diff --git "a/data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Module.SpanRankOperations.sym.json" "b/data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Module.SpanRankOperations.sym.json"
new file mode 100644--- /dev/null
+++ "b/data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Module.SpanRankOperations.sym.json"
@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["Submodule","spanRank_baseChange_le"],"typeFallback":"forall {R : Type.{u_1}} {A : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.SpanRankOperations.3154670743._hygCtx._hyg.4 : CommRing.{u_1} R] [inst._@.Mathlib.Algebra.Module.SpanRankOperations.3154670743._hygCtx._hyg.7 : CommRing.{u_2} A] [inst._@.Mathlib.Algebra.Module.SpanRankOperations.3154670743._hygCtx._hyg.10 : Algebra.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3154670743._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.3154670743._hygCtx._hyg.7))] {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.SpanRankOperations.3154670743._hygCtx._hyg.15 : AddCommGroup.{u_3} M] [inst._@.Mathlib.Algebra.Module.SpanRankOperations.3154670743._hygCtx._hyg.18 : Module.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3154670743._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.3154670743._hygCtx._hyg.15)] (N : Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3154670743._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.3154670743._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3154670743._hygCtx._hyg.18), LE.le.{max (succ u_2) (succ u_3)} Cardinal.{max u_2 u_3} Cardinal.instLE.{max u_2 u_3} (Submodule.spanRank.{max u_2 u_3, u_2} A (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3154670743._hygCtx._hyg.4) A M (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.3154670743._hygCtx._hyg.7))))) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.3154670743._hygCtx._hyg.15) (Algebra.toModule.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3154670743._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.3154670743._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3154670743._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3154670743._hygCtx._hyg.18) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.3154670743._hygCtx._hyg.7)) (TensorProduct.addCommMonoid.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3154670743._hygCtx._hyg.4) A M (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.3154670743._hygCtx._hyg.7))))) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.3154670743._hygCtx._hyg.15) (Algebra.toModule.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3154670743._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.3154670743._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3154670743._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3154670743._hygCtx._hyg.18) (TensorProduct.leftModule.{u_1, u_2, u_2, u_3} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3154670743._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.3154670743._hygCtx._hyg.7)) A M (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.3154670743._hygCtx._hyg.7))))) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.3154670743._hygCtx._hyg.15) (Semiring.toModule.{u_2} A (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.3154670743._hygCtx._hyg.7))) (Algebra.toModule.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3154670743._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.3154670743._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3154670743._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3154670743._hygCtx._hyg.18 (Algebra.to_smulCommClass.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3154670743._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.3154670743._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3154670743._hygCtx._hyg.10)) (Submodule.baseChange.{u_1, u_3, u_2} R M A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3154670743._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.3154670743._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3154670743._hygCtx._hyg.10 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.3154670743._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3154670743._hygCtx._hyg.18 N)) (Cardinal.lift.{u_2, u_3} (Submodule.spanRank.{u_3, u_1} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3154670743._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.3154670743._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3154670743._hygCtx._hyg.18 N))","typeFull":"∀ {R : Type u_1} {A : Type u_2} [inst : CommRing R] [inst_1 : CommRing A] [inst_2 : Algebra R A] {M : Type u_3}\n  [inst_3 : AddCommGroup M] [inst_4 : Module R M] (N : Submodule R M),\n  (Submodule.baseChange A N).spanRank ≤ Cardinal.lift.{u_2, u_3} N.spanRank","typeReadable":"∀ {R : Type u_1} {A : Type u_2} [inst : CommRing R] [inst_1 : CommRing A] [inst_2 : Algebra R A] {M : Type u_3}\n  [inst_3 : AddCommGroup M] [inst_4 : Module R M] (N : Submodule R M),\n  (Submodule.baseChange A N).spanRank ≤ Cardinal.lift.{u_2, u_3} N.spanRank","typeReferences":[["CommRing","toCommSemiring"],["TensorProduct"],["Cardinal","lift"],["Module"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["CommSemiring","toSemiring"],["Submodule","baseChange"],["AddCommGroup"],["TensorProduct","leftModule"],["CommRing"],["Algebra","to_smulCommClass"],["Cardinal"],["Algebra"],["Submodule"],["Semiring","toNonAssocSemiring"],["Algebra","toModule"],["Submodule","spanRank"],["LE","le"],["AddCommGroup","toAddCommMonoid"],["TensorProduct","addCommMonoid"],["Semiring","toModule"],["Cardinal","instLE"]],"valueReferences":[["le_refl"],["smulCommClass_self"],["PartialOrder","toPreorder"],["Cardinal","lift"],["AddGroupWithOne","toAddMonoidWithOne"],["Submodule","baseChange"],["Submodule","span"],["Algebra","to_smulCommClass"],["Set","Elem"],["Cardinal","partialOrder"],["Cardinal","mk"],["TensorProduct","instModule"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["Semiring","toNonAssocSemiring"],["Ring","toAddGroupWithOne"],["RingHom","id"],["TensorProduct","mk"],["eq_of_heq"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["Submodule","baseChange_span"],["Submodule","exists_span_set_card_eq_spanRank"],["Eq","symm"],["DistribMulAction","toMulAction"],["Eq","ndrec"],["Semiring","toModule"],["NonAssocSemiring","toAddCommMonoidWithOne"],["TensorProduct","smulCommClass_left"],["TensorProduct"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["TensorProduct","instSMul","_proof_1"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["LinearMap","module"],["And"],["TensorProduct","leftModule"],["LinearMap"],["CommSemiring","toCommMonoid"],["Exists","casesOn"],["Set","image"],["Submodule","spanRank_span_le_card"],["Eq","refl"],["Submodule","spanRank"],["AddMonoidWithOne","toOne"],["HEq"],["AddCommGroup","toAddCommMonoid"],["TensorProduct","addCommMonoid"],["Eq","mpr"],["LinearMap","instFunLike"],["CommMonoid","toMonoid"],["CommRing","toNonUnitalCommRing"],["AddCommMonoid","toAddMonoid"],["DFunLike","coe"],["Submodule"],["congrArg"],["Algebra","toModule"],["MonoidWithZero","toMonoid"],["Cardinal","lift_umax"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Eq"],["Cardinal","instLE"],["CommRing","toCommSemiring"],["HEq","refl"],["Set"],["Cardinal","mk_image_le_lift"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["CommSemiring","toSemiring"],["Semiring","toMonoidWithZero"],["Cardinal","lift_id'"],["Eq","casesOn"],["Cardinal"],["OfNat","ofNat"],["LinearMap","addCommMonoid"],["CommRing","toRing"],["Module","toDistribMulAction"],["One","toOfNat1"],["LE","le"],["le_imp_le_of_le_of_le"],["And","casesOn"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Algebra","Module","SpanRankOperations",0,"IsLocalRing","spanFinrank_eq_finrank_quotient","_proof_1_3"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4 : CommRing.{u_1} R] {M : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15 : AddCommGroup.{u_2} M] [inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)] [inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.25 : IsLocalRing.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))] (N : Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18), IsScalarTower.{u_1, u_1, u_2} R (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.25) (HasQuotient.Quotient.{u_2, u_2} (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.hasQuotient.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) (Submodule.addCommGroup.{u_1, u_2} R M (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (HSMul.hSMul.{u_1, u_2, u_2} (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (instHSMul.{u_1, u_2} (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.instSMul.{u_1, u_1, u_2} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.isScalarTower'.{u_1, u_1, u_2, u_1} R R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N R (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (instDistribSMul.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (instDistribSMul.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (DistribMulAction.toDistribSMul.{u_1, u_1} R R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))))) (Module.toDistribMulAction.{u_1, u_1} R R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))) (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (Submodule.module._proof_1.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (IsScalarTower.left.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (DistribMulAction.toMulAction.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18))) (Submodule.module._proof_1.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (IsLocalRing.maximalIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.25) (Top.top.{u_2} (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.instTop.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N))))) (Algebra.toSMul.{u_1, u_1} R (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.25) (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) (DivisionSemiring.toSemiring.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.25) (Semifield.toDivisionSemiring.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.25) (Field.toSemifield.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.25) (IsLocalRing.ResidueField.field.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.25)))) (IsLocalRing.ResidueField.algebra.{u_1, u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.25 R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4 (Algebra.id.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (SMulZeroClass.toSMul.{u_1, u_2} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.25) (HasQuotient.Quotient.{u_2, u_2} (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.hasQuotient.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) (Submodule.addCommGroup.{u_1, u_2} R M (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (HSMul.hSMul.{u_1, u_2, u_2} (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (instHSMul.{u_1, u_2} (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.instSMul.{u_1, u_1, u_2} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.isScalarTower'.{u_1, u_1, u_2, u_1} R R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N R (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (instDistribSMul.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (instDistribSMul.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (DistribMulAction.toDistribSMul.{u_1, u_1} R R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))))) (Module.toDistribMulAction.{u_1, u_1} R R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))) (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (Submodule.module._proof_1.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (IsScalarTower.left.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (DistribMulAction.toMulAction.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18))) (Submodule.module._proof_1.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (IsLocalRing.maximalIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.25) (Top.top.{u_2} (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.instTop.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N))))) (AddZero.toZero.{u_2} (HasQuotient.Quotient.{u_2, u_2} (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.hasQuotient.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) (Submodule.addCommGroup.{u_1, u_2} R M (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (HSMul.hSMul.{u_1, u_2, u_2} (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (instHSMul.{u_1, u_2} (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.instSMul.{u_1, u_1, u_2} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.isScalarTower'.{u_1, u_1, u_2, u_1} R R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N R (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (instDistribSMul.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (instDistribSMul.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (DistribMulAction.toDistribSMul.{u_1, u_1} R R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))))) (Module.toDistribMulAction.{u_1, u_1} R R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))) (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (Submodule.module._proof_1.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (IsScalarTower.left.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (DistribMulAction.toMulAction.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18))) (Submodule.module._proof_1.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (IsLocalRing.maximalIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.25) (Top.top.{u_2} (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.instTop.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N))))) (AddZeroClass.toAddZero.{u_2} (HasQuotient.Quotient.{u_2, u_2} (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.hasQuotient.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) (Submodule.addCommGroup.{u_1, u_2} R M (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (HSMul.hSMul.{u_1, u_2, u_2} (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (instHSMul.{u_1, u_2} (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.instSMul.{u_1, u_1, u_2} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.isScalarTower'.{u_1, u_1, u_2, u_1} R R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N R (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (instDistribSMul.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (instDistribSMul.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (DistribMulAction.toDistribSMul.{u_1, u_1} R R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))))) (Module.toDistribMulAction.{u_1, u_1} R R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))) (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (Submodule.module._proof_1.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (IsScalarTower.left.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (DistribMulAction.toMulAction.