l3lab/ntp-mathlib huggingface dataset:
/-- Given a subgroups basis, the basis elements as open additive subgroups in the associated
topology. -/
def openAddSubgroup (i : ι) : @OpenAddSubgroup A _ hB.topology :=
  -- Porting note: failed to synthesize instance `TopologicalSpace A`
  let _ := hB.topology
  { B i with
    isOpen' := 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
i : ι
x✝ : TopologicalSpace A := topology hB
src✝ : AddSubgroup A := B i
⊢ IsOpen
    { toAddSubmonoid := src✝.toAddSubmonoid,
            neg_mem' := (_ : ∀ {x : A}, x ∈ src✝.carrier → -x ∈ src✝.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier

Next tactic:
rw [isOpen_iff_mem_nhds]
/-- Given a subgroups basis, the basis elements as open additive subgroups in the associated
topology. -/
def openAddSubgroup (i : ι) : @OpenAddSubgroup A _ hB.topology :=
  -- Porting note: failed to synthesize instance `TopologicalSpace A`
  let _ := hB.topology
  { B i with
    isOpen' := by
      rw [isOpen_iff_mem_nhds]
      

Proof state:
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
i : ι
x✝ : TopologicalSpace A := topology hB
src✝ : AddSubgroup A := B i
⊢ ∀
    a ∈
      { toAddSubmonoid := src✝.toAddSubmonoid,
              neg_mem' :=
                (_ : ∀ {x : A}, x ∈ src✝.carrier → -x ∈ src✝.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier,
    { toAddSubmonoid := src✝.toAddSubmonoid,
              neg_mem' :=
                (_ : ∀ {x : A}, x ∈ src✝.carrier → -x ∈ src✝.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier ∈
      𝓝 a

Next tactic:
intro a a_in
/-- Given a subgroups basis, the basis elements as open additive subgroups in the associated
topology. -/
def openAddSubgroup (i : ι) : @OpenAddSubgroup A _ hB.topology :=
  -- Porting note: failed to synthesize instance `TopologicalSpace A`
  let _ := hB.topology
  { B i with
    isOpen' := by
      rw [isOpen_iff_mem_nhds]
      intro a a_in
      

Proof state:
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
i : ι
x✝ : TopologicalSpace A := topology hB
src✝ : AddSubgroup A := B i
a : A
a_in :
  a ∈
    { toAddSubmonoid := src✝.toAddSubmonoid,
            neg_mem' := (_ : ∀ {x : A}, x ∈ src✝.carrier → -x ∈ src✝.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier
⊢ { toAddSubmonoid := src✝.toAddSubmonoid,
            neg_mem' :=
              (_ : ∀ {x : A}, x ∈ src✝.carrier → -x ∈ src✝.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier ∈
    𝓝 a

Next tactic:
rw [(hB.hasBasis_nhds a).mem_iff]
/-- Given a subgroups basis, the basis elements as open additive subgroups in the associated
topology. -/
def openAddSubgroup (i : ι) : @OpenAddSubgroup A _ hB.topology :=
  -- Porting note: failed to synthesize instance `TopologicalSpace A`
  let _ := hB.topology
  { B i with
    isOpen' := by
      rw [isOpen_iff_mem_nhds]
      intro a a_in
      rw [(hB.hasBasis_nhds 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
i : ι
x✝ : TopologicalSpace A := topology hB
src✝ : AddSubgroup A := B i
a : A
a_in :
  a ∈
    { toAddSubmonoid := src✝.toAddSubmonoid,
            neg_mem' := (_ : ∀ {x : A}, x ∈ src✝.carrier → -x ∈ src✝.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier
⊢ ∃ i,
    True ∧
      {b | b - a ∈ B i} ⊆
        { toAddSubmonoid := src✝.toAddSubmonoid,
                neg_mem' :=
                  (_ : ∀ {x : A}, x ∈ src✝.carrier → -x ∈ src✝.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier

Next tactic:
use i, trivial
/-- Given a subgroups basis, the basis elements as open additive subgroups in the associated
topology. -/
def openAddSubgroup (i : ι) : @OpenAddSubgroup A _ hB.topology :=
  -- Porting note: failed to synthesize instance `TopologicalSpace A`
  let _ := hB.topology
  { B i with
    isOpen' := by
      rw [isOpen_iff_mem_nhds]
      intro a a_in
      rw [(hB.hasBasis_nhds a).mem_iff]
      use i, trivial
      

