diff --git "a/data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Module.Submodule.Pointwise.sym.json" "b/data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Module.Submodule.Pointwise.sym.json"
new file mode 100644--- /dev/null
+++ "b/data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Module.Submodule.Pointwise.sym.json"
@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["Submodule","pointwiseDistribMulAction","_proof_4"],"typeFallback":"forall {α : Type.{u_3}} {R : Type.{u_1}} {M : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.11 : Module.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.15 : Monoid.{u_3} α] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.18 : DistribMulAction.{u_3, u_2} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8)] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.22 : SMulCommClass.{u_3, u_1, u_2} α R M (SMulZeroClass.toSMul.{u_3, u_2} α M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_3, u_2} α M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_3, u_2} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.18))) (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 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8) (Module.toDistribMulAction.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.11))))] (_a : α) (_S₁ : Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.11) (_S₂ : Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.11), Eq.{succ u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.11) (Submodule.map.{u_1, u_1, u_2, u_2} R R M M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.11 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5)) (RingHomSurjective.ids.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5) (DistribSMul.toLinearMap.{u_1, u_3, u_2} R α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.11 (DistribMulAction.toDistribSMul.{u_3, u_2} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.18) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.22 _a) (Max.max.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.11) (SemilatticeSup.toMax.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.11) (Lattice.toSemilatticeSup.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.11) (ConditionallyCompleteLattice.toLattice.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.11) (CompleteLattice.toConditionallyCompleteLattice.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.11) (Submodule.completeLattice.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.11))))) _S₁ _S₂)) (Max.max.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.11) (SemilatticeSup.toMax.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.11) (Lattice.toSemilatticeSup.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.11) (ConditionallyCompleteLattice.toLattice.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.11) (CompleteLattice.toConditionallyCompleteLattice.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.11) (Submodule.completeLattice.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.11))))) (Submodule.map.{u_1, u_1, u_2, u_2} R R M M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.11 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5)) (RingHomSurjective.ids.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5) (DistribSMul.toLinearMap.{u_1, u_3, u_2} R α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.11 (DistribMulAction.toDistribSMul.{u_3, u_2} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.18) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.22 _a) _S₁) (Submodule.map.{u_1, u_1, u_2, u_2} R R M M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.11 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5)) (RingHomSurjective.ids.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5) (DistribSMul.toLinearMap.{u_1, u_3, u_2} R α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.11 (DistribMulAction.toDistribSMul.{u_3, u_2} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.18) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.22 _a) _S₂))","typeFull":"∀ {α : Type u_3} {R : Type u_1} {M : Type u_2} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  [inst_3 : Monoid α] [inst_4 : DistribMulAction α M] [inst_5 : SMulCommClass α R M] (_a : α) (_S₁ _S₂ : Submodule R M),\n  Submodule.map (DistribSMul.toLinearMap R M _a) (_S₁ ⊔ _S₂) =\n    Submodule.map (DistribSMul.toLinearMap R M _a) _S₁ ⊔ Submodule.map (DistribSMul.toLinearMap R M _a) _S₂","typeReadable":"∀ {α : Type u_3} {R : Type u_1} {M : Type u_2} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  [inst_3 : Monoid α] [inst_4 : DistribMulAction α M] [inst_5 : SMulCommClass α R M] (_a : α) (_S₁ _S₂ : Submodule R M),\n  Submodule.map (DistribSMul.toLinearMap R M _a) (_S₁ ⊔ _S₂) =\n    Submodule.map (DistribSMul.toLinearMap R M _a) _S₁ ⊔ Submodule.map (DistribSMul.toLinearMap R M _a) _S₂","typeReferences":[["Lattice","toSemilatticeSup"],["Module"],["Submodule","map"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Submodule"],["DistribSMul","toLinearMap"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["MonoidWithZero","toMonoid"],["Eq"],["CompleteLattice","toConditionallyCompleteLattice"],["DistribMulAction"],["DistribSMul","toSMulZeroClass"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["RingHomSurjective","ids"],["SMulCommClass"],["Module","toDistribMulAction"],["Max","max"],["SemilatticeSup","toMax"],["Submodule","completeLattice"],["Monoid"],["ConditionallyCompleteLattice","toLattice"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["Semiring","toNonAssocSemiring"],["RingHom","id"],["DistribMulAction","toDistribSMul"],["Submodule","map_sup"],["AddCommMonoid","toAddMonoid"],["DistribSMul","toLinearMap"],["RingHomSurjective","ids"]]},{"isProp":true,"kind":"theorem","name":["Submodule","pointwiseSetDistribMulAction","_proof_2"],"typeFallback":"forall {R : Type.{u_1}} {M : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11 : Module.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8] (s : Set.{u_1} R), Eq.{succ u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11) (HSMul.hSMul.{u_1, u_2, u_2} (Set.{u_1} R) (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11) (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11) (instHSMul.{u_1, u_2} (Set.{u_1} R) (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_1, u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11 R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5)) (Module.toDistribMulAction.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11))) s (Bot.bot.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11) (Submodule.instBot.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11))) (Bot.bot.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11) (Submodule.instBot.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11))","typeFull":"∀ {R : Type u_1} {M : Type u_2} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] (s : Set R),\n  s • ⊥ = ⊥","typeReadable":"∀ {R : Type u_1} {M : Type u_2} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] (s : Set R),\n  s • ⊥ = ⊥","typeReferences":[["Submodule","pointwiseSetSMul"],["Module"],["Set"],["Semiring","toMonoidWithZero"],["Bot","bot"],["Submodule"],["AddCommMonoid"],["Module","toDistribMulAction"],["Submodule","instBot"],["HSMul","hSMul"],["MonoidWithZero","toMonoid"],["instHSMul"],["Eq"],["Semiring"]],"valueReferences":[["Submodule","set_smul_bot"],["Module","toDistribMulAction"],["MonoidWithZero","toMonoid"],["Semiring","toMonoidWithZero"]]},{"isProp":false,"kind":"definition","name":["Submodule","pointwiseSetDistribMulAction"],"typeFallback":"forall {R : Type.{u_2}} {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 : AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11 : Module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.30 : SMulCommClass.{u_2, u_2, u_3} R R M (SMulZeroClass.toSMul.{u_2, u_3} R M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_2, u_3} R M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_2, u_3} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8) (Module.toDistribMulAction.{u_2, u_3} R M 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(Module.toDistribMulAction.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11))))], DistribMulAction.{u_2, u_3} (Set.{u_2} R) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11) (Set.monoid.{u_2} R (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5))) (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11))","typeFull":"{R : Type u_2} →\n  {M : Type u_3} →\n    [inst : Semiring R] →\n      [inst_1 : AddCommMonoid M] →\n        [inst_2 : Module R M] → [SMulCommClass R R M] → DistribMulAction (Set R) (Submodule R M)","typeReadable":"{R : Type u_2} →\n  {M : Type u_3} →\n    [inst : Semiring R] →\n      [inst_1 : AddCommMonoid M] →\n        [inst_2 : Module R M] → [SMulCommClass R R M] → DistribMulAction (Set R) (Submodule R M)","typeReferences":[["Set"],["Module"],["DistribMulAction","toDistribSMul"],["Set","monoid"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["Submodule"],["SMulCommClass"],["Module","toDistribMulAction"],["AddCommMonoid"],["MonoidWithZero","toMonoid"],["AddZero","toZero"],["Submodule","pointwiseAddCommMonoid"],["DistribMulAction"],["Semiring"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]],"valueReferences":[["Submodule","pointwiseSetMulAction"],["Submodule","pointwiseSetDistribMulAction","_proof_2"],["Submodule","pointwiseSetDistribMulAction","_proof_3"],["Set"],["MonoidWithZero","toMonoid"],["DistribMulAction","mk"],["Semiring","toMonoidWithZero"],["Set","monoid"],["AddCommMonoid","toAddMonoid"],["Submodule","pointwiseAddCommMonoid"],["Submodule"]]},{"isProp":true,"kind":"theorem","name":["Submodule","mem_set_smul_of_mem_mem"],"typeFallback":"forall {R : Type.{u_2}} {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1989521453._hygCtx._hyg.5 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1989521453._hygCtx._hyg.8 : AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1989521453._hygCtx._hyg.11 : Module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1989521453._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1989521453._hygCtx._hyg.8] {S : Type.{u_4}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1989521453._hygCtx._hyg.16 : Monoid.{u_4} S] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1989521453._hygCtx._hyg.19 : DistribMulAction.{u_4, u_3} S M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1989521453._hygCtx._hyg.16 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1989521453._hygCtx._hyg.8)] {s : Set.{u_4} S} {N : Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1989521453._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1989521453._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1989521453._hygCtx._hyg.11} {r : S} {m : M}, (Membership.mem.{u_4, u_4} S (Set.{u_4} S) (Set.instMembership.{u_4} S) s r) -> (Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1989521453._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1989521453._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1989521453._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1989521453._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1989521453._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1989521453._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1989521453._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1989521453._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1989521453._hygCtx._hyg.11)) N m) -> (Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1989521453._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1989521453._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1989521453._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1989521453._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1989521453._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1989521453._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1989521453._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1989521453._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1989521453._hygCtx._hyg.11)) (HSMul.hSMul.{u_4, u_3, u_3} (Set.{u_4} S) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1989521453._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1989521453._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1989521453._hygCtx._hyg.11) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1989521453._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1989521453._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1989521453._hygCtx._hyg.11) (instHSMul.{u_4, u_3} (Set.{u_4} S) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1989521453._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1989521453._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1989521453._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_3, u_4} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1989521453._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1989521453._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1989521453._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1989521453._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1989521453._hygCtx._hyg.19)) s N) (HSMul.hSMul.{u_4, u_3, u_3} S M M (instHSMul.{u_4, u_3} S M (SMulZeroClass.toSMul.{u_4, u_3} S M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1989521453._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_4, u_3} S M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1989521453._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_4, u_3} S M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1989521453._hygCtx._hyg.16 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1989521453._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1989521453._hygCtx._hyg.19)))) r m))","typeFull":"∀ {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {S : Type u_4}\n  [inst_3 : Monoid S] [inst_4 : DistribMulAction S M] {s : Set S} {N : Submodule R M} {r : S} {m : M},\n  r ∈ s → m ∈ N → r • m ∈ s • N","typeReadable":"∀ {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {S : Type u_4}\n  [inst_3 : Monoid S] [inst_4 : DistribMulAction S M] {s : Set S} {N : Submodule R M} {r : S} {m : M},\n  r ∈ s → m ∈ N → r • m ∈ s • N","typeReferences":[["Submodule","pointwiseSetSMul"],["SetLike","instMembership"],["Module"],["Set"],["Membership","mem"],["DistribMulAction","toDistribSMul"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["Submodule"],["Set","instMembership"],["AddCommMonoid"],["Submodule","setLike"],["HSMul","hSMul"],["Monoid"],["instHSMul"],["AddZero","toZero"],["DistribMulAction"],["DistribSMul","toSMulZeroClass"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["Submodule","instInfSet"],["Membership","mem"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Submodule","mem_set_smul_def"],["Submodule"],["congrArg"],["InfSet","sInf"],["instHSMul"],["Eq"],["propext"],["DistribSMul","toSMulZeroClass"],["Submodule","pointwiseSetSMul"],["SetLike","instMembership"],["Set"],["DistribMulAction","toDistribSMul"],["AddZeroClass","toAddZero"],["Set","instMembership"],["Submodule","mem_sInf"],["Submodule","setLike"],["HSMul","hSMul"],["id"],["Eq","mpr"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["setOf"]]},{"isProp":true,"kind":"theorem","name":["Submodule","mem_stabilizer_submodule_iff_map_eq"],"typeFallback":"forall {R : Type.{u_4}} {G : Type.{u_5}} {M : Type.{u_6}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4153314695._hygCtx._hyg.18 : Semiring.{u_4} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4153314695._hygCtx._hyg.21 : AddCommMonoid.{u_6} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4153314695._hygCtx._hyg.24 : Module.{u_4, u_6} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4153314695._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4153314695._hygCtx._hyg.21] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4153314695._hygCtx._hyg.28 : Group.{u_5} G] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4153314695._hygCtx._hyg.31 : DistribMulAction.{u_5, u_6} G M (DivInvMonoid.toMonoid.{u_5} G (Group.toDivInvMonoid.{u_5} G inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4153314695._hygCtx._hyg.28)) (AddCommMonoid.toAddMonoid.{u_6} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4153314695._hygCtx._hyg.21)] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4153314695._hygCtx._hyg.35 : SMulCommClass.{u_5, u_4, u_6} G R M (SMulZeroClass.toSMul.{u_5, u_6} G M (AddZero.toZero.{u_6} M (AddZeroClass.toAddZero.{u_6} M (AddMonoid.toAddZeroClass.{u_6} M (AddCommMonoid.toAddMonoid.{u_6} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4153314695._hygCtx._hyg.21)))) (DistribSMul.toSMulZeroClass.{u_5, u_6} G M (AddMonoid.toAddZeroClass.{u_6} M (AddCommMonoid.toAddMonoid.{u_6} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4153314695._hygCtx._hyg.21)) (DistribMulAction.toDistribSMul.{u_5, u_6} G M (DivInvMonoid.toMonoid.{u_5} G (Group.toDivInvMonoid.{u_5} G inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4153314695._hygCtx._hyg.28)) (AddCommMonoid.toAddMonoid.{u_6} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4153314695._hygCtx._hyg.21) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4153314695._hygCtx._hyg.31))) (SMulZeroClass.toSMul.{u_4, u_6} R M (AddZero.toZero.{u_6} M (AddZeroClass.toAddZero.{u_6} M (AddMonoid.toAddZeroClass.{u_6} M (AddCommMonoid.toAddMonoid.{u_6} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4153314695._hygCtx._hyg.21)))) (DistribSMul.toSMulZeroClass.{u_4, u_6} R M (AddMonoid.toAddZeroClass.{u_6} M (AddCommMonoid.toAddMonoid.{u_6} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4153314695._hygCtx._hyg.21)) (DistribMulAction.toDistribSMul.{u_4, u_6} R M (MonoidWithZero.toMonoid.{u_4} R (Semiring.toMonoidWithZero.{u_4} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4153314695._hygCtx._hyg.18)) (AddCommMonoid.toAddMonoid.{u_6} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4153314695._hygCtx._hyg.21) (Module.toDistribMulAction.{u_4, u_6} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4153314695._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4153314695._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4153314695._hygCtx._hyg.24))))] {S : Submodule.{u_4, u_6} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4153314695._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4153314695._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4153314695._hygCtx._hyg.24} {e : G}, Iff (Membership.mem.{u_5, u_5} G (Subgroup.{u_5} G inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4153314695._hygCtx._hyg.28) (SetLike.instMembership.{u_5, u_5} (Subgroup.{u_5} G inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4153314695._hygCtx._hyg.28) G (Subgroup.instSetLike.{u_5} G inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4153314695._hygCtx._hyg.28)) (MulAction.stabilizer.{u_5, u_6} G (Submodule.{u_4, u_6} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4153314695._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4153314695._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4153314695._hygCtx._hyg.24) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4153314695._hygCtx._hyg.28 (DistribMulAction.toMulAction.{u_5, u_6} G (Submodule.{u_4, u_6} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4153314695._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4153314695._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4153314695._hygCtx._hyg.24) (DivInvMonoid.toMonoid.{u_5} G (Group.toDivInvMonoid.{u_5} G inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4153314695._hygCtx._hyg.28)) (AddCommMonoid.toAddMonoid.{u_6} (Submodule.{u_4, u_6} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4153314695._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4153314695._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4153314695._hygCtx._hyg.24) (Submodule.pointwiseAddCommMonoid.{u_4, u_6} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4153314695._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4153314695._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4153314695._hygCtx._hyg.24)) (Submodule.pointwiseDistribMulAction.{u_5, u_4, u_6} G R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4153314695._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4153314695._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4153314695._hygCtx._hyg.24 (DivInvMonoid.toMonoid.{u_5} G (Group.toDivInvMonoid.{u_5} G inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4153314695._hygCtx._hyg.28)) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4153314695._hygCtx._hyg.31 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4153314695._hygCtx._hyg.35)) S) e) (Eq.{succ u_6} (Submodule.{u_4, u_6} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4153314695._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4153314695._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4153314695._hygCtx._hyg.24) (Submodule.map.{u_4, u_4, u_6, u_6} R R M M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4153314695._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4153314695._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4153314695._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4153314695._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4153314695._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4153314695._hygCtx._hyg.24 (RingHom.id.{u_4} R (Semiring.toNonAssocSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4153314695._hygCtx._hyg.18)) (RingHomSurjective.ids.{u_4} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4153314695._hygCtx._hyg.18) (DistribSMul.toLinearMap.{u_4, u_5, u_6} R G M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4153314695._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4153314695._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4153314695._hygCtx._hyg.24 (DistribMulAction.toDistribSMul.{u_5, u_6} G M (DivInvMonoid.toMonoid.{u_5} G (Group.toDivInvMonoid.{u_5} G inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4153314695._hygCtx._hyg.28)) (AddCommMonoid.toAddMonoid.{u_6} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4153314695._hygCtx._hyg.21) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4153314695._hygCtx._hyg.31) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4153314695._hygCtx._hyg.35 e) S) S)","typeFull":"∀ {R : Type u_4} {G : Type u_5} {M : Type u_6} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  [inst_3 : Group G] [inst_4 : DistribMulAction G M] [inst_5 : SMulCommClass G R M] {S : Submodule R M} {e : G},\n  e ∈ MulAction.stabilizer G S ↔ Submodule.map (DistribSMul.toLinearMap R M e) S = S","typeReadable":"∀ {R : Type u_4} {G : Type u_5} {M : Type u_6} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  [inst_3 : Group G] [inst_4 : DistribMulAction G M] [inst_5 : SMulCommClass G R M] {S : Submodule R M} {e : G},\n  e ∈ MulAction.stabilizer G S ↔ Submodule.map (DistribSMul.toLinearMap R M e) S = S","typeReferences":[["Module"],["Subgroup","instSetLike"],["Submodule","map"],["Membership","mem"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["DistribSMul","toLinearMap"],["Submodule"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["Eq"],["Group","toDivInvMonoid"],["DistribMulAction"],["DistribSMul","toSMulZeroClass"],["Group"],["SetLike","instMembership"],["MulAction","stabilizer"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["RingHomSurjective","ids"],["SMulCommClass"],["Module","toDistribMulAction"],["DivInvMonoid","toMonoid"],["Iff"],["Subgroup"],["AddZero","toZero"],["Submodule","pointwiseDistribMulAction"],["Submodule","pointwiseAddCommMonoid"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["SetLike","instMembership"],["MulAction","stabilizer"],["Subgroup","instSetLike"],["Membership","mem"],["Iff","rfl"],["AddCommMonoid","toAddMonoid"],["Submodule"],["DivInvMonoid","toMonoid"],["DistribMulAction","toMulAction"],["Subgroup"],["Group","toDivInvMonoid"],["Submodule","pointwiseAddCommMonoid"],["Submodule","pointwiseDistribMulAction"]]},{"isProp":false,"kind":"definition","name":["Submodule","pointwiseZero"],"typeFallback":"forall {R : Type.{u_2}} {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2511952736._hygCtx._hyg.5 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2511952736._hygCtx._hyg.8 : AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2511952736._hygCtx._hyg.11 : Module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2511952736._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2511952736._hygCtx._hyg.8], Zero.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2511952736._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2511952736._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2511952736._hygCtx._hyg.11)","typeFull":"{R : Type u_2} →\n  {M : Type u_3} → [inst : Semiring R] → [inst_1 : AddCommMonoid M] → [inst_2 : Module R M] → Zero (Submodule R M)","typeReadable":"{R : Type u_2} →\n  {M : Type u_3} → [inst : Semiring R] → [inst_1 : AddCommMonoid M] → [inst_2 : Module R M] → Zero (Submodule R M)","typeReferences":[["AddCommMonoid"],["Module"],["Zero"],["Submodule"],["Semiring"]],"valueReferences":[["Zero","mk"],["Submodule","instBot"],["Bot","bot"],["Submodule"]]},{"isProp":true,"kind":"theorem","name":["Submodule","pointwiseDistribMulAction","_proof_3"],"typeFallback":"forall {α : Type.{u_3}} {R : Type.{u_2}} {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8 : AddCommMonoid.{u_1} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.11 : Module.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.15 : Monoid.{u_3} α] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.18 : DistribMulAction.{u_3, u_1} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8)] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.22 : SMulCommClass.{u_3, u_2, u_1} α R M (SMulZeroClass.toSMul.{u_3, u_1} α M (AddZero.toZero.{u_1} M (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_3, u_1} α M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_3, u_1} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.18))) (SMulZeroClass.toSMul.{u_2, u_1} R M (AddZero.toZero.{u_1} M (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_2, u_1} R M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_2, u_1} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8) (Module.toDistribMulAction.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.11))))] (_a : α), Eq.{succ u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.11) (Submodule.map.{u_2, u_2, u_1, u_1} R R M M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.11 (RingHom.id.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5)) (RingHomSurjective.ids.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5) (DistribSMul.toLinearMap.{u_2, u_3, u_1} R α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.11 (DistribMulAction.toDistribSMul.{u_3, u_1} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.18) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.22 _a) (Bot.bot.{u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.11) (Submodule.instBot.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.11))) (Bot.bot.{u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.11) (Submodule.instBot.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.11))","typeFull":"∀ {α : Type u_3} {R : Type u_2} {M : Type u_1} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  [inst_3 : Monoid α] [inst_4 : DistribMulAction α M] [inst_5 : SMulCommClass α R M] (_a : α),\n  Submodule.map (DistribSMul.toLinearMap R M _a) ⊥ = ⊥","typeReadable":"∀ {α : Type u_3} {R : Type u_2} {M : Type u_1} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  [inst_3 : Monoid α] [inst_4 : DistribMulAction α M] [inst_5 : SMulCommClass α R M] (_a : α),\n  Submodule.map (DistribSMul.toLinearMap R M _a) ⊥ = ⊥","typeReferences":[["Module"],["Submodule","map"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Submodule"],["DistribSMul","toLinearMap"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["MonoidWithZero","toMonoid"],["Eq"],["DistribMulAction"],["DistribSMul","toSMulZeroClass"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["Bot","bot"],["RingHomSurjective","ids"],["SMulCommClass"],["Module","toDistribMulAction"],["Submodule","instBot"],["Monoid"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["Semiring","toNonAssocSemiring"],["RingHom","id"],["DistribMulAction","toDistribSMul"],["AddCommMonoid","toAddMonoid"],["Submodule","map_bot"],["DistribSMul","toLinearMap"],["RingHomSurjective","ids"]]},{"isProp":true,"kind":"theorem","name":["Submodule","pointwiseDistribMulAction","_proof_1"],"typeFallback":"forall {α : Type.{u_3}} {R : Type.{u_1}} {M : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.11 : Module.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.15 : Monoid.{u_3} α] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.18 : DistribMulAction.{u_3, u_2} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8)] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.22 : SMulCommClass.{u_3, u_1, u_2} α R M (SMulZeroClass.toSMul.{u_3, u_2} α M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_3, u_2} α M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_3, u_2} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.18))) (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 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8) (Module.toDistribMulAction.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.11))))] (_a₁ : α) (_a₂ : α) (S : Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.11), Eq.{succ u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.11) (Submodule.map.{u_1, u_1, u_2, u_2} R R M M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.11 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5)) (RingHomSurjective.ids.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5) (DistribSMul.toLinearMap.{u_1, u_3, u_2} R α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.11 (DistribMulAction.toDistribSMul.{u_3, u_2} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.18) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.22 (HMul.hMul.{u_3, u_3, u_3} α α α (instHMul.{u_3} α (Semigroup.toMul.{u_3} α (Monoid.toSemigroup.{u_3} α inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.15))) _a₁ _a₂)) S) (HSMul.hSMul.{u_3, u_2, u_2} α (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.11) (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.11) (instHSMul.{u_3, u_2} α (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.11) (SMul.mk.{u_3, u_2} α (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.11) (fun (a : α) (S : Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.11) => Submodule.map.{u_1, u_1, u_2, u_2} R R M M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.11 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5)) (RingHomSurjective.ids.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5) (DistribSMul.toLinearMap.{u_1, u_3, u_2} R α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.11 (DistribMulAction.toDistribSMul.{u_3, u_2} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.18) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.22 a) S))) _a₁ (HSMul.hSMul.{u_3, u_2, u_2} α (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.11) (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.11) (instHSMul.{u_3, u_2} α (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.11) (SMul.mk.{u_3, u_2} α (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.11) (fun (a : α) (S : Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.11) => Submodule.map.{u_1, u_1, u_2, u_2} R R M M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.11 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5)) 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SMulCommClass α R M] (_a₁ _a₂ : α) (S : Submodule R M),\n  Submodule.map (DistribSMul.toLinearMap R M (_a₁ * _a₂)) S = _a₁ • _a₂ • S","typeReadable":"∀ {α : Type u_3} {R : Type u_1} {M : Type u_2} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  [inst_3 : Monoid α] [inst_4 : DistribMulAction α M] [inst_5 : SMulCommClass α R M] (_a₁ _a₂ : α) (S : Submodule R M),\n  Submodule.map (DistribSMul.toLinearMap R M (_a₁ * _a₂)) S = _a₁ • _a₂ • S","typeReferences":[["Module"],["Submodule","map"],["HMul","hMul"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Semigroup","toMul"],["Submodule"],["DistribSMul","toLinearMap"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["MonoidWithZero","toMonoid"],["instHSMul"],["Monoid","toSemigroup"],["Eq"],["DistribMulAction"],["DistribSMul","toSMulZeroClass"],["SMul","mk"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["RingHomSurjective","ids"],["SMulCommClass"],["Module","toDistribMulAction"],["HSMul","hSMul"],["Monoid"],["instHMul"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["Eq","trans"],["RingHomCompTriple","ids"],["Submodule","map"],["HMul","hMul"],["LinearMap","comp"],["AddCommMonoid","toAddMonoid"],["Semigroup","toMul"],["DistribSMul","toLinearMap"],["Submodule"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["LinearMap","ext"],["DistribMulAction","toMulAction"],["instHSMul"],["Monoid","toSemigroup"],["SemigroupAction","mul_smul"],["SMul","mk"],["Module","End"],["Submodule","map_comp"],["congr_arg"],["DistribMulAction","toDistribSMul"],["RingHomSurjective","ids"],["HSMul","hSMul"],["instHMul"],["MulAction","toSemigroupAction"]]},{"isProp":true,"kind":"theorem","name":["Submodule","set_smul_le_of_le_le"],"typeFallback":"forall {R : Type.{u_2}} {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.408008347._hygCtx._hyg.5 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.408008347._hygCtx._hyg.8 : AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.408008347._hygCtx._hyg.11 : Module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.408008347._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.408008347._hygCtx._hyg.8] {S : Type.{u_4}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.408008347._hygCtx._hyg.16 : Monoid.{u_4} S] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.408008347._hygCtx._hyg.19 : DistribMulAction.{u_4, u_3} S M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.408008347._hygCtx._hyg.16 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.408008347._hygCtx._hyg.8)] {s : Set.{u_4} S} {t : Set.{u_4} S} {p : Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.408008347._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.408008347._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.408008347._hygCtx._hyg.11} {q : Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.408008347._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.408008347._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.408008347._hygCtx._hyg.11}, (LE.le.{u_4} (Set.{u_4} S) (Set.instLE.{u_4} S) s t) -> (LE.le.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.408008347._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.408008347._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.408008347._hygCtx._hyg.11) (Preorder.toLE.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.408008347._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.408008347._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.408008347._hygCtx._hyg.11) (PartialOrder.toPreorder.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.408008347._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.408008347._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.408008347._hygCtx._hyg.11) (Submodule.instPartialOrder.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.408008347._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.408008347._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.408008347._hygCtx._hyg.11))) p q) -> (LE.le.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.408008347._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.408008347._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.408008347._hygCtx._hyg.11) (Preorder.toLE.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.408008347._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.408008347._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.408008347._hygCtx._hyg.11) (PartialOrder.toPreorder.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.408008347._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.408008347._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.408008347._hygCtx._hyg.11) (Submodule.instPartialOrder.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.408008347._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.408008347._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.408008347._hygCtx._hyg.11))) (HSMul.hSMul.{u_4, u_3, u_3} (Set.{u_4} S) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.408008347._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.408008347._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.408008347._hygCtx._hyg.11) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.408008347._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.408008347._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.408008347._hygCtx._hyg.11) (instHSMul.{u_4, u_3} (Set.{u_4} S) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.408008347._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.408008347._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.408008347._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_3, u_4} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.408008347._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.408008347._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.408008347._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.408008347._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.408008347._hygCtx._hyg.19)) s p) (HSMul.hSMul.{u_4, u_3, u_3} (Set.{u_4} S) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.408008347._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.408008347._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.408008347._hygCtx._hyg.11) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.408008347._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.408008347._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.408008347._hygCtx._hyg.11) (instHSMul.{u_4, u_3} (Set.{u_4} S) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.408008347._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.408008347._hygCtx._hyg.8 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[inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3749028077._hygCtx._hyg.11 : Module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3749028077._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3749028077._hygCtx._hyg.8] {S : Type.{u_4}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3749028077._hygCtx._hyg.16 : Monoid.{u_4} S] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3749028077._hygCtx._hyg.19 : DistribMulAction.{u_4, u_3} S M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3749028077._hygCtx._hyg.16 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3749028077._hygCtx._hyg.8)] (s : Set.{u_4} S) (N : Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3749028077._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3749028077._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3749028077._hygCtx._hyg.11) (p : Submodule.{u_2, u_3} R M 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inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3749028077._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3749028077._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3749028077._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_3, u_4} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3749028077._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3749028077._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3749028077._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3749028077._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3749028077._hygCtx._hyg.19)) s N) p) (forall {{r : S}} {{n : M}}, (Membership.mem.