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18))) (Submodule.module._proof_1.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (IsLocalRing.maximalIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.25) (Top.top.{u_2} (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.instTop.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N))))) (AddMonoid.toAddZeroClass.{u_2} (HasQuotient.Quotient.{u_2, u_2} (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.hasQuotient.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) (Submodule.addCommGroup.{u_1, u_2} R M (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (HSMul.hSMul.{u_1, u_2, u_2} (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (instHSMul.{u_1, u_2} (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.instSMul.{u_1, u_1, u_2} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.isScalarTower'.{u_1, u_1, u_2, u_1} R R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N R (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (instDistribSMul.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (instDistribSMul.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (DistribMulAction.toDistribSMul.{u_1, u_1} R R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))))) (Module.toDistribMulAction.{u_1, u_1} R R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))) (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (Submodule.module._proof_1.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (IsScalarTower.left.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (DistribMulAction.toMulAction.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18))) (Submodule.module._proof_1.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (IsLocalRing.maximalIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.25) (Top.top.{u_2} (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.instTop.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N))))) (SubNegMonoid.toAddMonoid.{u_2} (HasQuotient.Quotient.{u_2, u_2} (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.hasQuotient.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) (Submodule.addCommGroup.{u_1, u_2} R M (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (HSMul.hSMul.{u_1, u_2, u_2} (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (instHSMul.{u_1, u_2} (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.instSMul.{u_1, u_1, u_2} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.isScalarTower'.{u_1, u_1, u_2, u_1} R R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N R (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (instDistribSMul.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (instDistribSMul.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (DistribMulAction.toDistribSMul.{u_1, u_1} R R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))))) (Module.toDistribMulAction.{u_1, u_1} R R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))) (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (Submodule.module._proof_1.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (IsScalarTower.left.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (DistribMulAction.toMulAction.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18))) (Submodule.module._proof_1.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (IsLocalRing.maximalIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.25) (Top.top.{u_2} (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.instTop.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N))))) (AddGroup.toSubNegMonoid.{u_2} (HasQuotient.Quotient.{u_2, u_2} (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.hasQuotient.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) (Submodule.addCommGroup.{u_1, u_2} R M (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (HSMul.hSMul.{u_1, u_2, u_2} (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (instHSMul.{u_1, u_2} (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.instSMul.{u_1, u_1, u_2} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.isScalarTower'.{u_1, u_1, u_2, u_1} R R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N R (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (instDistribSMul.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (instDistribSMul.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (DistribMulAction.toDistribSMul.{u_1, u_1} R R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))))) (Module.toDistribMulAction.{u_1, u_1} R R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))) (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (Submodule.module._proof_1.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (IsScalarTower.left.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (DistribMulAction.toMulAction.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18))) (Submodule.module._proof_1.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (IsLocalRing.maximalIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.25) (Top.top.{u_2} (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.instTop.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N))))) (AddCommGroup.toAddGroup.{u_2} (HasQuotient.Quotient.{u_2, u_2} (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.hasQuotient.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) (Submodule.addCommGroup.{u_1, u_2} R M (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (HSMul.hSMul.{u_1, u_2, u_2} (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (instHSMul.{u_1, u_2} (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.instSMul.{u_1, u_1, u_2} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.isScalarTower'.{u_1, u_1, u_2, u_1} R R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N R (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (instDistribSMul.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (instDistribSMul.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (DistribMulAction.toDistribSMul.{u_1, u_1} R R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))))) (Module.toDistribMulAction.{u_1, u_1} R R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))) (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (Submodule.module._proof_1.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (IsScalarTower.left.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (DistribMulAction.toMulAction.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18))) (Submodule.module._proof_1.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (IsLocalRing.maximalIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.25) (Top.top.{u_2} (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.instTop.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N))))) (Submodule.Quotient.addCommGroup.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) (Submodule.addCommGroup.{u_1, u_2} R M (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (HSMul.hSMul.{u_1, u_2, u_2} (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (instHSMul.{u_1, u_2} (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.instSMul.{u_1, u_1, u_2} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.isScalarTower'.{u_1, u_1, u_2, u_1} R R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N R (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (instDistribSMul.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (instDistribSMul.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (DistribMulAction.toDistribSMul.{u_1, u_1} R R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))))) (Module.toDistribMulAction.{u_1, u_1} R R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))) (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (Submodule.module._proof_1.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (IsScalarTower.left.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (DistribMulAction.toMulAction.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18))) (Submodule.module._proof_1.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (IsLocalRing.maximalIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.25) (Top.top.{u_2} (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.instTop.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N))))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.25) (HasQuotient.Quotient.{u_2, u_2} (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.hasQuotient.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) (Submodule.addCommGroup.{u_1, u_2} R M (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (HSMul.hSMul.{u_1, u_2, u_2} (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (instHSMul.{u_1, u_2} (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.instSMul.{u_1, u_1, u_2} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.isScalarTower'.{u_1, u_1, u_2, u_1} R R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N R (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (instDistribSMul.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (instDistribSMul.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (DistribMulAction.toDistribSMul.{u_1, u_1} R R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))))) (Module.toDistribMulAction.{u_1, u_1} R R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))) (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (Submodule.module._proof_1.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (IsScalarTower.left.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (DistribMulAction.toMulAction.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18))) (Submodule.module._proof_1.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (IsLocalRing.maximalIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.25) (Top.top.{u_2} (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.instTop.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N))))) (AddMonoid.toAddZeroClass.{u_2} (HasQuotient.Quotient.{u_2, u_2} (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.hasQuotient.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) (Submodule.addCommGroup.{u_1, u_2} R M (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (HSMul.hSMul.{u_1, u_2, u_2} (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (instHSMul.{u_1, u_2} (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.instSMul.{u_1, u_1, u_2} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.isScalarTower'.{u_1, u_1, u_2, u_1} R R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N R (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (instDistribSMul.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (instDistribSMul.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (DistribMulAction.toDistribSMul.{u_1, u_1} R R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))))) (Module.toDistribMulAction.{u_1, u_1} R R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))) (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (Submodule.module._proof_1.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (IsScalarTower.left.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (DistribMulAction.toMulAction.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18))) (Submodule.module._proof_1.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (IsLocalRing.maximalIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.25) (Top.top.{u_2} (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.instTop.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N))))) (SubNegMonoid.toAddMonoid.{u_2} (HasQuotient.Quotient.{u_2, u_2} (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.hasQuotient.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) (Submodule.addCommGroup.{u_1, u_2} R M (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (HSMul.hSMul.{u_1, u_2, u_2} (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (instHSMul.{u_1, u_2} (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.instSMul.{u_1, u_1, u_2} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.isScalarTower'.{u_1, u_1, u_2, u_1} R R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N R (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (instDistribSMul.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (instDistribSMul.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (DistribMulAction.toDistribSMul.{u_1, u_1} R R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))))) (Module.toDistribMulAction.{u_1, u_1} R R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))) (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (Submodule.module._proof_1.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (IsScalarTower.left.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (DistribMulAction.toMulAction.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18))) (Submodule.module._proof_1.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (IsLocalRing.maximalIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.25) (Top.top.{u_2} (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.instTop.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N))))) (AddGroup.toSubNegMonoid.{u_2} (HasQuotient.Quotient.{u_2, u_2} (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.hasQuotient.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) (Submodule.addCommGroup.{u_1, u_2} R M (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (HSMul.hSMul.{u_1, u_2, u_2} (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (instHSMul.{u_1, u_2} (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.instSMul.{u_1, u_1, u_2} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.isScalarTower'.{u_1, u_1, u_2, u_1} R R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N R (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (instDistribSMul.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (instDistribSMul.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (DistribMulAction.toDistribSMul.{u_1, u_1} R R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))))) (Module.toDistribMulAction.{u_1, u_1} R R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))) (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (Submodule.module._proof_1.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (IsScalarTower.left.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (DistribMulAction.toMulAction.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18))) (Submodule.module._proof_1.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (IsLocalRing.maximalIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.25) (Top.top.{u_2} (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.instTop.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N))))) (AddCommGroup.toAddGroup.{u_2} (HasQuotient.Quotient.{u_2, u_2} (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.hasQuotient.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) (Submodule.addCommGroup.{u_1, u_2} R M (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (HSMul.hSMul.{u_1, u_2, u_2} (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (instHSMul.