-- Mathlib/Topology/Algebra/LinearTopology.lean
/-- Assume that `M` is a module over two rings `R` and `R'`, and that its topology
is linear with respect to each of these rings. Then, it has a basis of neighborhoods of zero
made of sub-`(R, R')`-bimodules.

The proof is inspired by lemma 9 in [I. Kaplansky, *Topological Rings*](kaplansky_topological_1947).
TODO: Formalize the lemma in its full strength.

Note: due to the lack of a satisfying theory of sub-bimodules, we use `AddSubgroup`s with
extra conditions. -/
lemma hasBasis_subbimodule [IsLinearTopology R M] [IsLinearTopology R' M] :
    (𝓝 (0 : M)).HasBasis
      (fun I : AddSubgroup M ↦ (I : Set M) ∈ 𝓝 0 ∧
        (∀ r : R, ∀ x ∈ I, r • x ∈ I) ∧ (∀ r' : R', ∀ x ∈ I, r' • x ∈ I))
      (fun I : AddSubgroup M ↦ (I : Set M))

Proof state:
case intro.intro
R : Type u_1
R' : Type u_2
M : Type u_3
inst✝⁸ : Ring R
inst✝⁷ : Ring R'
inst✝⁶ : AddCommGroup M
inst✝⁵ : Module R M
inst✝⁴ : Module R' M
inst✝³ : SMulCommClass R R' M
inst✝² : TopologicalSpace M
inst✝¹ : IsLinearTopology R M
inst✝ : IsLinearTopology R' M
I : Submodule R M
hI : ↑I ∈ 𝓝 0
J : Submodule R' M
hJ : ↑J ∈ 𝓝 0
J_sub_I : ↑J ⊆ ↑I
uR : Set R := univ
uR' : Set R' := univ
hRR : uR * uR ⊆ uR
hRI : uR • ↑I ⊆ ↑I
hR'J : uR' • ↑J ⊆ ↑J
hRJ : uR • ↑J ⊆ ↑I
S : Set M := ↑J ∪ uR • ↑J
S_sub_I : S ⊆ ↑I
hRS : uR • S ⊆ S
hR'S : uR' • S ⊆ S
A : AddSubgroup M := AddSubgroup.closure S
⊢ ∃ i', (↑i' ∈ 𝓝 0 ∧ (∀ (r : R), ∀ x ∈ i', r • x ∈ i') ∧ ∀ (r' : R'), ∀ x ∈ i', r' • x ∈ i') ∧ ↑i' ⊆ ↑I

Tactic:
have hRA : ∀ r : R, ∀ i ∈ A, r • i ∈ A := fun r i hi ↦ by
  refine AddSubgroup.closure_induction (fun x hx => ?base) ?zero (fun x y _ _ hx hy ↦ ?add)
    (fun x _ hx ↦ ?neg) hi
  case base => exact AddSubgroup.subset_closure <| hRS <| Set.smul_mem_smul trivial hx
  case zero => simp_rw [smul_zero]; exact zero_mem _
  case add => simp_rw [smul_add]; exact add_mem hx hy
  case neg => simp_rw [smul_neg]; exact neg_mem hx

Result:
case intro.intro
R : Type u_1
R' : Type u_2
M : Type u_3
inst✝⁸ : Ring R
inst✝⁷ : Ring R'
inst✝⁶ : AddCommGroup M
inst✝⁵ : Module R M
inst✝⁴ : Module R' M
inst✝³ : SMulCommClass R R' M
inst✝² : TopologicalSpace M
inst✝¹ : IsLinearTopology R M
inst✝ : IsLinearTopology R' M
I : Submodule R M
hI : ↑I ∈ 𝓝 0
J : Submodule R' M
hJ : ↑J ∈ 𝓝 0
J_sub_I : ↑J ⊆ ↑I
uR : Set R := univ
uR' : Set R' := univ
hRR : uR * uR ⊆ uR
hRI : uR • ↑I ⊆ ↑I
hR'J : uR' • ↑J ⊆ ↑J
hRJ : uR • ↑J ⊆ ↑I
S : Set M := ↑J ∪ uR • ↑J
S_sub_I : S ⊆ ↑I
hRS : uR • S ⊆ S
hR'S : uR' • S ⊆ S
A : AddSubgroup M := AddSubgroup.closure S
hRA : ∀ (r : R), ∀ i ∈ A, r • i ∈ A
⊢ ∃ i', (↑i' ∈ 𝓝 0 ∧ (∀ (r : R), ∀ x ∈ i', r • x ∈ i') ∧ ∀ (r' : R'), ∀ x ∈ i', r' • x ∈ i') ∧ ↑i' ⊆ ↑I