{u_4, u_4} S (Set.{u_4} S) (Set.instMembership.{u_4} S) s r) -> (Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3749028077._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3749028077._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3749028077._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3749028077._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3749028077._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3749028077._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3749028077._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3749028077._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3749028077._hygCtx._hyg.11)) N n) -> (Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3749028077._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3749028077._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3749028077._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M 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{R : Type.{u_2}} {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.437601602._hygCtx._hyg.5 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.437601602._hygCtx._hyg.8 : AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.437601602._hygCtx._hyg.11 : Module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.437601602._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.437601602._hygCtx._hyg.8] {S : Type.{u_4}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.437601602._hygCtx._hyg.16 : Monoid.{u_4} S] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.437601602._hygCtx._hyg.19 : DistribMulAction.{u_4, u_3} S M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.437601602._hygCtx._hyg.16 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.437601602._hygCtx._hyg.8)] (s : Set.{u_4} S) (N : Submodule.{u_2, u_3} R M 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inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.437601602._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.437601602._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.437601602._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.437601602._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.437601602._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.437601602._hygCtx._hyg.11)) N n) -> (Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.437601602._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.437601602._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.437601602._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.437601602._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.437601602._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.437601602._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.437601602._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.437601602._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.437601602._hygCtx._hyg.11)) p (HSMul.hSMul.{u_4, u_3, u_3} S M M (instHSMul.{u_4, u_3} S M (SMulZeroClass.toSMul.{u_4, u_3} S M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.437601602._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_4, u_3} S M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.437601602._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_4, u_3} S M 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p","typeReadable":"∀ {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {S : Type u_4}\n  [inst_3 : Monoid S] [inst_4 : DistribMulAction S M] (s : Set S) (N p : Submodule R M),\n  (∀ ⦃r : S⦄ ⦃n : M⦄, r ∈ s → n ∈ N → r • n ∈ p) → s • N ≤ p","typeReferences":[["PartialOrder","toPreorder"],["Module"],["Membership","mem"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Submodule"],["AddCommMonoid"],["instHSMul"],["Preorder","toLE"],["DistribMulAction"],["DistribSMul","toSMulZeroClass"],["SetLike","instMembership"],["Submodule","pointwiseSetSMul"],["Set"],["Submodule","instPartialOrder"],["DistribMulAction","toDistribSMul"],["AddZeroClass","toAddZero"],["Set","instMembership"],["Submodule","setLike"],["HSMul","hSMul"],["LE","le"],["Monoid"],["AddZero","toZero"],["Semiring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["SetLike","instMembership"],["Set"],["Membership","mem"],["DistribMulAction","toDistribSMul"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["sInf_le"],["Submodule"],["Set","instMembership"],["Submodule","setLike"],["HSMul","hSMul"],["Submodule","completeLattice"],["instHSMul"],["CompleteLattice","toCompleteSemilatticeInf"],["AddZero","toZero"],["DistribSMul","toSMulZeroClass"],["AddMonoid","toAddZeroClass"],["setOf"]]},{"isProp":true,"kind":"theorem","name":["Submodule","smul_inductionOn_pointwise"],"typeFallback":"forall {R : Type.{u_2}} {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 : AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11 : Module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8] {S : Type.{u_4}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.16 : Monoid.{u_4} S] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.19 : DistribMulAction.{u_4, u_3} S M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.16 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8)] (N : Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.30 : SMulCommClass.{u_4, u_2, u_3} S R M (SMulZeroClass.toSMul.{u_4, u_3} S M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_4, u_3} S M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_4, u_3} S M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.16 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.19))) (SMulZeroClass.toSMul.{u_2, u_3} R M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_2, u_3} R M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_2, u_3} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8) (Module.toDistribMulAction.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11))))] {a : S} {p : forall (x : M), (Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11)) (HSMul.hSMul.{u_4, u_3, u_3} S (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (instHSMul.{u_4, u_3} S (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (SMulZeroClass.toSMul.{u_4, u_3} S (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (AddZero.toZero.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (AddZeroClass.toAddZero.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (AddMonoid.toAddZeroClass.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11))))) (DistribSMul.toSMulZeroClass.{u_4, u_3} S (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (AddMonoid.toAddZeroClass.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11))) (DistribMulAction.toDistribSMul.{u_4, u_3} S (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.16 (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11)) (Submodule.pointwiseDistribMulAction.{u_4, u_2, u_3} S R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.30))))) a N) x) -> Prop}, (forall (s : M) (hs : Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11)) N s), p (HSMul.hSMul.{u_4, u_3, u_3} S M M (instHSMul.{u_4, u_3} S M (SMulZeroClass.toSMul.{u_4, u_3} S M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_4, u_3} S M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_4, u_3} S M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.16 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.19)))) a s) (Submodule.smul_mem_pointwise_smul.{u_4, u_2, u_3} S R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.30 s a N hs)) -> (forall (r : R) (m : M) (mem : Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11)) (HSMul.hSMul.{u_4, u_3, u_3} S (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (instHSMul.{u_4, u_3} S (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (SMulZeroClass.toSMul.{u_4, u_3} S (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (AddZero.toZero.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (AddZeroClass.toAddZero.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (AddMonoid.toAddZeroClass.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11))))) (DistribSMul.toSMulZeroClass.{u_4, u_3} S (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (AddMonoid.toAddZeroClass.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11))) (DistribMulAction.toDistribSMul.{u_4, u_3} S (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.16 (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11)) (Submodule.pointwiseDistribMulAction.{u_4, u_2, u_3} S R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.30))))) a N) m), (p m mem) -> (p (HSMul.hSMul.{u_2, u_3, u_3} R M M (instHSMul.{u_2, u_3} R M (SMulZeroClass.toSMul.{u_2, u_3} R M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_2, u_3} R M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_2, u_3} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8) (Module.toDistribMulAction.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11))))) r m) (Submodule.smul_mem.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11 (HSMul.hSMul.{u_4, u_3, u_3} S (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (instHSMul.{u_4, u_3} S (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (SMulZeroClass.toSMul.{u_4, u_3} S (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (AddZero.toZero.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (AddZeroClass.toAddZero.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (AddMonoid.toAddZeroClass.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11))))) (DistribSMul.toSMulZeroClass.{u_4, u_3} S (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (AddMonoid.toAddZeroClass.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11))) (DistribMulAction.toDistribSMul.{u_4, u_3} S (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.16 (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11)) (Submodule.pointwiseDistribMulAction.{u_4, u_2, u_3} S R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.30))))) a N) m r mem))) -> (forall (x : M) (y : M) (hx : Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11)) (HSMul.hSMul.{u_4, u_3, u_3} S (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (instHSMul.{u_4, u_3} S (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (SMulZeroClass.toSMul.{u_4, u_3} S (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (AddZero.toZero.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (AddZeroClass.toAddZero.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (AddMonoid.toAddZeroClass.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11))))) (DistribSMul.toSMulZeroClass.{u_4, u_3} S (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (AddMonoid.toAddZeroClass.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11))) (DistribMulAction.toDistribSMul.{u_4, u_3} S (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.16 (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11)) (Submodule.pointwiseDistribMulAction.{u_4, u_2, u_3} S R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.30))))) a N) x) (hy : Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11)) (HSMul.hSMul.{u_4, u_3, u_3} S (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (instHSMul.{u_4, u_3} S (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (SMulZeroClass.toSMul.{u_4, u_3} S (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (AddZero.toZero.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (AddZeroClass.toAddZero.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (AddMonoid.toAddZeroClass.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11))))) (DistribSMul.toSMulZeroClass.{u_4, u_3} S (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (AddMonoid.toAddZeroClass.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11))) (DistribMulAction.toDistribSMul.{u_4, u_3} S (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.16 (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11)) (Submodule.pointwiseDistribMulAction.{u_4, u_2, u_3} S R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.30))))) a N) y), (p x hx) -> (p y hy) -> (p (HAdd.hAdd.{u_3, u_3, u_3} M M M (instHAdd.{u_3} M (AddCommMagma.toAdd.{u_3} M (AddCommSemigroup.toAddCommMagma.{u_3} M (AddCommMonoid.toAddCommSemigroup.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8)))) x y) (Submodule.add_mem.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11 (HSMul.hSMul.{u_4, u_3, u_3} S (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (instHSMul.{u_4, u_3} S (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (SMulZeroClass.toSMul.{u_4, u_3} S (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (AddZero.toZero.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (AddZeroClass.toAddZero.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (AddMonoid.toAddZeroClass.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11))))) (DistribSMul.toSMulZeroClass.{u_4, u_3} S (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (AddMonoid.toAddZeroClass.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11))) (DistribMulAction.toDistribSMul.{u_4, u_3} S (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.16 (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11)) (Submodule.pointwiseDistribMulAction.{u_4, u_2, u_3} S R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.30))))) a N) x y hx hy))) -> (p (OfNat.ofNat.{u_3} M 0 (Zero.toOfNat0.{u_3} M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8)))))) (Submodule.zero_mem.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11 (HSMul.hSMul.{u_4, u_3, u_3} S (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (instHSMul.{u_4, u_3} S (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (SMulZeroClass.toSMul.{u_4, u_3} S (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (AddZero.toZero.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (AddZeroClass.toAddZero.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (AddMonoid.toAddZeroClass.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11))))) (DistribSMul.toSMulZeroClass.{u_4, u_3} S (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (AddMonoid.toAddZeroClass.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11))) (DistribMulAction.toDistribSMul.{u_4, u_3} S (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.16 (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11)) (Submodule.pointwiseDistribMulAction.{u_4, u_2, u_3} S R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.30))))) a N))) -> (forall {x : M} (hx : Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11)) (HSMul.hSMul.{u_4, u_3, u_3} S (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (instHSMul.{u_4, u_3} S (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (SMulZeroClass.toSMul.{u_4, u_3} S (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (AddZero.toZero.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (AddZeroClass.toAddZero.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (AddMonoid.toAddZeroClass.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11))))) (DistribSMul.toSMulZeroClass.{u_4, u_3} S (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (AddMonoid.toAddZeroClass.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11))) (DistribMulAction.toDistribSMul.{u_4, u_3} S (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.16 (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11)) (Submodule.pointwiseDistribMulAction.{u_4, u_2, u_3} S R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3445666650._hygCtx._hyg.30))))) a N) x), p x hx)","typeFull":"∀ {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {S : Type u_4}\n  [inst_3 : Monoid S] [inst_4 : DistribMulAction S M] (N : Submodule R M) [inst_5 : SMulCommClass S R M] {a : S}\n  {p : (x : M) → x ∈ a • N → Prop},\n  (∀ (s : M) (hs : s ∈ N), p (a • s) ⋯) →\n    (∀ (r : R) (m : M) (mem : m ∈ a • N), p m mem → p (r • m) ⋯) →\n      (∀ (x y : M) (hx : x ∈ a • N) (hy : y ∈ a • N), p x hx → p y hy → p (x + y) ⋯) →\n        p 0 ⋯ → ∀ {x : M} (hx : x ∈ a • N), p x hx","typeReadable":"∀ {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {S : Type u_4}\n  [inst_3 : Monoid S] [inst_4 : DistribMulAction S M] (N : Submodule R M) [inst_5 : SMulCommClass S R M] {a : S}\n  {p : (x : M) → x ∈ a • N → Prop},\n  (∀ (s : M) (hs : s ∈ N), p (a • s) ⋯) →\n    (∀ (r : R) (m : M) (mem : m ∈ a • N), p m mem → p (r • m) ⋯) →\n      (∀ (x y : M) (hx : x ∈ a • N) (hy : y ∈ a • N), p x hx → p y hy → p (x + y) ⋯) →\n        p 0 ⋯ → ∀ {x : M} (hx : x ∈ a • N), p x hx","typeReferences":[["Module"],["Membership","mem"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Submodule","zero_mem"],["Submodule"],["AddCommMonoid"],["MonoidWithZero","toMonoid"],["instHSMul"],["Zero","toOfNat0"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["DistribMulAction"],["DistribSMul","toSMulZeroClass"],["SetLike","instMembership"],["instHAdd"],["Submodule","smul_mem"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["Submodule","add_mem"],["HAdd","hAdd"],["SMulCommClass"],["Module","toDistribMulAction"],["AddCommMonoid","toAddCommSemigroup"],["Submodule","setLike"],["HSMul","hSMul"],["Monoid"],["AddZero","toZero"],["Submodule","smul_mem_pointwise_smul"],["Submodule","pointwiseAddCommMonoid"],["Submodule","pointwiseDistribMulAction"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["Singleton","singleton"],["Membership","mem"],["SMulZeroClass","toSMul"],["Submodule","set_smul_inductionOn"],["forall_congr"],["Eq","symm"],["Submodule","singleton_set_smul"],["Eq","rec"],["Eq","ndrec"],["forall_prop_domain_congr"],["DistribSMul","toSMulZeroClass"],["Submodule","pointwiseSetSMul"],["SetLike","instMembership"],["Submodule","smul_mem"],["DistribMulAction","toDistribSMul"],["AddZeroClass","toAddZero"],["Set","instMembership"],["Eq","refl"],["HSMul","hSMul"],["id"],["Eq","mpr"],["AddZero","toZero"],["Submodule","pointwiseDistribMulAction"],["Submodule","pointwiseAddCommMonoid"],["AddMonoid","toAddZeroClass"],["Eq","substr"],["Eq","mp"],["AddCommMonoid","toAddMonoid"],["congrArg"],["Submodule"],["Submodule","zero_mem"],["MonoidWithZero","toMonoid"],["instHSMul"],["congrFun'"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["Zero","toOfNat0"],["Eq"],["Set"],["instHAdd"],["Semiring","toMonoidWithZero"],["_private","Mathlib","Algebra","Module","Submodule","Pointwise",0,"Submodule","smul_inductionOn_pointwise","_simp_1_1"],["Set","instSingletonSet"],["OfNat","ofNat"],["Submodule","add_mem"],["HAdd","hAdd"],["Module","toDistribMulAction"],["AddCommMonoid","toAddCommSemigroup"],["Submodule","setLike"],["Set","mem_singleton_iff","_simp_1"],["Submodule","smul_mem_pointwise_smul"],["Submodule","mem_set_smul_of_mem_mem"]]},{"isProp":true,"kind":"theorem","name":["Submodule","set_smul_inductionOn"],"typeFallback":"forall {R : Type.{u_2}} {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 : AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 : Module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8] {S : Type.{u_4}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 : Monoid.{u_4} S] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19 : DistribMulAction.{u_4, u_3} S M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8)] {s : Set.{u_4} S} {N : Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11} {motive : forall (x : M), (Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_4, u_3, u_3} (Set.{u_4} S) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_4, u_3} (Set.{u_4} S) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_3, u_4} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) x) -> Prop} (x : M) (hx : Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_4, u_3, u_3} (Set.{u_4} S) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_4, u_3} (Set.{u_4} S) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_3, u_4} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) x), (forall {{r : S}} {{n : M}} (mem₁ : Membership.mem.{u_4, u_4} S (Set.{u_4} S) (Set.instMembership.{u_4} S) s r) (mem₂ : Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) N n), motive (HSMul.hSMul.{u_4, u_3, u_3} S M M (instHSMul.{u_4, u_3} S M (SMulZeroClass.toSMul.{u_4, u_3} S M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_4, u_3} S M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_4, u_3} S M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)))) r n) (Submodule.mem_set_smul_of_mem_mem.{u_2, u_3, u_4} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19 s N r n mem₁ mem₂)) -> (forall (r : R) {{m : M}} (mem : Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_4, u_3, u_3} (Set.{u_4} S) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_4, u_3} (Set.{u_4} S) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_3, u_4} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) m), (motive m mem) -> (motive (HSMul.hSMul.{u_2, u_3, u_3} R M M (instHSMul.{u_2, u_3} R M (SMulZeroClass.toSMul.{u_2, u_3} R M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_2, u_3} R M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_2, u_3} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8) (Module.toDistribMulAction.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11))))) r m) (Submodule.smul_mem.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 (HSMul.hSMul.{u_4, u_3, u_3} (Set.{u_4} S) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_4, u_3} (Set.{u_4} S) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_3, u_4} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) m r mem))) -> (forall {{m₁ : M}} {{m₂ : M}} (mem₁ : Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_4, u_3, u_3} (Set.{u_4} S) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_4, u_3} (Set.{u_4} S) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_3, u_4} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) m₁) (mem₂ : Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_4, u_3, u_3} (Set.{u_4} S) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_4, u_3} (Set.{u_4} S) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_3, u_4} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) m₂), (motive m₁ mem₁) -> (motive m₂ mem₂) -> (motive (HAdd.hAdd.{u_3, u_3, u_3} M M M (instHAdd.{u_3} M (AddCommMagma.toAdd.{u_3} M (AddCommSemigroup.toAddCommMagma.{u_3} M (AddCommMonoid.toAddCommSemigroup.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8)))) m₁ m₂) (Submodule.add_mem.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 (HSMul.hSMul.{u_4, u_3, u_3} (Set.{u_4} S) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_4, u_3} (Set.{u_4} S) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_3, u_4} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) m₁ m₂ mem₁ mem₂))) -> (motive (OfNat.ofNat.{u_3} M 0 (Zero.toOfNat0.{u_3} M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8)))))) (Submodule.zero_mem.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 (HSMul.hSMul.{u_4, u_3, u_3} (Set.{u_4} S) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_4, u_3} (Set.{u_4} S) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_3, u_4} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N))) -> (motive x hx)","typeFull":"∀ {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {S : Type u_4}\n  [inst_3 : Monoid S] [inst_4 : DistribMulAction S M] {s : Set S} {N : Submodule R M}\n  {motive : (x : M) → x ∈ s • N → Prop} (x : M) (hx : x ∈ s • N),\n  (∀ ⦃r : S⦄ ⦃n : M⦄ (mem₁ : r ∈ s) (mem₂ : n ∈ N), motive (r • n) ⋯) →\n    (∀ (r : R) ⦃m : M⦄ (mem : m ∈ s • N), motive m mem → motive (r • m) ⋯) →\n      (∀ ⦃m₁ m₂ : M⦄ (mem₁ : m₁ ∈ s • N) (mem₂ : m₂ ∈ s • N), motive m₁ mem₁ → motive m₂ mem₂ → motive (m₁ + m₂) ⋯) →\n        motive 0 ⋯ → motive x hx","typeReadable":"∀ {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {S : Type u_4}\n  [inst_3 : Monoid S] [inst_4 : DistribMulAction S M] {s : Set S} {N : Submodule R M}\n  {motive : (x : M) → x ∈ s • N → Prop} (x : M) (hx : x ∈ s • N),\n  (∀ ⦃r : S⦄ ⦃n : M⦄ (mem₁ : r ∈ s) (mem₂ : n ∈ N), motive (r • n) ⋯) →\n    (∀ (r : R) ⦃m : M⦄ (mem : m ∈ s • N), motive m mem → motive (r • m) ⋯) →\n      (∀ ⦃m₁ m₂ : M⦄ (mem₁ : m₁ ∈ s • N) (mem₂ : m₂ ∈ s • N), motive m₁ mem₁ → motive m₂ mem₂ → motive (m₁ + m₂) ⋯) →\n        motive 0 ⋯ → motive x 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{α : Type.{u_1}} {ι : Sort.{u_4}} [inst._@.Mathlib.Order.CompleteLattice.Basic.10424780._hygCtx._hyg.13 : CompleteLattice.{u_1} α] {f : ι -> α} {a : α}, Eq.{1} Prop (LE.le.{u_1} α (Preorder.toLE.{u_1} α (PartialOrder.toPreorder.{u_1} α (CompleteSemilatticeInf.toPartialOrder.{u_1} α (CompleteLattice.toCompleteSemilatticeInf.{u_1} α inst._@.Mathlib.Order.CompleteLattice.Basic.10424780._hygCtx._hyg.13)))) (iSup.{u_1, u_4} α ι (CompleteSemilatticeSup.toSupSet.{u_1} α (CompleteLattice.toCompleteSemilatticeSup.{u_1} α inst._@.Mathlib.Order.CompleteLattice.Basic.10424780._hygCtx._hyg.13)) f) a) (forall (i : ι), LE.le.{u_1} α (Preorder.toLE.{u_1} α (PartialOrder.toPreorder.{u_1} α (CompleteSemilatticeInf.toPartialOrder.{u_1} α (CompleteLattice.toCompleteSemilatticeInf.{u_1} α inst._@.Mathlib.Order.CompleteLattice.Basic.10424780._hygCtx._hyg.13)))) (f i) a)","typeFull":"∀ {α : Type u_1} {ι : Sort u_4} [inst : CompleteLattice α] {f : ι → α} {a : α}, (iSup f ≤ a) = ∀ (i : ι), f i ≤ 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[inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1405222010._hygCtx._hyg.8 : AddCommGroup.{u_3} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1405222010._hygCtx._hyg.11 : Module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1405222010._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1405222010._hygCtx._hyg.8)] {ι : Sort.{u_4}} (S : ι -> (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1405222010._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1405222010._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1405222010._hygCtx._hyg.11)), Eq.{succ u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1405222010._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1405222010._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1405222010._hygCtx._hyg.11) (Neg.neg.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1405222010._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1405222010._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1405222010._hygCtx._hyg.11) (Submodule.pointwiseNeg.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1405222010._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1405222010._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1405222010._hygCtx._hyg.11) (iSup.{u_3, u_4} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1405222010._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1405222010._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1405222010._hygCtx._hyg.11) ι (ConditionallyCompletePartialOrderSup.toSupSet.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1405222010._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1405222010._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1405222010._hygCtx._hyg.11) (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1405222010._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1405222010._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1405222010._hygCtx._hyg.11) (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1405222010._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1405222010._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1405222010._hygCtx._hyg.11) (CompleteLattice.toConditionallyCompleteLattice.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1405222010._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1405222010._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1405222010._hygCtx._hyg.11) (Submodule.completeLattice.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1405222010._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1405222010._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1405222010._hygCtx._hyg.11))))) (fun (i : ι) => S i))) (iSup.{u_3, u_4} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1405222010._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1405222010._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1405222010._hygCtx._hyg.11) ι (ConditionallyCompletePartialOrderSup.toSupSet.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1405222010._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1405222010._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1405222010._hygCtx._hyg.11) (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1405222010._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1405222010._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1405222010._hygCtx._hyg.11) (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1405222010._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1405222010._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1405222010._hygCtx._hyg.11) (CompleteLattice.toConditionallyCompleteLattice.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1405222010._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1405222010._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1405222010._hygCtx._hyg.11) (Submodule.completeLattice.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1405222010._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1405222010._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1405222010._hygCtx._hyg.11))))) (fun (i : ι) => Neg.neg.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1405222010._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1405222010._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1405222010._hygCtx._hyg.11) (Submodule.pointwiseNeg.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1405222010._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1405222010._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1405222010._hygCtx._hyg.11) (S i)))","typeFull":"∀ {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommGroup M] [inst_2 : Module R M] {ι : Sort u_4}\n  (S : ι → Submodule R M), -⨆ i, S i = ⨆ i, -S i","typeReadable":"∀ {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommGroup M] [inst_2 : Module R M] {ι : Sort u_4}\n  (S : ι → Submodule R M), -⨆ i, S i = ⨆ i, -S 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[inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.113869382._hygCtx._hyg.11 : Module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.113869382._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.113869382._hygCtx._hyg.8] {S : Type.{u_4}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.113869382._hygCtx._hyg.16 : Monoid.{u_4} S] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.113869382._hygCtx._hyg.19 : DistribMulAction.{u_4, u_3} S M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.113869382._hygCtx._hyg.16 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.113869382._hygCtx._hyg.8)] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.113869382._hygCtx._hyg.30 : SMulCommClass.{u_4, u_2, u_3} S R M (SMulZeroClass.toSMul.{u_4, u_3} S M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M 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inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.113869382._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_2, u_3} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.113869382._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.113869382._hygCtx._hyg.8) (Module.toDistribMulAction.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.113869382._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.113869382._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.113869382._hygCtx._hyg.11))))] (s : Set.{u_4} S) (t : Set.{u_3} M), Eq.{succ u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.113869382._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.113869382._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.113869382._hygCtx._hyg.11) (HSMul.hSMul.{u_4, u_3, u_3} (Set.{u_4} S) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.113869382._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.113869382._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.113869382._hygCtx._hyg.11) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.113869382._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.113869382._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.113869382._hygCtx._hyg.11) (instHSMul.{u_4, u_3} (Set.{u_4} S) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.113869382._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.113869382._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.113869382._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_3, u_4} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.113869382._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.113869382._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.113869382._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.113869382._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.113869382._hygCtx._hyg.19)) s (Submodule.span.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.113869382._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.113869382._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.113869382._hygCtx._hyg.11 t)) (Submodule.span.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.113869382._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.113869382._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.113869382._hygCtx._hyg.11 (HSMul.hSMul.{u_4, u_3, u_3} (Set.{u_4} S) (Set.{u_3} M) (Set.{u_3} M) (instHSMul.{u_4, u_3} (Set.{u_4} S) (Set.{u_3} M) (Set.smul.{u_4, u_3} S M 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inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3285923494._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3285923494._hygCtx._hyg.11) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3285923494._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3285923494._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3285923494._hygCtx._hyg.11) (instHSMul.{u_4, u_3} (Set.{u_4} S) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3285923494._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3285923494._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3285923494._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_3, u_4} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3285923494._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3285923494._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3285923494._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3285923494._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3285923494._hygCtx._hyg.19)) (Singleton.singleton.{u_4, u_4} S (Set.{u_4} S) (Set.instSingletonSet.{u_4} S) r) N) x) (Exists.{succ u_3} M (fun (m : M) => And (Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3285923494._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3285923494._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3285923494._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3285923494._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3285923494._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3285923494._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3285923494._hygCtx._hyg.5 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{R : Type.{u_1}} {M : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11 : Module.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8] (s : Set.{u_1} R) (t : Set.{u_1} R) (x : Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11) {{r : R}} {{n : M}}, (Membership.mem.{u_1, u_1} R (Set.{u_1} R) (Set.instMembership.{u_1} R) (HMul.hMul.{u_1, u_1, u_1} (Set.{u_1} R) (Set.{u_1} R) (Set.{u_1} R) (instHMul.{u_1} (Set.{u_1} R) (Semigroup.toMul.{u_1} (Set.{u_1} R) (Monoid.toSemigroup.{u_1} (Set.{u_1} R) (Set.monoid.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5)))))) s t) r) -> (Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11) M (Submodule.setLike.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11)) x n) -> (Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11) M (Submodule.setLike.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11)) (HSMul.hSMul.{u_1, u_2, u_2} (Set.{u_1} R) (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11) (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11) (instHSMul.{u_1, u_2} (Set.{u_1} R) (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_1, u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11 R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5)) (Module.toDistribMulAction.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11))) s (HSMul.hSMul.{u_1, u_2, u_2} (Set.{u_1} R) (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11) (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11) (instHSMul.{u_1, u_2} (Set.{u_1} R) (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_1, u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11 R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5)) (Module.toDistribMulAction.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11))) t x)) (HSMul.hSMul.{u_1, u_2, u_2} R M M (instHSMul.{u_1, u_2} R M (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 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8) (Module.toDistribMulAction.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11))))) r n))","typeFull":"∀ {R : Type u_1} {M : Type u_2} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] (s t : Set R)\n  (x : Submodule R M) ⦃r : R⦄ ⦃n : M⦄, r ∈ s * t → n ∈ x → r • n ∈ s • t • x","typeReadable":"∀ {R : Type u_1} {M : Type u_2} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] (s t : Set R)\n  (x : Submodule R M) ⦃r : R⦄ ⦃n : M⦄, r ∈ s * t → n ∈ x → r • n ∈ s • t • x","typeReferences":[["Module"],["Membership","mem"],["HMul","hMul"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Semigroup","toMul"],["Submodule"],["AddCommMonoid"],["MonoidWithZero","toMonoid"],["instHSMul"],["Monoid","toSemigroup"],["DistribSMul","toSMulZeroClass"],["SetLike","instMembership"],["Submodule","pointwiseSetSMul"],["Set"],["DistribMulAction","toDistribSMul"],["Set","monoid"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["Set","instMembership"],["Module","toDistribMulAction"],["Submodule","setLike"],["HSMul","hSMul"],["instHMul"],["AddZero","toZero"],["Semiring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["Membership","mem"],["NonUnitalSemiring","toSemigroupWithZero"],["SemigroupAction","toSMul"],["HMul","hMul"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["congrArg"],["Semigroup","toMul"],["Submodule"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["instHSMul"],["Monoid","toSemigroup"],["Eq","ndrec"],["Eq"],["Semiring","toNonUnitalSemiring"],["SemigroupAction","mul_smul"],["DistribSMul","toSMulZeroClass"],["Exists"],["SetLike","instMembership"],["Submodule","pointwiseSetSMul"],["Set"],["DistribMulAction","toDistribSMul"],["And"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["Set","instMembership"],["Exists","casesOn"],["Module","toDistribMulAction"],["Submodule","setLike"],["HSMul","hSMul"],["id"],["instHMul"],["Eq","mpr"],["SemigroupWithZero","toSemigroup"],["MulAction","toSemigroupAction"],["AddZero","toZero"],["Submodule","mem_set_smul_of_mem_mem"],["And","casesOn"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Submodule","mem_set_smul"],"typeFallback":"forall {R : Type.{u_2}} {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.8 : AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.11 : Module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.8] (sR : Set.{u_2} R) (N : Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.11) (x : M) [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.31 : SMulCommClass.