Proof state:
case right
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
i : ι
x✝ : TopologicalSpace A := topology hB
src✝ : AddSubgroup A := B i
a : A
a_in :
  a ∈
    { toAddSubmonoid := src✝.toAddSubmonoid,
            neg_mem' := (_ : ∀ {x : A}, x ∈ src✝.carrier → -x ∈ src✝.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier
⊢ {b | b - a ∈ B i} ⊆
    { toAddSubmonoid := src✝.toAddSubmonoid,
            neg_mem' := (_ : ∀ {x : A}, x ∈ src✝.carrier → -x ∈ src✝.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier

Next tactic:
rintro b b_in
/-- Given a subgroups basis, the basis elements as open additive subgroups in the associated
topology. -/
def openAddSubgroup (i : ι) : @OpenAddSubgroup A _ hB.topology :=
  -- Porting note: failed to synthesize instance `TopologicalSpace A`
  let _ := hB.topology
  { B i with
    isOpen' := by
      rw [isOpen_iff_mem_nhds]
      intro a a_in
      rw [(hB.hasBasis_nhds a).mem_iff]
      use i, trivial
      rintro b b_in
      

Proof state:
case right
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
i : ι
x✝ : TopologicalSpace A := topology hB
src✝ : AddSubgroup A := B i
a : A
a_in :
  a ∈
    { toAddSubmonoid := src✝.toAddSubmonoid,
            neg_mem' := (_ : ∀ {x : A}, x ∈ src✝.carrier → -x ∈ src✝.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier
b : A
b_in : b ∈ {b | b - a ∈ B i}
⊢ b ∈
    { toAddSubmonoid := src✝.toAddSubmonoid,
            neg_mem' := (_ : ∀ {x : A}, x ∈ src✝.carrier → -x ∈ src✝.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier

Next tactic:
simpa using (B i).add_mem a_in b_in
theorem nonarchimedean : @NonarchimedeanRing A _ hB.topology := 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
⊢ NonarchimedeanRing A

Next tactic:
letI := hB.topology
theorem nonarchimedean : @NonarchimedeanRing A _ hB.topology := by
  letI := hB.topology
  

Proof state:
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
this : TopologicalSpace A := topology hB
⊢ NonarchimedeanRing A

Next tactic:
constructor
theorem nonarchimedean : @NonarchimedeanRing A _ hB.topology := by
  letI := hB.topology
  constructor
  

Proof state:
case is_nonarchimedean
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
this : TopologicalSpace A := topology hB
⊢ ∀ U ∈ 𝓝 0, ∃ V, ↑V ⊆ U

Next tactic:
intro U hU
theorem nonarchimedean : @NonarchimedeanRing A _ hB.topology := by
  letI := hB.topology
  constructor
  intro U hU
  

Proof state:
case is_nonarchimedean
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
this : TopologicalSpace A := topology hB
U : Set A
hU : U ∈ 𝓝 0
⊢ ∃ V, ↑V ⊆ U

Next tactic:
obtain ⟨i, -, hi : (B i : Set A) ⊆ U⟩ := hB.hasBasis_nhds_zero.mem_iff.mp hU
theorem nonarchimedean : @NonarchimedeanRing A _ hB.topology := by
  letI := hB.topology
  constructor
  intro U hU
  obtain ⟨i, -, hi : (B i : Set A) ⊆ U⟩ := hB.hasBasis_nhds_zero.mem_iff.mp hU
  

Proof state:
case is_nonarchimedean.intro.intro
A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
this : TopologicalSpace A := topology hB
U : Set A
hU : U ∈ 𝓝 0
i : ι
hi : ↑(B i) ⊆ U
⊢ ∃ V, ↑V ⊆ U

Next tactic:
exact ⟨hB.openAddSubgroup i, hi⟩
theorem toRing_subgroups_basis (hB : SubmodulesRingBasis B) :
    RingSubgroupsBasis fun i => (B i).toAddSubgroup := by
  