{u_1, u_2} (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.instSMul.{u_1, u_1, u_2} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.isScalarTower'.{u_1, u_1, u_2, u_1} R R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N R (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (instDistribSMul.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (instDistribSMul.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (DistribMulAction.toDistribSMul.{u_1, u_1} R R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))))) (Module.toDistribMulAction.{u_1, u_1} R R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))) (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (Submodule.module._proof_1.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (IsScalarTower.left.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (DistribMulAction.toMulAction.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18))) (Submodule.module._proof_1.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (IsLocalRing.maximalIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.25) (Top.top.{u_2} (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.instTop.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N))))) (Submodule.Quotient.addCommGroup.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) (Submodule.addCommGroup.{u_1, u_2} R M (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (HSMul.hSMul.{u_1, u_2, u_2} (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (instHSMul.{u_1, u_2} (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.instSMul.{u_1, u_1, u_2} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.isScalarTower'.{u_1, u_1, u_2, u_1} R R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N R (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (instDistribSMul.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (instDistribSMul.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (DistribMulAction.toDistribSMul.{u_1, u_1} R R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))))) (Module.toDistribMulAction.{u_1, u_1} R R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))) (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (Submodule.module._proof_1.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (IsScalarTower.left.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (DistribMulAction.toMulAction.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18))) (Submodule.module._proof_1.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (IsLocalRing.maximalIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.25) (Top.top.{u_2} (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.instTop.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N))))))))) (DistribMulAction.toDistribSMul.{u_1, u_2} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.25) (HasQuotient.Quotient.{u_2, u_2} (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.hasQuotient.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) (Submodule.addCommGroup.{u_1, u_2} R M (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (HSMul.hSMul.{u_1, u_2, u_2} (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (instHSMul.{u_1, u_2} (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.instSMul.{u_1, u_1, u_2} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.isScalarTower'.{u_1, u_1, u_2, u_1} R R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N R (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (instDistribSMul.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (instDistribSMul.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (DistribMulAction.toDistribSMul.{u_1, u_1} R R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))))) (Module.toDistribMulAction.{u_1, u_1} R R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))) (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (Submodule.module._proof_1.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (IsScalarTower.left.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (DistribMulAction.toMulAction.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18))) (Submodule.module._proof_1.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (IsLocalRing.maximalIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.25) (Top.top.{u_2} (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.instTop.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N))))) (MonoidWithZero.toMonoid.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.25) (Semiring.toMonoidWithZero.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.25) (DivisionSemiring.toSemiring.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.25) (Semifield.toDivisionSemiring.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.25) (Field.toSemifield.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.25) (IsLocalRing.ResidueField.field.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.25)))))) (SubNegMonoid.toAddMonoid.{u_2} (HasQuotient.Quotient.{u_2, u_2} (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.hasQuotient.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) (Submodule.addCommGroup.{u_1, u_2} R M (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (HSMul.hSMul.{u_1, u_2, u_2} (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (instHSMul.{u_1, u_2} (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.instSMul.{u_1, u_1, u_2} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.isScalarTower'.{u_1, u_1, u_2, u_1} R R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N R (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (instDistribSMul.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (instDistribSMul.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (DistribMulAction.toDistribSMul.{u_1, u_1} R R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))))) (Module.toDistribMulAction.{u_1, u_1} R R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))) (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (Submodule.module._proof_1.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (IsScalarTower.left.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (DistribMulAction.toMulAction.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18))) (Submodule.module._proof_1.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (IsLocalRing.maximalIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.25) (Top.top.{u_2} (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.instTop.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N))))) (AddGroup.toSubNegMonoid.{u_2} (HasQuotient.Quotient.{u_2, u_2} (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.hasQuotient.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) (Submodule.addCommGroup.{u_1, u_2} R M (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (HSMul.hSMul.{u_1, u_2, u_2} (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (instHSMul.{u_1, u_2} (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.instSMul.{u_1, u_1, u_2} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.isScalarTower'.{u_1, u_1, u_2, u_1} R R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N R (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (instDistribSMul.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (instDistribSMul.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (DistribMulAction.toDistribSMul.{u_1, u_1} R R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))))) (Module.toDistribMulAction.{u_1, u_1} R R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))) (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (Submodule.module._proof_1.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (IsScalarTower.left.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (DistribMulAction.toMulAction.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18))) (Submodule.module._proof_1.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (IsLocalRing.maximalIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.25) (Top.top.{u_2} (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.instTop.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N))))) (AddCommGroup.toAddGroup.{u_2} (HasQuotient.Quotient.{u_2, u_2} (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.hasQuotient.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) (Submodule.addCommGroup.{u_1, u_2} R M (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (HSMul.hSMul.{u_1, u_2, u_2} (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (instHSMul.{u_1, u_2} (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.instSMul.{u_1, u_1, u_2} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.isScalarTower'.{u_1, u_1, u_2, u_1} R R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N R (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (instDistribSMul.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (instDistribSMul.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (DistribMulAction.toDistribSMul.{u_1, u_1} R R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))))) (Module.toDistribMulAction.{u_1, u_1} R R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))) (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (Submodule.module._proof_1.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (IsScalarTower.left.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (DistribMulAction.toMulAction.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18))) (Submodule.module._proof_1.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (IsLocalRing.maximalIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.25) (Top.top.{u_2} (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.instTop.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N))))) (Submodule.Quotient.addCommGroup.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) (Submodule.addCommGroup.{u_1, u_2} R M (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (HSMul.hSMul.{u_1, u_2, u_2} (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (instHSMul.{u_1, u_2} (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.instSMul.{u_1, u_1, u_2} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.isScalarTower'.{u_1, u_1, u_2, u_1} R R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N R (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (instDistribSMul.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (instDistribSMul.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (DistribMulAction.toDistribSMul.{u_1, u_1} R R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))))) (Module.toDistribMulAction.{u_1, u_1} R R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))) (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (Submodule.module._proof_1.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (IsScalarTower.left.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (DistribMulAction.toMulAction.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18))) (Submodule.module._proof_1.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (IsLocalRing.maximalIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.25) (Top.top.{u_2} (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.instTop.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)))))))) (Module.toDistribMulAction.{u_1, u_2} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.25) (HasQuotient.Quotient.{u_2, u_2} (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.hasQuotient.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) (Submodule.addCommGroup.{u_1, u_2} R M (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (HSMul.hSMul.{u_1, u_2, u_2} (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (instHSMul.{u_1, u_2} (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.instSMul.{u_1, u_1, u_2} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.isScalarTower'.{u_1, u_1, u_2, u_1} R R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N R (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (instDistribSMul.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (instDistribSMul.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (DistribMulAction.toDistribSMul.{u_1, u_1} R R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))))) (Module.toDistribMulAction.{u_1, u_1} R R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))) (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (Submodule.module._proof_1.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (IsScalarTower.left.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (DistribMulAction.toMulAction.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18))) (Submodule.module._proof_1.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (IsLocalRing.maximalIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.25) (Top.top.{u_2} (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.instTop.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N))))) (DivisionSemiring.toSemiring.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.25) (Semifield.toDivisionSemiring.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.25) (Field.toSemifield.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.25) (IsLocalRing.ResidueField.field.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.25)))) (AddCommGroup.toAddCommMonoid.{u_2} (HasQuotient.Quotient.{u_2, u_2} (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.hasQuotient.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) (Submodule.addCommGroup.{u_1, u_2} R M (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (HSMul.hSMul.{u_1, u_2, u_2} (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (instHSMul.{u_1, u_2} (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.instSMul.{u_1, u_1, u_2} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.isScalarTower'.{u_1, u_1, u_2, u_1} R R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N R (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (instDistribSMul.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (instDistribSMul.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (DistribMulAction.toDistribSMul.{u_1, u_1} R R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))))) (Module.toDistribMulAction.{u_1, u_1} R R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))) (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (Submodule.module._proof_1.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (IsScalarTower.left.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (DistribMulAction.toMulAction.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18))) (Submodule.module._proof_1.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (IsLocalRing.maximalIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.25) (Top.top.{u_2} (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.instTop.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N))))) (Submodule.Quotient.addCommGroup.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) (Submodule.addCommGroup.{u_1, u_2} R M (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (HSMul.hSMul.{u_1, u_2, u_2} (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (instHSMul.