{u_2, u_2, u_3} R R (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.11)) N x)) (Submodule.smul.{u_2, u_2, u_3} R R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.11 N (SMulZeroClass.toSMul.{u_2, u_2} R R (AddZero.toZero.{u_2} R (AddZeroClass.toAddZero.{u_2} R (AddMonoid.toAddZeroClass.{u_2} R (AddMonoidWithOne.toAddMonoid.{u_2} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_2} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5))))))) (DistribSMul.toSMulZeroClass.{u_2, u_2} R R (AddMonoid.toAddZeroClass.{u_2} R (AddMonoidWithOne.toAddMonoid.{u_2} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_2} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5))))) (instDistribSMul.{u_2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5))))) (SMulZeroClass.toSMul.{u_2, u_3} R M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_2, u_3} R M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_2, u_3} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.8) (Module.toDistribMulAction.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.11)))) (IsScalarTower.left.{u_2, u_3} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5)) (DistribMulAction.toMulAction.{u_2, u_3} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.8) (Module.toDistribMulAction.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.11)))) (Submodule.smul.{u_2, u_2, u_3} R R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.11 N (SMulZeroClass.toSMul.{u_2, u_2} R R (AddZero.toZero.{u_2} R (AddZeroClass.toAddZero.{u_2} R (AddMonoid.toAddZeroClass.{u_2} R (AddMonoidWithOne.toAddMonoid.{u_2} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_2} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5))))))) (DistribSMul.toSMulZeroClass.{u_2, u_2} R R (AddMonoid.toAddZeroClass.{u_2} R (AddMonoidWithOne.toAddMonoid.{u_2} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_2} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5))))) (instDistribSMul.{u_2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5))))) (SMulZeroClass.toSMul.{u_2, u_3} R M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_2, u_3} R M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_2, u_3} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.8) (Module.toDistribMulAction.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.11)))) (IsScalarTower.left.{u_2, u_3} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5)) (DistribMulAction.toMulAction.{u_2, u_3} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.8) (Module.toDistribMulAction.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.11))))], Iff (Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.11)) (HSMul.hSMul.{u_2, u_3, u_3} (Set.{u_2} R) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.11) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.11) (instHSMul.{u_2, u_3} (Set.{u_2} R) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_3, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.11 R (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5)) (Module.toDistribMulAction.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.11))) sR N) x) (Exists.{max (succ u_2) (succ u_3)} (Finsupp.{u_2, u_3} R (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.11)) N x)) (Submodule.zero.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.11 N)) (fun (c : Finsupp.{u_2, u_3} R (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.11)) N x)) (Submodule.zero.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.11 N)) => And (HasSubset.Subset.{u_2} (Set.{u_2} R) (Set.instHasSubset.{u_2} R) (SetLike.coe.{u_2, u_2} (Finset.{u_2} R) R (Finset.instSetLike.{u_2} R) (Finsupp.support.{u_2, u_3} R (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.11)) N x)) (Submodule.zero.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.11 N) c)) sR) (Eq.{succ u_3} M x (Subtype.val.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.11)) N x) (Finsupp.sum.{u_2, u_3, u_3} R (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.11)) N x)) (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.11)) N x)) (Submodule.zero.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.11 N) (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.11 N) c (fun (r : R) (m : Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.11)) N x)) => HSMul.hSMul.{u_2, u_3, u_3} R (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.11)) N x)) (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.11)) N x)) (instHSMul.{u_2, u_3} R (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.11)) N x)) (Submodule.smul.{u_2, u_2, u_3} R R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.11 N (SMulZeroClass.toSMul.{u_2, u_2} R R (AddZero.toZero.{u_2} R (AddZeroClass.toAddZero.{u_2} R (AddMonoid.toAddZeroClass.{u_2} R (AddMonoidWithOne.toAddMonoid.{u_2} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_2} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5))))))) (DistribSMul.toSMulZeroClass.{u_2, u_2} R R (AddMonoid.toAddZeroClass.{u_2} R (AddMonoidWithOne.toAddMonoid.{u_2} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_2} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5))))) (instDistribSMul.{u_2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5))))) (SMulZeroClass.toSMul.{u_2, u_3} R M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_2, u_3} R M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_2, u_3} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.5)) 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inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.500620249._hygCtx._hyg.11))))) r m))))))","typeFull":"∀ {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] (sR : Set R)\n  (N : Submodule R M) (x : M) [SMulCommClass R R ↥N], x ∈ sR • N ↔ ∃ c, ↑c.support ⊆ sR ∧ x = ↑(c.sum fun r m => r • m)","typeReadable":"∀ {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] (sR : Set R)\n  (N : Submodule R M) (x : M) [SMulCommClass R R ↥N], x ∈ sR • N ↔ ∃ c, ↑c.support ⊆ sR ∧ x = ↑(c.sum fun r m => r • 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{R : Type.{u_2}} {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3202808869._hygCtx._hyg.5 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3202808869._hygCtx._hyg.8 : AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3202808869._hygCtx._hyg.11 : Module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3202808869._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3202808869._hygCtx._hyg.8] {S : Type.{u_4}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3202808869._hygCtx._hyg.16 : Monoid.{u_4} S] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3202808869._hygCtx._hyg.19 : DistribMulAction.{u_4, u_3} S M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3202808869._hygCtx._hyg.16 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3202808869._hygCtx._hyg.8)] (N : Submodule.{u_2, u_3} R M 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inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3202808869._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3202808869._hygCtx._hyg.11) (instHSMul.{u_4, u_3} (Set.{u_4} S) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3202808869._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3202808869._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3202808869._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_3, u_4} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3202808869._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3202808869._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3202808869._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3202808869._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3202808869._hygCtx._hyg.19)) (Max.max.{u_4} (Set.{u_4} S) (SemilatticeSup.toMax.{u_4} (Set.{u_4} S) (Lattice.toSemilatticeSup.{u_4} 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inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3202808869._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_3, u_4} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3202808869._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3202808869._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3202808869._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3202808869._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3202808869._hygCtx._hyg.19)) s N) (HSMul.hSMul.{u_4, u_3, u_3} (Set.{u_4} S) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3202808869._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3202808869._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3202808869._hygCtx._hyg.11) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3202808869._hygCtx._hyg.5 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Module R M] {S : Type u_4}\n  [inst_3 : Monoid S] [inst_4 : DistribMulAction S M] (N : Submodule R M) (s t : Set S), (s ⊔ t) • N = s • N ⊔ t • N","typeReadable":"∀ {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {S : Type u_4}\n  [inst_3 : Monoid S] [inst_4 : DistribMulAction S M] (N : Submodule R M) (s t : Set S), (s ⊔ t) • N = s • N ⊔ t • 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inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3542245458._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3542245458._hygCtx._hyg.11) (Submodule.instPartialOrder.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3542245458._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3542245458._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3542245458._hygCtx._hyg.11))) (Neg.neg.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3542245458._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3542245458._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3542245458._hygCtx._hyg.11) (Submodule.pointwiseNeg.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3542245458._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3542245458._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3542245458._hygCtx._hyg.11) S) (Neg.neg.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3542245458._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3542245458._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3542245458._hygCtx._hyg.11) (Submodule.pointwiseNeg.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3542245458._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3542245458._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3542245458._hygCtx._hyg.11) T)) (LE.le.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3542245458._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3542245458._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3542245458._hygCtx._hyg.11) (Preorder.toLE.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3542245458._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3542245458._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3542245458._hygCtx._hyg.11) (PartialOrder.toPreorder.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3542245458._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3542245458._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3542245458._hygCtx._hyg.11) (Submodule.instPartialOrder.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3542245458._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3542245458._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3542245458._hygCtx._hyg.11))) S T)","typeFull":"∀ {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommGroup M] [inst_2 : Module R M]\n  {S T : Submodule R M}, (-S ≤ -T) = (S ≤ T)","typeReadable":"∀ {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommGroup M] [inst_2 : Module R M]\n  {S T : Submodule R M}, (-S ≤ -T) = (S ≤ T)","typeReferences":[["PartialOrder","toPreorder"],["Neg","neg"],["Module"],["Submodule","instPartialOrder"],["LE","le"],["AddCommGroup","toAddCommMonoid"],["Submodule","pointwiseNeg"],["AddCommGroup"],["Preorder","toLE"],["Eq"],["Submodule"],["Semiring"]],"valueReferences":[["PartialOrder","toPreorder"],["Neg","neg"],["Submodule","instPartialOrder"],["LE","le"],["AddCommGroup","toAddCommMonoid"],["Submodule","pointwiseNeg"],["Preorder","toLE"],["Submodule","neg_le_neg"],["propext"],["Submodule"]]},{"isProp":true,"kind":"theorem","name":["Submodule","coe_set_neg"],"typeFallback":"forall {R : Type.{u_2}} {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.170536826._hygCtx._hyg.5 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.170536826._hygCtx._hyg.8 : AddCommGroup.{u_3} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.170536826._hygCtx._hyg.11 : Module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.170536826._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.170536826._hygCtx._hyg.8)] (S : Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.170536826._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.170536826._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.170536826._hygCtx._hyg.11), Eq.{succ u_3} (Set.{u_3} M) (SetLike.coe.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.170536826._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.170536826._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.170536826._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.170536826._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.170536826._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.170536826._hygCtx._hyg.11) (Neg.neg.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.170536826._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.170536826._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.170536826._hygCtx._hyg.11) (Submodule.pointwiseNeg.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.170536826._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.170536826._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.170536826._hygCtx._hyg.11) S)) (Neg.neg.{u_3} (Set.{u_3} M) (Set.neg.{u_3} M (NegZeroClass.toNeg.{u_3} M (SubNegZeroMonoid.toNegZeroClass.{u_3} M (SubtractionMonoid.toSubNegZeroMonoid.{u_3} M (SubtractionCommMonoid.toSubtractionMonoid.{u_3} M (AddCommGroup.toDivisionAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.170536826._hygCtx._hyg.8)))))) (SetLike.coe.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.170536826._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.170536826._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.170536826._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.170536826._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.170536826._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.170536826._hygCtx._hyg.11) S))","typeFull":"∀ {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommGroup M] [inst_2 : Module R M] (S : Submodule R M),\n  ↑(-S) = -↑S","typeReadable":"∀ {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommGroup M] [inst_2 : Module R M] (S : Submodule R M),\n  ↑(-S) = -↑S","typeReferences":[["SubtractionMonoid","toSubNegZeroMonoid"],["Set","neg"],["Module"],["Set"],["Neg","neg"],["SubtractionCommMonoid","toSubtractionMonoid"],["AddCommGroup"],["SubNegZeroMonoid","toNegZeroClass"],["Submodule"],["SetLike","coe"],["NegZeroClass","toNeg"],["AddCommGroup","toDivisionAddCommMonoid"],["Submodule","setLike"],["AddCommGroup","toAddCommMonoid"],["Submodule","pointwiseNeg"],["Eq"],["Semiring"]],"valueReferences":[["rfl"],["SetLike","coe"],["Neg","neg"],["Submodule","setLike"],["Set"],["AddCommGroup","toAddCommMonoid"],["Submodule","pointwiseNeg"],["Submodule"]]},{"isProp":true,"kind":"theorem","name":["Submodule","pointwiseDistribMulAction","_proof_2"],"typeFallback":"forall {α : Type.{u_3}} {R : Type.{u_1}} {M : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.11 : Module.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.15 : Monoid.{u_3} α] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.18 : DistribMulAction.{u_3, u_2} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8)] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.22 : SMulCommClass.{u_3, u_1, u_2} α R M (SMulZeroClass.toSMul.{u_3, u_2} α M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_3, u_2} α M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_3, u_2} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.18))) (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 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8) (Module.toDistribMulAction.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.11))))] (S : Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.11), Eq.{succ u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.11) (Submodule.map.{u_1, u_1, u_2, u_2} R R M M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.11 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5)) (RingHomSurjective.ids.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5) (DistribSMul.toLinearMap.{u_1, u_3, u_2} R α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.11 (DistribMulAction.toDistribSMul.{u_3, u_2} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.18) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.22 (OfNat.ofNat.{u_3} α 1 (One.toOfNat1.{u_3} α (MulOne.toOne.{u_3} α (MulOneClass.toMulOne.{u_3} α (Monoid.toMulOneClass.{u_3} α inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.15)))))) S) S","typeFull":"∀ {α : Type u_3} {R : Type u_1} {M : Type u_2} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  [inst_3 : Monoid α] [inst_4 : DistribMulAction α M] [inst_5 : SMulCommClass α R M] (S : Submodule R M),\n  Submodule.map (DistribSMul.toLinearMap R M 1) S = S","typeReadable":"∀ {α : Type u_3} {R : Type u_1} {M : Type u_2} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  [inst_3 : Monoid α] [inst_4 : DistribMulAction α M] [inst_5 : SMulCommClass α R M] (S : Submodule R M),\n  Submodule.map (DistribSMul.toLinearMap R M 1) S = S","typeReferences":[["MulOneClass","toMulOne"],["Module"],["Submodule","map"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Submodule"],["DistribSMul","toLinearMap"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["Monoid","toMulOneClass"],["MonoidWithZero","toMonoid"],["Eq"],["DistribMulAction"],["DistribSMul","toSMulZeroClass"],["MulOne","toOne"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["RingHomSurjective","ids"],["SMulCommClass"],["Module","toDistribMulAction"],["One","toOfNat1"],["Monoid"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["MulOneClass","toMulOne"],["Module","End"],["Eq","trans"],["MulOne","toOne"],["congr_arg"],["Submodule","map"],["DistribMulAction","toDistribSMul"],["AddCommMonoid","toAddMonoid"],["LinearMap","id"],["OfNat","ofNat"],["Submodule"],["RingHomSurjective","ids"],["DistribSMul","toLinearMap"],["Semiring","toNonAssocSemiring"],["One","toOfNat1"],["RingHom","id"],["LinearMap","ext"],["one_smul"],["Monoid","toMulOneClass"],["DistribMulAction","toMulAction"],["Submodule","map_id"]]},{"isProp":false,"kind":"definition","name":["Submodule","involutivePointwiseNeg"],"typeFallback":"forall {R : Type.{u_2}} {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.596932550._hygCtx._hyg.5 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.596932550._hygCtx._hyg.8 : AddCommGroup.{u_3} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.596932550._hygCtx._hyg.11 : Module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.596932550._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.596932550._hygCtx._hyg.8)], InvolutiveNeg.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.596932550._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.596932550._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.596932550._hygCtx._hyg.11)","typeFull":"{R : Type u_2} →\n  {M : Type u_3} →\n    [inst : Semiring R] → [inst_1 : AddCommGroup M] → [inst_2 : Module R M] → InvolutiveNeg (Submodule R M)","typeReadable":"{R : Type u_2} →\n  {M : Type u_3} →\n    [inst : Semiring R] → [inst_1 : AddCommGroup M] → [inst_2 : Module R M] → InvolutiveNeg (Submodule R M)","typeReferences":[["InvolutiveNeg"],["Module"],["AddCommGroup","toAddCommMonoid"],["AddCommGroup"],["Submodule"],["Semiring"]],"valueReferences":[["Submodule","involutivePointwiseNeg","_proof_1"],["AddCommGroup","toAddCommMonoid"],["Submodule","pointwiseNeg"],["InvolutiveNeg","mk"],["Submodule"]]},{"isProp":true,"kind":"theorem","name":["Submodule","pointwiseAddCommMonoid","_proof_6"],"typeFallback":"forall {R : Type.{u_1}} {M : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11 : Module.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8] (a : Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11) (b : Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11), Eq.{succ u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11) (Max.max.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11) (SemilatticeSup.toMax.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11) (Lattice.toSemilatticeSup.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11) (ConditionallyCompleteLattice.toLattice.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11) (CompleteLattice.toConditionallyCompleteLattice.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11) (Submodule.completeLattice.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11))))) a b) (Max.max.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11) (SemilatticeSup.toMax.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11) (Lattice.toSemilatticeSup.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11) (ConditionallyCompleteLattice.toLattice.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11) (CompleteLattice.toConditionallyCompleteLattice.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11) (Submodule.completeLattice.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11))))) b a)","typeFull":"∀ {R : Type u_1} {M : Type u_2} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  (a b : Submodule R M), a ⊔ b = b ⊔ a","typeReadable":"∀ {R : Type u_1} {M : Type u_2} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  (a b : Submodule R M), a ⊔ b = b ⊔ a","typeReferences":[["Lattice","toSemilatticeSup"],["Max","max"],["AddCommMonoid"],["Module"],["SemilatticeSup","toMax"],["Submodule","completeLattice"],["ConditionallyCompleteLattice","toLattice"],["Eq"],["CompleteLattice","toConditionallyCompleteLattice"],["Submodule"],["Semiring"]],"valueReferences":[["Lattice","toSemilatticeSup"],["Submodule","completeLattice"],["ConditionallyCompleteLattice","toLattice"],["sup_comm"],["CompleteLattice","toConditionallyCompleteLattice"],["Submodule"]]},{"isProp":true,"kind":"theorem","name":["Submodule","smul_iSup'"],"typeFallback":"forall {α : Type.{u_1}} {R : Type.{u_2}} {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.5 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.8 : AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.11 : Module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.8] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.15 : Monoid.{u_1} α] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.18 : DistribMulAction.{u_1, u_3} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.8)] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.22 : SMulCommClass.{u_1, u_2, u_3} α R M (SMulZeroClass.toSMul.{u_1, u_3} α M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_1, u_3} α M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_1, u_3} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.18))) (SMulZeroClass.toSMul.{u_2, u_3} R M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_2, u_3} R M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_2, u_3} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.8) (Module.toDistribMulAction.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.11))))] (a : α) {ι : Sort.{u_4}} (f : ι -> (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.11)), Eq.{succ u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.11) (HSMul.hSMul.{u_1, u_3, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.11) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.11) (instHSMul.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.11) (SMulZeroClass.toSMul.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.11) (AddZero.toZero.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.11) (AddZeroClass.toAddZero.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.11) (AddMonoid.toAddZeroClass.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.11) (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.11))))) (DistribSMul.toSMulZeroClass.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.11) (AddMonoid.toAddZeroClass.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.11) (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.11))) (DistribMulAction.toDistribSMul.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.11) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.11)) (Submodule.pointwiseDistribMulAction.{u_1, u_2, u_3} α R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.22))))) a (iSup.{u_3, u_4} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.11) ι (ConditionallyCompletePartialOrderSup.toSupSet.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.11) (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.11) (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.11) (CompleteLattice.toConditionallyCompleteLattice.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.11) (Submodule.completeLattice.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.11))))) (fun (i : ι) => f i))) (iSup.{u_3, u_4} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.11) ι (ConditionallyCompletePartialOrderSup.toSupSet.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.11) (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.11) (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.11) (CompleteLattice.toConditionallyCompleteLattice.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.11) (Submodule.completeLattice.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.11))))) (fun (i : ι) => HSMul.hSMul.{u_1, u_3, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.11) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.11) (instHSMul.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.11) (SMulZeroClass.toSMul.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.11) (AddZero.toZero.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.11) (AddZeroClass.toAddZero.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.11) (AddMonoid.toAddZeroClass.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.11) (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.11))))) (DistribSMul.toSMulZeroClass.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.11) (AddMonoid.toAddZeroClass.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.11) (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.11))) (DistribMulAction.toDistribSMul.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.11) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.11)) (Submodule.pointwiseDistribMulAction.{u_1, u_2, u_3} α R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1593450553._hygCtx._hyg.22))))) a (f i)))","typeFull":"∀ {α : Type u_1} {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  [inst_3 : Monoid α] [inst_4 : DistribMulAction α M] [inst_5 : SMulCommClass α R M] (a : α) {ι : Sort u_4}\n  (f : ι → Submodule R M), a • ⨆ i, f i = ⨆ i, a • f i","typeReadable":"∀ {α : Type u_1} {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  [inst_3 : Monoid α] [inst_4 : DistribMulAction α M] [inst_5 : SMulCommClass α R M] (a : α) {ι : Sort u_4}\n  (f : ι → Submodule R M), a • ⨆ i, f i = ⨆ i, a • f i","typeReferences":[["Module"],["ConditionallyCompleteLattice","toConditionallyCompletePartialOrder"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Submodule"],["AddCommMonoid"],["ConditionallyCompletePartialOrder","toConditionallyCompletePartialOrderSup"],["iSup"],["ConditionallyCompletePartialOrderSup","toSupSet"],["MonoidWithZero","toMonoid"],["instHSMul"],["Eq"],["CompleteLattice","toConditionallyCompleteLattice"],["DistribMulAction"],["DistribSMul","toSMulZeroClass"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["SMulCommClass"],["Module","toDistribMulAction"],["HSMul","hSMul"],["Submodule","completeLattice"],["Monoid"],["AddZero","toZero"],["Submodule","pointwiseAddCommMonoid"],["Submodule","pointwiseDistribMulAction"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["Submodule","map_iSup"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["DistribMulAction","toDistribSMul"],["AddCommMonoid","toAddMonoid"],["DistribSMul","toLinearMap"],["RingHomSurjective","ids"]]},{"isProp":true,"kind":"theorem","name":["Submodule","pointwise_smul_toAddSubgroup"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.15 : Monoid.{u_1} α] {R : Type.{u_4}} {M : Type.{u_5}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.29 : Ring.{u_4} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.32 : AddCommGroup.{u_5} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.35 : DistribMulAction.{u_1, u_5} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.15 (SubNegMonoid.toAddMonoid.{u_5} M (AddGroup.toSubNegMonoid.{u_5} M (AddCommGroup.toAddGroup.{u_5} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.32)))] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.39 : Module.{u_4, u_5} R M (Ring.toSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.29) (AddCommGroup.toAddCommMonoid.{u_5} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.32)] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.43 : SMulCommClass.{u_1, u_4, u_5} α R M (SMulZeroClass.toSMul.{u_1, u_5} α M (AddZero.toZero.{u_5} M (AddZeroClass.toAddZero.{u_5} M (AddMonoid.toAddZeroClass.{u_5} M (SubNegMonoid.toAddMonoid.{u_5} M (AddGroup.toSubNegMonoid.{u_5} M (AddCommGroup.toAddGroup.{u_5} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.32)))))) (DistribSMul.toSMulZeroClass.{u_1, u_5} α M (AddMonoid.toAddZeroClass.{u_5} M (SubNegMonoid.toAddMonoid.{u_5} M (AddGroup.toSubNegMonoid.{u_5} M (AddCommGroup.toAddGroup.{u_5} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.32)))) (DistribMulAction.toDistribSMul.{u_1, u_5} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.15 (SubNegMonoid.toAddMonoid.{u_5} M (AddGroup.toSubNegMonoid.{u_5} M (AddCommGroup.toAddGroup.{u_5} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.32))) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.35))) (SMulZeroClass.toSMul.{u_4, u_5} R M (AddZero.toZero.{u_5} M (AddZeroClass.toAddZero.{u_5} M (AddMonoid.toAddZeroClass.{u_5} M (SubNegMonoid.toAddMonoid.{u_5} M (AddGroup.toSubNegMonoid.{u_5} M (AddCommGroup.toAddGroup.{u_5} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.32)))))) (DistribSMul.toSMulZeroClass.{u_4, u_5} R M (AddMonoid.toAddZeroClass.{u_5} M (SubNegMonoid.toAddMonoid.{u_5} M (AddGroup.toSubNegMonoid.{u_5} M (AddCommGroup.toAddGroup.{u_5} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.32)))) (DistribMulAction.toDistribSMul.{u_4, u_5} R M (MonoidWithZero.toMonoid.{u_4} R (Semiring.toMonoidWithZero.{u_4} R (Ring.toSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.29))) (SubNegMonoid.toAddMonoid.{u_5} M (AddGroup.toSubNegMonoid.{u_5} M (AddCommGroup.toAddGroup.{u_5} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.32))) (Module.toDistribMulAction.{u_4, u_5} R M (Ring.toSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.29) (AddCommGroup.toAddCommMonoid.{u_5} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.32) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.39))))] (a : α) (S : Submodule.{u_4, u_5} R M (Ring.toSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.29) (AddCommGroup.toAddCommMonoid.{u_5} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.32) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.39), Eq.{succ u_5} (AddSubgroup.{u_5} M (AddCommGroup.toAddGroup.{u_5} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.32)) (Submodule.toAddSubgroup.{u_4, u_5} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.32 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.39 (HSMul.hSMul.{u_1, u_5, u_5} α (Submodule.{u_4, u_5} R M (Ring.toSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.29) (AddCommGroup.toAddCommMonoid.{u_5} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.32) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.39) (Submodule.{u_4, u_5} R M (Ring.toSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.29) (AddCommGroup.toAddCommMonoid.{u_5} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.32) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.39) (instHSMul.{u_1, u_5} α (Submodule.{u_4, u_5} R M (Ring.toSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.29) (AddCommGroup.toAddCommMonoid.{u_5} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.32) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.39) (SMulZeroClass.toSMul.{u_1, u_5} α (Submodule.{u_4, u_5} R M (Ring.toSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.29) (AddCommGroup.toAddCommMonoid.{u_5} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.32) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.39) (AddZero.toZero.{u_5} (Submodule.{u_4, u_5} R M (Ring.toSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.29) (AddCommGroup.toAddCommMonoid.{u_5} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.32) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.39) (AddZeroClass.toAddZero.{u_5} (Submodule.{u_4, u_5} R M (Ring.toSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.29) (AddCommGroup.toAddCommMonoid.{u_5} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.32) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.39) (AddMonoid.toAddZeroClass.{u_5} (Submodule.{u_4, u_5} R M (Ring.toSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.29) (AddCommGroup.toAddCommMonoid.{u_5} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.32) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.39) (AddCommMonoid.toAddMonoid.{u_5} (Submodule.{u_4, u_5} R M (Ring.toSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.29) (AddCommGroup.toAddCommMonoid.{u_5} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.32) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.39) (Submodule.pointwiseAddCommMonoid.{u_4, u_5} R M (Ring.toSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.29) (AddCommGroup.toAddCommMonoid.{u_5} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.32) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.39))))) (DistribSMul.toSMulZeroClass.{u_1, u_5} α (Submodule.{u_4, u_5} R M (Ring.toSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.29) (AddCommGroup.toAddCommMonoid.{u_5} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.32) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.39) (AddMonoid.toAddZeroClass.{u_5} (Submodule.{u_4, u_5} R M (Ring.toSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.29) (AddCommGroup.toAddCommMonoid.{u_5} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.32) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.39) (AddCommMonoid.toAddMonoid.{u_5} (Submodule.{u_4, u_5} R M (Ring.toSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.29) (AddCommGroup.toAddCommMonoid.{u_5} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.32) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.39) (Submodule.pointwiseAddCommMonoid.{u_4, u_5} R M (Ring.toSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.29) (AddCommGroup.toAddCommMonoid.{u_5} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.32) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.39))) (DistribMulAction.toDistribSMul.{u_1, u_5} α (Submodule.{u_4, u_5} R M (Ring.toSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.29) (AddCommGroup.toAddCommMonoid.{u_5} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.32) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.39) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_5} (Submodule.{u_4, u_5} R M (Ring.toSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.29) (AddCommGroup.toAddCommMonoid.{u_5} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.32) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.39) (Submodule.pointwiseAddCommMonoid.{u_4, u_5} R M (Ring.toSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.29) (AddCommGroup.toAddCommMonoid.{u_5} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.32) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.39)) (Submodule.pointwiseDistribMulAction.{u_1, u_4, u_5} α R M (Ring.toSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.29) (AddCommGroup.toAddCommMonoid.{u_5} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.32) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.35 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.43))))) a S)) (HSMul.hSMul.{u_1, u_5, u_5} α (AddSubgroup.{u_5} M (AddCommGroup.toAddGroup.{u_5} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.32)) (AddSubgroup.{u_5} M (AddCommGroup.toAddGroup.{u_5} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.32)) (instHSMul.{u_1, u_5} α (AddSubgroup.{u_5} M (AddCommGroup.toAddGroup.{u_5} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.32)) (SemigroupAction.toSMul.{u_1, u_5} α (AddSubgroup.{u_5} M (AddCommGroup.toAddGroup.{u_5} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.32)) (Monoid.toSemigroup.{u_1} α inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.15) (MulAction.toSemigroupAction.{u_1, u_5} α (AddSubgroup.{u_5} M (AddCommGroup.toAddGroup.{u_5} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.32)) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.15 (AddSubgroup.pointwiseMulAction.{u_1, u_5} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.15 (AddCommGroup.toAddGroup.{u_5} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.32) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.35)))) a (Submodule.toAddSubgroup.{u_4, u_5} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.32 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.756791647._hygCtx._hyg.39 S))","typeFull":"∀ {α : Type u_1} [inst : Monoid α] {R : Type u_4} {M : Type u_5} [inst_1 : Ring R] [inst_2 : AddCommGroup M]\n  [inst_3 : DistribMulAction α M] [inst_4 : Module R M] [inst_5 : SMulCommClass α R M] (a : α) (S : Submodule R M),\n  (a • S).toAddSubgroup = a • S.toAddSubgroup","typeReadable":"∀ {α : Type u_1} [inst : Monoid α] {R : Type u_4} {M : Type u_5} [inst_1 : Ring R] [inst_2 : AddCommGroup M]\n  [inst_3 : DistribMulAction α M] [inst_4 : Module R M] [inst_5 : SMulCommClass α R M] (a : α) (S : Submodule R M),\n  (a • S).toAddSubgroup = a • 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{R : Type.{u_2}} {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.5 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.8 : AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.11 : Module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.8] {S : Type.{u_4}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.16 : Monoid.{u_4} S] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.19 : DistribMulAction.{u_4, u_3} S M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.16 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.8)] (s : Set.{u_4} S) (N : Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.11) (x : M), Iff (Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.11)) (HSMul.hSMul.{u_4, u_3, u_3} (Set.{u_4} S) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.11) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.11) (instHSMul.{u_4, u_3} (Set.{u_4} S) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_3, u_4} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.19)) s N) x) (Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.11)) (InfSet.sInf.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.11) (Submodule.instInfSet.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.11) (setOf.