Proof state:
ι : Type u_1
R : Type u_2
A : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing A
inst✝ : Algebra R A
B : ι → Submodule R A
hB✝ hB : SubmodulesRingBasis B
⊢ RingSubgroupsBasis fun i => Submodule.toAddSubgroup (B i)

Next tactic:
apply RingSubgroupsBasis.of_comm (fun i => (B i).toAddSubgroup) hB.inter hB.mul
theorem toRing_subgroups_basis (hB : SubmodulesRingBasis B) :
    RingSubgroupsBasis fun i => (B i).toAddSubgroup := by
  apply RingSubgroupsBasis.of_comm (fun i => (B i).toAddSubgroup) hB.inter hB.mul
  

Proof state:
ι : Type u_1
R : Type u_2
A : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing A
inst✝ : Algebra R A
B : ι → Submodule R A
hB✝ hB : SubmodulesRingBasis B
⊢ ∀ (x : A) (i : ι), ∃ j, ↑(Submodule.toAddSubgroup (B j)) ⊆ (fun y => x * y) ⁻¹' ↑(Submodule.toAddSubgroup (B i))

Next tactic:
intro a i
theorem toRing_subgroups_basis (hB : SubmodulesRingBasis B) :
    RingSubgroupsBasis fun i => (B i).toAddSubgroup := by
  apply RingSubgroupsBasis.of_comm (fun i => (B i).toAddSubgroup) hB.inter hB.mul
  intro a i
  

Proof state:
ι : Type u_1
R : Type u_2
A : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing A
inst✝ : Algebra R A
B : ι → Submodule R A
hB✝ hB : SubmodulesRingBasis B
a : A
i : ι
⊢ ∃ j, ↑(Submodule.toAddSubgroup (B j)) ⊆ (fun y => a * y) ⁻¹' ↑(Submodule.toAddSubgroup (B i))

Next tactic:
rcases hB.leftMul a i with ⟨j, hj⟩
theorem toRing_subgroups_basis (hB : SubmodulesRingBasis B) :
    RingSubgroupsBasis fun i => (B i).toAddSubgroup := by
  apply RingSubgroupsBasis.of_comm (fun i => (B i).toAddSubgroup) hB.inter hB.mul
  intro a i
  rcases hB.leftMul a i with ⟨j, hj⟩
  

Proof state:
case intro
ι : Type u_1
R : Type u_2
A : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing A
inst✝ : Algebra R A
B : ι → Submodule R A
hB✝ hB : SubmodulesRingBasis B
a : A
i j : ι
hj : a • B j ≤ B i
⊢ ∃ j, ↑(Submodule.toAddSubgroup (B j)) ⊆ (fun y => a * y) ⁻¹' ↑(Submodule.toAddSubgroup (B i))

Next tactic:
use j
theorem toRing_subgroups_basis (hB : SubmodulesRingBasis B) :
    RingSubgroupsBasis fun i => (B i).toAddSubgroup := by
  apply RingSubgroupsBasis.of_comm (fun i => (B i).toAddSubgroup) hB.inter hB.mul
  intro a i
  rcases hB.leftMul a i with ⟨j, hj⟩
  use j
  

Proof state:
case h
ι : Type u_1
R : Type u_2
A : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing A
inst✝ : Algebra R A
B : ι → Submodule R A
hB✝ hB : SubmodulesRingBasis B
a : A
i j : ι
hj : a • B j ≤ B i
⊢ ↑(Submodule.toAddSubgroup (B j)) ⊆ (fun y => a * y) ⁻¹' ↑(Submodule.toAddSubgroup (B i))

Next tactic:
rintro b (b_in : b ∈ B j)
theorem toRing_subgroups_basis (hB : SubmodulesRingBasis B) :
    RingSubgroupsBasis fun i => (B i).toAddSubgroup := by
  apply RingSubgroupsBasis.of_comm (fun i => (B i).toAddSubgroup) hB.inter hB.mul
  intro a i
  rcases hB.leftMul a i with ⟨j, hj⟩
  use j
  rintro b (b_in : b ∈ B j)
  

