i} ⊆ s
⊢ ↑(B i) ∈ RingFilterBasis.toAddGroupFilterBasis

Next tactic:
use i
theorem hasBasis_nhds (a : A) :
    HasBasis (@nhds A hB.topology a) (fun _ => True) fun i => { b | b - a ∈ B i } :=
  ⟨by
    intro s
    rw [(hB.toRingFilterBasis.toAddGroupFilterBasis.nhds_hasBasis a).mem_iff]
    simp only [true_and]
    constructor
    · rintro ⟨-, ⟨i, rfl⟩, hi⟩
      use i
      suffices h : { b : A | b - a ∈ B i } = (fun y => a + y) '' ↑(B i)
      · rw [h]
        assumption
      simp only [image_add_left, neg_add_eq_sub]
      ext b
      simp
    · rintro ⟨i, hi⟩
      use B i
      constructor
      · use i
      · 

Proof state:
case h.right
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
a : A
s : Set A
i : ι
hi : {b | b - a ∈ B i} ⊆ s
⊢ (fun y => a + y) '' ↑(B i) ⊆ s

Next tactic:
rw [image_subset_iff]
theorem hasBasis_nhds (a : A) :
    HasBasis (@nhds A hB.topology a) (fun _ => True) fun i => { b | b - a ∈ B i } :=
  ⟨by
    intro s
    rw [(hB.toRingFilterBasis.toAddGroupFilterBasis.nhds_hasBasis a).mem_iff]
    simp only [true_and]
    constructor
    · rintro ⟨-, ⟨i, rfl⟩, hi⟩
      use i
      suffices h : { b : A | b - a ∈ B i } = (fun y => a + y) '' ↑(B i)
      · rw [h]
        assumption
      simp only [image_add_left, neg_add_eq_sub]
      ext b
      simp
    · rintro ⟨i, hi⟩
      use B i
      constructor
      · use i
      · rw [image_subset_iff]
        

Proof state:
case h.right
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
a : A
s : Set A
i : ι
hi : {b | b - a ∈ B i} ⊆ s
⊢ ↑(B i) ⊆ (fun y => a + y) ⁻¹' s

Next tactic:
rintro b b_in
theorem hasBasis_nhds (a : A) :
    HasBasis (@nhds A hB.topology a) (fun _ => True) fun i => { b | b - a ∈ B i } :=
  ⟨by
    intro s
    rw [(hB.toRingFilterBasis.toAddGroupFilterBasis.nhds_hasBasis a).mem_iff]
    simp only [true_and]
    constructor
    · rintro ⟨-, ⟨i, rfl⟩, hi⟩
      use i
      suffices h : { b : A | b - a ∈ B i } = (fun y => a + y) '' ↑(B i)
      · rw [h]
        assumption
      simp only [image_add_left, neg_add_eq_sub]
      ext b
      simp
    · rintro ⟨i, hi⟩
      use B i
      constructor
      · use i
      · rw [image_subset_iff]
        rintro b b_in
        

Proof state:
case h.right
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
a : A
s : Set A
i : ι
hi : {b | b - a ∈ B i} ⊆ s
b : A
b_in : b ∈ ↑(B i)
⊢ b ∈ (fun y => a + y) ⁻¹' s

Next tactic:
apply hi
theorem hasBasis_nhds (a : A) :
    HasBasis (@nhds A hB.topology a) (fun _ => True) fun i => { b | b - a ∈ B i } :=
  ⟨by
    intro s
    rw [(hB.toRingFilterBasis.toAddGroupFilterBasis.nhds_hasBasis a).mem_iff]
    simp only [true_and]
    constructor
    · rintro ⟨-, ⟨i, rfl⟩, hi⟩
      use i
      suffices h : { b : A | b - a ∈ B i } = (fun y => a + y) '' ↑(B i)
      · rw [h]
        assumption
      simp only [image_add_left, neg_add_eq_sub]
      ext b
      simp
    · rintro ⟨i, hi⟩
      use B i
      constructor
      · use i
      · rw [image_subset_iff]
        rintro b b_in
        apply hi
        

Proof state:
case h.right.a
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
a : A
s : Set A
i : ι
hi : {b | b - a ∈ B i} ⊆ s
b : A
b_in : b ∈ ↑(B i)
⊢ (fun y => a + y) b ∈ {b | b - a ∈ B i}

Next tactic:
simpa using b_in