Proof state:
R : Type u_1
R' : Type u_2
M : Type u_3
inst✝⁸ : Ring R
inst✝⁷ : Ring R'
inst✝⁶ : AddCommGroup M
inst✝⁵ : Module R M
inst✝⁴ : Module R' M
inst✝³ : SMulCommClass R R' M
inst✝² : TopologicalSpace M
inst✝¹ : IsLinearTopology R M
inst✝ : IsLinearTopology R' M
I : Submodule R M
hI : ↑I ∈ 𝓝 0
J : Submodule R' M
hJ : ↑J ∈ 𝓝 0
J_sub_I : ↑J ⊆ ↑I
uR : Set R := univ
uR' : Set R' := univ
hRR : uR * uR ⊆ uR
hRI : uR • ↑I ⊆ ↑I
hR'J : uR' • ↑J ⊆ ↑J
hRJ : uR • ↑J ⊆ ↑I
S : Set M := ↑J ∪ uR • ↑J
S_sub_I : S ⊆ ↑I
hRS : uR • S ⊆ S
hR'S : uR' • S ⊆ S
A : AddSubgroup M := AddSubgroup.closure S
r : R
i : M
hi : i ∈ A
⊢ r • i ∈ A

Tactic:
refine AddSubgroup.closure_induction (fun x hx => ?base) ?zero (fun x y _ _ hx hy ↦ ?add)
  (fun x _ hx ↦ ?neg) hi

Result:
case base
R : Type u_1
R' : Type u_2
M : Type u_3
inst✝⁸ : Ring R
inst✝⁷ : Ring R'
inst✝⁶ : AddCommGroup M
inst✝⁵ : Module R M
inst✝⁴ : Module R' M
inst✝³ : SMulCommClass R R' M
inst✝² : TopologicalSpace M
inst✝¹ : IsLinearTopology R M
inst✝ : IsLinearTopology R' M
I : Submodule R M
hI : ↑I ∈ 𝓝 0
J : Submodule R' M
hJ : ↑J ∈ 𝓝 0
J_sub_I : ↑J ⊆ ↑I
uR : Set R := univ
uR' : Set R' := univ
hRR : uR * uR ⊆ uR
hRI : uR • ↑I ⊆ ↑I
hR'J : uR' • ↑J ⊆ ↑J
hRJ : uR • ↑J ⊆ ↑I
S : Set M := ↑J ∪ uR • ↑J
S_sub_I : S ⊆ ↑I
hRS : uR • S ⊆ S
hR'S : uR' • S ⊆ S
A : AddSubgroup M := AddSubgroup.closure S
r : R
i : M
hi : i ∈ A
x : M
hx : x ∈ S
⊢ r • x ∈ A

case zero
R : Type u_1
R' : Type u_2
M : Type u_3
inst✝⁸ : Ring R
inst✝⁷ : Ring R'
inst✝⁶ : AddCommGroup M
inst✝⁵ : Module R M
inst✝⁴ : Module R' M
inst✝³ : SMulCommClass R R' M
inst✝² : TopologicalSpace M
inst✝¹ : IsLinearTopology R M
inst✝ : IsLinearTopology R' M
I : Submodule R M
hI : ↑I ∈ 𝓝 0
J : Submodule R' M
hJ : ↑J ∈ 𝓝 0
J_sub_I : ↑J ⊆ ↑I
uR : Set R := univ
uR' : Set R' := univ
hRR : uR * uR ⊆ uR
hRI : uR • ↑I ⊆ ↑I
hR'J : uR' • ↑J ⊆ ↑J
hRJ : uR • ↑J ⊆ ↑I
S : Set M := ↑J ∪ uR • ↑J
S_sub_I : S ⊆ ↑I
hRS : uR • S ⊆ S
hR'S : uR' • S ⊆ S
A : AddSubgroup M := AddSubgroup.closure S
r : R
i : M
hi : i ∈ A
⊢ r • 0 ∈ A