{u_1, u_2} (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.instSMul.{u_1, u_1, u_2} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.isScalarTower'.{u_1, u_1, u_2, u_1} R R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N R (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (instDistribSMul.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (instDistribSMul.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (DistribMulAction.toDistribSMul.{u_1, u_1} R R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))))) (Module.toDistribMulAction.{u_1, u_1} R R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))) (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (Submodule.module._proof_1.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (IsScalarTower.left.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (DistribMulAction.toMulAction.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18))) (Submodule.module._proof_1.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (IsLocalRing.maximalIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.25) (Top.top.{u_2} (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.instTop.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)))))) (_private.Mathlib.Algebra.Module.SpanRankOperations.0.IsLocalRing.spanFinrank_eq_finrank_quotient._aux_1_1.{u_1, u_2} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4 M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.25 N))))) (Submodule.Quotient.instSMul.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) (Submodule.addCommGroup.{u_1, u_2} R M (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (HSMul.hSMul.{u_1, u_2, u_2} (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (instHSMul.{u_1, u_2} (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.instSMul.{u_1, u_1, u_2} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.isScalarTower'.{u_1, u_1, u_2, u_1} R R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N R (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (instDistribSMul.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (instDistribSMul.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R 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{R : Type.{u_1}} {A : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4 : CommRing.{u_1} R] [inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.7 : CommRing.{u_2} A] [inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.10 : Algebra.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.7))] {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.15 : AddCommGroup.{u_3} M] [inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.18 : Module.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.15)] (N : Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.18), (Submodule.FG.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.18 N) -> (LE.le.{0} Nat instLENat (Submodule.spanFinrank.{max u_2 u_3, u_2} A (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4) A (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.18) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.18)) N x)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} A (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} A (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} A (CommRing.toNonUnitalCommRing.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.7))))) (Submodule.addCommMonoid.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.18 N) (Algebra.toModule.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.10) (Submodule.module.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.18 N)) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.7)) (TensorProduct.addCommMonoid.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4) A (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.18) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.18)) N x)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} A (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} A (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} A (CommRing.toNonUnitalCommRing.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.7))))) (Submodule.addCommMonoid.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.18 N) (Algebra.toModule.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.10) (Submodule.module.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.18 N)) (TensorProduct.leftModule.{u_1, u_2, u_2, u_3} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.7)) A (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.18) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.18)) N x)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} A (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} A (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} A (CommRing.toNonUnitalCommRing.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.7))))) (Submodule.addCommMonoid.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.18 N) (Semiring.toModule.{u_2} A (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.7))) (Algebra.toModule.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.10) (Submodule.module.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.18 N) (Algebra.to_smulCommClass.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.10)) (Top.top.{max u_2 u_3} (Submodule.{u_2, max u_3 u_2} A (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4) A (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.18) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.18)) N x)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} A (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} A (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} A (CommRing.toNonUnitalCommRing.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.7))))) (Submodule.addCommMonoid.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.18 N) (Algebra.toModule.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.10) (Submodule.module.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.18 N)) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.7)) (TensorProduct.addCommMonoid.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4) A (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.18) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.18)) N x)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} A (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} A (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} A (CommRing.toNonUnitalCommRing.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.7))))) (Submodule.addCommMonoid.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.18 N) (Algebra.toModule.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.10) (Submodule.module.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.18 N)) (TensorProduct.leftModule.{u_1, u_2, u_2, u_3} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.7)) A (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.18) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.18)) N x)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} A (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} A (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} A (CommRing.toNonUnitalCommRing.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.7))))) (Submodule.addCommMonoid.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.18 N) (Semiring.toModule.{u_2} A (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.7))) (Algebra.toModule.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.10) (Submodule.module.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.18 N) (Algebra.to_smulCommClass.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.10))) (Submodule.instTop.{u_2, max u_2 u_3} A (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4) A (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.18) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.18)) N x)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} A (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} A (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} A (CommRing.toNonUnitalCommRing.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.7))))) (Submodule.addCommMonoid.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.18 N) (Algebra.toModule.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.10) (Submodule.module.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.18 N)) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.7)) (TensorProduct.addCommMonoid.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4) A (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.18) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.18)) N x)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} A (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} A (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} A (CommRing.toNonUnitalCommRing.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.7))))) (Submodule.addCommMonoid.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.18 N) (Algebra.toModule.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.10) (Submodule.module.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.18 N)) (TensorProduct.leftModule.{u_1, u_2, u_2, u_3} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.7)) A (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.18) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.18)) N x)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} A (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} A (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} A (CommRing.toNonUnitalCommRing.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.7))))) (Submodule.addCommMonoid.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.18 N) (Semiring.toModule.{u_2} A (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.7))) (Algebra.toModule.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.10) (Submodule.module.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.18 N) (Algebra.to_smulCommClass.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.10))))) (Submodule.spanFinrank.{u_3, u_1} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3069699780._hygCtx._hyg.18 N))","typeFull":"∀ {R : Type u_1} {A : Type u_2} [inst : CommRing R] [inst_1 : CommRing A] [inst_2 : Algebra R A] {M : Type u_3}\n  [inst_3 : AddCommGroup M] [inst_4 : Module R M] (N : Submodule R M), N.FG → ⊤.spanFinrank ≤ N.spanFinrank","typeReadable":"∀ {R : Type u_1} {A : Type u_2} [inst : CommRing R] [inst_1 : CommRing A] [inst_2 : Algebra R A] {M : Type u_3}\n  [inst_3 : AddCommGroup M] [inst_4 : Module R M] (N : Submodule R M), N.FG → ⊤.spanFinrank ≤ N.spanFinrank","typeReferences":[["Subtype"],["Module"],["Submodule","module"],["Membership","mem"],["CommRing","toNonUnitalCommRing"],["Algebra","to_smulCommClass"],["Submodule"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["Algebra","toModule"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["Semiring","toModule"],["Submodule","spanFinrank"],["CommRing","toCommSemiring"],["TensorProduct"],["SetLike","instMembership"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["Submodule","instTop"],["CommSemiring","toSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["AddCommGroup"],["TensorProduct","leftModule"],["CommRing"],["Submodule","FG"],["Algebra"],["Nat"],["Submodule","setLike"],["LE","le"],["Top","top"],["AddCommGroup","toAddCommMonoid"],["TensorProduct","addCommMonoid"],["Submodule","addCommMonoid"],["instLENat"]],"valueReferences":[["le_refl"],["Subtype"],["Submodule","module"],["Membership","mem"],["Submodule","baseChange"],["CommRing","toNonUnitalCommRing"],["Algebra","to_smulCommClass"],["Nat","instPreorder"],["congrArg"],["Submodule"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["Semiring","toNonAssocSemiring"],["Submodule","fg_top"],["Algebra","toModule"],["Submodule","FG","spanFinrank_baseChange_le"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["Eq","symm"],["Semiring","toModule"],["Submodule","spanFinrank_top"],["Submodule","spanFinrank"],["CommRing","toCommSemiring"],["TensorProduct"],["SetLike","instMembership"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["Submodule","addCommGroup"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Submodule","instTop"],["CommSemiring","toSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["TensorProduct","leftModule"],["Submodule","baseChange_top"],["Submodule","FG"],["CommRing","toRing"],["Nat"],["Submodule","setLike"],["Iff","mpr"],["LE","le"],["Top","top"],["AddCommGroup","toAddCommMonoid"],["le_imp_le_of_le_of_le"],["Eq","mpr"],["TensorProduct","addCommMonoid"],["Submodule","addCommMonoid"],["instLENat"]]},{"isProp":true,"kind":"theorem","name":["IsLocalRing","spanFinrank_maximalIdeal_eq_finrank_cotangentSpace_of_fg"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.SpanRankOperations.2865327830._hygCtx._hyg.4 : CommRing.{u_1} R] [inst._@.Mathlib.Algebra.Module.SpanRankOperations.2865327830._hygCtx._hyg.25 : IsLocalRing.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2865327830._hygCtx._hyg.4))], (Ideal.FG.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2865327830._hygCtx._hyg.4)) (IsLocalRing.maximalIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2865327830._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2865327830._hygCtx._hyg.25)) -> (Eq.{1} Nat (Submodule.spanFinrank.{u_1, u_1} R R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2865327830._hygCtx._hyg.4)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2865327830._hygCtx._hyg.4))))) (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2865327830._hygCtx._hyg.4))) (IsLocalRing.maximalIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2865327830._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2865327830._hygCtx._hyg.25)) (Module.finrank.{u_1, u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2865327830._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.2865327830._hygCtx._hyg.25) (IsLocalRing.CotangentSpace.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2865327830._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.2865327830._hygCtx._hyg.25) (DivisionSemiring.toSemiring.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2865327830._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.2865327830._hygCtx._hyg.25) (Semifield.toDivisionSemiring.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2865327830._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.2865327830._hygCtx._hyg.25) (Field.toSemifield.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2865327830._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.2865327830._hygCtx._hyg.25) (IsLocalRing.ResidueField.field.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2865327830._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.2865327830._hygCtx._hyg.25)))) (AddCommGroup.toAddCommMonoid.{u_1} (IsLocalRing.CotangentSpace.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2865327830._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.2865327830._hygCtx._hyg.25) (Ideal.instAddCommGroupCotangent.