Proof state:
case h
ι : Type u_1
R : Type u_2
A : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing A
inst✝ : Algebra R A
B : ι → Submodule R A
hB✝ hB : SubmodulesRingBasis B
a : A
i j : ι
hj : a • B j ≤ B i
b : A
b_in : b ∈ B j
⊢ b ∈ (fun y => a * y) ⁻¹' ↑(Submodule.toAddSubgroup (B i))

Next tactic:
exact hj ⟨b, b_in, rfl⟩
/-- The image of a submodules basis is a module filter basis. -/
def toModuleFilterBasis : ModuleFilterBasis R M where
  sets := { U | ∃ i, U = B i }
  nonempty := by
    

Proof state:
ι : Type u_1
R : Type u_2
A : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
M : Type u_4
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : TopologicalSpace R
inst✝ : Nonempty ι
B : ι → Submodule R M
hB : SubmodulesBasis B
⊢ Set.Nonempty {U | ∃ i, U = ↑(B i)}

Next tactic:
inhabit ι
/-- The image of a submodules basis is a module filter basis. -/
def toModuleFilterBasis : ModuleFilterBasis R M where
  sets := { U | ∃ i, U = B i }
  nonempty := by
    inhabit ι
    

Proof state:
ι : Type u_1
R : Type u_2
A : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
M : Type u_4
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : TopologicalSpace R
inst✝ : Nonempty ι
B : ι → Submodule R M
hB : SubmodulesBasis B
inhabited_h : Inhabited ι
⊢ Set.Nonempty {U | ∃ i, U = ↑(B i)}

Next tactic:
exact ⟨B default, default, rfl⟩
/-- The image of a submodules basis is a module filter basis. -/
def toModuleFilterBasis : ModuleFilterBasis R M where
  sets := { U | ∃ i, U = B i }
  nonempty := by
    inhabit ι
    exact ⟨B default, default, rfl⟩
  inter_sets := by
    

Proof state:
ι : Type u_1
R : Type u_2
A : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
M : Type u_4
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : TopologicalSpace R
inst✝ : Nonempty ι
B : ι → Submodule R M
hB : SubmodulesBasis B
⊢ ∀ {x y : Set M}, x ∈ {U | ∃ i, U = ↑(B i)} → y ∈ {U | ∃ i, U = ↑(B i)} → ∃ z ∈ {U | ∃ i, U = ↑(B i)}, z ⊆ x ∩ y

Next tactic:
rintro _ _ ⟨i, rfl⟩ ⟨j, rfl⟩
/-- The image of a submodules basis is a module filter basis. -/
def toModuleFilterBasis : ModuleFilterBasis R M where
  sets := { U | ∃ i, U = B i }
  nonempty := by
    inhabit ι
    exact ⟨B default, default, rfl⟩
  inter_sets := by
    rintro _ _ ⟨i, rfl⟩ ⟨j, rfl⟩
    

Proof state:
case intro.intro
ι : Type u_1
R : Type u_2
A : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
M : Type u_4
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : TopologicalSpace R
inst✝ : Nonempty ι
B : ι → Submodule R M
hB : SubmodulesBasis B
i j : ι
⊢ ∃ z ∈ {U | ∃ i, U = ↑(B i)}, z ⊆ ↑(B i) ∩ ↑(B j)

Next tactic:
cases' hB.inter i j with k hk
/-- The image of a submodules basis is a module filter basis. -/
def toModuleFilterBasis : ModuleFilterBasis R M 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
    

Proof state:
case intro.intro.intro
ι : Type u_1
R : Type u_2
A : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
M : Type u_4
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : TopologicalSpace R
inst✝ : Nonempty ι
B : ι → Submodule R M
hB : SubmodulesBasis B
i j k : ι
hk : B k ≤ B i ⊓ B j
⊢ ∃ z ∈ {U | ∃ i, U = ↑(B i)}, z ⊆ ↑(B i) ∩ ↑(B j)

Next tactic:
use B k
/-- The image of a submodules basis is a module filter basis. -/
def toModuleFilterBasis : ModuleFilterBasis R M 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
    

Proof state:
case h
ι : Type u_1
R : Type u_2
A : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
M : Type u_4
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : TopologicalSpace R
inst✝ : Nonempty ι
B : ι → Submodule R M
hB : SubmodulesBasis B
i j k : ι
hk : B k ≤ B i ⊓ B j
⊢ ↑(B k) ∈ {U | ∃ i, U = ↑(B i)} ∧ ↑(B k) ⊆ ↑(B i) ∩ ↑(B j)

