l3lab/ntp-mathlib huggingface dataset:
/-- 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⟩
      exact (B i).add_mem y_in z_in
  neg' := 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 ⊆ (fun x => -x) ⁻¹' 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⟩
    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⟩
    

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 ⊆ (fun x => -x) ⁻¹' ↑(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
    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
    

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) ⊆ (fun x => -x) ⁻¹' ↑(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
    · 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
    · 

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
    · 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
    · 

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) ⊆ (fun x => -x) ⁻¹' ↑(B i)

Next tactic:
intro x x_in
/-- 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⟩
      exact (B i).add_mem y_in z_in
  neg' := by
    rintro _ ⟨i, rfl⟩
    use B i
    constructor
    · use i
    · intro x x_in
      

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
x : M
x_in : x ∈ ↑(B i)
⊢ x ∈ (fun x => -x) ⁻¹' ↑(B i)

Next tactic:
exact (B i).neg_mem x_in
/-- 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⟩
      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
    

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
⊢ ∀ (x₀ : M) {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 ⊆ (fun x => x₀ + x + -x₀) ⁻¹' U

Next tactic:
rintro x₀ _ ⟨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⟩
    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⟩
    

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
x₀ : M
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 ⊆ (fun x => x₀ + x + -x₀) ⁻¹' ↑(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
    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
    

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
x₀ : M
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) ⊆ (fun x => x₀ + x + -x₀) ⁻¹' ↑(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
    · 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
    · 

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
x₀ : M
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
    · 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
    · 

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
x₀ : M
i : ι
⊢ ↑(B i) ⊆ (fun x => x₀ + x + -x₀) ⁻¹' ↑(B i)

Next tactic:
simp
/-- 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⟩
      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
  smul' := 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 ∈ AddGroupFilterBasis.toFilterBasis.sets → ∃ V ∈ 𝓝 0, ∃ W ∈ AddGroupFilterBasis.toFilterBasis.sets, V • W ⊆ 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⟩
    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
  smul' := 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 ∈ 𝓝 0, ∃ W ∈ AddGroupFilterBasis.toFilterBasis.sets, V • W ⊆ ↑(B i)

Next tactic:
use univ
/-- 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⟩
      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
  smul' := by
    rintro _ ⟨i, rfl⟩
    use univ
    

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
⊢ univ ∈ 𝓝 0 ∧ ∃ W ∈ AddGroupFilterBasis.toFilterBasis.sets, univ • W ⊆ ↑(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
    · 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
  smul' := by
    rintro _ ⟨i, rfl⟩
    use univ
    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
⊢ univ ∈ 𝓝 0

Next tactic:
exact univ_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
    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
  smul' := by
    rintro _ ⟨i, rfl⟩
    use univ
    constructor
    · exact univ_mem
    · 

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
⊢ ∃ W ∈ AddGroupFilterBasis.toFilterBasis.sets, univ • W ⊆ ↑(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
    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
  smul' := by
    rintro _ ⟨i, rfl⟩
    use univ
    constructor
    · exact univ_mem
    · 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) ∈ AddGroupFilterBasis.toFilterBasis.sets ∧ univ • ↑(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
    · 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
  smul' := by
    rintro _ ⟨i, rfl⟩
    use univ
    constructor
    · exact univ_mem
    · 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) ∈ AddGroupFilterBasis.toFilterBasis.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
    · 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
  smul' := by
    rintro _ ⟨i, rfl⟩
    use univ
    constructor
    · exact univ_mem
    · 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
⊢ univ • ↑(B i) ⊆ ↑(B i)

Next tactic:
rintro _ ⟨a, m, -, hm, rfl⟩