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2865327830._hygCtx._hyg.4 (IsLocalRing.maximalIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2865327830._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2865327830._hygCtx._hyg.25))) (IsLocalRing.instModuleResidueFieldCotangentSpace.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2865327830._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.2865327830._hygCtx._hyg.25)))","typeFull":"∀ {R : Type u_1} [inst : CommRing R] [inst_1 : IsLocalRing R],\n  (IsLocalRing.maximalIdeal R).FG →\n    Submodule.spanFinrank (IsLocalRing.maximalIdeal R) =\n      Module.finrank (IsLocalRing.ResidueField R) (IsLocalRing.CotangentSpace R)","typeReadable":"∀ {R : Type u_1} [inst : CommRing R] [inst_1 : IsLocalRing R],\n  (IsLocalRing.maximalIdeal R).FG →\n    Submodule.spanFinrank (IsLocalRing.maximalIdeal R) =\n      Module.finrank (IsLocalRing.ResidueField R) (IsLocalRing.CotangentSpace R)","typeReferences":[["Submodule","spanFinrank"],["IsLocalRing","CotangentSpace"],["CommRing","toCommSemiring"],["IsLocalRing","instModuleResidueFieldCotangentSpace"],["CommSemiring","toSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["IsLocalRing","ResidueField"],["DivisionSemiring","toSemiring"],["IsLocalRing"],["CommRing"],["IsLocalRing","maximalIdeal"],["Ideal","FG"],["Nat"],["Ideal","instAddCommGroupCotangent"],["Module","finrank"],["Semiring","toNonAssocSemiring"],["AddCommGroup","toAddCommMonoid"],["Field","toSemifield"],["Semifield","toDivisionSemiring"],["Eq"],["Semiring","toModule"],["IsLocalRing","ResidueField","field"]],"valueReferences":[["Ring","toAddCommGroup"],["CommRing","toRing"],["CommRing","toCommSemiring"],["IsLocalRing","spanFinrank_eq_finrank_quotient"],["CommSemiring","toSemiring"],["Semiring","toModule"],["IsLocalRing","maximalIdeal"]]},{"isProp":true,"kind":"theorem","name":["IsLocalRing","spanFinrank_maximalIdeal_eq_finrank_cotangentSpace"],"typeFallback":"forall (R : Type.{u_1}) [inst._@.Mathlib.Algebra.Module.SpanRankOperations.3851403358._hygCtx._hyg.4 : CommRing.{u_1} R] [inst._@.Mathlib.Algebra.Module.SpanRankOperations.3851403358._hygCtx._hyg.25 : IsLocalRing.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3851403358._hygCtx._hyg.4))] [inst._@.Mathlib.Algebra.Module.SpanRankOperations.3851403358._hygCtx._hyg.28 : IsNoetherianRing.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3851403358._hygCtx._hyg.4))], Eq.{1} Nat (Submodule.spanFinrank.{u_1, u_1} R R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3851403358._hygCtx._hyg.4)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3851403358._hygCtx._hyg.4))))) (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3851403358._hygCtx._hyg.4))) (IsLocalRing.maximalIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3851403358._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3851403358._hygCtx._hyg.25)) (Module.finrank.{u_1, u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3851403358._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.3851403358._hygCtx._hyg.25) (IsLocalRing.CotangentSpace.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3851403358._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.3851403358._hygCtx._hyg.25) (DivisionSemiring.toSemiring.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3851403358._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.3851403358._hygCtx._hyg.25) (Semifield.toDivisionSemiring.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3851403358._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.3851403358._hygCtx._hyg.25) (Field.toSemifield.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3851403358._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.3851403358._hygCtx._hyg.25) (IsLocalRing.ResidueField.field.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3851403358._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.3851403358._hygCtx._hyg.25)))) (AddCommGroup.toAddCommMonoid.{u_1} (IsLocalRing.CotangentSpace.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3851403358._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.3851403358._hygCtx._hyg.25) (Ideal.instAddCommGroupCotangent.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3851403358._hygCtx._hyg.4 (IsLocalRing.maximalIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3851403358._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Module.SpanRankOperations.3851403358._hygCtx._hyg.25))) (IsLocalRing.instModuleResidueFieldCotangentSpace.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.3851403358._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.3851403358._hygCtx._hyg.25))","typeFull":"∀ (R : Type u_1) [inst : CommRing R] [inst_1 : IsLocalRing R] [IsNoetherianRing R],\n  Submodule.spanFinrank (IsLocalRing.maximalIdeal R) =\n    Module.finrank (IsLocalRing.ResidueField R) (IsLocalRing.CotangentSpace R)","typeReadable":"∀ (R : Type u_1) [inst : CommRing R] [inst_1 : IsLocalRing R] [IsNoetherianRing R],\n  Submodule.spanFinrank (IsLocalRing.maximalIdeal R) =\n    Module.finrank (IsLocalRing.ResidueField R) (IsLocalRing.CotangentSpace R)","typeReferences":[["Submodule","spanFinrank"],["IsLocalRing","CotangentSpace"],["CommRing","toCommSemiring"],["IsLocalRing","instModuleResidueFieldCotangentSpace"],["CommSemiring","toSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["IsLocalRing","ResidueField"],["DivisionSemiring","toSemiring"],["IsLocalRing"],["CommRing"],["IsLocalRing","maximalIdeal"],["IsNoetherianRing"],["Nat"],["Ideal","instAddCommGroupCotangent"],["Module","finrank"],["Semiring","toNonAssocSemiring"],["AddCommGroup","toAddCommMonoid"],["Field","toSemifield"],["Semifield","toDivisionSemiring"],["Eq"],["Semiring","toModule"],["IsLocalRing","ResidueField","field"]],"valueReferences":[["CommRing","toCommSemiring"],["Ideal","fg_of_isNoetherianRing"],["CommSemiring","toSemiring"],["IsLocalRing","spanFinrank_maximalIdeal_eq_finrank_cotangentSpace_of_fg"],["IsLocalRing","maximalIdeal"]]},{"isProp":true,"kind":"theorem","name":["IsLocalRing","spanFinrank_eq_finrank_quotient"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4 : CommRing.{u_1} R] {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15 : AddCommGroup.{u_3} M] [inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 : Module.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)] [inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.25 : IsLocalRing.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))] (N : Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18), (Submodule.FG.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) -> (Eq.{1} Nat (Submodule.spanFinrank.{u_3, u_1} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Module.finrank.{u_1, u_3} (HasQuotient.Quotient.{u_1, u_1} R (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Ideal.instHasQuotient_1.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) (IsLocalRing.maximalIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.25)) (HasQuotient.Quotient.{u_3, u_3} (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (Submodule.{u_1, u_3} R (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.hasQuotient.{u_1, u_3} R (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) (Submodule.addCommGroup.{u_1, u_3} R M (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (HSMul.hSMul.{u_1, u_3, u_3} (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Submodule.{u_1, u_3} R (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.{u_1, u_3} R (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (instHSMul.{u_1, u_3} (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Submodule.{u_1, u_3} R (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.instSMul.{u_1, u_1, u_3} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (Submodule.addCommMonoid.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.isScalarTower'.{u_1, u_1, u_3, u_1} R R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N R (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (instDistribSMul.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (SMulZeroClass.toSMul.{u_1, u_3} R M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))))) (DistribSMul.toSMulZeroClass.{u_1, u_3} R M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))) (DistribMulAction.toDistribSMul.{u_1, u_3} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_3} M (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (instDistribSMul.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (SMulZeroClass.toSMul.{u_1, u_3} R M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))))) (DistribSMul.toSMulZeroClass.{u_1, u_3} R M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))) (DistribMulAction.toDistribSMul.{u_1, u_3} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_3} M (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (DistribMulAction.toDistribSMul.{u_1, u_1} R R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))))) (Module.toDistribMulAction.{u_1, u_1} R R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))) (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (Submodule.module._proof_1.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (IsScalarTower.left.{u_1, u_3} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (DistribMulAction.toMulAction.{u_1, u_3} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_3} M (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18))) (Submodule.module._proof_1.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (IsLocalRing.maximalIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.25) (Top.top.{u_3} (Submodule.{u_1, u_3} R (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.instTop.{u_1, u_3} R (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N))))) (Ideal.Quotient.semiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4 (IsLocalRing.maximalIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.25)) (AddCommGroup.toAddCommMonoid.{u_3} (HasQuotient.Quotient.{u_3, u_3} (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (Submodule.{u_1, u_3} R (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.hasQuotient.{u_1, u_3} R (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) (Submodule.addCommGroup.{u_1, u_3} R M (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (HSMul.hSMul.{u_1, u_3, u_3} (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Submodule.{u_1, u_3} R (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.{u_1, u_3} R (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (instHSMul.{u_1, u_3} (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Submodule.{u_1, u_3} R (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.instSMul.{u_1, u_1, u_3} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (Submodule.addCommMonoid.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.isScalarTower'.{u_1, u_1, u_3, u_1} R R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N R (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (instDistribSMul.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (SMulZeroClass.toSMul.{u_1, u_3} R M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))))) (DistribSMul.toSMulZeroClass.{u_1, u_3} R M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))) (DistribMulAction.toDistribSMul.{u_1, u_3} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_3} M (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (instDistribSMul.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (SMulZeroClass.toSMul.{u_1, u_3} R M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))))) (DistribSMul.toSMulZeroClass.{u_1, u_3} R M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))) (DistribMulAction.toDistribSMul.{u_1, u_3} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_3} M (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (DistribMulAction.toDistribSMul.{u_1, u_1} R R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))))) (Module.toDistribMulAction.{u_1, u_1} R R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))) (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (Submodule.module._proof_1.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (IsScalarTower.left.{u_1, u_3} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (DistribMulAction.toMulAction.{u_1, u_3} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_3} M (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18))) (Submodule.module._proof_1.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (IsLocalRing.maximalIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.25) (Top.top.{u_3} (Submodule.{u_1, u_3} R (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.instTop.{u_1, u_3} R (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N))))) (Submodule.Quotient.addCommGroup.{u_1, u_3} R (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) (Submodule.addCommGroup.{u_1, u_3} R M (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (HSMul.hSMul.{u_1, u_3, u_3} (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Submodule.{u_1, u_3} R (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.{u_1, u_3} R (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (instHSMul.{u_1, u_3} (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Submodule.{u_1, u_3} R (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.instSMul.{u_1, u_1, u_3} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (Submodule.addCommMonoid.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.isScalarTower'.{u_1, u_1, u_3, u_1} R R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N R (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (instDistribSMul.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (SMulZeroClass.toSMul.{u_1, u_3} R M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))))) (DistribSMul.toSMulZeroClass.{u_1, u_3} R M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))) (DistribMulAction.toDistribSMul.{u_1, u_3} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_3} M (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (instDistribSMul.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (SMulZeroClass.toSMul.{u_1, u_3} R M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))))) (DistribSMul.toSMulZeroClass.{u_1, u_3} R M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))) (DistribMulAction.toDistribSMul.{u_1, u_3} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_3} M (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (DistribMulAction.toDistribSMul.{u_1, u_1} R R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))))) (Module.toDistribMulAction.{u_1, u_1} R R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))) (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (Submodule.module._proof_1.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (IsScalarTower.left.{u_1, u_3} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (DistribMulAction.toMulAction.{u_1, u_3} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_3} M (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18))) (Submodule.module._proof_1.