Next tactic:
constructor
/-- The image of a submodules basis is a module filter basis. -/
def toModuleFilterBasis : ModuleFilterBasis R M 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
    · 

Proof state:
case h.left
ι : Type u_1
R : Type u_2
A : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
M : Type u_4
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : TopologicalSpace R
inst✝ : Nonempty ι
B : ι → Submodule R M
hB : SubmodulesBasis B
i j k : ι
hk : B k ≤ B i ⊓ B j
⊢ ↑(B k) ∈ {U | ∃ i, U = ↑(B i)}

Next tactic:
use k
/-- The image of a submodules basis is a module filter basis. -/
def toModuleFilterBasis : ModuleFilterBasis R M 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
    · 

Proof state:
case h.right
ι : Type u_1
R : Type u_2
A : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
M : Type u_4
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : TopologicalSpace R
inst✝ : Nonempty ι
B : ι → Submodule R M
hB : SubmodulesBasis B
i j k : ι
hk : B k ≤ B i ⊓ B j
⊢ ↑(B k) ⊆ ↑(B i) ∩ ↑(B j)

Next tactic:
exact hk
/-- The image of a submodules basis is a module filter basis. -/
def toModuleFilterBasis : ModuleFilterBasis R M 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
    

Proof state:
ι : Type u_1
R : Type u_2
A : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
M : Type u_4
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : TopologicalSpace R
inst✝ : Nonempty ι
B : ι → Submodule R M
hB : SubmodulesBasis B
⊢ ∀ {U : Set M},
    U ∈
        { sets := {U | ∃ i, U = ↑(B i)}, nonempty := (_ : ∃ x, x ∈ {U | ∃ i, U = ↑(B i)}),
            inter_sets :=
              (_ :
                ∀ {x y : Set M},
                  x ∈ {U | ∃ i, U = ↑(B i)} →
                    y ∈ {U | ∃ i, U = ↑(B i)} → ∃ z ∈ {U | ∃ i, U = ↑(B i)}, z ⊆ x ∩ y) }.sets →
      0 ∈ U

Next tactic:
rintro _ ⟨i, rfl⟩
/-- The image of a submodules basis is a module filter basis. -/
def toModuleFilterBasis : ModuleFilterBasis R M 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⟩
    

Proof state:
case intro
ι : Type u_1
R : Type u_2
A : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
M : Type u_4
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : TopologicalSpace R
inst✝ : Nonempty ι
B : ι → Submodule R M
hB : SubmodulesBasis B
i : ι
⊢ 0 ∈ ↑(B i)

Next tactic:
exact (B i).zero_mem
/-- The image of a submodules basis is a module filter basis. -/
def toModuleFilterBasis : ModuleFilterBasis R M 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
    

Proof state:
ι : Type u_1
R : Type u_2
A : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
M : Type u_4
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : TopologicalSpace R
inst✝ : Nonempty ι
B : ι → Submodule R M
hB : SubmodulesBasis B
⊢ ∀ {U : Set M},
    U ∈
        { sets := {U | ∃ i, U = ↑(B i)}, nonempty := (_ : ∃ x, x ∈ {U | ∃ i, U = ↑(B i)}),
            inter_sets :=
              (_ :
                ∀ {x y : Set M},
                  x ∈ {U | ∃ i, U = ↑(B i)} →
                    y ∈ {U | ∃ i, U = ↑(B i)} → ∃ z ∈ {U | ∃ i, U = ↑(B i)}, z ⊆ x ∩ y) }.sets →
      ∃
        V ∈
          { sets := {U | ∃ i, U = ↑(B i)}, nonempty := (_ : ∃ x, x ∈ {U | ∃ i, U = ↑(B i)}),
              inter_sets :=
                (_ :
                  ∀ {x y : Set M},
                    x ∈ {U | ∃ i, U = ↑(B i)} →
                      y ∈ {U | ∃ i, U = ↑(B i)} → ∃ z ∈ {U | ∃ i, U = ↑(B i)}, z ⊆ x ∩ y) }.sets,
        V + V ⊆ U