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.11) (fun (p : Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.11) => forall {{r : S}} {n : M}, (Membership.mem.{u_4, u_4} S (Set.{u_4} S) (Set.instMembership.{u_4} S) s r) -> (Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.11)) N n) -> (Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.11)) p (HSMul.hSMul.{u_4, u_3, u_3} S M M (instHSMul.{u_4, u_3} S M (SMulZeroClass.toSMul.{u_4, u_3} S M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_4, u_3} S M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_4, u_3} S M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.16 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3994801020._hygCtx._hyg.19)))) r n))))) x)","typeFull":"∀ {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {S : Type u_4}\n  [inst_3 : Monoid S] [inst_4 : DistribMulAction S M] (s : Set S) (N : Submodule R M) (x : M),\n  x ∈ s • N ↔ x ∈ sInf {p | ∀ ⦃r : S⦄ {n : M}, r ∈ s → n ∈ N → r • n ∈ p}","typeReadable":"∀ {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {S : Type u_4}\n  [inst_3 : Monoid S] [inst_4 : DistribMulAction S M] (s : Set S) (N : Submodule R M) (x : M),\n  x ∈ s • N ↔ x ∈ sInf {p | ∀ ⦃r : S⦄ {n : M}, r ∈ s → n ∈ N → r • n ∈ p}","typeReferences":[["Submodule","instInfSet"],["Module"],["Membership","mem"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Submodule"],["AddCommMonoid"],["InfSet","sInf"],["instHSMul"],["DistribMulAction"],["DistribSMul","toSMulZeroClass"],["SetLike","instMembership"],["Submodule","pointwiseSetSMul"],["Set"],["DistribMulAction","toDistribSMul"],["AddZeroClass","toAddZero"],["Set","instMembership"],["Submodule","setLike"],["Iff"],["HSMul","hSMul"],["Monoid"],["AddZero","toZero"],["Semiring"],["setOf"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["Submodule","pointwiseSetSMul"],["SetLike","instMembership"],["Set"],["Submodule","setLike"],["HSMul","hSMul"],["Membership","mem"],["instHSMul"],["Iff","rfl"],["Submodule"]]},{"isProp":true,"kind":"theorem","name":["Submodule","pointwiseSetMulAction","_proof_4"],"typeFallback":"forall {R : Type.{u_2}} {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 : AddCommMonoid.{u_1} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11 : Module.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8] (x : Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11) (m : M), (Exists.{succ u_1} M (fun (m_1 : M) => And (Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11)) x m_1) (Eq.{succ u_1} M m (HSMul.hSMul.{u_2, u_1, u_1} R M M (instHSMul.{u_2, u_1} R M (SMulZeroClass.toSMul.{u_2, u_1} R M (AddZero.toZero.{u_1} M (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_2, u_1} R M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_2, u_1} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8) (Module.toDistribMulAction.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11))))) (OfNat.ofNat.{u_2} R 1 (One.toOfNat1.{u_2} R (AddMonoidWithOne.toOne.{u_2} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_2} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5)))))) m_1)))) -> (Membership.mem.{u_1, u_1} M 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inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1550418663._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1550418663._hygCtx._hyg.11)) N n) -> (Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1550418663._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1550418663._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1550418663._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1550418663._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1550418663._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1550418663._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1550418663._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1550418663._hygCtx._hyg.8 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inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1550418663._hygCtx._hyg.11))) p (HSMul.hSMul.{u_4, u_3, u_3} (Set.{u_4} S) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1550418663._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1550418663._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1550418663._hygCtx._hyg.11) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1550418663._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1550418663._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1550418663._hygCtx._hyg.11) (instHSMul.{u_4, u_3} (Set.{u_4} S) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1550418663._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1550418663._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1550418663._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_3, u_4} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1550418663._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1550418663._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1550418663._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1550418663._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1550418663._hygCtx._hyg.19)) s N)) -> (Eq.{succ u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1550418663._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1550418663._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1550418663._hygCtx._hyg.11) (HSMul.hSMul.{u_4, u_3, u_3} (Set.{u_4} S) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1550418663._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1550418663._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1550418663._hygCtx._hyg.11) (Submodule.{u_2, u_3} R 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Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {S : Type u_4}\n  [inst_3 : Monoid S] [inst_4 : DistribMulAction S M] (s : Set S) (N p : Submodule R M),\n  (∀ ⦃r : S⦄ ⦃n : M⦄, r ∈ s → n ∈ N → r • n ∈ p) → p ≤ s • N → s • N = p","typeReadable":"∀ {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {S : Type u_4}\n  [inst_3 : Monoid S] [inst_4 : DistribMulAction S M] (s : Set S) (N p : Submodule R M),\n  (∀ ⦃r : S⦄ ⦃n : M⦄, r ∈ s → n ∈ N → r • n ∈ p) → p ≤ s • N → s • N = 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[inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.5 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.8 : AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.11 : Module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.8] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.15 : Monoid.{u_1} α] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.18 : DistribMulAction.{u_1, u_3} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.8)] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.22 : SMulCommClass.{u_1, u_2, u_3} α R M (SMulZeroClass.toSMul.{u_1, u_3} α M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_1, u_3} α M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_1, u_3} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.18))) (SMulZeroClass.toSMul.{u_2, u_3} R M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_2, u_3} R M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_2, u_3} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.8) (Module.toDistribMulAction.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.11))))] (m : M) (a : α) (S : Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.11), (Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.11)) S m) -> (Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.11)) (HSMul.hSMul.{u_1, u_3, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.11) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.11) (instHSMul.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.11) (SMulZeroClass.toSMul.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.11) (AddZero.toZero.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.11) (AddZeroClass.toAddZero.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.11) (AddMonoid.toAddZeroClass.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.11) (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.11))))) (DistribSMul.toSMulZeroClass.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.11) (AddMonoid.toAddZeroClass.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.11) (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.11))) (DistribMulAction.toDistribSMul.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.11) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.11)) (Submodule.pointwiseDistribMulAction.{u_1, u_2, u_3} α R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.22))))) a S) (HSMul.hSMul.{u_1, u_3, u_3} α M M (instHSMul.{u_1, u_3} α M (SMulZeroClass.toSMul.{u_1, u_3} α M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_1, u_3} α M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_1, u_3} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.878961345._hygCtx._hyg.18)))) a m))","typeFull":"∀ {α : Type u_1} {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  [inst_3 : Monoid α] [inst_4 : DistribMulAction α M] [inst_5 : SMulCommClass α R M] (m : M) (a : α)\n  (S : Submodule R M), m ∈ S → a • m ∈ a • S","typeReadable":"∀ {α : Type u_1} {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  [inst_3 : Monoid α] [inst_4 : DistribMulAction α M] [inst_5 : SMulCommClass α R M] (m : M) (a : α)\n  (S : Submodule R M), m ∈ S → a • m ∈ a • S","typeReferences":[["SetLike","instMembership"],["Module"],["Membership","mem"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["Submodule"],["SMulCommClass"],["Module","toDistribMulAction"],["AddCommMonoid"],["Submodule","setLike"],["HSMul","hSMul"],["MonoidWithZero","toMonoid"],["Monoid"],["instHSMul"],["AddZero","toZero"],["Submodule","pointwiseDistribMulAction"],["Submodule","pointwiseAddCommMonoid"],["DistribMulAction"],["Semiring"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]],"valueReferences":[["Set","smul_mem_smul_set"],["SetLike","coe"],["Submodule","setLike"],["DistribMulAction","toDistribSMul"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["AddZero","toZero"],["Submodule"],["DistribSMul","toSMulZeroClass"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Submodule","smul_span"],"typeFallback":"forall {α : Type.{u_1}} {R : Type.{u_2}} {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.5 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.8 : AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.11 : Module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.8] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.15 : Monoid.{u_1} α] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.18 : DistribMulAction.{u_1, u_3} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.8)] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.22 : SMulCommClass.{u_1, u_2, u_3} α R M (SMulZeroClass.toSMul.{u_1, u_3} α M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_1, u_3} α M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_1, u_3} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.18))) (SMulZeroClass.toSMul.{u_2, u_3} R M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_2, u_3} R M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_2, u_3} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.8) (Module.toDistribMulAction.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.11))))] (a : α) (s : Set.{u_3} M), Eq.{succ u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.11) (HSMul.hSMul.{u_1, u_3, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.11) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.11) (instHSMul.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.11) (SMulZeroClass.toSMul.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.11) (AddZero.toZero.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.11) (AddZeroClass.toAddZero.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.11) (AddMonoid.toAddZeroClass.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.11) (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.11))))) (DistribSMul.toSMulZeroClass.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.11) (AddMonoid.toAddZeroClass.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.11) (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.11))) (DistribMulAction.toDistribSMul.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.11) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.11)) (Submodule.pointwiseDistribMulAction.{u_1, u_2, u_3} α R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.22))))) a (Submodule.span.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.11 s)) (Submodule.span.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.11 (HSMul.hSMul.{u_1, u_3, u_3} α (Set.{u_3} M) (Set.{u_3} M) (instHSMul.{u_1, u_3} α (Set.{u_3} M) (Set.smulSet.{u_1, u_3} α M (SMulZeroClass.toSMul.{u_1, u_3} α M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_1, u_3} α M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_1, u_3} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.18))))) a s))","typeFull":"∀ {α : Type u_1} {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  [inst_3 : Monoid α] [inst_4 : DistribMulAction α M] [inst_5 : SMulCommClass α R M] (a : α) (s : Set M),\n  a • Submodule.span R s = Submodule.span R (a • s)","typeReadable":"∀ {α : Type u_1} {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  [inst_3 : Monoid α] [inst_4 : DistribMulAction α M] [inst_5 : SMulCommClass α R M] (a : α) (s : Set M),\n  a • Submodule.span R s = Submodule.span R (a • s)","typeReferences":[["Module"],["SMulZeroClass","toSMul"],["Submodule","span"],["AddCommMonoid","toAddMonoid"],["Submodule"],["AddCommMonoid"],["MonoidWithZero","toMonoid"],["Set","smulSet"],["instHSMul"],["Eq"],["DistribMulAction"],["DistribSMul","toSMulZeroClass"],["Set"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["SMulCommClass"],["Module","toDistribMulAction"],["HSMul","hSMul"],["Monoid"],["AddZero","toZero"],["Submodule","pointwiseAddCommMonoid"],["Submodule","pointwiseDistribMulAction"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["Submodule","map_span"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["DistribMulAction","toDistribSMul"],["AddCommMonoid","toAddMonoid"],["DistribSMul","toLinearMap"],["RingHomSurjective","ids"]]},{"isProp":true,"kind":"theorem","name":["Submodule","pointwiseCentralScalar"],"typeFallback":"forall {α : Type.{u_1}} {R : Type.{u_2}} {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.5 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.8 : AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.11 : Module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.8] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.15 : Monoid.{u_1} α] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.18 : DistribMulAction.{u_1, u_3} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.8)] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.22 : SMulCommClass.{u_1, u_2, u_3} α R M (SMulZeroClass.toSMul.{u_1, u_3} α M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_1, u_3} α M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_1, u_3} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.18))) (SMulZeroClass.toSMul.{u_2, u_3} R M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_2, u_3} R M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_2, u_3} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.8) (Module.toDistribMulAction.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.11))))] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.27 : DistribMulAction.{u_1, u_3} (MulOpposite.{u_1} α) M (MulOpposite.instMonoid.{u_1} α inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.15) (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.8)] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.34 : SMulCommClass.{u_1, u_2, u_3} (MulOpposite.{u_1} α) R M (SMulZeroClass.toSMul.{u_1, u_3} (MulOpposite.{u_1} α) M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_1, u_3} (MulOpposite.{u_1} α) M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_1, u_3} (MulOpposite.{u_1} α) M (MulOpposite.instMonoid.{u_1} α inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.15) (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.27))) (SMulZeroClass.toSMul.{u_2, u_3} R M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_2, u_3} R M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_2, u_3} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.8) (Module.toDistribMulAction.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.11))))] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.42 : IsCentralScalar.{u_1, u_3} α M (SMulZeroClass.toSMul.{u_1, u_3} α M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_1, u_3} α M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_1, u_3} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.18))) (SMulZeroClass.toSMul.{u_1, u_3} (MulOpposite.{u_1} α) M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_1, u_3} (MulOpposite.{u_1} α) M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_1, u_3} (MulOpposite.{u_1} α) M (MulOpposite.instMonoid.{u_1} α inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.15) (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.27)))], IsCentralScalar.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.11) (SMulZeroClass.toSMul.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.11) (AddZero.toZero.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.11) (AddZeroClass.toAddZero.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.11) (AddMonoid.toAddZeroClass.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.11) (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.11))))) (DistribSMul.toSMulZeroClass.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.11) (AddMonoid.toAddZeroClass.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.11) (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.11))) (DistribMulAction.toDistribSMul.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.11) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.11)) (Submodule.pointwiseDistribMulAction.{u_1, u_2, u_3} α R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.22)))) (SMulZeroClass.toSMul.{u_1, u_3} (MulOpposite.{u_1} α) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.11) (AddZero.toZero.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.11) (AddZeroClass.toAddZero.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.11) (AddMonoid.toAddZeroClass.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.11) (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.11))))) (DistribSMul.toSMulZeroClass.{u_1, u_3} (MulOpposite.{u_1} α) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.11) (AddMonoid.toAddZeroClass.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.11) (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.11))) (DistribMulAction.toDistribSMul.{u_1, u_3} (MulOpposite.{u_1} α) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.11) (MulOpposite.instMonoid.{u_1} α inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.15) (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.11)) (Submodule.pointwiseDistribMulAction.{u_1, u_2, u_3} (MulOpposite.{u_1} α) R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.11 (MulOpposite.instMonoid.{u_1} α inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.15) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3508568758._hygCtx._hyg.34))))","typeFull":"∀ {α : Type u_1} {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  [inst_3 : Monoid α] [inst_4 : DistribMulAction α M] [inst_5 : SMulCommClass α R M] [inst_6 : DistribMulAction αᵐᵒᵖ M]\n  [inst_7 : SMulCommClass αᵐᵒᵖ R M] [IsCentralScalar α M], IsCentralScalar α (Submodule R M)","typeReadable":"∀ {α : Type u_1} {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  [inst_3 : Monoid α] [inst_4 : DistribMulAction α M] [inst_5 : SMulCommClass α R M] [inst_6 : DistribMulAction αᵐᵒᵖ M]\n  [inst_7 : SMulCommClass αᵐᵒᵖ R M] [IsCentralScalar α M], IsCentralScalar α (Submodule R M)","typeReferences":[["Module"],["DistribMulAction","toDistribSMul"],["MulOpposite","instMonoid"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["Submodule"],["SMulCommClass"],["MulOpposite"],["Module","toDistribMulAction"],["AddCommMonoid"],["IsCentralScalar"],["MonoidWithZero","toMonoid"],["Monoid"],["AddZero","toZero"],["Submodule","pointwiseDistribMulAction"],["Submodule","pointwiseAddCommMonoid"],["DistribMulAction"],["Semiring"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]],"valueReferences":[["Module","End"],["Submodule","map"],["congr_arg"],["DistribMulAction","toDistribSMul"],["MulOpposite","instMonoid"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["RingHomSurjective","ids"],["DistribSMul","toLinearMap"],["Submodule"],["MulOpposite"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["LinearMap","ext"],["MulOpposite","op"],["IsCentralScalar","mk"],["AddZero","toZero"],["Submodule","pointwiseAddCommMonoid"],["Submodule","pointwiseDistribMulAction"],["IsCentralScalar","op_smul_eq_smul"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Submodule","pointwiseSetDistribMulAction","_proof_3"],"typeFallback":"forall {R : Type.{u_1}} {M : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11 : Module.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.30 : SMulCommClass.{u_1, u_1, u_2} R R M (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 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8) (Module.toDistribMulAction.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11)))) (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 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8) (Module.toDistribMulAction.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11))))] (s : Set.{u_1} R) (x : Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11) (y : 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inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11) (instHSMul.{u_1, u_2} (Set.{u_1} R) (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11) (SemigroupAction.toSMul.{u_1, u_2} (Set.{u_1} R) (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11) (Monoid.toSemigroup.{u_1} (Set.{u_1} R) (Set.monoid.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5)))) 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inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11))))) x y)) (HAdd.hAdd.{u_2, u_2, u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11) (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11) (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11) (instHAdd.{u_2} 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inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11))))) (HSMul.hSMul.{u_1, u_2, u_2} (Set.{u_1} R) (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11) (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 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inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11) (Set.monoid.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5))) (Submodule.pointwiseSetMulAction.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.30)))) s x) (HSMul.hSMul.{u_1, u_2, u_2} (Set.{u_1} R) (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11) (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11) (instHSMul.{u_1, u_2} (Set.{u_1} R) (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11) (SemigroupAction.toSMul.{u_1, u_2} (Set.{u_1} R) (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11) (Monoid.toSemigroup.{u_1} (Set.{u_1} R) (Set.monoid.{u_1} R 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{R : Type.{u_2}} {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 : AddCommMonoid.{u_1} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 : Module.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8] {S : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 : Monoid.{u_3} S] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19 : DistribMulAction.{u_3, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8)] {s : Set.{u_3} S} {N : Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11} {motive : forall (x : M), (Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) x) -> Prop} {b._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.208 : M} (motive_1 : (Membership.mem.{u_1, u_1} M (Set.{u_1} M) (Set.instMembership.{u_1} M) (setOf.{u_1} M (fun (m : M) => Exists.{0} (Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) m) (fun (mem : Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) m) => motive m mem))) b._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.208) -> Prop) (x._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx.204.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.227 : Membership.mem.{u_1, u_1} M (Set.{u_1} M) (Set.instMembership.{u_1} M) (setOf.{u_1} M (fun (m : M) => Exists.{0} (Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) m) (fun (mem : Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) m) => motive m mem))) b._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.208), (forall (mem' : Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) b._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.208) (h' : motive b._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.208 mem'), motive_1 (Exists.intro.{0} (Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) b._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.208) (fun (mem : Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) b._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.208) => motive b._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.208 mem) mem' h')) -> (motive_1 x._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx.204.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.227)","typeFull":"∀ {R : Type u_2} {M : Type u_1} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {S : Type u_3}\n  [inst_3 : Monoid S] [inst_4 : DistribMulAction S M] {s : Set S} {N : Submodule R M}\n  {motive : (x : M) → x ∈ s • N → Prop} {b : M} (motive_1 : b ∈ {m | ∃ (mem : m ∈ s • N), motive m mem} → Prop)\n  (x : b ∈ {m | ∃ (mem : m ∈ s • N), motive m mem}),\n  (∀ (mem' : b ∈ s • N) (h' : motive b mem'), motive_1 ⋯) → motive_1 x","typeReadable":"∀ {R : Type u_2} {M : Type u_1} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {S : Type u_3}\n  [inst_3 : Monoid S] [inst_4 : DistribMulAction S M] {s : Set S} {N : Submodule R M}\n  {motive : (x : M) → x ∈ s • N → Prop} {b : M} (motive_1 : b ∈ {m | ∃ (mem : m ∈ s • N), motive m mem} → Prop)\n  (x : b ∈ {m | ∃ (mem : m ∈ s • N), motive m mem}),\n  (∀ (mem' : b ∈ s • N) (h' : motive b mem'), motive_1 ⋯) → motive_1 x","typeReferences":[["Exists"],["Submodule","pointwiseSetSMul"],["SetLike","instMembership"],["Module"],["Set"],["Membership","mem"],["Exists","intro"],["AddCommMonoid","toAddMonoid"],["Set","instMembership"],["Submodule"],["AddCommMonoid"],["Submodule","setLike"],["HSMul","hSMul"],["Monoid"],["instHSMul"],["DistribMulAction"],["setOf"],["Semiring"]],"valueReferences":[["Exists","casesOn"],["Submodule","pointwiseSetSMul"],["SetLike","instMembership"],["Set"],["Submodule","setLike"],["HSMul","hSMul"],["Membership","mem"],["instHSMul"],["Submodule"]]},{"isProp":true,"kind":"theorem","name":["Submodule","pointwiseSetMulAction","_proof_3"],"typeFallback":"forall {R : Type.{u_2}} {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 : AddCommMonoid.{u_1} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11 : Module.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.30 : SMulCommClass.{u_2, u_2, u_1} R R M (SMulZeroClass.toSMul.{u_2, u_1} R M (AddZero.toZero.{u_1} M (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_2, u_1} R M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_2, u_1} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8) (Module.toDistribMulAction.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11)))) (SMulZeroClass.toSMul.{u_2, u_1} R M (AddZero.toZero.{u_1} M (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_2, u_1} R M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_2, u_1} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8) (Module.toDistribMulAction.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11))))] (s : Set.{u_2} R) (t : Set.{u_2} R) (x : Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11) (r : R) (m : M), (Membership.mem.{u_2, u_2} R (Set.{u_2} R) (Set.instMembership.{u_2} R) s r) -> (Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11)) (HSMul.hSMul.{u_2, u_1, u_1} (Set.{u_2} R) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11) (instHSMul.{u_2, u_1} (Set.{u_2} R) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11 R (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5)) (Module.toDistribMulAction.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11))) t x) m) -> (Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11)) (HSMul.hSMul.{u_2, u_1, u_1} (Set.{u_2} R) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11) (instHSMul.{u_2, u_1} (Set.{u_2} R) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11 R (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5)) (Module.toDistribMulAction.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11))) (HMul.hMul.{u_2, u_2, u_2} (Set.{u_2} R) (Set.{u_2} R) (Set.{u_2} R) (instHMul.{u_2} (Set.{u_2} R) (Semigroup.toMul.{u_2} (Set.{u_2} R) (Monoid.toSemigroup.{u_2} (Set.{u_2} R) (Set.monoid.{u_2} R 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R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11))))) r m))","typeFull":"∀ {R : Type u_2} {M : Type u_1} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  [SMulCommClass R R M] (s t : Set R) (x : Submodule R M) (r : R) (m : M), r ∈ s → m ∈ t • x → r • m ∈ (s * t) • x","typeReadable":"∀ {R : Type u_2} {M : Type u_1} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  [SMulCommClass R R M] (s t : Set R) (x : Submodule R M) (r : R) (m : M), r ∈ s → m ∈ t • x → r • m ∈ (s * t) • 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{R : Type.{u_2}} {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.5 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.8 : AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.11 : Module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.8] {S : Type.{u_4}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.16 : Monoid.{u_4} S] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.19 : DistribMulAction.{u_4, u_3} S M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.16 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.8)] (N : Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.11) [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.30 : SMulCommClass.{u_4, u_2, u_3} S R M (SMulZeroClass.toSMul.{u_4, u_3} S M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_4, u_3} S M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_4, u_3} S M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.16 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.19))) (SMulZeroClass.toSMul.{u_2, u_3} R M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_2, u_3} R M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_2, u_3} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.8) (Module.toDistribMulAction.{u_2, u_3} R M 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inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.11) (AddMonoid.toAddZeroClass.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.11) (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.11))))) (DistribSMul.toSMulZeroClass.{u_4, u_3} S (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.11) (AddMonoid.toAddZeroClass.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.11) (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.8 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inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.11)) (Submodule.pointwiseDistribMulAction.{u_4, u_2, u_3} S R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.30))))) r N) (HSMul.hSMul.{u_4, u_3, u_3} (Set.{u_4} S) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.11) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.11) (instHSMul.{u_4, u_3} (Set.{u_4} S) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_3, u_4} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.19)) (Singleton.singleton.{u_4, u_4} S (Set.{u_4} S) (Set.instSingletonSet.{u_4} S) r) N)","typeFull":"∀ {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {S : Type u_4}\n  [inst_3 : Monoid S] [inst_4 : DistribMulAction S M] (N : Submodule R M) [inst_5 : SMulCommClass S R M] (r : S),\n  r • N = {r} • N","typeReadable":"∀ {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {S : Type u_4}\n  [inst_3 : Monoid S] [inst_4 : DistribMulAction S M] (N : Submodule R M) [inst_5 : SMulCommClass S R M] (r : S),\n  r • N = {r} • N","typeReferences":[["Module"],["Singleton","singleton"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Submodule"],["AddCommMonoid"],["MonoidWithZero","toMonoid"],["instHSMul"],["Eq"],["DistribMulAction"],["DistribSMul","toSMulZeroClass"],["Submodule","pointwiseSetSMul"],["Set"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["Set","instSingletonSet"],["AddZeroClass","toAddZero"],["SMulCommClass"],["Module","toDistribMulAction"],["HSMul","hSMul"],["Monoid"],["AddZero","toZero"],["Submodule","pointwiseAddCommMonoid"],["Submodule","pointwiseDistribMulAction"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["Submodule","pointwiseSetSMul"],["Set"],["Singleton","singleton"],["DistribMulAction","toDistribSMul"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Set","instSingletonSet"],["AddZeroClass","toAddZero"],["Submodule"],["HSMul","hSMul"],["Eq","symm"],["instHSMul"],["Submodule","singleton_set_smul"],["AddZero","toZero"],["Submodule","pointwiseDistribMulAction"],["Submodule","pointwiseAddCommMonoid"],["DistribSMul","toSMulZeroClass"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Submodule","stabilizer_coe"],"typeFallback":"forall {R : Type.{u_4}} {G : Type.{u_5}} {M : Type.{u_6}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.534955780._hygCtx._hyg.18 : Semiring.{u_4} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.534955780._hygCtx._hyg.21 : AddCommMonoid.{u_6} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.534955780._hygCtx._hyg.24 : Module.{u_4, u_6} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.534955780._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.534955780._hygCtx._hyg.21] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.534955780._hygCtx._hyg.28 : Group.{u_5} G] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.534955780._hygCtx._hyg.31 : DistribMulAction.{u_5, u_6} G M (DivInvMonoid.toMonoid.{u_5} G (Group.toDivInvMonoid.{u_5} G inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.534955780._hygCtx._hyg.28)) (AddCommMonoid.toAddMonoid.{u_6} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.534955780._hygCtx._hyg.21)] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.534955780._hygCtx._hyg.35 : SMulCommClass.{u_5, u_4, u_6} G R M (SMulZeroClass.toSMul.{u_5, u_6} G M (AddZero.toZero.{u_6} M (AddZeroClass.toAddZero.{u_6} M (AddMonoid.toAddZeroClass.{u_6} M (AddCommMonoid.toAddMonoid.{u_6} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.534955780._hygCtx._hyg.21)))) (DistribSMul.toSMulZeroClass.{u_5, u_6} G M (AddMonoid.toAddZeroClass.{u_6} M (AddCommMonoid.toAddMonoid.{u_6} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.534955780._hygCtx._hyg.21)) (DistribMulAction.toDistribSMul.{u_5, u_6} G M (DivInvMonoid.toMonoid.{u_5} G (Group.toDivInvMonoid.{u_5} G inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.534955780._hygCtx._hyg.28)) (AddCommMonoid.toAddMonoid.{u_6} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.534955780._hygCtx._hyg.21) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.534955780._hygCtx._hyg.31))) (SMulZeroClass.toSMul.{u_4, u_6} R M (AddZero.toZero.{u_6} M 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inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.534955780._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.534955780._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.534955780._hygCtx._hyg.24}, Eq.{succ u_5} (Subgroup.{u_5} G inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.534955780._hygCtx._hyg.28) (MulAction.stabilizer.{u_5, u_6} G (Submodule.{u_4, u_6} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.534955780._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.534955780._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.534955780._hygCtx._hyg.24) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.534955780._hygCtx._hyg.28 (DistribMulAction.toMulAction.{u_5, u_6} G (Submodule.{u_4, u_6} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.534955780._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.534955780._hygCtx._hyg.21 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inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.534955780._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.534955780._hygCtx._hyg.24 (DivInvMonoid.toMonoid.{u_5} G (Group.toDivInvMonoid.{u_5} G inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.534955780._hygCtx._hyg.28)) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.534955780._hygCtx._hyg.31 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.534955780._hygCtx._hyg.35)) S) (MulAction.stabilizer.{u_5, u_6} G (Set.{u_6} M) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.534955780._hygCtx._hyg.28 (Set.mulActionSet.{u_5, u_6} G M (DivInvMonoid.toMonoid.{u_5} G (Group.toDivInvMonoid.{u_5} G inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.534955780._hygCtx._hyg.28)) (DistribMulAction.toMulAction.{u_5, u_6} G M (DivInvMonoid.toMonoid.{u_5} G (Group.toDivInvMonoid.{u_5} G inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.534955780._hygCtx._hyg.28)) (AddCommMonoid.toAddMonoid.{u_6} M 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{R : Type.{u_2}} {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.5 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.8 : AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.11 : Module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.8] {S : Type.{u_4}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.16 : Monoid.{u_4} S] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.19 : DistribMulAction.{u_4, u_3} S M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.16 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.8)] (N : Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.11) [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.30 : SMulCommClass.{u_4, u_2, u_3} S R M (SMulZeroClass.toSMul.{u_4, u_3} S M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_4, u_3} S M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_4, u_3} S M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.16 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.19))) (SMulZeroClass.toSMul.{u_2, u_3} R M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_2, u_3} R M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_2, u_3} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.8) (Module.toDistribMulAction.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.11))))] (r : S), Eq.{succ u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.11) (HSMul.hSMul.{u_4, u_3, u_3} (Set.{u_4} S) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.11) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4243363593._hygCtx._hyg.8 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inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.899056964._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.899056964._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.899056964._hygCtx._hyg.11) (Submodule.instBot.