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (IsLocalRing.maximalIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.25) (Top.top.{u_3} (Submodule.{u_1, u_3} R (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.instTop.{u_1, u_3} R (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)))))) (Module.instQuotientIdealSubmoduleHSMulTop.{u_1, u_3} R (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) (Submodule.addCommGroup.{u_1, u_3} R M (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (IsLocalRing.maximalIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.25) (Ideal.instIsTwoSided_1.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4 (IsLocalRing.maximalIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.25)))))","typeFull":"∀ {R : Type u_1} [inst : CommRing R] {M : Type u_3} [inst_1 : AddCommGroup M] [inst_2 : Module R M]\n  [inst_3 : IsLocalRing R] (N : Submodule R M),\n  N.FG → N.spanFinrank = Module.finrank (R ⧸ IsLocalRing.maximalIdeal R) (↥N ⧸ IsLocalRing.maximalIdeal R • ⊤)","typeReadable":"∀ {R : Type u_1} [inst : CommRing R] {M : Type u_3} [inst_1 : AddCommGroup M] [inst_2 : Module R M]\n  [inst_3 : IsLocalRing R] (N : Submodule R M),\n  N.FG → N.spanFinrank = Module.finrank (R ⧸ IsLocalRing.maximalIdeal R) (↥N ⧸ IsLocalRing.maximalIdeal R • ⊤)","typeReferences":[["Membership","mem"],["SMulZeroClass","toSMul"],["AddMonoidWithOne","toAddMonoid"],["instDistribSMul"],["Ideal","instHasQuotient_1"],["IsLocalRing","maximalIdeal"],["Module","finrank"],["Semiring","toNonAssocSemiring"],["Ideal","instIsTwoSided_1"],["DistribMulAction","toMulAction"],["Semiring","toModule"],["Submodule","instSMul"],["NonAssocSemiring","toAddCommMonoidWithOne"],["DistribSMul","toSMulZeroClass"],["Submodule","spanFinrank"],["Ideal","Quotient","semiring"],["SetLike","instMembership"],["Submodule","addCommGroup"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Submodule","module","_proof_1"],["DistribMulAction","toDistribSMul"],["AddZeroClass","toAddZero"],["Submodule","FG"],["Nat"],["HSMul","hSMul"],["Top","top"],["AddCommGroup","toAddCommMonoid"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Submodule","hasQuotient"],["Subtype"],["HasQuotient","Quotient"],["Module"],["Submodule","module"],["AddCommMonoid","toAddMonoid"],["Submodule"],["Submodule","Quotient","addCommGroup"],["Ideal"],["MonoidWithZero","toMonoid"],["instHSMul"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Eq"],["CommRing","toCommSemiring"],["IsScalarTower","left"],["Submodule","instTop"],["CommSemiring","toSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["IsLocalRing"],["CommRing"],["Submodule","isScalarTower'"],["CommRing","toRing"],["Module","toDistribMulAction"],["Submodule","setLike"],["Module","instQuotientIdealSubmoduleHSMulTop"],["Submodule","addCommMonoid"]],"valueReferences":[["IsLocalRing","instCommRingResidueField"],["Submodule","Quotient","module"],["LinearEquiv","extendScalarsOfSurjective"],["Membership","mem"],["SMulZeroClass","toSMul"],["LinearEquiv","finrank_eq"],["AddMonoidWithOne","toAddMonoid"],["instDistribSMul"],["Algebra","to_smulCommClass"],["Algebra","id"],["Ideal","instHasQuotient_1"],["TensorProduct","instModule"],["IsLocalRing","maximalIdeal"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["Module","finrank"],["Semiring","toNonAssocSemiring"],["Ideal","instIsTwoSided_1"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["Eq","symm"],["DistribMulAction","toMulAction"],["IsScalarTower","right"],["Semifield","toDivisionSemiring"],["Semiring","toModule"],["_private","Mathlib","Algebra","Module","SpanRankOperations",0,"IsLocalRing","spanFinrank_eq_finrank_quotient","_proof_1_3"],["Submodule","instSMul"],["NonAssocSemiring","toAddCommMonoidWithOne"],["DistribSMul","toSMulZeroClass"],["Submodule","spanFinrank"],["TensorProduct"],["Ideal","Quotient","semiring"],["SetLike","instMembership"],["Module","finrank_eq_spanFinrank_of_free"],["TensorProduct","spanFinrank_top_eq_of_residueField"],["TensorProduct","instSMul","_proof_1"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["Ideal","Quotient","commRing"],["Submodule","addCommGroup"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Submodule","module","_proof_1"],["DistribMulAction","toDistribSMul"],["TensorProduct","leftModule"],["DivisionSemiring","toSemiring"],["AddZeroClass","toAddZero"],["Nat"],["HSMul","hSMul"],["id"],["Top","top"],["AddCommGroup","toAddCommMonoid"],["TensorProduct","addCommMonoid"],["Eq","mpr"],["_private","Mathlib","Algebra","Module","SpanRankOperations",0,"IsLocalRing","spanFinrank_eq_finrank_quotient","_aux_1_1"],["Field","instIsLocalRing"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["IsLocalRing","residue_surjective"],["IsLocalRing","ResidueField","algebra"],["Submodule","hasQuotient"],["TensorProduct","quotTensorEquivQuotSMul"],["IsLocalRing","instModuleResidueFieldOfAlgebra"],["Subtype"],["HasQuotient","Quotient"],["Submodule","module"],["CommRing","toNonUnitalCommRing"],["AddCommMonoid","toAddMonoid"],["congrArg"],["Submodule"],["Submodule","Quotient","addCommGroup"],["commRing_strongRankCondition"],["Ideal"],["TensorProduct","isScalarTower_left"],["Algebra","toSMul"],["MonoidWithZero","toMonoid"],["Module","Free","of_divisionRing"],["instHSMul"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Eq"],["CommRing","toCommSemiring"],["IsScalarTower","left"],["DivisionRing","toRing"],["Submodule","instTop"],["CommSemiring","toSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["IsLocalRing","ResidueField"],["Field","toDivisionRing"],["IsLocalRing","toNontrivial"],["Semiring","toMonoidWithZero"],["Submodule","isScalarTower'"],["Ring","toAddCommGroup"],["CommRing","toRing"],["Module","toDistribMulAction"],["Submodule","setLike"],["TensorProduct","addCommGroup"],["Field","toSemifield"],["Module","instQuotientIdealSubmoduleHSMulTop"],["Submodule","addCommMonoid"],["IsLocalRing","ResidueField","field"]]},{"isProp":true,"kind":"theorem","name":["Submodule","FG","spanFinrank_baseChange_le"],"typeFallback":"forall {R : Type.{u_1}} {A : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.SpanRankOperations.384370287._hygCtx._hyg.4 : CommRing.{u_1} R] [inst._@.Mathlib.Algebra.Module.SpanRankOperations.384370287._hygCtx._hyg.7 : CommRing.{u_2} A] [inst._@.Mathlib.Algebra.Module.SpanRankOperations.384370287._hygCtx._hyg.10 : Algebra.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.384370287._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.384370287._hygCtx._hyg.7))] {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.SpanRankOperations.384370287._hygCtx._hyg.15 : AddCommGroup.{u_3} M] [inst._@.Mathlib.Algebra.Module.SpanRankOperations.384370287._hygCtx._hyg.18 : Module.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.384370287._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.384370287._hygCtx._hyg.15)] (N : Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.384370287._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.384370287._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.384370287._hygCtx._hyg.18), (Submodule.FG.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.384370287._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.384370287._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.384370287._hygCtx._hyg.18 N) -> (LE.le.{0} Nat instLENat (Submodule.spanFinrank.{max u_2 u_3, u_2} A (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.384370287._hygCtx._hyg.4) A M (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.384370287._hygCtx._hyg.7))))) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.384370287._hygCtx._hyg.15) (Algebra.toModule.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.384370287._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.384370287._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.SpanRankOperations.384370287._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Module.SpanRankOperations.384370287._hygCtx._hyg.18) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.384370287._hygCtx._hyg.7)) (TensorProduct.addCommMonoid.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.384370287._hygCtx._hyg.4) A M (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.384370287._hygCtx._hyg.7))))) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.384370287._hygCtx._hyg.15) (Algebra.toModule.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.384370287._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.384370287._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.SpanRankOperations.384370287._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Module.SpanRankOperations.384370287._hygCtx._hyg.18) (TensorProduct.leftModule.{u_1, u_2, u_2, u_3} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.384370287._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.384370287._hygCtx._hyg.7)) A M (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} A (Semiring.toNonAssocSemiring.{u_2} A (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.384370287._hygCtx._hyg.7))))) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.384370287._hygCtx._hyg.15) (Semiring.toModule.{u_2} A (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.384370287._hygCtx._hyg.7))) (Algebra.toModule.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.384370287._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.384370287._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.SpanRankOperations.384370287._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Module.SpanRankOperations.384370287._hygCtx._hyg.18 (Algebra.to_smulCommClass.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.384370287._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.384370287._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.SpanRankOperations.384370287._hygCtx._hyg.10)) (Submodule.baseChange.{u_1, u_3, u_2} R M A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.384370287._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.384370287._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.SpanRankOperations.384370287._hygCtx._hyg.10 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.384370287._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.384370287._hygCtx._hyg.18 N)) (Submodule.spanFinrank.{u_3, u_1} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.384370287._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.384370287._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.384370287._hygCtx._hyg.18 N))","typeFull":"∀ {R : Type u_1} {A : Type u_2} [inst : CommRing R] [inst_1 : CommRing A] [inst_2 : Algebra R A] {M : Type u_3}\n  [inst_3 : AddCommGroup M] [inst_4 : Module R M] (N : Submodule R M),\n  N.FG → (Submodule.baseChange A N).spanFinrank ≤ N.spanFinrank","typeReadable":"∀ {R : Type u_1} {A : Type u_2} [inst : CommRing R] [inst_1 : CommRing A] [inst_2 : Algebra R A] {M : Type u_3}\n  [inst_3 : AddCommGroup M] [inst_4 : Module R M] (N : Submodule R M),\n  N.FG → (Submodule.baseChange A N).spanFinrank ≤ N.spanFinrank","typeReferences":[["Submodule","spanFinrank"],["TensorProduct"],["CommRing","toCommSemiring"],["Module"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["CommSemiring","toSemiring"],["Submodule","baseChange"],["AddCommGroup"],["TensorProduct","leftModule"],["CommRing"],["Algebra","to_smulCommClass"],["Submodule","FG"],["Algebra"],["Submodule"],["Nat"],["Semiring","toNonAssocSemiring"],["Algebra","toModule"],["LE","le"],["AddCommGroup","toAddCommMonoid"],["TensorProduct","addCommMonoid"],["Semiring","toModule"],["instLENat"]],"valueReferences":[["le_refl"],["Submodule","spanFinrank","eq_1"],["PartialOrder","toPreorder"],["MonoidWithZeroHom","funLike"],["Cardinal","lift"],["Eq","trans"],["Preorder","toLT"],["eq_true"],["Submodule","baseChange"],["GT","gt"],["Algebra","to_smulCommClass"],["Cardinal","partialOrder"],["Cardinal","commSemiring"],["Submodule","spanRank_baseChange_le"],["Semiring","toNonAssocSemiring"],["MonoidWithZeroHom"],["Semiring","toModule"],["NonAssocSemiring","toMulZeroOneClass"],["Submodule","spanFinrank"],["TensorProduct"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["TensorProduct","leftModule"],["Submodule","FG"],["Nat"],["Submodule","spanRank"],["Iff","mpr"],["gt_iff_lt","_simp_1"],["AddCommGroup","toAddCommMonoid"],["TensorProduct","addCommMonoid"],["Eq","mpr"],["Cardinal","toNat_lift"],["Cardinal","toNat_le_toNat"],["Cardinal","aleph0"],["DFunLike","coe"],["congrArg"],["Nat","instPreorder"],["Algebra","toModule"],["CommRing","toCommSemiring"],["Cardinal","toNat"],["Nat","instMulZeroOneClass"],["True"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["CommSemiring","toSemiring"],["Cardinal"],["LT","lt"],["of_eq_true"],["LE","le"],["le_imp_le_of_le_of_le"],["Submodule","spanRank_finite_iff_fg"],["instLENat"],["Cardinal","lift_lt_aleph0","_simp_1"]]},{"isProp":true,"kind":"theorem","name":["TensorProduct","spanRank_top_le"],"typeFallback":"forall {R : Type.{u_1}} {A : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4 : CommRing.{u_1} R] [inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.7 : CommRing.{u_2} A] [inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.10 : Algebra.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.7))] {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.15 : AddCommGroup.{u_3} M] [inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.18 : Module.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.15)] (N : Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.18), LE.le.{max (succ u_2) (succ u_3)} Cardinal.{max u_2 u_3} Cardinal.instLE.{max u_2 u_3} (Submodule.spanRank.{max u_2 u_3, u_2} A (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4) A (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.18) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.18)) N x)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} A (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} A (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} A (CommRing.toNonUnitalCommRing.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.7))))) (Submodule.addCommMonoid.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.18 N) (Algebra.toModule.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.10) (Submodule.module.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.18 N)) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.7)) (TensorProduct.addCommMonoid.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4) A (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.18) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.18)) N x)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} A (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} A (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} A (CommRing.toNonUnitalCommRing.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.7))))) (Submodule.addCommMonoid.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.18 N) (Algebra.toModule.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.10) (Submodule.module.