Next tactic:
rintro _ ⟨i, rfl⟩
/-- The image of a submodules basis is a module filter basis. -/
def toModuleFilterBasis : ModuleFilterBasis R M 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⟩
    

Proof state:
case intro
ι : Type u_1
R : Type u_2
A : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
M : Type u_4
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : TopologicalSpace R
inst✝ : Nonempty ι
B : ι → Submodule R M
hB : SubmodulesBasis B
i : ι
⊢ ∃
    V ∈
      { sets := {U | ∃ i, U = ↑(B i)}, nonempty := (_ : ∃ x, x ∈ {U | ∃ i, U = ↑(B i)}),
          inter_sets :=
            (_ :
              ∀ {x y : Set M},
                x ∈ {U | ∃ i, U = ↑(B i)} → y ∈ {U | ∃ i, U = ↑(B i)} → ∃ z ∈ {U | ∃ i, U = ↑(B i)}, z ⊆ x ∩ y) }.sets,
    V + V ⊆ ↑(B i)

Next tactic:
use B i
/-- The image of a submodules basis is a module filter basis. -/
def toModuleFilterBasis : ModuleFilterBasis R M 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
    

Proof state:
case h
ι : Type u_1
R : Type u_2
A : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
M : Type u_4
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : TopologicalSpace R
inst✝ : Nonempty ι
B : ι → Submodule R M
hB : SubmodulesBasis B
i : ι
⊢ ↑(B i) ∈
      { sets := {U | ∃ i, U = ↑(B i)}, nonempty := (_ : ∃ x, x ∈ {U | ∃ i, U = ↑(B i)}),
          inter_sets :=
            (_ :
              ∀ {x y : Set M},
                x ∈ {U | ∃ i, U = ↑(B i)} → y ∈ {U | ∃ i, U = ↑(B i)} → ∃ z ∈ {U | ∃ i, U = ↑(B i)}, z ⊆ x ∩ y) }.sets ∧
    ↑(B i) + ↑(B i) ⊆ ↑(B i)

Next tactic:
constructor
/-- The image of a submodules basis is a module filter basis. -/
def toModuleFilterBasis : ModuleFilterBasis R M 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
    · 

Proof state:
case h.left
ι : Type u_1
R : Type u_2
A : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
M : Type u_4
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : TopologicalSpace R
inst✝ : Nonempty ι
B : ι → Submodule R M
hB : SubmodulesBasis B
i : ι
⊢ ↑(B i) ∈
    { sets := {U | ∃ i, U = ↑(B i)}, nonempty := (_ : ∃ x, x ∈ {U | ∃ i, U = ↑(B i)}),
        inter_sets :=
          (_ :
            ∀ {x y : Set M},
              x ∈ {U | ∃ i, U = ↑(B i)} → y ∈ {U | ∃ i, U = ↑(B i)} → ∃ z ∈ {U | ∃ i, U = ↑(B i)}, z ⊆ x ∩ y) }.sets

Next tactic:
use i
/-- The image of a submodules basis is a module filter basis. -/
def toModuleFilterBasis : ModuleFilterBasis R M 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
    · 

Proof state:
case h.right
ι : Type u_1
R : Type u_2
A : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
M : Type u_4
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : TopologicalSpace R
inst✝ : Nonempty ι
B : ι → Submodule R M
hB : SubmodulesBasis B
i : ι
⊢ ↑(B i) + ↑(B i) ⊆ ↑(B i)

Next tactic:
rintro x ⟨y, z, y_in, z_in, rfl⟩
/-- The image of a submodules basis is a module filter basis. -/
def toModuleFilterBasis : ModuleFilterBasis R M 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⟩
      

Proof state:
case h.right.intro.intro.intro.intro
ι : Type u_1
R : Type u_2
A : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
M : Type u_4
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : TopologicalSpace R
inst✝ : Nonempty ι
B : ι → Submodule R M
hB : SubmodulesBasis B
i : ι
y z : M
y_in : y ∈ ↑(B i)
z_in : z ∈ ↑(B i)
⊢ (fun x x_1 => x + x_1) y z ∈ ↑(B i)

Next tactic:
exact (B i).add_mem y_in z_in