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.899056964._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.899056964._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.899056964._hygCtx._hyg.11))","typeFull":"∀ {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {S : Type u_4}\n  [inst_3 : Monoid S] [inst_4 : DistribMulAction S M] (N : Submodule R M), ∅ • N = ⊥","typeReadable":"∀ {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {S : Type u_4}\n  [inst_3 : Monoid S] [inst_4 : DistribMulAction S M] (N : Submodule R M), ∅ • N = 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{R : Type.{u_2}} {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314969261._hygCtx._hyg.5 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314969261._hygCtx._hyg.8 : AddCommGroup.{u_3} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314969261._hygCtx._hyg.11 : Module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314969261._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314969261._hygCtx._hyg.8)] (S : Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314969261._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314969261._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314969261._hygCtx._hyg.11) (T : Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314969261._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314969261._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314969261._hygCtx._hyg.11), Eq.{succ u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314969261._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314969261._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314969261._hygCtx._hyg.11) (Neg.neg.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314969261._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314969261._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314969261._hygCtx._hyg.11) (Submodule.pointwiseNeg.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314969261._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314969261._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314969261._hygCtx._hyg.11) (Min.min.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314969261._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314969261._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314969261._hygCtx._hyg.11) (Submodule.instMin.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314969261._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314969261._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314969261._hygCtx._hyg.11) S T)) (Min.min.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314969261._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314969261._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314969261._hygCtx._hyg.11) (Submodule.instMin.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314969261._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314969261._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314969261._hygCtx._hyg.11) (Neg.neg.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314969261._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314969261._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314969261._hygCtx._hyg.11) (Submodule.pointwiseNeg.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314969261._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314969261._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314969261._hygCtx._hyg.11) S) (Neg.neg.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314969261._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314969261._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314969261._hygCtx._hyg.11) (Submodule.pointwiseNeg.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314969261._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314969261._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314969261._hygCtx._hyg.11) T))","typeFull":"∀ {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommGroup M] [inst_2 : Module R M]\n  (S T : Submodule R M), -(S ⊓ T) = -S ⊓ -T","typeReadable":"∀ {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommGroup M] [inst_2 : Module R M]\n  (S T : Submodule R M), -(S ⊓ T) = -S ⊓ -T","typeReferences":[["Min","min"],["Neg","neg"],["Module"],["AddCommGroup","toAddCommMonoid"],["Submodule","pointwiseNeg"],["AddCommGroup"],["Submodule","instMin"],["Eq"],["Submodule"],["Semiring"]],"valueReferences":[["rfl"],["Min","min"],["Neg","neg"],["AddCommGroup","toAddCommMonoid"],["Submodule","pointwiseNeg"],["Submodule","instMin"],["Submodule"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Algebra","Module","Submodule","Pointwise",0,"Submodule","smul_def","_simp_1_1"],"typeFallback":"forall {α : Type.{u_1}} {R : Type.{u_2}} {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.5 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.8 : AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.11 : Module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.8] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.15 : Monoid.{u_1} α] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.18 : DistribMulAction.{u_1, u_3} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.8)] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.22 : SMulCommClass.{u_1, u_2, u_3} α R M (SMulZeroClass.toSMul.{u_1, u_3} α M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_1, u_3} α M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M 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inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.8) (Module.toDistribMulAction.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.11))))] (a : α) (s : Set.{u_3} M), Eq.{succ u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.11) (Submodule.span.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.11 (HSMul.hSMul.{u_1, u_3, u_3} α (Set.{u_3} M) (Set.{u_3} M) (instHSMul.{u_1, u_3} α (Set.{u_3} M) (Set.smulSet.{u_1, u_3} α M (SMulZeroClass.toSMul.{u_1, u_3} α M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_1, u_3} α M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_1, u_3} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.18))))) a s)) (HSMul.hSMul.{u_1, u_3, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.11) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.11) (instHSMul.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.11) (SMulZeroClass.toSMul.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.11) (AddZero.toZero.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.11) (AddZeroClass.toAddZero.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.11) (AddMonoid.toAddZeroClass.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.11) (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.11))))) (DistribSMul.toSMulZeroClass.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.11) (AddMonoid.toAddZeroClass.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.11) (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.11))) (DistribMulAction.toDistribSMul.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.11) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.11)) (Submodule.pointwiseDistribMulAction.{u_1, u_2, u_3} α R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.22))))) a (Submodule.span.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2864546998._hygCtx._hyg.11 s))","typeFull":"∀ {α : Type u_1} {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  [inst_3 : Monoid α] [inst_4 : DistribMulAction α M] [inst_5 : SMulCommClass α R M] (a : α) (s : Set M),\n  Submodule.span R (a • s) = a • Submodule.span R s","typeReadable":"∀ {α : Type u_1} {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  [inst_3 : Monoid α] [inst_4 : DistribMulAction α M] [inst_5 : SMulCommClass α R M] (a : α) (s : Set M),\n  Submodule.span R (a • s) = a • Submodule.span R s","typeReferences":[["Module"],["SMulZeroClass","toSMul"],["Submodule","span"],["AddCommMonoid","toAddMonoid"],["Submodule"],["AddCommMonoid"],["MonoidWithZero","toMonoid"],["Set","smulSet"],["instHSMul"],["Eq"],["DistribMulAction"],["DistribSMul","toSMulZeroClass"],["Set"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["SMulCommClass"],["Module","toDistribMulAction"],["HSMul","hSMul"],["Monoid"],["AddZero","toZero"],["Submodule","pointwiseAddCommMonoid"],["Submodule","pointwiseDistribMulAction"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["Submodule","smul_span"],["Set"],["DistribMulAction","toDistribSMul"],["Submodule","span"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["Submodule"],["HSMul","hSMul"],["Set","smulSet"],["Eq","symm"],["instHSMul"],["AddZero","toZero"],["Submodule","pointwiseAddCommMonoid"],["Submodule","pointwiseDistribMulAction"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Submodule","smul_le_self_of_tower","_simp_1"],"typeFallback":"forall {R : Type.{u_2}} {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.5 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.8 : AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.11 : Module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.8] {α : Type.{u_4}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.28 : Monoid.{u_4} α] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.31 : SMul.{u_4, u_2} α R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.35 : DistribMulAction.{u_4, u_3} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.28 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.8)] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.39 : SMulCommClass.{u_4, u_2, u_3} α R M (SMulZeroClass.toSMul.{u_4, u_3} α M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_4, u_3} α M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_4, u_3} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.28 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.35))) (SMulZeroClass.toSMul.{u_2, u_3} R M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_2, u_3} R M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_2, u_3} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.8) (Module.toDistribMulAction.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.11))))] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.44 : IsScalarTower.{u_4, u_2, u_3} α R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.31 (SMulZeroClass.toSMul.{u_2, u_3} R M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_2, u_3} R M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_2, u_3} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.8) (Module.toDistribMulAction.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.11)))) (SMulZeroClass.toSMul.{u_4, u_3} α M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_4, u_3} α M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_4, u_3} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.28 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.35)))] (a : α) (S : Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.11), Eq.{1} Prop (LE.le.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.11) (Preorder.toLE.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.11) (PartialOrder.toPreorder.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.11) (Submodule.instPartialOrder.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.11))) (HSMul.hSMul.{u_4, u_3, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.11) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.11) (instHSMul.{u_4, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.11) (SMulZeroClass.toSMul.{u_4, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.11) (AddZero.toZero.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.11) (AddZeroClass.toAddZero.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.11) (AddMonoid.toAddZeroClass.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.11) (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.11))))) (DistribSMul.toSMulZeroClass.{u_4, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.11) (AddMonoid.toAddZeroClass.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.11) (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.11))) (DistribMulAction.toDistribSMul.{u_4, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.11) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.28 (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.11)) (Submodule.pointwiseDistribMulAction.{u_4, u_2, u_3} α R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.28 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.35 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.39))))) a S) S) True","typeFull":"∀ {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {α : Type u_4}\n  [inst_3 : Monoid α] [inst_4 : SMul α R] [inst_5 : DistribMulAction α M] [inst_6 : SMulCommClass α R M]\n  [IsScalarTower α R M] (a : α) (S : Submodule R M), (a • S ≤ S) = True","typeReadable":"∀ {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {α : Type u_4}\n  [inst_3 : Monoid α] [inst_4 : SMul α R] [inst_5 : DistribMulAction α M] [inst_6 : SMulCommClass α R M]\n  [IsScalarTower α R M] (a : α) (S : Submodule R M), (a • S ≤ S) = True","typeReferences":[["PartialOrder","toPreorder"],["Module"],["IsScalarTower"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Submodule"],["SMul"],["AddCommMonoid"],["MonoidWithZero","toMonoid"],["instHSMul"],["Eq"],["Preorder","toLE"],["DistribMulAction"],["DistribSMul","toSMulZeroClass"],["True"],["Submodule","instPartialOrder"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["SMulCommClass"],["Module","toDistribMulAction"],["LE","le"],["HSMul","hSMul"],["Monoid"],["AddZero","toZero"],["Submodule","pointwiseDistribMulAction"],["Submodule","pointwiseAddCommMonoid"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["PartialOrder","toPreorder"],["Submodule","instPartialOrder"],["DistribMulAction","toDistribSMul"],["eq_true"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["Submodule"],["Submodule","smul_le_self_of_tower"],["LE","le"],["HSMul","hSMul"],["instHSMul"],["Preorder","toLE"],["AddZero","toZero"],["Submodule","pointwiseDistribMulAction"],["Submodule","pointwiseAddCommMonoid"],["DistribSMul","toSMulZeroClass"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Submodule","neg_toAddSubmonoid"],"typeFallback":"forall {R : Type.{u_2}} {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1959437638._hygCtx._hyg.5 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1959437638._hygCtx._hyg.8 : AddCommGroup.{u_3} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1959437638._hygCtx._hyg.11 : Module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1959437638._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1959437638._hygCtx._hyg.8)] (S : Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1959437638._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1959437638._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1959437638._hygCtx._hyg.11), Eq.{succ u_3} (AddSubmonoid.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1959437638._hygCtx._hyg.8)))) (Submodule.toAddSubmonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1959437638._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1959437638._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1959437638._hygCtx._hyg.11 (Neg.neg.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1959437638._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1959437638._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1959437638._hygCtx._hyg.11) (Submodule.pointwiseNeg.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1959437638._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1959437638._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1959437638._hygCtx._hyg.11) S)) (Neg.neg.{u_3} (AddSubmonoid.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1959437638._hygCtx._hyg.8)))) (AddSubmonoid.neg.{u_3} M (AddCommGroup.toAddGroup.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1959437638._hygCtx._hyg.8)) (Submodule.toAddSubmonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1959437638._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1959437638._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1959437638._hygCtx._hyg.11 S))","typeFull":"∀ {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommGroup M] [inst_2 : Module R M] (S : Submodule R M),\n  (-S).toAddSubmonoid = -S.toAddSubmonoid","typeReadable":"∀ {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommGroup M] [inst_2 : Module R M] (S : Submodule R M),\n  (-S).toAddSubmonoid = -S.toAddSubmonoid","typeReferences":[["Submodule","toAddSubmonoid"],["Module"],["Neg","neg"],["AddCommGroup","toAddGroup"],["AddCommGroup"],["AddCommMonoid","toAddMonoid"],["Submodule"],["AddSubmonoid"],["AddSubmonoid","neg"],["AddCommGroup","toAddCommMonoid"],["Submodule","pointwiseNeg"],["Eq"],["Semiring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["rfl"],["Submodule","toAddSubmonoid"],["Neg","neg"],["AddSubmonoid"],["AddCommGroup","toAddCommMonoid"],["Submodule","pointwiseNeg"],["AddCommMonoid","toAddMonoid"],["Submodule"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Submodule","mem_smul_pointwise_iff_exists"],"typeFallback":"forall {α : Type.{u_1}} {R : Type.{u_2}} {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.5 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.8 : AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.11 : Module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.8] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.15 : Monoid.{u_1} α] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.18 : DistribMulAction.{u_1, u_3} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.8)] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.22 : SMulCommClass.{u_1, u_2, u_3} α R M (SMulZeroClass.toSMul.{u_1, u_3} α M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_1, u_3} α M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_1, u_3} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.18))) (SMulZeroClass.toSMul.{u_2, u_3} R M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_2, u_3} R M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_2, u_3} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.8) (Module.toDistribMulAction.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.11))))] (m : M) (a : α) (S : Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.11), Iff (Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.11)) (HSMul.hSMul.{u_1, u_3, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.11) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.11) (instHSMul.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.11) (SMulZeroClass.toSMul.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.11) (AddZero.toZero.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.11) (AddZeroClass.toAddZero.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.11) (AddMonoid.toAddZeroClass.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.11) (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.11))))) (DistribSMul.toSMulZeroClass.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.11) (AddMonoid.toAddZeroClass.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.11) (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.11))) (DistribMulAction.toDistribSMul.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.11) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.11)) (Submodule.pointwiseDistribMulAction.{u_1, u_2, u_3} α R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.22))))) a S) m) (Exists.{succ u_3} M (fun (b : M) => And (Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.11)) S b) (Eq.{succ u_3} M (HSMul.hSMul.{u_1, u_3, u_3} α M M (instHSMul.{u_1, u_3} α M (SMulZeroClass.toSMul.{u_1, u_3} α M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_1, u_3} α M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_1, u_3} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3174668242._hygCtx._hyg.18)))) a b) m)))","typeFull":"∀ {α : Type u_1} {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  [inst_3 : Monoid α] [inst_4 : DistribMulAction α M] [inst_5 : SMulCommClass α R M] (m : M) (a : α)\n  (S : Submodule R M), m ∈ a • S ↔ ∃ b ∈ S, a • b = m","typeReadable":"∀ {α : Type u_1} {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  [inst_3 : Monoid α] [inst_4 : DistribMulAction α M] [inst_5 : SMulCommClass α R M] (m : M) (a : α)\n  (S : Submodule R M), m ∈ a • S ↔ ∃ b ∈ S, a • b = m","typeReferences":[["Module"],["Membership","mem"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Submodule"],["AddCommMonoid"],["MonoidWithZero","toMonoid"],["instHSMul"],["Eq"],["DistribMulAction"],["DistribSMul","toSMulZeroClass"],["Exists"],["SetLike","instMembership"],["And"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["SMulCommClass"],["Module","toDistribMulAction"],["Submodule","setLike"],["Iff"],["HSMul","hSMul"],["Monoid"],["AddZero","toZero"],["Submodule","pointwiseAddCommMonoid"],["Submodule","pointwiseDistribMulAction"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["SetLike","coe"],["Submodule","setLike"],["DistribMulAction","toDistribSMul"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["AddZero","toZero"],["Set","mem_smul_set"],["Submodule"],["DistribSMul","toSMulZeroClass"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Submodule","set_smul_mono_left"],"typeFallback":"forall {R : Type.{u_2}} {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.336126294._hygCtx._hyg.5 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.336126294._hygCtx._hyg.8 : AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.336126294._hygCtx._hyg.11 : Module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.336126294._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.336126294._hygCtx._hyg.8] {S : Type.{u_4}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.336126294._hygCtx._hyg.16 : Monoid.{u_4} S] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.336126294._hygCtx._hyg.19 : DistribMulAction.{u_4, u_3} S M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.336126294._hygCtx._hyg.16 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.336126294._hygCtx._hyg.8)] (N : Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.336126294._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.336126294._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.336126294._hygCtx._hyg.11) {s : Set.{u_4} S} {t : Set.{u_4} S}, (LE.le.{u_4} (Set.{u_4} S) (Set.instLE.{u_4} S) s t) -> (LE.le.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.336126294._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.336126294._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.336126294._hygCtx._hyg.11) (Preorder.toLE.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.336126294._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.336126294._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.336126294._hygCtx._hyg.11) (PartialOrder.toPreorder.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.336126294._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.336126294._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.336126294._hygCtx._hyg.11) (Submodule.instPartialOrder.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.336126294._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.336126294._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.336126294._hygCtx._hyg.11))) (HSMul.hSMul.{u_4, u_3, u_3} (Set.{u_4} S) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.336126294._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.336126294._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.336126294._hygCtx._hyg.11) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.336126294._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.336126294._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.336126294._hygCtx._hyg.11) (instHSMul.{u_4, u_3} (Set.{u_4} S) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.336126294._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.336126294._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.336126294._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_3, u_4} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.336126294._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.336126294._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.336126294._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.336126294._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.336126294._hygCtx._hyg.19)) s N) (HSMul.hSMul.{u_4, u_3, u_3} (Set.{u_4} S) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.336126294._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.336126294._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.336126294._hygCtx._hyg.11) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.336126294._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.336126294._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.336126294._hygCtx._hyg.11) (instHSMul.{u_4, u_3} (Set.{u_4} S) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.336126294._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.336126294._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.336126294._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_3, u_4} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.336126294._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.336126294._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.336126294._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.336126294._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.336126294._hygCtx._hyg.19)) t N))","typeFull":"∀ {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {S : Type u_4}\n  [inst_3 : Monoid S] [inst_4 : DistribMulAction S M] (N : Submodule R M) {s t : Set S}, s ≤ t → s • N ≤ t • N","typeReadable":"∀ {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {S : Type u_4}\n  [inst_3 : Monoid S] [inst_4 : DistribMulAction S M] (N : Submodule R M) {s t : Set S}, s ≤ t → s • N ≤ t • N","typeReferences":[["Submodule","pointwiseSetSMul"],["PartialOrder","toPreorder"],["Module"],["Set"],["Submodule","instPartialOrder"],["AddCommMonoid","toAddMonoid"],["Submodule"],["AddCommMonoid"],["HSMul","hSMul"],["LE","le"],["Monoid"],["instHSMul"],["Preorder","toLE"],["DistribMulAction"],["Semiring"],["Set","instLE"]],"valueReferences":[["Submodule","pointwiseSetSMul"],["Set"],["HSMul","hSMul"],["instHSMul"],["Submodule","set_smul_le"],["Submodule","mem_set_smul_of_mem_mem"],["Submodule"]]},{"isProp":true,"kind":"theorem","name":["Submodule","smul_bot'"],"typeFallback":"forall {α : Type.{u_1}} {R : Type.{u_2}} {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.5 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.8 : AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.11 : Module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.8] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.15 : Monoid.{u_1} α] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.18 : DistribMulAction.{u_1, u_3} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.8)] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.22 : SMulCommClass.{u_1, u_2, u_3} α R M (SMulZeroClass.toSMul.{u_1, u_3} α M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_1, u_3} α M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_1, u_3} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.18))) (SMulZeroClass.toSMul.{u_2, u_3} R M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_2, u_3} R M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_2, u_3} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.8) (Module.toDistribMulAction.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.11))))] (a : α), Eq.{succ u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.11) (HSMul.hSMul.{u_1, u_3, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.11) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.11) (instHSMul.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.11) (SMulZeroClass.toSMul.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.11) (AddZero.toZero.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.11) (AddZeroClass.toAddZero.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.11) (AddMonoid.toAddZeroClass.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.11) (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.11))))) (DistribSMul.toSMulZeroClass.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.11) (AddMonoid.toAddZeroClass.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.11) (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.11))) (DistribMulAction.toDistribSMul.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.11) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.11)) (Submodule.pointwiseDistribMulAction.{u_1, u_2, u_3} α R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.22))))) a (Bot.bot.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.11) (Submodule.instBot.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.11))) (Bot.bot.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2880551806._hygCtx._hyg.8 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[inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.596932550._hygCtx._hyg.5 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.596932550._hygCtx._hyg.8 : AddCommGroup.{u_2} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.596932550._hygCtx._hyg.11 : Module.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.596932550._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.596932550._hygCtx._hyg.8)] (_S : Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.596932550._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.596932550._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.596932550._hygCtx._hyg.11), Eq.{succ u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.596932550._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_2} M 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AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1158311364._hygCtx._hyg.11 : Module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1158311364._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1158311364._hygCtx._hyg.8] {S : Type.{u_4}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1158311364._hygCtx._hyg.16 : Monoid.{u_4} S] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1158311364._hygCtx._hyg.19 : DistribMulAction.{u_4, u_3} S M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1158311364._hygCtx._hyg.16 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1158311364._hygCtx._hyg.8)] (s : Set.{u_4} S), Eq.{succ u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1158311364._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1158311364._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1158311364._hygCtx._hyg.11) 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{R : Type.{u_2}} {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1299264539._hygCtx._hyg.5 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1299264539._hygCtx._hyg.8 : AddCommGroup.{u_3} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1299264539._hygCtx._hyg.11 : Module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1299264539._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1299264539._hygCtx._hyg.8)] {S : Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1299264539._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1299264539._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1299264539._hygCtx._hyg.11} {T : Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1299264539._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1299264539._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1299264539._hygCtx._hyg.11}, Iff (LE.le.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1299264539._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1299264539._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1299264539._hygCtx._hyg.11) (Preorder.toLE.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1299264539._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1299264539._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1299264539._hygCtx._hyg.11) (PartialOrder.toPreorder.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1299264539._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1299264539._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1299264539._hygCtx._hyg.11) (Submodule.instPartialOrder.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1299264539._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1299264539._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1299264539._hygCtx._hyg.11))) (Neg.neg.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1299264539._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1299264539._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1299264539._hygCtx._hyg.11) (Submodule.pointwiseNeg.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1299264539._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1299264539._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1299264539._hygCtx._hyg.11) S) T) (LE.le.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1299264539._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1299264539._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1299264539._hygCtx._hyg.11) (Preorder.toLE.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1299264539._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1299264539._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1299264539._hygCtx._hyg.11) (PartialOrder.toPreorder.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1299264539._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1299264539._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1299264539._hygCtx._hyg.11) (Submodule.instPartialOrder.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1299264539._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1299264539._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1299264539._hygCtx._hyg.11))) S (Neg.neg.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1299264539._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1299264539._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1299264539._hygCtx._hyg.11) (Submodule.pointwiseNeg.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1299264539._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1299264539._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1299264539._hygCtx._hyg.11) T))","typeFull":"∀ {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommGroup M] [inst_2 : Module R M]\n  {S T : Submodule R M}, -S ≤ T ↔ S ≤ -T","typeReadable":"∀ {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommGroup M] [inst_2 : Module R M]\n  {S T : Submodule R M}, -S ≤ T ↔ S ≤ 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{α : Type.{u_1}} [inst._@.Mathlib.Order.Interval.Set.Defs.1398200156._hygCtx._hyg.3 : Preorder.{u_1} α] (b : α), Eq.{succ u_1} (Set.{u_1} α) (Set.Ici.{u_1} α inst._@.Mathlib.Order.Interval.Set.Defs.1398200156._hygCtx._hyg.3 b) (setOf.{u_1} α (fun (x : α) => LE.le.{u_1} α (Preorder.toLE.{u_1} α inst._@.Mathlib.Order.Interval.Set.Defs.1398200156._hygCtx._hyg.3) b x))","typeFull":"∀ {α : Type u_1} [inst : Preorder α] (b : α), Set.Ici b = {x | b ≤ x}","typeReadable":"∀ {α : Type u_1} [inst : Preorder α] (b : α), Set.Ici b = {x | b ≤ x}","typeReferences":[["Set","Ici"],["Preorder"],["Set"],["LE","le"],["Preorder","toLE"],["Eq"],["setOf"]],"valueReferences":[["Set","Ici"],["Set","Ici_def"],["Set"],["LE","le"],["Eq","symm"],["Preorder","toLE"],["setOf"]]},{"isProp":true,"kind":"theorem","name":["Submodule","smul_def"],"typeFallback":"forall {α : Type.{u_1}} {R : Type.{u_2}} {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.5 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.8 : AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.11 : Module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.8] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.15 : Monoid.{u_1} α] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.18 : DistribMulAction.{u_1, u_3} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.8)] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.22 : SMulCommClass.{u_1, u_2, u_3} α R M (SMulZeroClass.toSMul.{u_1, u_3} α M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_1, u_3} α M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_1, u_3} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.18))) (SMulZeroClass.toSMul.{u_2, u_3} R M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_2, u_3} R M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_2, u_3} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.8) (Module.toDistribMulAction.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.11))))] (a : α) (S : Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.11), Eq.{succ u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.11) (HSMul.hSMul.{u_1, u_3, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.11) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.11) (instHSMul.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.11) (SMulZeroClass.toSMul.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.11) (AddZero.toZero.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.11) (AddZeroClass.toAddZero.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.11) (AddMonoid.toAddZeroClass.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.11) (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.11))))) (DistribSMul.toSMulZeroClass.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.11) (AddMonoid.toAddZeroClass.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.11) (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.11))) (DistribMulAction.toDistribSMul.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.11) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.11)) (Submodule.pointwiseDistribMulAction.{u_1, u_2, u_3} α R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.22))))) a S) (Submodule.span.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.11 (HSMul.hSMul.{u_1, u_3, u_3} α (Set.{u_3} M) (Set.{u_3} M) (instHSMul.{u_1, u_3} α (Set.{u_3} M) (Set.smulSet.{u_1, u_3} α M (SMulZeroClass.toSMul.{u_1, u_3} α M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_1, u_3} α M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_1, u_3} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.18))))) a (SetLike.coe.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1925582909._hygCtx._hyg.11) S)))","typeFull":"∀ {α : Type u_1} {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  [inst_3 : Monoid α] [inst_4 : DistribMulAction α M] [inst_5 : SMulCommClass α R M] (a : α) (S : Submodule R M),\n  a • S = Submodule.span R (a • ↑S)","typeReadable":"∀ {α : Type u_1} {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  [inst_3 : Monoid α] [inst_4 : DistribMulAction α M] [inst_5 : SMulCommClass α R M] (a : α) (S : Submodule R M),\n  a • S = Submodule.span R (a • 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{R : Type.{u_2}} {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.617408854._hygCtx._hyg.5 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.617408854._hygCtx._hyg.8 : AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.617408854._hygCtx._hyg.11 : Module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.617408854._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.617408854._hygCtx._hyg.8] (p : Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.617408854._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.617408854._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.617408854._hygCtx._hyg.11) (q : Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.617408854._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.617408854._hygCtx._hyg.8 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q","typeReferences":[["Lattice","toSemilatticeSup"],["Module"],["instHAdd"],["Submodule"],["HAdd","hAdd"],["Submodule","pointwiseAdd"],["AddCommMonoid"],["Max","max"],["SemilatticeSup","toMax"],["Submodule","completeLattice"],["ConditionallyCompleteLattice","toLattice"],["Eq"],["CompleteLattice","toConditionallyCompleteLattice"],["Semiring"]],"valueReferences":[["HAdd","hAdd"],["rfl"],["Submodule","pointwiseAdd"],["instHAdd"],["Submodule"]]},{"isProp":false,"kind":"definition","name":["Submodule","pointwiseMulActionWithZero"],"typeFallback":"forall {α : Type.