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.18 N)) (TensorProduct.leftModule.{u_1, u_2, u_2, u_3} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.7)) A (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.18) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.18)) N x)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} A (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} A (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} A (CommRing.toNonUnitalCommRing.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.7))))) (Submodule.addCommMonoid.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.18 N) (Semiring.toModule.{u_2} A (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.7))) (Algebra.toModule.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.10) (Submodule.module.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.18 N) (Algebra.to_smulCommClass.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.10)) (Top.top.{max u_2 u_3} (Submodule.{u_2, max u_3 u_2} A (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4) A (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.18) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.18)) N x)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} A (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} A (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} A (CommRing.toNonUnitalCommRing.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.7))))) (Submodule.addCommMonoid.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.18 N) (Algebra.toModule.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.10) (Submodule.module.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.18 N)) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.7)) (TensorProduct.addCommMonoid.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4) A (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.18) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.18)) N x)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} A (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} A (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} A (CommRing.toNonUnitalCommRing.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.7))))) (Submodule.addCommMonoid.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.18 N) (Algebra.toModule.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.10) (Submodule.module.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.18 N)) (TensorProduct.leftModule.{u_1, u_2, u_2, u_3} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.7)) A (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.18) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.18)) N x)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} A (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} A (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} A (CommRing.toNonUnitalCommRing.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.7))))) (Submodule.addCommMonoid.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.18 N) (Semiring.toModule.{u_2} A (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.7))) (Algebra.toModule.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.10) (Submodule.module.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.18 N) (Algebra.to_smulCommClass.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.10))) (Submodule.instTop.{u_2, max u_2 u_3} A (TensorProduct.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4) A (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.18) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.18)) N x)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} A (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} A (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} A (CommRing.toNonUnitalCommRing.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.7))))) (Submodule.addCommMonoid.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.18 N) (Algebra.toModule.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.10) (Submodule.module.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.18 N)) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.7)) (TensorProduct.addCommMonoid.{u_1, u_2, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4) A (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.18) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.18)) N x)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} A (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} A (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} A (CommRing.toNonUnitalCommRing.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.7))))) (Submodule.addCommMonoid.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.18 N) (Algebra.toModule.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.10) (Submodule.module.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.18 N)) (TensorProduct.leftModule.{u_1, u_2, u_2, u_3} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.7)) A (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.18) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.18)) N x)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} A (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_2} A (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_2} A (CommRing.toNonUnitalCommRing.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.7))))) (Submodule.addCommMonoid.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.18 N) (Semiring.toModule.{u_2} A (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.7))) (Algebra.toModule.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.10) (Submodule.module.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.18 N) (Algebra.to_smulCommClass.{u_1, u_2} R A (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4) (CommSemiring.toSemiring.{u_2} A (CommRing.toCommSemiring.{u_2} A inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.7)) inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.10))))) (Cardinal.lift.{u_2, u_3} (Submodule.spanRank.{u_3, u_1} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.479750512._hygCtx._hyg.18 N))","typeFull":"∀ {R : Type u_1} {A : Type u_2} [inst : CommRing R] [inst_1 : CommRing A] [inst_2 : Algebra R A] {M : Type u_3}\n  [inst_3 : AddCommGroup M] [inst_4 : Module R M] (N : Submodule R M), ⊤.spanRank ≤ Cardinal.lift.{u_2, u_3} N.spanRank","typeReadable":"∀ {R : Type u_1} {A : Type u_2} [inst : CommRing R] [inst_1 : CommRing A] [inst_2 : Algebra R A] {M : Type u_3}\n  [inst_3 : AddCommGroup M] [inst_4 : Module R M] (N : Submodule R M), ⊤.spanRank ≤ Cardinal.lift.{u_2, u_3} N.spanRank","typeReferences":[["Subtype"],["Cardinal","lift"],["Module"],["Submodule","module"],["Membership","mem"],["CommRing","toNonUnitalCommRing"],["Algebra","to_smulCommClass"],["Submodule"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["Algebra","toModule"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["Semiring","toModule"],["Cardinal","instLE"],["TensorProduct"],["CommRing","toCommSemiring"],["SetLike","instMembership"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["Submodule","instTop"],["CommSemiring","toSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["AddCommGroup"],["TensorProduct","leftModule"],["CommRing"],["Cardinal"],["Algebra"],["Submodule","setLike"],["Submodule","spanRank"],["LE","le"],["Top","top"],["AddCommGroup","toAddCommMonoid"],["TensorProduct","addCommMonoid"],["Submodule","addCommMonoid"]],"valueReferences":[["le_refl"],["PartialOrder","toPreorder"],["Subtype"],["Cardinal","lift"],["Submodule","module"],["Membership","mem"],["Submodule","baseChange"],["CommRing","toNonUnitalCommRing"],["Algebra","to_smulCommClass"],["Cardinal","partialOrder"],["congrArg"],["Submodule"],["Submodule","spanRank_baseChange_le"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["Semiring","toNonAssocSemiring"],["Algebra","toModule"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["Eq","symm"],["Semiring","toModule"],["Cardinal","instLE"],["Submodule","spanRank_top"],["CommRing","toCommSemiring"],["TensorProduct"],["SetLike","instMembership"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["Submodule","addCommGroup"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Submodule","instTop"],["CommSemiring","toSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["TensorProduct","leftModule"],["Submodule","baseChange_top"],["Cardinal"],["CommRing","toRing"],["Submodule","setLike"],["Submodule","spanRank"],["LE","le"],["Top","top"],["AddCommGroup","toAddCommMonoid"],["le_imp_le_of_le_of_le"],["Eq","mpr"],["TensorProduct","addCommMonoid"],["Submodule","addCommMonoid"]]},{"isProp":false,"kind":"definition","name":["_private","Mathlib","Algebra","Module","SpanRankOperations",0,"_aux_Mathlib_Algebra_Module_SpanRankOperations___unexpand_IsLocalRing_ResidueField_1"],"typeFallback":"Lean.PrettyPrinter.Unexpander","typeFull":"Lean.PrettyPrinter.Unexpander","typeReadable":"Lean.PrettyPrinter.Unexpander","typeReferences":[["Lean","PrettyPrinter","Unexpander"]],"valueReferences":[["Lean","Name"],["Lean","TSyntax","raw"],["Bool","false"],["Lean","MonadQuotation","getCurrMacroScope"],["Lean","MacroScope"],["Lean","SourceInfo"],["Lean","MonadQuotation","toMonadRef"],["Lean","MonadRef","mkInfoFromRefPos"],["ReaderT","instApplicativeOfMonad"],["List","cons"],["Bool","true"],["Unit","unit"],["MonadExcept","throw"],["Lean","Name","mkStr4"],["Lean","SyntaxNodeKind"],["Lean","TSyntax","mk"],["EStateM","instMonad"],["Lean","PrettyPrinter","instMonadQuotationUnexpandM"],["instMonadExceptOfMonadExceptOf"],["Lean","Syntax"],["Bool","or"],["cond"],["Lean","Name","mkStr"],["Unit"],["instDecidableEqBool"],["Nat"],["Monad","toBind"],["Lean","Syntax","atom"],["Lean","withRef"],["Bool"],["Lean","Name","anonymous"],["Applicative","toPure"],["ReaderT","instMonadExceptOf"],["Lean","PrettyPrinter","UnexpandM"],["EStateM"],["Lean","Syntax","isOfKind"],["Lean","Syntax","matchesNull"],["PUnit"],["instOfNatNat"],["Lean","Name","mkNum"],["Lean","MonadQuotation","getContext"],["Eq"],["EStateM","instMonadExceptOfOfBacktrackable"],["List","nil"],["Bind","bind"],["Lean","Name","mkStr1"],["ite"],["Lean","Syntax","matchesIdent"],["Lean","Syntax","node1"],["ReaderT","instMonad"],["OfNat","ofNat"],["EStateM","nonBacktrackable"],["Lean","Syntax","getArg"],["Pure","pure"]]},{"isProp":true,"kind":"theorem","name":["TensorProduct","spanFinrank_top_eq_of_residueField"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 : CommRing.{u_1} R] {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.15 : AddCommGroup.{u_3} M] [inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.18 : Module.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.15)] (N : Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.18) [inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25 : IsLocalRing.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4))], (Submodule.FG.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.18 N) -> (Eq.{1} Nat (Submodule.spanFinrank.{max u_1 u_3, u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (TensorProduct.{u_1, u_1, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4) (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.18) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.18)) N x)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (CommRing.toNonUnitalCommRing.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (IsLocalRing.instCommRingResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25)))))) (Submodule.addCommMonoid.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.18 N) (IsLocalRing.instModuleResidueFieldOfAlgebra.{u_1, u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25 R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 (Algebra.id.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4))) (Submodule.module.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.18 N)) (DivisionSemiring.toSemiring.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (Semifield.toDivisionSemiring.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (Field.toSemifield.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (IsLocalRing.ResidueField.field.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25)))) (TensorProduct.addCommMonoid.{u_1, u_1, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4) (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.18) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.18)) N x)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (CommRing.toNonUnitalCommRing.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (IsLocalRing.instCommRingResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25)))))) (Submodule.addCommMonoid.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.18 N) (IsLocalRing.instModuleResidueFieldOfAlgebra.{u_1, u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25 R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 (Algebra.id.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4))) (Submodule.module.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.18 N)) (TensorProduct.leftModule.{u_1, u_1, u_1, u_3} R (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4) (DivisionSemiring.toSemiring.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (Semifield.toDivisionSemiring.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (Field.toSemifield.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (IsLocalRing.ResidueField.field.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25)))) (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.18) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.18)) N x)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (CommRing.toNonUnitalCommRing.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (IsLocalRing.instCommRingResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25)))))) (Submodule.addCommMonoid.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.18 N) (Semiring.toModule.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (DivisionSemiring.toSemiring.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (Semifield.toDivisionSemiring.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (Field.toSemifield.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (IsLocalRing.ResidueField.field.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25))))) (IsLocalRing.instModuleResidueFieldOfAlgebra.{u_1, u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25 R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 (Algebra.id.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4))) (Submodule.module.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.18 N) (Algebra.to_smulCommClass.{u_1, u_1} R (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4) (DivisionSemiring.toSemiring.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (Semifield.toDivisionSemiring.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (Field.toSemifield.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (IsLocalRing.ResidueField.field.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25)))) (IsLocalRing.ResidueField.algebra.