feat(Algebra/Module/Submodule): Symmetric submodules are modular elements in the lattice of submodules over a semiring (#36689)

Add two lemmas that are replacements for modularity on the lattice of submodules that are not necessarily over a ring, but a semiring:

- `sup_inf_assoc_of_le_of_neg_le`: if `s ≤ p` and `-s ≤ p` then `(s ⊔ t) ⊓ p = s ⊔ (t ⊓ p)`
- `inf_sup_assoc_of_le_of_neg_le`: if `p ≤ s` and `-p ≤ s` then `(s ⊓ t) ⊔ p = s ⊓ (t ⊔ p)`

This allows to shorten the proof that the lattice of submodules over a ring is modular.

I also add specialized versions for cones.

Co-authored-by: Martin Winter <martin.winter.math@gmail.com>

Author: martinwintermath

Mathlib/Geometry/Convex/Cone/Pointed.lean

@@ -284,7 +284,22 @@ theorem toConvexCone_positive : ↑(positive R E) = ConvexCone.positive R E :=
 
 end PositiveCone
 
+section AddCommGroup
+
+variable {R M : Type*} [Ring R] [PartialOrder R] [IsOrderedRing R] [AddCommGroup E] [Module R E]
+
+lemma sup_inf_assoc_of_le_submodule {C : PointedCone R E} (D : PointedCone R E)
+    {S : Submodule R E} (hCS : C ≤ S) : (C ⊔ D) ⊓ S = C ⊔ (D ⊓ S) :=
+  sup_inf_assoc_of_le_of_neg_le _ hCS (fun _ hx => by simpa using hCS hx)
+
+lemma inf_sup_assoc_of_le_of_submodule_le {C : PointedCone R E} (D : PointedCone R E)
+    {S : Submodule R E} (hSC : S ≤ C) : (C ⊓ D) ⊔ S = C ⊓ (D ⊔ S) :=
+  inf_sup_assoc_of_le_of_neg_le _ hSC (fun _ hx => by apply hSC; simpa [hSC] using hx)
+
+end AddCommGroup
+
 section OrderedAddCommGroup
+
 variable [Ring R] [PartialOrder R] [IsOrderedRing R] [AddCommGroup E] [PartialOrder E]
   [IsOrderedAddMonoid E] [Module R E]
 

Mathlib/LinearAlgebra/Span/Basic.lean

@@ -457,7 +457,27 @@ end AddCommMonoid
 
 section AddCommGroup
 
-variable [Ring R] [AddCommGroup M] [Module R M]
+variable {R M : Type*} [Semiring R] [AddCommGroup M] [Module R M]
+
+lemma sup_inf_assoc_of_le_of_neg_le {s : Submodule R M} (t : Submodule R M)
+    {p : Submodule R M} (hsp : s ≤ p) (hnsp : ∀ x ∈ s, -x ∈ p) :
+    (s ⊔ t) ⊓ p = s ⊔ (t ⊓ p) := by
+  ext x; simp only [mem_sup, mem_inf]
+  constructor
+  · rintro ⟨⟨y, hy, z, hz, hyzx⟩, hx⟩
+    refine ⟨y, hy, z, ⟨hz, ?_⟩, hyzx⟩
+    rw [← add_right_inj, neg_add_cancel_left] at hyzx
+    simpa [hyzx] using p.add_mem (hnsp y hy) hx
+  · rintro ⟨y, hy, z, ⟨hz, hz'⟩, hyzx⟩
+    refine ⟨⟨y, hy, z, hz, hyzx⟩, ?_⟩
+    simpa [← hyzx] using p.add_mem (hsp hy) hz'
+
+lemma inf_sup_assoc_of_le_of_neg_le {s : Submodule R M} (t : Submodule R M)
+    {p : Submodule R M} (hps : p ≤ s) (hnps : ∀ x ∈ p, -x ∈ s) :
+    (s ⊓ t) ⊔ p = s ⊓ (t ⊔ p) := by
+  rw [sup_comm, inf_comm, ← sup_inf_assoc_of_le_of_neg_le t hps hnps, inf_comm, sup_comm]
+
+variable {R M : Type*} [Ring R] [AddCommGroup M] [Module R M]
 
 lemma _root_.AddSubgroup.toIntSubmodule_closure (s : Set M) :
     (AddSubgroup.closure s).toIntSubmodule = .span ℤ s :=
@@ -471,14 +491,8 @@ theorem span_neg (s : Set M) : span R (-s) = span R s :=
     _ = map (-LinearMap.id) (span R s) := (map_span (-LinearMap.id) _).symm
     _ = span R s := by simp
 
-instance : IsModularLattice (Submodule R M) :=
-  ⟨fun y z xz a ha => by
-    rw [mem_inf, mem_sup] at ha
-    rcases ha with ⟨⟨b, hb, c, hc, rfl⟩, haz⟩
-    rw [mem_sup]
-    refine ⟨b, hb, c, mem_inf.2 ⟨hc, ?_⟩, rfl⟩
-    rw [← add_sub_cancel_right c b, add_comm]
-    apply z.sub_mem haz (xz hb)⟩
+instance : IsModularLattice (Submodule R M) := ⟨fun x _ hxy _ _ => by
+    rwa [← sup_inf_assoc_of_le_of_neg_le x hxy (fun _ hy => neg_mem (hxy hy))]⟩
 
 lemma isCompl_comap_subtype_of_isCompl_of_le {p q r : Submodule R M}
     (h₁ : IsCompl q r) (h₂ : q ≤ p) :