{u_1}} {R : Type.{u_2}} {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.5 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.8 : AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.11 : Module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.8] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.15 : Semiring.{u_1} α] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.18 : Module.{u_1, u_3} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.8] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.22 : SMulCommClass.{u_1, u_2, u_3} α R M (SMulZeroClass.toSMul.{u_1, u_3} α M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_1, u_3} α M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.8)) 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inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.15) (Submodule.pointwiseZero.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.11)","typeFull":"{α : Type u_1} →\n  {R : Type u_2} →\n    {M : Type u_3} →\n      [inst : Semiring R] →\n        [inst_1 : AddCommMonoid M] →\n          [inst_2 : Module R M] →\n            [inst_3 : Semiring α] → [inst_4 : Module α M] → [SMulCommClass α R M] → MulActionWithZero α (Submodule R M)","typeReadable":"{α : Type u_1} →\n  {R : Type u_2} →\n    {M : Type u_3} →\n      [inst : Semiring R] →\n        [inst_1 : AddCommMonoid M] →\n          [inst_2 : Module R M] →\n            [inst_3 : Semiring α] → [inst_4 : Module α M] → [SMulCommClass α R M] → MulActionWithZero α (Submodule R 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(SubNegZeroMonoid.toNegZeroClass.{u_3} M (SubtractionMonoid.toSubNegZeroMonoid.{u_3} M (SubtractionCommMonoid.toSubtractionMonoid.{u_3} M (AddCommGroup.toDivisionAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1854497662._hygCtx._hyg.8))))) g))","typeFull":"∀ {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommGroup M] [inst_2 : Module R M] {g : M}\n  {S : Submodule R M}, g ∈ -S ↔ -g ∈ S","typeReadable":"∀ {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommGroup M] [inst_2 : Module R M] {g : M}\n  {S : Submodule R M}, g ∈ -S ↔ -g ∈ 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(AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3079363428._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3079363428._hygCtx._hyg.11) (Submodule.pointwiseNeg.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3079363428._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3079363428._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3079363428._hygCtx._hyg.11) (S i)))","typeFull":"∀ {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommGroup M] [inst_2 : Module R M] {ι : Sort u_4}\n  (S : ι → Submodule R M), -⨅ i, S i = ⨅ i, -S i","typeReadable":"∀ {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommGroup M] [inst_2 : Module R M] {ι : Sort u_4}\n  (S : ι → Submodule R M), -⨅ i, S i = ⨅ i, -S 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inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2769122245._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2769122245._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2769122245._hygCtx._hyg.11) (Submodule.instPartialOrder.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2769122245._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2769122245._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2769122245._hygCtx._hyg.11)))","typeFull":"{R : Type u_2} →\n  {M : Type u_3} →\n    [inst : Semiring R] → [inst_1 : AddCommGroup M] → [inst_2 : Module R M] → Submodule R M ≃o Submodule R M","typeReadable":"{R : Type u_2} →\n  {M : Type u_3} →\n    [inst : Semiring R] → [inst_1 : AddCommGroup M] → [inst_2 : Module R M] → Submodule R M ≃o Submodule R M","typeReferences":[["PartialOrder","toPreorder"],["Module"],["Submodule","instPartialOrder"],["OrderIso"],["AddCommGroup","toAddCommMonoid"],["AddCommGroup"],["Preorder","toLE"],["Submodule"],["Semiring"]],"valueReferences":[["RelIso","mk"],["PartialOrder","toPreorder"],["Submodule","instPartialOrder"],["LE","le"],["AddCommGroup","toAddCommMonoid"],["Equiv","neg"],["Submodule","involutivePointwiseNeg"],["Preorder","toLE"],["Submodule","neg_le_neg"],["Submodule"]]},{"isProp":false,"kind":"definition","name":["Submodule","pointwiseAddCommMonoid"],"typeFallback":"forall {R : Type.{u_2}} {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 : AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11 : Module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8], AddCommMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11)","typeFull":"{R : Type u_2} →\n  {M : Type u_3} →\n    [inst : Semiring R] → [inst_1 : AddCommMonoid M] → [inst_2 : Module R M] → AddCommMonoid (Submodule R M)","typeReadable":"{R : Type u_2} →\n  {M : Type u_3} →\n    [inst : Semiring R] → [inst_1 : AddCommMonoid M] → [inst_2 : Module R M] → AddCommMonoid (Submodule R M)","typeReferences":[["AddCommMonoid"],["Module"],["Submodule"],["Semiring"]],"valueReferences":[["Submodule","pointwiseAddCommMonoid","_proof_5"],["AddCommMonoid","mk"],["Submodule","pointwiseAddCommMonoid","_proof_4"],["Submodule","pointwiseAddCommMonoid","_proof_1"],["Submodule","pointwiseZero"],["nsmulRec"],["Submodule"],["AddMonoid","mk"],["Submodule","pointwiseAddCommMonoid","_proof_2"],["Submodule","pointwiseAdd"],["Submodule","pointwiseAddCommMonoid","_proof_3"],["Submodule","pointwiseAddCommMonoid","_proof_6"],["AddSemigroup","mk"]]},{"isProp":true,"kind":"theorem","name":["Submodule","span_set_smul"],"typeFallback":"forall {R : Type.{u_2}} {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.578112082._hygCtx._hyg.5 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.578112082._hygCtx._hyg.8 : AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.578112082._hygCtx._hyg.11 : Module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.578112082._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.578112082._hygCtx._hyg.8] {S : Type.{u_4}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.578112082._hygCtx._hyg.16 : Monoid.{u_4} S] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.578112082._hygCtx._hyg.19 : DistribMulAction.{u_4, u_3} S M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.578112082._hygCtx._hyg.16 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.578112082._hygCtx._hyg.8)] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.578112082._hygCtx._hyg.30 : SMulCommClass.{u_4, u_2, u_3} S R M (SMulZeroClass.toSMul.{u_4, u_3} S M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.578112082._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_4, u_3} S M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.578112082._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_4, u_3} S M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.578112082._hygCtx._hyg.16 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.578112082._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.578112082._hygCtx._hyg.19))) (SMulZeroClass.toSMul.{u_2, u_3} R M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.578112082._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_2, u_3} R M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.578112082._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_2, u_3} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.578112082._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.578112082._hygCtx._hyg.8) (Module.toDistribMulAction.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.578112082._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.578112082._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.578112082._hygCtx._hyg.11))))] (s : Set.{u_4} S) (t : Set.{u_3} M), Eq.{succ u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.578112082._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.578112082._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.578112082._hygCtx._hyg.11) (Submodule.span.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.578112082._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.578112082._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.578112082._hygCtx._hyg.11 (HSMul.hSMul.{u_4, u_3, u_3} (Set.{u_4} S) (Set.{u_3} M) (Set.{u_3} M) (instHSMul.{u_4, u_3} (Set.{u_4} S) (Set.{u_3} M) (Set.smul.{u_4, u_3} S M (SMulZeroClass.toSMul.{u_4, u_3} S M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.578112082._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_4, u_3} S M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.578112082._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_4, u_3} S M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.578112082._hygCtx._hyg.16 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.578112082._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.578112082._hygCtx._hyg.19))))) s t)) (HSMul.hSMul.{u_4, u_3, u_3} (Set.{u_4} S) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.578112082._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.578112082._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.578112082._hygCtx._hyg.11) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.578112082._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.578112082._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.578112082._hygCtx._hyg.11) (instHSMul.{u_4, u_3} (Set.{u_4} S) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.578112082._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.578112082._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.578112082._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_3, u_4} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.578112082._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.578112082._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.578112082._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.578112082._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.578112082._hygCtx._hyg.19)) s (Submodule.span.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.578112082._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.578112082._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.578112082._hygCtx._hyg.11 t))","typeFull":"∀ {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {S : Type u_4}\n  [inst_3 : Monoid S] [inst_4 : DistribMulAction S M] [SMulCommClass S R M] (s : Set S) (t : Set M),\n  Submodule.span R (s • t) = s • Submodule.span R t","typeReadable":"∀ {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {S : Type u_4}\n  [inst_3 : Monoid S] [inst_4 : DistribMulAction S M] [SMulCommClass S R M] (s : Set S) (t : Set M),\n  Submodule.span R (s • t) = s • Submodule.span R t","typeReferences":[["Submodule","pointwiseSetSMul"],["Set"],["Module"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["Submodule","span"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["Set","smul"],["Submodule"],["SMulCommClass"],["Module","toDistribMulAction"],["AddCommMonoid"],["HSMul","hSMul"],["MonoidWithZero","toMonoid"],["Monoid"],["instHSMul"],["Eq"],["AddZero","toZero"],["DistribMulAction"],["Semiring"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]],"valueReferences":[["Submodule","pointwiseSetSMul"],["Set"],["DistribMulAction","toDistribSMul"],["Submodule","span"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["Submodule","set_smul_span"],["Submodule"],["Set","smul"],["HSMul","hSMul"],["Eq","symm"],["instHSMul"],["AddZero","toZero"],["DistribSMul","toSMulZeroClass"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Submodule","neg_sup"],"typeFallback":"forall {R : Type.{u_2}} {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.290753590._hygCtx._hyg.5 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.290753590._hygCtx._hyg.8 : AddCommGroup.{u_3} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.290753590._hygCtx._hyg.11 : Module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.290753590._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.290753590._hygCtx._hyg.8)] (S : Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.290753590._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.290753590._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.290753590._hygCtx._hyg.11) (T : Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.290753590._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.290753590._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.290753590._hygCtx._hyg.11), Eq.{succ u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.290753590._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.290753590._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.290753590._hygCtx._hyg.11) (Neg.neg.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.290753590._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.290753590._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.290753590._hygCtx._hyg.11) (Submodule.pointwiseNeg.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.290753590._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.290753590._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.290753590._hygCtx._hyg.11) (Max.max.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.290753590._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.290753590._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.290753590._hygCtx._hyg.11) (SemilatticeSup.toMax.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.290753590._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.290753590._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.290753590._hygCtx._hyg.11) (Lattice.toSemilatticeSup.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.290753590._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.290753590._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.290753590._hygCtx._hyg.11) (ConditionallyCompleteLattice.toLattice.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.290753590._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.290753590._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.290753590._hygCtx._hyg.11) (CompleteLattice.toConditionallyCompleteLattice.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.290753590._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.290753590._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.290753590._hygCtx._hyg.11) (Submodule.completeLattice.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.290753590._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.290753590._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.290753590._hygCtx._hyg.11))))) S T)) (Max.max.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.290753590._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.290753590._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.290753590._hygCtx._hyg.11) (SemilatticeSup.toMax.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.290753590._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.290753590._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.290753590._hygCtx._hyg.11) (Lattice.toSemilatticeSup.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.290753590._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.290753590._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.290753590._hygCtx._hyg.11) (ConditionallyCompleteLattice.toLattice.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.290753590._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.290753590._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.290753590._hygCtx._hyg.11) (CompleteLattice.toConditionallyCompleteLattice.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.290753590._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.290753590._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.290753590._hygCtx._hyg.11) (Submodule.completeLattice.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.290753590._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.290753590._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.290753590._hygCtx._hyg.11))))) (Neg.neg.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.290753590._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.290753590._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.290753590._hygCtx._hyg.11) (Submodule.pointwiseNeg.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.290753590._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.290753590._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.290753590._hygCtx._hyg.11) S) (Neg.neg.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.290753590._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.290753590._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.290753590._hygCtx._hyg.11) (Submodule.pointwiseNeg.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.290753590._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.290753590._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.290753590._hygCtx._hyg.11) T))","typeFull":"∀ {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommGroup M] [inst_2 : Module R M]\n  (S T : Submodule R M), -(S ⊔ T) = -S ⊔ -T","typeReadable":"∀ {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommGroup M] [inst_2 : Module R M]\n  (S T : Submodule R M), -(S ⊔ T) = -S ⊔ -T","typeReferences":[["Lattice","toSemilatticeSup"],["Module"],["Neg","neg"],["AddCommGroup"],["Submodule"],["Max","max"],["SemilatticeSup","toMax"],["Submodule","completeLattice"],["AddCommGroup","toAddCommMonoid"],["Submodule","pointwiseNeg"],["ConditionallyCompleteLattice","toLattice"],["Eq"],["CompleteLattice","toConditionallyCompleteLattice"],["Semiring"]],"valueReferences":[["Submodule","negOrderIso"],["Lattice","toSemilatticeSup"],["Submodule","completeLattice"],["AddCommGroup","toAddCommMonoid"],["OrderIso","map_sup"],["ConditionallyCompleteLattice","toLattice"],["CompleteLattice","toConditionallyCompleteLattice"],["Submodule"]]},{"isProp":true,"kind":"theorem","name":["Submodule","pointwiseMulActionWithZero","_proof_2"],"typeFallback":"forall {α : Type.{u_3}} {R : Type.{u_1}} {M : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.5 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.8 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.11 : Module.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.8] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.15 : Semiring.{u_3} α] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.18 : Module.{u_3, u_2} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.8] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.22 : SMulCommClass.{u_3, u_1, u_2} α R M (SMulZeroClass.toSMul.{u_3, u_2} α M (AddZero.toZero.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_3, u_2} α M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_3, u_2} α M (MonoidWithZero.toMonoid.{u_3} α (Semiring.toMonoidWithZero.{u_3} α inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.15)) (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.8) (Module.toDistribMulAction.{u_3, u_2} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.18)))) (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 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.8) (Module.toDistribMulAction.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.11))))] (S : Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.11), Eq.{succ u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.11) (Submodule.map.{u_1, u_1, u_2, u_2} R R M M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.11 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.5)) (RingHomSurjective.ids.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.5) (DistribSMul.toLinearMap.{u_1, u_3, u_2} R α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.11 (DistribMulAction.toDistribSMul.{u_3, u_2} α M (MonoidWithZero.toMonoid.{u_3} α (Semiring.toMonoidWithZero.{u_3} α inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.15)) (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.8) (Module.toDistribMulAction.{u_3, u_2} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.18)) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.22 (OfNat.ofNat.{u_3} α 0 (Zero.toOfNat0.{u_3} α (MulZeroClass.toZero.{u_3} α (MulZeroOneClass.toMulZeroClass.{u_3} α (MonoidWithZero.toMulZeroOneClass.{u_3} α (Semiring.toMonoidWithZero.{u_3} α inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.15))))))) S) (OfNat.ofNat.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.11) 0 (Zero.toOfNat0.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.11) (Submodule.pointwiseZero.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.11)))","typeFull":"∀ {α : Type u_3} {R : Type u_1} {M : Type u_2} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  [inst_3 : Semiring α] [inst_4 : Module α M] [inst_5 : SMulCommClass α R M] (S : Submodule R M),\n  Submodule.map (DistribSMul.toLinearMap R M 0) S = 0","typeReadable":"∀ {α : Type u_3} {R : Type u_1} {M : Type u_2} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  [inst_3 : Semiring α] [inst_4 : Module α M] [inst_5 : SMulCommClass α R M] (S : Submodule R M),\n  Submodule.map (DistribSMul.toLinearMap R M 0) S = 0","typeReferences":[["Module"],["Submodule","map"],["MonoidWithZero","toMulZeroOneClass"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Submodule"],["DistribSMul","toLinearMap"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["MonoidWithZero","toMonoid"],["Zero","toOfNat0"],["Eq"],["DistribSMul","toSMulZeroClass"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["Submodule","pointwiseZero"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["RingHomSurjective","ids"],["SMulCommClass"],["Module","toDistribMulAction"],["MulZeroOneClass","toMulZeroClass"],["MulZeroClass","toZero"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["Eq","trans"],["Submodule","map"],["MonoidWithZero","toMulZeroOneClass"],["AddCommMonoid","toAddMonoid"],["DistribSMul","toLinearMap"],["Submodule"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["LinearMap","ext"],["MonoidWithZero","toMonoid"],["Zero","toOfNat0"],["MulActionWithZero","toSMulWithZero"],["LinearMap","instZero"],["congr_arg"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["Module","toMulActionWithZero"],["Submodule","pointwiseZero"],["Submodule","map_zero"],["AddZeroClass","toAddZero"],["LinearMap"],["OfNat","ofNat"],["RingHomSurjective","ids"],["Module","toDistribMulAction"],["MulZeroOneClass","toMulZeroClass"],["MulZeroClass","toZero"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["zero_smul"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]]},{"isProp":false,"kind":"definition","name":["Submodule","pointwiseSetSMul"],"typeFallback":"forall {R : Type.{u_2}} {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3282838664._hygCtx._hyg.5 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3282838664._hygCtx._hyg.8 : AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3282838664._hygCtx._hyg.11 : Module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3282838664._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3282838664._hygCtx._hyg.8] {S : Type.{u_4}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3282838664._hygCtx._hyg.16 : Monoid.{u_4} S] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3282838664._hygCtx._hyg.19 : DistribMulAction.{u_4, u_3} S M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3282838664._hygCtx._hyg.16 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3282838664._hygCtx._hyg.8)], SMul.{u_4, u_3} (Set.{u_4} S) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3282838664._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3282838664._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3282838664._hygCtx._hyg.11)","typeFull":"{R : Type u_2} →\n  {M : Type u_3} →\n    [inst : Semiring R] →\n      [inst_1 : AddCommMonoid M] →\n        [inst_2 : Module R M] →\n          {S : Type u_4} → [inst_3 : Monoid S] → [DistribMulAction S M] → SMul (Set S) (Submodule R M)","typeReadable":"{R : Type u_2} →\n  {M : Type u_3} →\n    [inst : Semiring R] →\n      [inst_1 : AddCommMonoid M] →\n        [inst_2 : Module R M] →\n          {S : Type u_4} → [inst_3 : Monoid S] → [DistribMulAction S M] → SMul (Set S) (Submodule R M)","typeReferences":[["SMul"],["AddCommMonoid"],["Set"],["Module"],["Monoid"],["AddCommMonoid","toAddMonoid"],["DistribMulAction"],["Submodule"],["Semiring"]],"valueReferences":[["SMul","mk"],["SetLike","instMembership"],["Submodule","instInfSet"],["Set"],["Membership","mem"],["DistribMulAction","toDistribSMul"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["Submodule"],["Set","instMembership"],["Submodule","setLike"],["InfSet","sInf"],["HSMul","hSMul"],["instHSMul"],["AddZero","toZero"],["DistribSMul","toSMulZeroClass"],["AddMonoid","toAddZeroClass"],["setOf"]]},{"isProp":true,"kind":"theorem","name":["Submodule","pointwise_smul_def"],"typeFallback":"forall {α : Type.{u_1}} {R : Type.{u_2}} {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.5 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.8 : AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.11 : Module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.8] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.15 : Monoid.{u_1} α] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.18 : DistribMulAction.{u_1, u_3} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.8)] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.22 : SMulCommClass.{u_1, u_2, u_3} α R M (SMulZeroClass.toSMul.{u_1, u_3} α M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_1, u_3} α M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_1, u_3} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.18))) (SMulZeroClass.toSMul.{u_2, u_3} R M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_2, u_3} R M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_2, u_3} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.8) (Module.toDistribMulAction.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.11))))] {a : α} {S : Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.11}, Eq.{succ u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.11) (HSMul.hSMul.{u_1, u_3, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.11) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.11) (instHSMul.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.11) (SMulZeroClass.toSMul.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.11) (AddZero.toZero.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.11) (AddZeroClass.toAddZero.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.11) (AddMonoid.toAddZeroClass.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.11) (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.11))))) (DistribSMul.toSMulZeroClass.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.11) (AddMonoid.toAddZeroClass.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.11) (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.11))) (DistribMulAction.toDistribSMul.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.11) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.11)) (Submodule.pointwiseDistribMulAction.{u_1, u_2, u_3} α R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.22))))) a S) (Submodule.map.{u_2, u_2, u_3, u_3} R R M M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.11 (RingHom.id.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.5)) (RingHomSurjective.ids.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.5) (DistribSMul.toLinearMap.{u_2, u_1, u_3} R α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.11 (DistribMulAction.toDistribSMul.{u_1, u_3} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.18) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3816989933._hygCtx._hyg.22 a) S)","typeFull":"∀ {α : Type u_1} {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  [inst_3 : Monoid α] [inst_4 : DistribMulAction α M] [inst_5 : SMulCommClass α R M] {a : α} {S : Submodule R M},\n  a • S = Submodule.map (DistribSMul.toLinearMap R M a) S","typeReadable":"∀ {α : Type u_1} {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  [inst_3 : Monoid α] [inst_4 : DistribMulAction α M] [inst_5 : SMulCommClass α R M] {a : α} {S : Submodule R M},\n  a • S = Submodule.map (DistribSMul.toLinearMap R M a) S","typeReferences":[["Module"],["Submodule","map"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["DistribSMul","toLinearMap"],["Submodule"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["MonoidWithZero","toMonoid"],["instHSMul"],["Eq"],["DistribMulAction"],["DistribSMul","toSMulZeroClass"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["RingHomSurjective","ids"],["SMulCommClass"],["Module","toDistribMulAction"],["HSMul","hSMul"],["Monoid"],["AddZero","toZero"],["Submodule","pointwiseAddCommMonoid"],["Submodule","pointwiseDistribMulAction"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["rfl"],["DistribMulAction","toDistribSMul"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["Submodule"],["HSMul","hSMul"],["instHSMul"],["AddZero","toZero"],["Submodule","pointwiseAddCommMonoid"],["Submodule","pointwiseDistribMulAction"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Submodule","pointwise_smul_toAddSubmonoid"],"typeFallback":"forall {α : Type.{u_1}} {R : Type.{u_2}} {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.5 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.8 : AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.11 : Module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.8] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.15 : Monoid.{u_1} α] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.18 : DistribMulAction.{u_1, u_3} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.8)] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.22 : SMulCommClass.{u_1, u_2, u_3} α R M (SMulZeroClass.toSMul.{u_1, u_3} α M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_1, u_3} α M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_1, u_3} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.18))) (SMulZeroClass.toSMul.{u_2, u_3} R M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_2, u_3} R M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_2, u_3} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.8) (Module.toDistribMulAction.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.11))))] (a : α) (S : Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.11), Eq.{succ u_3} (AddSubmonoid.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.8))) (Submodule.toAddSubmonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.11 (HSMul.hSMul.{u_1, u_3, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.11) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.11) (instHSMul.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.11) (SMulZeroClass.toSMul.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.11) (AddZero.toZero.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.11) (AddZeroClass.toAddZero.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.11) (AddMonoid.toAddZeroClass.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.11) (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.11))))) (DistribSMul.toSMulZeroClass.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.11) (AddMonoid.toAddZeroClass.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.11) (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.11))) (DistribMulAction.toDistribSMul.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.11) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.11)) (Submodule.pointwiseDistribMulAction.{u_1, u_2, u_3} α R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.22))))) a S)) (HSMul.hSMul.{u_1, u_3, u_3} α (AddSubmonoid.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.8))) (AddSubmonoid.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.8))) (instHSMul.{u_1, u_3} α (AddSubmonoid.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.8))) (SemigroupAction.toSMul.{u_1, u_3} α (AddSubmonoid.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.8))) (Monoid.toSemigroup.{u_1} α inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.15) (MulAction.toSemigroupAction.{u_1, u_3} α (AddSubmonoid.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.8))) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.15 (AddSubmonoid.pointwiseMulAction.{u_1, u_3} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.18)))) a (Submodule.toAddSubmonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1355522107._hygCtx._hyg.11 S))","typeFull":"∀ {α : Type u_1} {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  [inst_3 : Monoid α] [inst_4 : DistribMulAction α M] [inst_5 : SMulCommClass α R M] (a : α) (S : Submodule R M),\n  (a • S).toAddSubmonoid = a • S.toAddSubmonoid","typeReadable":"∀ {α : Type u_1} {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  [inst_3 : Monoid α] [inst_4 : DistribMulAction α M] [inst_5 : SMulCommClass α R M] (a : α) (S : Submodule R M),\n  (a • S).toAddSubmonoid = a • S.toAddSubmonoid","typeReferences":[["AddSubmonoid","pointwiseMulAction"],["Submodule","toAddSubmonoid"],["Module"],["SemigroupAction","toSMul"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Submodule"],["AddCommMonoid"],["MonoidWithZero","toMonoid"],["instHSMul"],["Monoid","toSemigroup"],["Eq"],["DistribMulAction"],["DistribSMul","toSMulZeroClass"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["SMulCommClass"],["Module","toDistribMulAction"],["AddSubmonoid"],["HSMul","hSMul"],["Monoid"],["MulAction","toSemigroupAction"],["AddZero","toZero"],["Submodule","pointwiseAddCommMonoid"],["Submodule","pointwiseDistribMulAction"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["rfl"],["Submodule","toAddSubmonoid"],["DistribMulAction","toDistribSMul"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["Submodule"],["AddSubmonoid"],["HSMul","hSMul"],["instHSMul"],["AddZero","toZero"],["Submodule","pointwiseDistribMulAction"],["Submodule","pointwiseAddCommMonoid"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Submodule","set_smul_eq_map"],"typeFallback":"forall {R : Type.{u_2}} {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 : AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 : Module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8] (sR : Set.{u_2} R) (N : Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.30 : SMulCommClass.{u_2, u_2, u_3} R R (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (Submodule.smul.{u_2, u_2, u_3} R R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N (SMulZeroClass.toSMul.{u_2, u_2} R R (AddZero.toZero.{u_2} R (AddZeroClass.toAddZero.{u_2} R (AddMonoid.toAddZeroClass.{u_2} R (AddMonoidWithOne.toAddMonoid.{u_2} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_2} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5))))))) (DistribSMul.toSMulZeroClass.{u_2, u_2} R R (AddMonoid.toAddZeroClass.{u_2} R (AddMonoidWithOne.toAddMonoid.{u_2} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_2} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5))))) (instDistribSMul.{u_2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5))))) (SMulZeroClass.toSMul.{u_2, u_3} R M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_2, u_3} R M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_2, u_3} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8) (Module.toDistribMulAction.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)))) (IsScalarTower.left.{u_2, u_3} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5)) (DistribMulAction.toMulAction.{u_2, u_3} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8) (Module.toDistribMulAction.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)))) (Submodule.smul.{u_2, u_2, u_3} R R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N (SMulZeroClass.toSMul.{u_2, u_2} R R (AddZero.toZero.{u_2} R (AddZeroClass.toAddZero.{u_2} R (AddMonoid.toAddZeroClass.{u_2} R (AddMonoidWithOne.toAddMonoid.{u_2} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_2} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5))))))) (DistribSMul.toSMulZeroClass.{u_2, u_2} R R (AddMonoid.toAddZeroClass.{u_2} R (AddMonoidWithOne.toAddMonoid.{u_2} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_2} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5))))) (instDistribSMul.{u_2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5))))) (SMulZeroClass.toSMul.{u_2, u_3} R M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_2, u_3} R M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_2, u_3} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8) (Module.toDistribMulAction.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)))) (IsScalarTower.left.{u_2, u_3} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5)) (DistribMulAction.toMulAction.{u_2, u_3} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8) (Module.toDistribMulAction.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11))))], Eq.{succ u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (HSMul.hSMul.{u_2, u_3, u_3} (Set.{u_2} R) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (instHSMul.{u_2, u_3} (Set.{u_2} R) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_3, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 R (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5)) (Module.toDistribMulAction.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11))) sR N) (Submodule.map.{u_2, u_2, max u_3 u_2, u_3} R R (Finsupp.{u_2, u_3} R (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (AddZero.toZero.{u_3} (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (AddZeroClass.toAddZero.{u_3} (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (AddMonoid.toAddZeroClass.{u_3} (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (AddCommMonoid.toAddMonoid.{u_3} (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N)))))) M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 (Finsupp.instAddCommMonoid.{u_2, u_3} R (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N)) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 (Finsupp.module.{u_2, u_3, u_2} R (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Submodule.module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N)) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 (RingHom.id.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5)) 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inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N)))))) (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) M 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(Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Submodule.module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Submodule.module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N))) (LinearMap.{u_2, u_2, max u_3 u_2, u_3} R R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 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inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (AddZero.toZero.{u_3} (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (AddZeroClass.toAddZero.{u_3} (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (AddMonoid.toAddZeroClass.{u_3} (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (AddCommMonoid.toAddMonoid.{u_3} (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N)))))) (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (Finsupp.instAddCommMonoid.{u_2, u_3} R (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N)) (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Finsupp.module.