{u_1, u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25 R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 (Algebra.id.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4))))) (Top.top.{max u_1 u_3} (Submodule.{u_1, max u_3 u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (TensorProduct.{u_1, u_1, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4) (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.18) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.18)) N x)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (CommRing.toNonUnitalCommRing.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (IsLocalRing.instCommRingResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25)))))) (Submodule.addCommMonoid.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.18 N) (IsLocalRing.instModuleResidueFieldOfAlgebra.{u_1, u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25 R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 (Algebra.id.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4))) (Submodule.module.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.18 N)) (DivisionSemiring.toSemiring.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (Semifield.toDivisionSemiring.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (Field.toSemifield.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (IsLocalRing.ResidueField.field.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25)))) (TensorProduct.addCommMonoid.{u_1, u_1, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4) (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.18) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.18)) N x)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (CommRing.toNonUnitalCommRing.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (IsLocalRing.instCommRingResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25)))))) (Submodule.addCommMonoid.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.18 N) (IsLocalRing.instModuleResidueFieldOfAlgebra.{u_1, u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25 R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 (Algebra.id.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4))) (Submodule.module.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.18 N)) (TensorProduct.leftModule.{u_1, u_1, u_1, u_3} R (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4) (DivisionSemiring.toSemiring.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (Semifield.toDivisionSemiring.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (Field.toSemifield.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (IsLocalRing.ResidueField.field.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25)))) (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.18) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.18)) N x)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (CommRing.toNonUnitalCommRing.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (IsLocalRing.instCommRingResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25)))))) (Submodule.addCommMonoid.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.18 N) (Semiring.toModule.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (DivisionSemiring.toSemiring.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (Semifield.toDivisionSemiring.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (Field.toSemifield.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (IsLocalRing.ResidueField.field.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25))))) (IsLocalRing.instModuleResidueFieldOfAlgebra.{u_1, u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25 R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 (Algebra.id.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4))) (Submodule.module.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.18 N) (Algebra.to_smulCommClass.{u_1, u_1} R (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4) (DivisionSemiring.toSemiring.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (Semifield.toDivisionSemiring.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (Field.toSemifield.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (IsLocalRing.ResidueField.field.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25)))) (IsLocalRing.ResidueField.algebra.{u_1, u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25 R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 (Algebra.id.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4)))))) (Submodule.instTop.{u_1, max u_1 u_3} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (TensorProduct.{u_1, u_1, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4) (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.18) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.18)) N x)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (CommRing.toNonUnitalCommRing.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (IsLocalRing.instCommRingResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25)))))) (Submodule.addCommMonoid.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.18 N) (IsLocalRing.instModuleResidueFieldOfAlgebra.{u_1, u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25 R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 (Algebra.id.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4))) (Submodule.module.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.18 N)) (DivisionSemiring.toSemiring.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (Semifield.toDivisionSemiring.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (Field.toSemifield.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (IsLocalRing.ResidueField.field.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25)))) (TensorProduct.addCommMonoid.{u_1, u_1, u_3} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4) (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.18) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.18)) N x)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (CommRing.toNonUnitalCommRing.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (IsLocalRing.instCommRingResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25)))))) (Submodule.addCommMonoid.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.18 N) (IsLocalRing.instModuleResidueFieldOfAlgebra.{u_1, u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25 R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 (Algebra.id.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4))) (Submodule.module.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.18 N)) (TensorProduct.leftModule.{u_1, u_1, u_1, u_3} R (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4) (DivisionSemiring.toSemiring.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (Semifield.toDivisionSemiring.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (Field.toSemifield.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (IsLocalRing.ResidueField.field.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25)))) (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.18) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.18)) N x)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (CommRing.toNonUnitalCommRing.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (IsLocalRing.instCommRingResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25)))))) (Submodule.addCommMonoid.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.18 N) (Semiring.toModule.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (DivisionSemiring.toSemiring.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (Semifield.toDivisionSemiring.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (Field.toSemifield.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (IsLocalRing.ResidueField.field.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25))))) (IsLocalRing.instModuleResidueFieldOfAlgebra.{u_1, u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25 R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 (Algebra.id.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4))) (Submodule.module.{u_1, u_3} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.18 N) (Algebra.to_smulCommClass.{u_1, u_1} R (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4) (DivisionSemiring.toSemiring.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (Semifield.toDivisionSemiring.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (Field.toSemifield.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25) (IsLocalRing.ResidueField.field.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25)))) (IsLocalRing.ResidueField.algebra.{u_1, u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.25 R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4 (Algebra.id.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4)))))))) (Submodule.spanFinrank.{u_3, u_1} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.458448670._hygCtx._hyg.18 N))","typeFull":"∀ {R : Type u_1} [inst : CommRing R] {M : Type u_3} [inst_1 : AddCommGroup M] [inst_2 : Module R M] (N : Submodule R M)\n  [inst_3 : IsLocalRing R], N.FG → ⊤.spanFinrank = N.spanFinrank","typeReadable":"∀ {R : Type u_1} [inst : CommRing R] {M : Type u_3} [inst_1 : AddCommGroup M] [inst_2 : Module R M] (N : Submodule R M)\n  [inst_3 : IsLocalRing R], N.FG → ⊤.spanFinrank = 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{R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4 : CommRing.{u_1} R] {M : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15 : AddCommGroup.{u_2} M] [inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)] [inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.25 : IsLocalRing.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))] (N : Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R 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inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (instHSMul.{u_1, u_2} (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.instSMul.{u_1, u_1, u_2} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.isScalarTower'.{u_1, u_1, u_2, u_1} R R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N R (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (instDistribSMul.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (instDistribSMul.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (DistribMulAction.toDistribSMul.{u_1, u_1} R R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))))) (Module.toDistribMulAction.{u_1, u_1} R R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))) (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (Submodule.module._proof_1.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (IsScalarTower.left.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (DistribMulAction.toMulAction.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18))) (Submodule.module._proof_1.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (IsLocalRing.maximalIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.25) (Top.top.{u_2} (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.instTop.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N))))) (DivisionSemiring.toSemiring.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.25) (Semifield.toDivisionSemiring.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.25) (Field.toSemifield.{u_1} (IsLocalRing.ResidueField.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.25) (IsLocalRing.ResidueField.field.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.25)))) (AddCommGroup.toAddCommMonoid.{u_2} (HasQuotient.Quotient.{u_2, u_2} (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.hasQuotient.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) (Submodule.addCommGroup.{u_1, u_2} R M (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (HSMul.hSMul.{u_1, u_2, u_2} (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (instHSMul.{u_1, u_2} (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.instSMul.{u_1, u_1, u_2} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.isScalarTower'.{u_1, u_1, u_2, u_1} R R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N R (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (instDistribSMul.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (instDistribSMul.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (DistribMulAction.toDistribSMul.{u_1, u_1} R R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))))) (Module.toDistribMulAction.{u_1, u_1} R R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))) (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (Submodule.module._proof_1.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (IsScalarTower.left.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (DistribMulAction.toMulAction.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18))) (Submodule.module._proof_1.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (IsLocalRing.maximalIdeal.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.25) (Top.top.{u_2} (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.instTop.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N))))) (Submodule.Quotient.addCommGroup.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) (Submodule.addCommGroup.{u_1, u_2} R M (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (HSMul.hSMul.{u_1, u_2, u_2} (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (instHSMul.{u_1, u_2} (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Submodule.{u_1, u_2} R (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N)) (Submodule.instSMul.{u_1, u_1, u_2} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))) (Subtype.{succ u_2} M (fun (x : M) => Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) M (Submodule.setLike.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)) N x)) (Submodule.addCommMonoid.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N) (Submodule.isScalarTower'.{u_1, u_1, u_2, u_1} R R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18 N R (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (instDistribSMul.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (instDistribSMul.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (SMulZeroClass.toSMul.{u_1, u_2} R M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15))) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18)))) (SMulZeroClass.toSMul.{u_1, u_1} R R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4))))))) (DistribMulAction.toDistribSMul.{u_1, u_1} R R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R 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(AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.18) (IsScalarTower.left.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (DistribMulAction.toMulAction.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.4)))) (AddCommMonoid.toAddMonoid.{u_2} M (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.SpanRankOperations.2755579173._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R 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