case add
R : Type u_1
R' : Type u_2
M : Type u_3
inst✝⁸ : Ring R
inst✝⁷ : Ring R'
inst✝⁶ : AddCommGroup M
inst✝⁵ : Module R M
inst✝⁴ : Module R' M
inst✝³ : SMulCommClass R R' M
inst✝² : TopologicalSpace M
inst✝¹ : IsLinearTopology R M
inst✝ : IsLinearTopology R' M
I : Submodule R M
hI : ↑I ∈ 𝓝 0
J : Submodule R' M
hJ : ↑J ∈ 𝓝 0
J_sub_I : ↑J ⊆ ↑I
uR : Set R := univ
uR' : Set R' := univ
hRR : uR * uR ⊆ uR
hRI : uR • ↑I ⊆ ↑I
hR'J : uR' • ↑J ⊆ ↑J
hRJ : uR • ↑J ⊆ ↑I
S : Set M := ↑J ∪ uR • ↑J
S_sub_I : S ⊆ ↑I
hRS : uR • S ⊆ S
hR'S : uR' • S ⊆ S
A : AddSubgroup M := AddSubgroup.closure S
r : R
i : M
hi : i ∈ A
x y : M
x✝¹ : x ∈ AddSubgroup.closure S
x✝ : y ∈ AddSubgroup.closure S
hx : r • x ∈ A
hy : r • y ∈ A
⊢ r • (x + y) ∈ A

case neg
R : Type u_1
R' : Type u_2
M : Type u_3
inst✝⁸ : Ring R
inst✝⁷ : Ring R'
inst✝⁶ : AddCommGroup M
inst✝⁵ : Module R M
inst✝⁴ : Module R' M
inst✝³ : SMulCommClass R R' M
inst✝² : TopologicalSpace M
inst✝¹ : IsLinearTopology R M
inst✝ : IsLinearTopology R' M
I : Submodule R M
hI : ↑I ∈ 𝓝 0
J : Submodule R' M
hJ : ↑J ∈ 𝓝 0
J_sub_I : ↑J ⊆ ↑I
uR : Set R := univ
uR' : Set R' := univ
hRR : uR * uR ⊆ uR
hRI : uR • ↑I ⊆ ↑I
hR'J : uR' • ↑J ⊆ ↑J
hRJ : uR • ↑J ⊆ ↑I
S : Set M := ↑J ∪ uR • ↑J
S_sub_I : S ⊆ ↑I
hRS : uR • S ⊆ S
hR'S : uR' • S ⊆ S
A : AddSubgroup M := AddSubgroup.closure S
r : R
i : M
hi : i ∈ A
x : M
x✝ : x ∈ AddSubgroup.closure S
hx : r • x ∈ A
⊢ r • -x ∈ A

Proof state:
R : Type u_1
R' : Type u_2
M : Type u_3
inst✝⁸ : Ring R
inst✝⁷ : Ring R'
inst✝⁶ : AddCommGroup M
inst✝⁵ : Module R M
inst✝⁴ : Module R' M
inst✝³ : SMulCommClass R R' M
inst✝² : TopologicalSpace M
inst✝¹ : IsLinearTopology R M
inst✝ : IsLinearTopology R' M
I : Submodule R M
hI : ↑I ∈ 𝓝 0
J : Submodule R' M
hJ : ↑J ∈ 𝓝 0
J_sub_I : ↑J ⊆ ↑I
uR : Set R := univ
uR' : Set R' := univ
hRR : uR * uR ⊆ uR
hRI : uR • ↑I ⊆ ↑I
hR'J : uR' • ↑J ⊆ ↑J
hRJ : uR • ↑J ⊆ ↑I
S : Set M := ↑J ∪ uR • ↑J
S_sub_I : S ⊆ ↑I
hRS : uR • S ⊆ S
hR'S : uR' • S ⊆ S
A : AddSubgroup M := AddSubgroup.closure S
r : R
i : M
hi : i ∈ A
x : M
x✝ : x ∈ AddSubgroup.closure S
hx : r • x ∈ A
⊢ -(r • x) ∈ A

Tactic:
exact neg_mem hx

Result:
no goals