{u_2, u_3, u_2} R (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Submodule.module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N)) (Submodule.module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N)) (Pi.addCommMonoid.{u_2, u_3} R (fun (a._@._internal._hyg.0 : R) => LinearMap.{u_2, u_2, u_3, u_3} R R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 (RingHom.id.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5)) (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Submodule.module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Submodule.module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N)) (fun (i : R) => LinearMap.addCommMonoid.{u_2, u_2, u_3, u_3} R R (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Submodule.module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Submodule.module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (RingHom.id.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5)))) (LinearMap.addCommMonoid.{u_2, u_2, max u_2 u_3, u_3} R R (Finsupp.{u_2, u_3} R (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (AddZero.toZero.{u_3} (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (AddZeroClass.toAddZero.{u_3} (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (AddMonoid.toAddZeroClass.{u_3} (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (AddCommMonoid.toAddMonoid.{u_3} (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) 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inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 (Finsupp.instAddCommMonoid.{u_2, u_3} R (Subtype.{succ 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inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N)) (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Finsupp.module.{u_2, u_3, u_2} R (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Submodule.module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N)) (Submodule.module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (RingHom.id.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5))) (Pi.Function.module.{u_2, u_2, u_3} R R (LinearMap.{u_2, u_2, u_3, u_3} R R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 (RingHom.id.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5)) (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} 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inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Submodule.module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Submodule.module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N)) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 (LinearMap.addCommMonoid.{u_2, u_2, u_3, u_3} R R (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} 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inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Submodule.module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Submodule.module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (RingHom.id.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5))) (LinearMap.module.{u_2, u_2, u_2, u_3, u_3} R R R (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Submodule.module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Submodule.module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (RingHom.id.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5)) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 (Submodule.module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.30)) (LinearMap.module.{u_2, u_2, u_2, max u_2 u_3, u_3} R R R (Finsupp.{u_2, u_3} R (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (AddZero.toZero.{u_3} (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (AddZeroClass.toAddZero.{u_3} (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (AddMonoid.toAddZeroClass.{u_3} (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (AddCommMonoid.toAddMonoid.{u_3} (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N)))))) (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 (Finsupp.instAddCommMonoid.{u_2, u_3} R (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N)) (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Finsupp.module.{u_2, u_3, u_2} R (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Submodule.module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N)) (Submodule.module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (RingHom.id.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5)) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 (Submodule.module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.30)) (R -> (LinearMap.{u_2, u_2, u_3, u_3} R R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 (RingHom.id.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5)) (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Submodule.module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Submodule.module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N))) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : R -> (LinearMap.{u_2, u_2, u_3, u_3} R R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 (RingHom.id.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5)) (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Submodule.module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Submodule.module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N))) => LinearMap.{u_2, u_2, max u_3 u_2, u_3} R R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 (RingHom.id.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5)) (Finsupp.{u_2, u_3} R (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (AddZero.toZero.{u_3} (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (AddZeroClass.toAddZero.{u_3} (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (AddMonoid.toAddZeroClass.{u_3} (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (AddCommMonoid.toAddMonoid.{u_3} (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N)))))) (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (Finsupp.instAddCommMonoid.{u_2, u_3} R (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N)) (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Finsupp.module.{u_2, u_3, u_2} R (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Submodule.module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N)) (Submodule.module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N)) (EquivLike.toFunLike.{max (succ u_3) (succ u_2), max (succ u_3) (succ u_2), max (succ u_3) (succ u_2)} (LinearEquiv.{u_2, u_2, max u_2 u_3, max u_3 u_2} R R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 (RingHom.id.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5)) (RingHom.id.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5)) (RingHomInvPair.ids.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5) (RingHomInvPair.ids.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5) (R -> (LinearMap.{u_2, u_2, u_3, u_3} R R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 (RingHom.id.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5)) (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Submodule.module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Submodule.module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N))) (LinearMap.{u_2, u_2, max u_3 u_2, u_3} R R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 (RingHom.id.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5)) (Finsupp.{u_2, u_3} R (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (AddZero.toZero.{u_3} (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (AddZeroClass.toAddZero.{u_3} (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (AddMonoid.toAddZeroClass.{u_3} (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (AddCommMonoid.toAddMonoid.{u_3} (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N)))))) (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (Finsupp.instAddCommMonoid.{u_2, u_3} R (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N)) (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Finsupp.module.{u_2, u_3, u_2} R (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Submodule.module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N)) (Submodule.module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N)) (Pi.addCommMonoid.{u_2, u_3} R (fun (a._@._internal._hyg.0 : R) => LinearMap.{u_2, u_2, u_3, u_3} R R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 (RingHom.id.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5)) (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Submodule.module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Submodule.module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N)) (fun (i : R) => LinearMap.addCommMonoid.{u_2, u_2, u_3, u_3} R R (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Submodule.module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Submodule.module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (RingHom.id.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5)))) (LinearMap.addCommMonoid.{u_2, u_2, max u_2 u_3, u_3} R R (Finsupp.{u_2, u_3} R (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (AddZero.toZero.{u_3} (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) 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inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N)))))) (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 (Finsupp.instAddCommMonoid.{u_2, u_3} R (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N)) (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Finsupp.module.{u_2, u_3, u_2} R (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) 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inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N)) (Submodule.module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (RingHom.id.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5))) (Pi.Function.module.{u_2, u_2, u_3} R R (LinearMap.{u_2, u_2, u_3, u_3} R R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 (RingHom.id.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5)) 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inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Submodule.module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Submodule.module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N)) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 (LinearMap.addCommMonoid.{u_2, u_2, u_3, 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inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Submodule.module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Submodule.module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (RingHom.id.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5))) (LinearMap.module.{u_2, u_2, u_2, u_3, u_3} R R R (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Submodule.module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 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inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 (Finsupp.instAddCommMonoid.{u_2, u_3} R (Subtype.{succ 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inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Submodule.module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N)) (Submodule.module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (RingHom.id.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5)) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 (Submodule.module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.30)) (R -> (LinearMap.{u_2, u_2, u_3, u_3} R R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 (RingHom.id.{u_2} R 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(Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Submodule.module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Submodule.module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N))) (LinearMap.{u_2, u_2, max u_3 u_2, u_3} R R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 (RingHom.id.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5)) (Finsupp.{u_2, u_3} R (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (AddZero.toZero.{u_3} (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (AddZeroClass.toAddZero.{u_3} (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (AddMonoid.toAddZeroClass.{u_3} (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (AddCommMonoid.toAddMonoid.{u_3} (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N)))))) (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (Finsupp.instAddCommMonoid.{u_2, u_3} R (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N)) (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Finsupp.module.{u_2, u_3, u_2} R (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Submodule.module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N)) (Submodule.module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N)) (LinearEquiv.instEquivLike.{u_2, u_2, max u_3 u_2, max u_3 u_2} R R (R -> (LinearMap.{u_2, u_2, u_3, u_3} R R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 (RingHom.id.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5)) (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Submodule.module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Submodule.module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N))) (LinearMap.{u_2, u_2, max u_3 u_2, u_3} R R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 (RingHom.id.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5)) (Finsupp.{u_2, u_3} R (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (AddZero.toZero.{u_3} (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (AddZeroClass.toAddZero.{u_3} (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (AddMonoid.toAddZeroClass.{u_3} (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (AddCommMonoid.toAddMonoid.{u_3} (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N)))))) (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (Finsupp.instAddCommMonoid.{u_2, u_3} R (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N)) (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Finsupp.module.{u_2, u_3, u_2} R (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Submodule.module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N)) (Submodule.module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N)) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 (Pi.addCommMonoid.{u_2, u_3} R (fun (a._@._internal._hyg.0 : R) => LinearMap.{u_2, u_2, u_3, u_3} R R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 (RingHom.id.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5)) (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Submodule.module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Submodule.module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N)) (fun (i : R) => LinearMap.addCommMonoid.{u_2, u_2, u_3, u_3} R R (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Submodule.module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Submodule.module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (RingHom.id.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5)))) (LinearMap.addCommMonoid.{u_2, u_2, max u_2 u_3, u_3} R R (Finsupp.{u_2, u_3} R (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (AddZero.toZero.{u_3} (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (AddZeroClass.toAddZero.{u_3} (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (AddMonoid.toAddZeroClass.{u_3} (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (AddCommMonoid.toAddMonoid.{u_3} (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N)))))) (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 (Finsupp.instAddCommMonoid.{u_2, u_3} R (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N)) (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Finsupp.module.{u_2, u_3, u_2} R (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Submodule.module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N)) (Submodule.module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (RingHom.id.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5))) (Pi.Function.module.{u_2, u_2, u_3} R R (LinearMap.{u_2, u_2, u_3, u_3} R R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 (RingHom.id.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5)) (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Submodule.module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Submodule.module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N)) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 (LinearMap.addCommMonoid.{u_2, u_2, u_3, u_3} R R (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Submodule.module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Submodule.module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (RingHom.id.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5))) (LinearMap.module.{u_2, u_2, u_2, u_3, u_3} R R R (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Submodule.module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Submodule.module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (RingHom.id.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5)) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 (Submodule.module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.30)) (LinearMap.module.{u_2, u_2, u_2, max u_2 u_3, u_3} R R R (Finsupp.{u_2, u_3} R (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (AddZero.toZero.{u_3} (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (AddZeroClass.toAddZero.{u_3} (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (AddMonoid.toAddZeroClass.{u_3} (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N 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inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N)))))) (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 (Finsupp.instAddCommMonoid.{u_2, u_3} R (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N)) (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Finsupp.module.{u_2, u_3, u_2} R (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (Submodule.module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N)) (Submodule.module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) (RingHom.id.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5)) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 (Submodule.module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11 N) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.30) (RingHom.id.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5)) (RingHom.id.{u_2} R (Semiring.toNonAssocSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5)) (RingHomInvPair.ids.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5) (RingHomInvPair.ids.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5))) (Finsupp.lsum.{u_2, u_3, u_3, u_2, u_2, u_2} R (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) (Subtype.{succ u_3} M (fun (x : M) => Membership.mem.{u_3, u_3} M (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) (SetLike.instMembership.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.11)) N x)) R R R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 (Submodule.addCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.694225953._hygCtx._hyg.5 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inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3461669942._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3461669942._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3461669942._hygCtx._hyg.11))","typeFull":"∀ {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M], 0 = ⊥","typeReadable":"∀ {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M], 0 = ⊥","typeReferences":[["AddCommMonoid"],["Module"],["Submodule","instBot"],["Submodule","pointwiseZero"],["Zero","toOfNat0"],["Eq"],["Bot","bot"],["OfNat","ofNat"],["Submodule"],["Semiring"]],"valueReferences":[["rfl"],["Submodule","pointwiseZero"],["Zero","toOfNat0"],["OfNat","ofNat"],["Submodule"]]},{"isProp":true,"kind":"theorem","name":["Submodule","smul_sup'"],"typeFallback":"forall {α : Type.{u_1}} {R : Type.{u_2}} {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.5 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.8 : AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.11 : Module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.8] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.15 : Monoid.{u_1} α] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.18 : DistribMulAction.{u_1, u_3} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.8)] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.22 : SMulCommClass.{u_1, u_2, u_3} α R M (SMulZeroClass.toSMul.{u_1, u_3} α M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_1, u_3} α M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_1, u_3} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.18))) (SMulZeroClass.toSMul.{u_2, u_3} R M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_2, u_3} R M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_2, u_3} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.8) (Module.toDistribMulAction.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.11))))] (a : α) (S : Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.11) (T : Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.11), Eq.{succ u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.11) (HSMul.hSMul.{u_1, u_3, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.11) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.11) (instHSMul.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.11) (SMulZeroClass.toSMul.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.11) (AddZero.toZero.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.11) (AddZeroClass.toAddZero.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.11) (AddMonoid.toAddZeroClass.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.11) (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.11))))) (DistribSMul.toSMulZeroClass.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.11) (AddMonoid.toAddZeroClass.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.11) (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.11))) (DistribMulAction.toDistribSMul.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.11) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.11)) (Submodule.pointwiseDistribMulAction.{u_1, u_2, u_3} α R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.22))))) a (Max.max.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.11) (SemilatticeSup.toMax.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.11) (Lattice.toSemilatticeSup.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.11) (ConditionallyCompleteLattice.toLattice.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.11) (CompleteLattice.toConditionallyCompleteLattice.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.11) (Submodule.completeLattice.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.11))))) S T)) (Max.max.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.11) (SemilatticeSup.toMax.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.11) (Lattice.toSemilatticeSup.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.11) (ConditionallyCompleteLattice.toLattice.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.11) (CompleteLattice.toConditionallyCompleteLattice.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.11) (Submodule.completeLattice.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.11))))) (HSMul.hSMul.{u_1, u_3, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.11) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.11) (instHSMul.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.11) (SMulZeroClass.toSMul.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.11) (AddZero.toZero.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.11) (AddZeroClass.toAddZero.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.11) (AddMonoid.toAddZeroClass.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.11) (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.11))))) (DistribSMul.toSMulZeroClass.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.11) (AddMonoid.toAddZeroClass.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.11) (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.11))) (DistribMulAction.toDistribSMul.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.11) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.11)) (Submodule.pointwiseDistribMulAction.{u_1, u_2, u_3} α R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.22))))) a S) (HSMul.hSMul.{u_1, u_3, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.8 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inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.11) (AddZeroClass.toAddZero.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.11) (AddMonoid.toAddZeroClass.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.11) (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.5 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inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.11) (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.11))) (DistribMulAction.toDistribSMul.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.11) 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inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.945924243._hygCtx._hyg.22))))) a T))","typeFull":"∀ {α : Type u_1} {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  [inst_3 : Monoid α] [inst_4 : DistribMulAction α M] [inst_5 : SMulCommClass α R M] (a : α) (S T : Submodule R M),\n  a • (S ⊔ T) = a • S ⊔ a • T","typeReadable":"∀ {α : Type u_1} {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  [inst_3 : Monoid α] [inst_4 : DistribMulAction α M] [inst_5 : SMulCommClass α R M] (a : α) (S T : Submodule R M),\n  a • (S ⊔ T) = a • S ⊔ a • 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(R : Type.{u_1}) {S : Type.{u_3}} (M : Type.{u_4}) [inst._@.Mathlib.Algebra.Module.LinearMap.End.3803786946._hygCtx._hyg.10 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.LinearMap.End.3803786946._hygCtx._hyg.13 : AddCommMonoid.{u_4} M] [inst._@.Mathlib.Algebra.Module.LinearMap.End.3803786946._hygCtx._hyg.16 : Module.{u_1, u_4} R M inst._@.Mathlib.Algebra.Module.LinearMap.End.3803786946._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.LinearMap.End.3803786946._hygCtx._hyg.13] [inst._@.Mathlib.Algebra.Module.LinearMap.End.3803786946._hygCtx._hyg.23 : DistribSMul.{u_3, u_4} S M (AddMonoid.toAddZeroClass.{u_4} M (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.End.3803786946._hygCtx._hyg.13))] [inst._@.Mathlib.Algebra.Module.LinearMap.End.3803786946._hygCtx._hyg.27 : SMulCommClass.{u_3, u_1, u_4} S R M (SMulZeroClass.toSMul.{u_3, u_4} S M (AddZero.toZero.{u_4} M (AddZeroClass.toAddZero.{u_4} M (AddMonoid.toAddZeroClass.{u_4} M (AddCommMonoid.toAddMonoid.{u_4} 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inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8] {S : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 : Monoid.{u_3} S] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19 : DistribMulAction.{u_3, u_1} S M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8)] {s : Set.{u_3} S} {N : Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11} {motive : forall (x : M), (Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) x) -> Prop} (x : M) (smul₁ : forall (r : R) {{m : M}} (mem : Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) m), (motive m mem) -> (motive (HSMul.hSMul.{u_2, u_1, u_1} R M M (instHSMul.{u_2, u_1} R M (SMulZeroClass.toSMul.{u_2, u_1} R M (AddZero.toZero.{u_1} M (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_2, u_1} R M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_2, u_1} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8) (Module.toDistribMulAction.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11))))) r m) (Submodule.smul_mem.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) m r mem))) (add : forall {{m₁ : M}} {{m₂ : M}} (mem₁ : Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) m₁) (mem₂ : Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) m₂), (motive m₁ mem₁) -> (motive m₂ mem₂) -> (motive (HAdd.hAdd.{u_1, u_1, u_1} M M M (instHAdd.{u_1} M (AddCommMagma.toAdd.{u_1} M (AddCommSemigroup.toAddCommMagma.{u_1} M (AddCommMonoid.toAddCommSemigroup.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8)))) m₁ m₂) (Submodule.add_mem.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) m₁ m₂ mem₁ mem₂))) (zero : motive (OfNat.ofNat.{u_1} M 0 (Zero.toOfNat0.{u_1} M (AddZero.toZero.{u_1} M (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8)))))) (Submodule.zero_mem.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N))) (motive_1 : (Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (Submodule.mk.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 (AddSubmonoid.mk.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8)) (AddSubsemigroup.mk.{u_1} M (AddZero.toAdd.{u_1} M (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8)))) (setOf.{u_1} M (fun (m : M) => Exists.{0} (Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) m) (fun (mem : Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) m) => motive m mem))) (fun {a._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.207 : M} {b._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.208 : M} (x._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.203 : Membership.mem.{u_1, u_1} M (Set.{u_1} M) (Set.instMembership.{u_1} M) (setOf.{u_1} M (fun (m : M) => Exists.{0} (Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) m) (fun (mem : Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) m) => motive m mem))) a._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.207) (x._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.204 : Membership.mem.{u_1, u_1} M (Set.{u_1} M) (Set.instMembership.{u_1} M) (setOf.{u_1} M (fun (m : M) => Exists.{0} (Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) m) (fun (mem : Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) m) => motive m mem))) b._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.208) => _private.Mathlib.Algebra.Module.Submodule.Pointwise.0.Submodule.set_smul_inductionOn.match_1_1.{u_1, u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19 s N motive a._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.207 (fun (x._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx.203.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.215 : Membership.mem.{u_1, u_1} M (Set.{u_1} M) (Set.instMembership.{u_1} M) (setOf.{u_1} M (fun (m : M) => Exists.{0} (Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) m) (fun (mem : Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) m) => motive m mem))) a._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.207) => Membership.mem.{u_1, u_1} M (Set.{u_1} M) (Set.instMembership.{u_1} M) (setOf.{u_1} M (fun (m : M) => Exists.{0} (Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) m) (fun (mem : Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) m) => motive m mem))) (HAdd.hAdd.{u_1, u_1, u_1} M M M (instHAdd.{u_1} M (AddZero.toAdd.{u_1} M (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8))))) a._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.207 b._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.208)) x._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.203 (fun (mem : Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) a._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.207) (h : motive a._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.207 mem) => _private.Mathlib.Algebra.Module.Submodule.Pointwise.0.Submodule.set_smul_inductionOn.match_1_1.{u_1, u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19 s N motive b._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.208 (fun (x._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx.204.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.227 : Membership.mem.{u_1, u_1} M (Set.{u_1} M) (Set.instMembership.{u_1} M) (setOf.{u_1} M (fun (m : M) => Exists.{0} (Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) m) (fun (mem : Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) m) => motive m mem))) b._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.208) => Membership.mem.{u_1, u_1} M (Set.{u_1} M) (Set.instMembership.{u_1} M) (setOf.{u_1} M (fun (m : M) => Exists.{0} (Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) m) (fun (mem : Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) m) => motive m mem))) (HAdd.hAdd.{u_1, u_1, u_1} M M M (instHAdd.{u_1} M (AddZero.toAdd.{u_1} M (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8))))) a._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.207 b._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.208)) x._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.204 (fun (mem' : Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) b._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.208) (h' : motive b._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.208 mem') => Exists.intro.{0} (Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) (HAdd.hAdd.{u_1, u_1, u_1} M M M (instHAdd.{u_1} M (AddZero.toAdd.{u_1} M (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8))))) a._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.207 b._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.208)) (fun (mem : Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) (HAdd.hAdd.{u_1, u_1, u_1} M M M (instHAdd.{u_1} M (AddZero.toAdd.{u_1} M (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8))))) a._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.207 b._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.208)) => motive (HAdd.hAdd.{u_1, u_1, u_1} M M M (instHAdd.{u_1} M (AddZero.toAdd.{u_1} M (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8))))) a._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.207 b._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.208) mem) (Submodule.add_mem.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) a._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.207 b._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.208 mem mem') (add a._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.207 b._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.208 mem mem' h h'))))) (Exists.intro.{0} (Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) (OfNat.ofNat.{u_1} M 0 (Zero.toOfNat0.{u_1} M (AddZero.toZero.{u_1} M (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8))))))) (fun (mem : Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) (OfNat.ofNat.{u_1} M 0 (Zero.toOfNat0.{u_1} M (AddZero.toZero.{u_1} M (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8))))))) => motive (OfNat.ofNat.{u_1} M 0 (Zero.toOfNat0.{u_1} M (AddZero.toZero.{u_1} M (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8)))))) mem) (Submodule.zero_mem.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N)) zero)) (fun (r : R) (x._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.260 : M) (x._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.256 : Membership.mem.{u_1, u_1} M (Set.{u_1} M) (Set.instMembership.{u_1} M) (setOf.{u_1} M (fun (m : M) => Exists.{0} (Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) m) (fun (mem : Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) m) => motive m mem))) x._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.260) => _private.Mathlib.Algebra.Module.Submodule.Pointwise.0.Submodule.set_smul_inductionOn.match_1_1.{u_1, u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19 s N motive x._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.260 (fun (x._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx.256.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.267 : Membership.mem.{u_1, u_1} M (Set.{u_1} M) (Set.instMembership.{u_1} M) (setOf.{u_1} M (fun (m : M) => Exists.{0} (Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) m) (fun (mem : Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) m) => motive m mem))) x._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.260) => Membership.mem.{u_1, u_1} M (Set.{u_1} M) (Set.instMembership.{u_1} M) (setOf.{u_1} M (fun (m : M) => Exists.{0} (Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) m) (fun (mem : Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) m) => motive m mem))) (HSMul.hSMul.{u_2, u_1, u_1} R M M (instHSMul.{u_2, u_1} R M (SMulZeroClass.toSMul.{u_2, u_1} R M (AddZero.toZero.{u_1} M (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_2, u_1} R M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_2, u_1} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8) (Module.toDistribMulAction.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11))))) r x._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.260)) x._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.256 (fun (mem : Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) x._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.260) (h : motive x._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.260 mem) => Exists.intro.{0} (Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) (HSMul.hSMul.{u_2, u_1, u_1} R M M (instHSMul.{u_2, u_1} R M (SMulZeroClass.toSMul.{u_2, u_1} R M (AddZero.toZero.{u_1} M (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_2, u_1} R M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_2, u_1} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8) (Module.toDistribMulAction.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11))))) r x._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.260)) (fun (mem : Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) (HSMul.hSMul.{u_2, u_1, u_1} R M M (instHSMul.{u_2, u_1} R M (SMulZeroClass.toSMul.{u_2, u_1} R M (AddZero.toZero.{u_1} M (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_2, u_1} R M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_2, u_1} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8) (Module.toDistribMulAction.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11))))) r x._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.260)) => motive (HSMul.hSMul.{u_2, u_1, u_1} R M M (instHSMul.{u_2, u_1} R M (SMulZeroClass.toSMul.{u_2, u_1} R M (AddZero.toZero.{u_1} M (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_2, u_1} R M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_2, u_1} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8) (Module.toDistribMulAction.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11))))) r x._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.260) mem) (Submodule.smul_mem.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) x._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.260 r mem) (smul₁ r x._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.260 mem h)))) x) -> Prop) (x._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.308 : Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (Submodule.mk.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 (AddSubmonoid.mk.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8)) (AddSubsemigroup.mk.{u_1} M (AddZero.toAdd.{u_1} M (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8)))) (setOf.{u_1} M (fun (m : M) => Exists.{0} (Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) m) (fun (mem : Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) m) => motive m mem))) (fun {a._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.207 : M} {b._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.208 : M} (x._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.203 : Membership.mem.{u_1, u_1} M (Set.{u_1} M) (Set.instMembership.{u_1} M) (setOf.{u_1} M (fun (m : M) => Exists.{0} (Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) m) (fun (mem : Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) m) => motive m mem))) a._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.207) (x._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.204 : Membership.mem.{u_1, u_1} M (Set.{u_1} M) (Set.instMembership.{u_1} M) (setOf.{u_1} M (fun (m : M) => Exists.{0} (Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) m) (fun (mem : Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) m) => motive m mem))) b._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.208) => _private.Mathlib.Algebra.Module.Submodule.Pointwise.0.Submodule.set_smul_inductionOn.match_1_1.{u_1, u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19 s N motive a._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.207 (fun (x._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx.203.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.215 : Membership.mem.{u_1, u_1} M (Set.{u_1} M) (Set.instMembership.{u_1} M) (setOf.{u_1} M (fun (m : M) => Exists.{0} (Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) m) (fun (mem : Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) m) => motive m mem))) a._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.207) => Membership.mem.{u_1, u_1} M (Set.{u_1} M) (Set.instMembership.{u_1} M) (setOf.{u_1} M (fun (m : M) => Exists.{0} (Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) m) (fun (mem : Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) m) => motive m mem))) (HAdd.hAdd.{u_1, u_1, u_1} M M M (instHAdd.{u_1} M (AddZero.toAdd.{u_1} M (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8))))) a._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.207 b._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.208)) x._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.203 (fun (mem : Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) a._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.207) (h : motive a._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.207 mem) => _private.Mathlib.Algebra.Module.Submodule.Pointwise.0.Submodule.set_smul_inductionOn.match_1_1.{u_1, u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19 s N motive b._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.208 (fun (x._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx.204.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.227 : Membership.mem.{u_1, u_1} M (Set.{u_1} M) (Set.instMembership.{u_1} M) (setOf.{u_1} M (fun (m : M) => Exists.{0} (Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) m) (fun (mem : Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) m) => motive m mem))) b._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.208) => Membership.mem.{u_1, u_1} M (Set.{u_1} M) (Set.instMembership.{u_1} M) (setOf.{u_1} M (fun (m : M) => Exists.{0} (Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) m) (fun (mem : Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) m) => motive m mem))) (HAdd.hAdd.{u_1, u_1, u_1} M M M (instHAdd.{u_1} M (AddZero.toAdd.{u_1} M (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8))))) a._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.207 b._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.208)) x._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.204 (fun (mem' : Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) b._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.208) (h' : motive b._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.208 mem') => Exists.intro.{0} (Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) (HAdd.hAdd.{u_1, u_1, u_1} M M M (instHAdd.{u_1} M (AddZero.toAdd.{u_1} M (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8))))) a._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.207 b._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.208)) (fun (mem : Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) (HAdd.hAdd.{u_1, u_1, u_1} M M M (instHAdd.{u_1} M (AddZero.toAdd.{u_1} M (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8))))) a._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.207 b._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.208)) => motive (HAdd.hAdd.{u_1, u_1, u_1} M M M (instHAdd.{u_1} M (AddZero.toAdd.{u_1} M (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8))))) a._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.207 b._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.208) mem) (Submodule.add_mem.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) a._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.207 b._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.208 mem mem') (add a._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.207 b._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.208 mem mem' h h'))))) (Exists.intro.{0} (Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) (OfNat.ofNat.{u_1} M 0 (Zero.toOfNat0.{u_1} M (AddZero.toZero.{u_1} M (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8))))))) (fun (mem : Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) (OfNat.ofNat.{u_1} M 0 (Zero.toOfNat0.{u_1} M (AddZero.toZero.{u_1} M (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8))))))) => motive (OfNat.ofNat.{u_1} M 0 (Zero.toOfNat0.{u_1} M (AddZero.toZero.{u_1} M (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8)))))) mem) (Submodule.zero_mem.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N)) zero)) (fun (r : R) (x._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.260 : M) (x._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.256 : Membership.mem.{u_1, u_1} M (Set.{u_1} M) (Set.instMembership.{u_1} M) (setOf.{u_1} M (fun (m : M) => Exists.{0} (Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) m) (fun (mem : Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) m) => motive m mem))) x._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.260) => _private.Mathlib.Algebra.Module.Submodule.Pointwise.0.Submodule.set_smul_inductionOn.match_1_1.{u_1, u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19 s N motive x._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.260 (fun (x._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx.256.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.267 : Membership.mem.{u_1, u_1} M (Set.{u_1} M) (Set.instMembership.{u_1} M) (setOf.{u_1} M (fun (m : M) => Exists.{0} (Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) m) (fun (mem : Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) m) => motive m mem))) x._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.260) => Membership.mem.{u_1, u_1} M (Set.{u_1} M) (Set.instMembership.{u_1} M) (setOf.{u_1} M (fun (m : M) => Exists.{0} (Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) m) (fun (mem : Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) m) => motive m mem))) (HSMul.hSMul.{u_2, u_1, u_1} R M M (instHSMul.{u_2, u_1} R M (SMulZeroClass.toSMul.{u_2, u_1} R M (AddZero.toZero.{u_1} M (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_2, u_1} R M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_2, u_1} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8) (Module.toDistribMulAction.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11))))) r x._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.260)) x._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.256 (fun (mem : Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) x._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.260) (h : motive x._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.260 mem) => Exists.intro.{0} (Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) (HSMul.hSMul.{u_2, u_1, u_1} R M M (instHSMul.{u_2, u_1} R M (SMulZeroClass.toSMul.{u_2, u_1} R M (AddZero.toZero.{u_1} M (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_2, u_1} R M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_2, u_1} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8) (Module.toDistribMulAction.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11))))) r x._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.260)) (fun (mem : Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) (HSMul.hSMul.{u_2, u_1, u_1} R M M (instHSMul.{u_2, u_1} R M (SMulZeroClass.toSMul.{u_2, u_1} R M (AddZero.toZero.{u_1} M (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_2, u_1} R M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_2, u_1} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8) (Module.toDistribMulAction.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11))))) r x._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.260)) => motive (HSMul.hSMul.{u_2, u_1, u_1} R M M (instHSMul.{u_2, u_1} R M (SMulZeroClass.toSMul.{u_2, u_1} R M (AddZero.toZero.{u_1} M (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_2, u_1} R M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_2, u_1} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8) (Module.toDistribMulAction.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11))))) r x._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.260) mem) (Submodule.smul_mem.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) x._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.260 r mem) (smul₁ r x._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.260 mem h)))) x), (forall (w._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.317 : Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) x) (h : motive x w._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.317), motive_1 (Exists.intro.{0} (Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) x) (fun (mem : Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11)) (HSMul.hSMul.{u_3, u_1, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (instHSMul.{u_3, u_1} (Set.{u_3} S) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.19)) s N) x) => motive x mem) w._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.317 h)) -> (motive_1 x._@.Mathlib.Algebra.Module.Submodule.Pointwise.1108009345._hygCtx._hyg.308)","typeFull":"∀ {R : Type u_2} {M : Type u_1} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {S : Type u_3}\n  [inst_3 : Monoid S] [inst_4 : DistribMulAction S M] {s : Set S} {N : Submodule R M}\n  {motive : (x : M) → x ∈ s • N → Prop} (x : M)\n  (smul₁ : ∀ (r : R) ⦃m : M⦄ (mem : m ∈ s • N), motive m mem → motive (r • m) ⋯)\n  (add : ∀ ⦃m₁ m₂ : M⦄ (mem₁ : m₁ ∈ s • N) (mem₂ : m₂ ∈ s • N), motive m₁ mem₁ → motive m₂ mem₂ → motive (m₁ + m₂) ⋯)\n  (zero : motive 0 ⋯)\n  (motive_1 :\n    x ∈ { carrier := {m | ∃ (mem : m ∈ s • N), motive m mem}, add_mem' := ⋯, zero_mem' := ⋯, smul_mem' := ⋯ } → Prop)\n  (x_1 : x ∈ { carrier := {m | ∃ (mem : m ∈ s • N), motive m mem}, add_mem' := ⋯, zero_mem' := ⋯, smul_mem' := ⋯ }),\n  (∀ (w : x ∈ s • N) (h : motive x w), motive_1 ⋯) → motive_1 x_1","typeReadable":"∀ {R : Type u_2} {M : Type u_1} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {S : Type u_3}\n  [inst_3 : Monoid S] [inst_4 : DistribMulAction S M] {s : Set S} {N : Submodule R M}\n  {motive : (x : M) → x ∈ s • N → Prop} (x : M)\n  (smul₁ : ∀ (r : R) ⦃m : M⦄ (mem : m ∈ s • N), motive m mem → motive (r • m) ⋯)\n  (add : ∀ ⦃m₁ m₂ : M⦄ (mem₁ : m₁ ∈ s • N) (mem₂ : m₂ ∈ s • N), motive m₁ mem₁ → motive m₂ mem₂ → motive (m₁ + m₂) ⋯)\n  (zero : motive 0 ⋯)\n  (motive_1 :\n    x ∈ { carrier := {m | ∃ (mem : m ∈ s • N), motive m mem}, add_mem' := ⋯, zero_mem' := ⋯, smul_mem' := ⋯ } → Prop)\n  (x_1 : x ∈ { carrier := {m | ∃ (mem : m ∈ s • N), motive m mem}, add_mem' := ⋯, zero_mem' := ⋯, smul_mem' := ⋯ }),\n  (∀ (w : x ∈ s • N) (h : motive x w), motive_1 ⋯) → motive_1 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{α : Type.{u_1}} {R : Type.{u_2}} {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.5 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.8 : AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.11 : Module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.8] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.15 : Monoid.{u_1} α] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.18 : DistribMulAction.{u_1, u_3} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.8)] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.22 : SMulCommClass.{u_1, u_2, u_3} α R M (SMulZeroClass.toSMul.{u_1, u_3} α M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_1, u_3} α M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_1, u_3} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.18))) (SMulZeroClass.toSMul.{u_2, u_3} R M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_2, u_3} R M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_2, u_3} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.8) (Module.toDistribMulAction.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.11))))] (a : α) (S : Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.11), Eq.{succ u_3} (Set.{u_3} M) (HSMul.hSMul.{u_1, u_3, u_3} α (Set.{u_3} M) (Set.{u_3} M) (instHSMul.{u_1, u_3} α (Set.{u_3} M) (Set.smulSet.{u_1, u_3} α M (SMulZeroClass.toSMul.{u_1, u_3} α M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_1, u_3} α M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_1, u_3} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.18))))) a (SetLike.coe.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.11) S)) (SetLike.coe.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.11) (HSMul.hSMul.{u_1, u_3, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.11) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.11) (instHSMul.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.11) (SMulZeroClass.toSMul.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.11) (AddZero.toZero.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.11) (AddZeroClass.toAddZero.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.11) (AddMonoid.toAddZeroClass.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.11) (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.11))))) (DistribSMul.toSMulZeroClass.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.11) (AddMonoid.toAddZeroClass.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.11) (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.11))) (DistribMulAction.toDistribSMul.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.11) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.11)) (Submodule.pointwiseDistribMulAction.{u_1, u_2, u_3} α R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.22))))) a S))","typeFull":"∀ {α : Type u_1} {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  [inst_3 : Monoid α] [inst_4 : DistribMulAction α M] [inst_5 : SMulCommClass α R M] (a : α) (S : Submodule R M),\n  a • ↑S = ↑(a • S)","typeReadable":"∀ {α : Type u_1} {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  [inst_3 : Monoid α] [inst_4 : DistribMulAction α M] [inst_5 : SMulCommClass α R M] (a : α) (S : Submodule R M),\n  a • ↑S = ↑(a • S)","typeReferences":[["Module"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Submodule"],["AddCommMonoid"],["MonoidWithZero","toMonoid"],["Set","smulSet"],["instHSMul"],["Eq"],["DistribMulAction"],["DistribSMul","toSMulZeroClass"],["Set"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["SMulCommClass"],["Module","toDistribMulAction"],["SetLike","coe"],["Submodule","setLike"],["HSMul","hSMul"],["Monoid"],["AddZero","toZero"],["Submodule","pointwiseDistribMulAction"],["Submodule","pointwiseAddCommMonoid"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["Set"],["DistribMulAction","toDistribSMul"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["Submodule"],["Submodule","coe_pointwise_smul"],["SetLike","coe"],["Submodule","setLike"],["HSMul","hSMul"],["Set","smulSet"],["Eq","symm"],["instHSMul"],["AddZero","toZero"],["Submodule","pointwiseDistribMulAction"],["Submodule","pointwiseAddCommMonoid"],["DistribSMul","toSMulZeroClass"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Submodule","pointwiseSetMulAction","_proof_1"],"typeFallback":"forall {R : Type.{u_1}} {M : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11 : Module.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.30 : SMulCommClass.{u_1, u_1, u_2} R R M (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 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8) (Module.toDistribMulAction.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11)))) (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 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8) (Module.toDistribMulAction.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11))))] (s : Set.{u_1} R) (t : Set.{u_1} R) (x : Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11), Eq.{succ u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11) (HSMul.hSMul.{u_1, u_2, u_2} (Set.{u_1} R) (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11) (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11) (instHSMul.{u_1, u_2} (Set.{u_1} R) (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_1, u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11 R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5)) (Module.toDistribMulAction.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11))) (HMul.hMul.{u_1, u_1, u_1} (Set.{u_1} R) (Set.{u_1} R) (Set.{u_1} R) (instHMul.{u_1} (Set.{u_1} R) (Semigroup.toMul.{u_1} (Set.{u_1} R) 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inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_1, u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11 R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5)) (Module.toDistribMulAction.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11))) s (HSMul.hSMul.{u_1, u_2, u_2} (Set.{u_1} R) (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11) (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11) (instHSMul.{u_1, u_2} (Set.{u_1} R) (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_1, u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11 R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5)) (Module.toDistribMulAction.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11))) t x))","typeFull":"∀ {R : Type u_1} {M : Type u_2} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  [SMulCommClass R R M] (s t : Set R) (x : Submodule R M), (s * t) • x = s • t • x","typeReadable":"∀ {R : Type u_1} {M : Type u_2} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  [SMulCommClass R R M] (s t : Set R) (x : Submodule R M), (s * t) • x = s • t • x","typeReferences":[["Module"],["HMul","hMul"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Submodule"],["Semigroup","toMul"],["AddCommMonoid"],["MonoidWithZero","toMonoid"],["instHSMul"],["Monoid","toSemigroup"],["Eq"],["DistribSMul","toSMulZeroClass"],["Submodule","pointwiseSetSMul"],["Set"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["Set","monoid"],["AddZeroClass","toAddZero"],["SMulCommClass"],["Module","toDistribMulAction"],["HSMul","hSMul"],["instHMul"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["Submodule","pointwiseSetSMul"],["Set"],["Submodule","instPartialOrder"],["Set","monoid"],["Semiring","toMonoidWithZero"],["HMul","hMul"],["Submodule","pointwiseSetMulAction","_proof_2"],["Submodule","pointwiseSetMulAction","_proof_3"],["Submodule"],["Semigroup","toMul"],["Module","toDistribMulAction"],["HSMul","hSMul"],["MonoidWithZero","toMonoid"],["instHSMul"],["instHMul"],["Submodule","set_smul_le"],["Monoid","toSemigroup"],["le_antisymm"]]},{"isProp":true,"kind":"theorem","name":["Submodule","pointwiseNeg","_proof_1"],"typeFallback":"forall {R : Type.{u_2}} {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3972992742._hygCtx._hyg.5 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3972992742._hygCtx._hyg.8 : AddCommGroup.{u_1} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3972992742._hygCtx._hyg.11 : Module.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3972992742._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3972992742._hygCtx._hyg.8)] (p : Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3972992742._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3972992742._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3972992742._hygCtx._hyg.11) (r : R) (m : M), (Membership.mem.{u_1, u_1} M (Set.{u_1} M) (Set.instMembership.{u_1} M) (AddSubsemigroup.carrier.{u_1} M (AddZero.toAdd.{u_1} M (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3972992742._hygCtx._hyg.8))))) (AddSubmonoid.toAddSubsemigroup.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3972992742._hygCtx._hyg.8))) (Neg.neg.{u_1} (AddSubmonoid.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3972992742._hygCtx._hyg.8)))) (AddSubmonoid.neg.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3972992742._hygCtx._hyg.8)) (Submodule.toAddSubmonoid.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3972992742._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3972992742._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3972992742._hygCtx._hyg.11 p)))) m) -> (Membership.mem.{u_1, u_1} M (Set.{u_1} M) (Set.instMembership.{u_1} M) (Neg.neg.{u_1} (Set.{u_1} M) (Set.neg.{u_1} M (NegZeroClass.toNeg.{u_1} M (SubNegZeroMonoid.toNegZeroClass.{u_1} M (SubtractionMonoid.toSubNegZeroMonoid.{u_1} M (AddGroup.toSubtractionMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3972992742._hygCtx._hyg.8)))))) (SetLike.coe.{u_1, u_1} (AddSubmonoid.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3972992742._hygCtx._hyg.8))))) M (AddSubmonoid.instSetLike.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3972992742._hygCtx._hyg.8))))) (Submodule.toAddSubmonoid.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3972992742._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3972992742._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3972992742._hygCtx._hyg.11 p))) (HSMul.hSMul.{u_2, u_1, u_1} R M M (instHSMul.{u_2, u_1} R M (SMulZeroClass.toSMul.{u_2, u_1} R M (AddZero.toZero.{u_1} M (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3972992742._hygCtx._hyg.8))))) (DistribSMul.toSMulZeroClass.{u_2, u_1} R M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3972992742._hygCtx._hyg.8))) (DistribMulAction.toDistribSMul.{u_2, u_1} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3972992742._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_1} M (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3972992742._hygCtx._hyg.8)) (Module.toDistribMulAction.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3972992742._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3972992742._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3972992742._hygCtx._hyg.11))))) r m))","typeFull":"∀ {R : Type u_2} {M : Type u_1} [inst : Semiring R] [inst_1 : AddCommGroup M] [inst_2 : Module R M] (p : Submodule R M)\n  (r : R), ∀ m ∈ (-p.toAddSubmonoid).carrier, r • m ∈ -↑p.toAddSubmonoid","typeReadable":"∀ {R : Type u_2} {M : Type u_1} [inst : Semiring R] [inst_1 : AddCommGroup M] [inst_2 : Module R M] (p : Submodule R M)\n  (r : R), ∀ m ∈ (-p.toAddSubmonoid).carrier, r • m ∈ -↑p.toAddSubmonoid","typeReferences":[["SubtractionMonoid","toSubNegZeroMonoid"],["Submodule","toAddSubmonoid"],["Set","neg"],["Module"],["AddCommGroup","toAddGroup"],["Membership","mem"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["SubNegZeroMonoid","toNegZeroClass"],["AddGroup","toSubtractionMonoid"],["Submodule"],["AddSubsemigroup","carrier"],["MonoidWithZero","toMonoid"],["instHSMul"],["AddGroup","toSubNegMonoid"],["DistribSMul","toSMulZeroClass"],["Set"],["Neg","neg"],["AddSubmonoid","instSetLike"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["AddZeroClass","toAddZero"],["AddZero","toAdd"],["Set","instMembership"],["Module","toDistribMulAction"],["NegZeroClass","toNeg"],["SetLike","coe"],["SubNegMonoid","toAddMonoid"],["AddSubmonoid"],["AddSubmonoid","neg"],["HSMul","hSMul"],["AddCommGroup","toAddCommMonoid"],["AddSubmonoid","toAddSubsemigroup"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["SubtractionMonoid","toSubNegZeroMonoid"],["Set","neg"],["Submodule","toAddSubmonoid"],["Set","mem_neg"],["AddCommGroup","toAddGroup"],["Membership","mem"],["Iff","mp"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["SubNegZeroMonoid","toNegZeroClass"],["AddGroup","toSubtractionMonoid"],["Submodule"],["MonoidWithZero","toMonoid"],["instHSMul"],["AddGroup","toSubNegMonoid"],["Eq","rec"],["DistribSMul","toSMulZeroClass"],["smul_neg"],["SetLike","instMembership"],["Neg","neg"],["Set"],["Submodule","smul_mem"],["AddSubmonoid","instSetLike"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["Set","instMembership"],["Module","toDistribMulAction"],["SetLike","coe"],["NegZeroClass","toNeg"],["SubNegMonoid","toAddMonoid"],["Submodule","setLike"],["AddSubmonoid"],["Iff","mpr"],["HSMul","hSMul"],["AddCommGroup","toAddCommMonoid"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Submodule","coe_pointwise_smul"],"typeFallback":"forall {α : Type.{u_1}} {R : Type.{u_2}} {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.5 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.8 : AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.11 : Module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.8] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.15 : Monoid.{u_1} α] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.18 : DistribMulAction.{u_1, u_3} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.8)] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.22 : SMulCommClass.{u_1, u_2, u_3} α R M (SMulZeroClass.toSMul.{u_1, u_3} α M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_1, u_3} α M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_1, u_3} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.18))) (SMulZeroClass.toSMul.{u_2, u_3} R M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_2, u_3} R M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_2, u_3} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.8) (Module.toDistribMulAction.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.11))))] (a : α) (S : Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.11), Eq.{succ u_3} (Set.{u_3} M) (SetLike.coe.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.11) (HSMul.hSMul.{u_1, u_3, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.11) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.11) (instHSMul.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.11) (SMulZeroClass.toSMul.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.11) (AddZero.toZero.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.11) (AddZeroClass.toAddZero.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.11) (AddMonoid.toAddZeroClass.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.11) (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.11))))) (DistribSMul.toSMulZeroClass.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.11) (AddMonoid.toAddZeroClass.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.11) (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.11))) (DistribMulAction.toDistribSMul.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.11) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.11)) (Submodule.pointwiseDistribMulAction.{u_1, u_2, u_3} α R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.22))))) a S)) (HSMul.hSMul.{u_1, u_3, u_3} α (Set.{u_3} M) (Set.{u_3} M) (instHSMul.{u_1, u_3} α (Set.{u_3} M) (Set.smulSet.{u_1, u_3} α M (SMulZeroClass.toSMul.{u_1, u_3} α M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_1, u_3} α M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_1, u_3} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.18))))) a (SetLike.coe.{u_3, u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.11) M (Submodule.setLike.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2051223567._hygCtx._hyg.11) S))","typeFull":"∀ {α : Type u_1} {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  [inst_3 : Monoid α] [inst_4 : DistribMulAction α M] [inst_5 : SMulCommClass α R M] (a : α) (S : Submodule R M),\n  ↑(a • S) = a • ↑S","typeReadable":"∀ {α : Type u_1} {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  [inst_3 : Monoid α] [inst_4 : DistribMulAction α M] [inst_5 : SMulCommClass α R M] (a : α) (S : Submodule R M),\n  ↑(a • S) = a • 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{α : Type.{u_1}} {R : Type.{u_2}} {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8 : AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.11 : Module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.15 : Monoid.{u_1} α] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.18 : DistribMulAction.{u_1, u_3} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8)] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.22 : SMulCommClass.{u_1, u_2, u_3} α R M (SMulZeroClass.toSMul.{u_1, u_3} α M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_1, u_3} α M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_1, u_3} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.18))) (SMulZeroClass.toSMul.{u_2, u_3} R M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_2, u_3} R M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_2, u_3} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8) (Module.toDistribMulAction.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.11))))], DistribMulAction.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.11) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.4112518777._hygCtx._hyg.11))","typeFull":"{α : Type u_1} →\n  {R : Type u_2} →\n    {M : Type u_3} →\n      [inst : Semiring R] →\n        [inst_1 : AddCommMonoid M] →\n          [inst_2 : Module R M] →\n            [inst_3 : Monoid α] →\n              [inst_4 : DistribMulAction α M] → [SMulCommClass α R M] → DistribMulAction α (Submodule R M)","typeReadable":"{α : Type u_1} →\n  {R : Type u_2} →\n    {M : Type u_3} →\n      [inst : Semiring R] →\n        [inst_1 : AddCommMonoid M] →\n          [inst_2 : Module R M] →\n            [inst_3 : Monoid α] →\n              [inst_4 : DistribMulAction α M] → [SMulCommClass α R M] → DistribMulAction α (Submodule R M)","typeReferences":[["Module"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["Submodule"],["SMulCommClass"],["Module","toDistribMulAction"],["AddCommMonoid"],["MonoidWithZero","toMonoid"],["Monoid"],["AddZero","toZero"],["Submodule","pointwiseAddCommMonoid"],["DistribMulAction"],["Semiring"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]],"valueReferences":[["SMul","mk"],["Submodule","pointwiseDistribMulAction","_proof_1"],["SemigroupAction","mk"],["Submodule","map"],["DistribMulAction","toDistribSMul"],["DistribMulAction","mk"],["AddCommMonoid","toAddMonoid"],["Submodule"],["RingHomSurjective","ids"],["DistribSMul","toLinearMap"],["Submodule","pointwiseDistribMulAction","_proof_2"],["Submodule","pointwiseDistribMulAction","_proof_3"],["Submodule","pointwiseDistribMulAction","_proof_4"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["Monoid","toSemigroup"],["Submodule","pointwiseAddCommMonoid"],["MulAction","mk"]]},{"isProp":true,"kind":"theorem","name":["Submodule","pointwiseAddCommMonoid","_proof_4"],"typeFallback":"forall {R : Type.{u_1}} {M : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11 : Module.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8] (a : Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11), Eq.{succ u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11) (Max.max.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11) (SemilatticeSup.toMax.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11) (Lattice.toSemilatticeSup.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11) (ConditionallyCompleteLattice.toLattice.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11) (CompleteLattice.toConditionallyCompleteLattice.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11) (Submodule.completeLattice.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11))))) (Bot.bot.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11) (OrderBot.toBot.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11) (Preorder.toLE.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11) (PartialOrder.toPreorder.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11) (SemilatticeSup.toPartialOrder.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11) (Lattice.toSemilatticeSup.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11) (ConditionallyCompleteLattice.toLattice.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11) (CompleteLattice.toConditionallyCompleteLattice.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11) (Submodule.completeLattice.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11))))))) (Submodule.instOrderBot.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11))) a) a","typeFull":"∀ {R : Type u_1} {M : Type u_2} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  (a : Submodule R M), ⊥ ⊔ a = a","typeReadable":"∀ {R : Type u_1} {M : Type u_2} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  (a : Submodule R M), ⊥ ⊔ a = a","typeReferences":[["Lattice","toSemilatticeSup"],["PartialOrder","toPreorder"],["Module"],["SemilatticeSup","toPartialOrder"],["Bot","bot"],["Submodule"],["AddCommMonoid"],["Max","max"],["SemilatticeSup","toMax"],["Submodule","completeLattice"],["ConditionallyCompleteLattice","toLattice"],["OrderBot","toBot"],["Submodule","instOrderBot"],["Preorder","toLE"],["Eq"],["CompleteLattice","toConditionallyCompleteLattice"],["Semiring"]],"valueReferences":[["Lattice","toSemilatticeSup"],["Submodule","completeLattice"],["ConditionallyCompleteLattice","toLattice"],["bot_sup_eq"],["Submodule","instOrderBot"],["CompleteLattice","toConditionallyCompleteLattice"],["Submodule"]]},{"isProp":false,"kind":"definition","name":["Submodule","pointwiseSetMulAction"],"typeFallback":"forall {R : Type.{u_2}} {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 : AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11 : Module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.30 : SMulCommClass.{u_2, u_2, u_3} R R M (SMulZeroClass.toSMul.{u_2, u_3} R M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_2, u_3} R M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_2, u_3} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8) (Module.toDistribMulAction.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11)))) (SMulZeroClass.toSMul.{u_2, u_3} R M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_2, u_3} R M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_2, u_3} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8) (Module.toDistribMulAction.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11))))], MulAction.{u_2, u_3} (Set.{u_2} R) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11) (Set.monoid.{u_2} R (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5)))","typeFull":"{R : Type u_2} →\n  {M : Type u_3} →\n    [inst : Semiring R] →\n      [inst_1 : AddCommMonoid M] → [inst_2 : Module R M] → [SMulCommClass R R M] → MulAction (Set R) (Submodule R M)","typeReadable":"{R : Type u_2} →\n  {M : Type u_3} →\n    [inst : Semiring R] →\n      [inst_1 : AddCommMonoid M] → [inst_2 : Module R M] → [SMulCommClass R R M] → MulAction (Set R) (Submodule R M)","typeReferences":[["Set"],["Module"],["DistribMulAction","toDistribSMul"],["Set","monoid"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["Submodule"],["SMulCommClass"],["Module","toDistribMulAction"],["AddCommMonoid"],["MulAction"],["MonoidWithZero","toMonoid"],["AddZero","toZero"],["Semiring"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]],"valueReferences":[["Module","toDistribMulAction"],["Submodule","pointwiseSetSMul"],["Submodule","pointwiseSetMulAction","_proof_1"],["SemigroupAction","mk"],["Set"],["MonoidWithZero","toMonoid"],["Submodule","pointwiseSetMulAction","_proof_5"],["Semiring","toMonoidWithZero"],["Set","monoid"],["Monoid","toSemigroup"],["Submodule"],["MulAction","mk"]]},{"isProp":true,"kind":"theorem","name":["Submodule","instCovariantClassHSMulLe"],"typeFallback":"forall {α : Type.{u_1}} {R : Type.{u_2}} {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.5 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.8 : AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.11 : Module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.8] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.15 : Monoid.{u_1} α] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.18 : DistribMulAction.{u_1, u_3} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.8)] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.22 : SMulCommClass.{u_1, u_2, u_3} α R M (SMulZeroClass.toSMul.{u_1, u_3} α M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_1, u_3} α M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_1, u_3} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.18))) (SMulZeroClass.toSMul.{u_2, u_3} R M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_2, u_3} R M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_2, u_3} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.8) (Module.toDistribMulAction.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.11))))], CovariantClass.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.11) (HSMul.hSMul.{u_1, u_3, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.11) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.11) (instHSMul.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.11) (SMulZeroClass.toSMul.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.11) (AddZero.toZero.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.11) (AddZeroClass.toAddZero.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.11) (AddMonoid.toAddZeroClass.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.11) (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.11))))) (DistribSMul.toSMulZeroClass.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.11) (AddMonoid.toAddZeroClass.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.11) (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.11))) (DistribMulAction.toDistribSMul.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.11) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.11)) (Submodule.pointwiseDistribMulAction.{u_1, u_2, u_3} α R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.22)))))) (LE.le.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.11) (Preorder.toLE.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.11) (PartialOrder.toPreorder.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.11) (Submodule.instPartialOrder.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3105796765._hygCtx._hyg.11))))","typeFull":"∀ {α : Type u_1} {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  [inst_3 : Monoid α] [inst_4 : DistribMulAction α M] [inst_5 : SMulCommClass α R M],\n  CovariantClass α (Submodule R M) HSMul.hSMul LE.le","typeReadable":"∀ {α : Type u_1} {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  [inst_3 : Monoid α] [inst_4 : DistribMulAction α M] [inst_5 : SMulCommClass α R M],\n  CovariantClass α (Submodule R M) HSMul.hSMul 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{α : Type.{u_1}} {R : Type.{u_2}} {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.5 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.8 : AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.11 : Module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.8] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.15 : Monoid.{u_1} α] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.18 : DistribMulAction.{u_1, u_3} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.8)] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.22 : SMulCommClass.{u_1, u_2, u_3} α R M (SMulZeroClass.toSMul.{u_1, u_3} α M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_1, u_3} α M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_1, u_3} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.18))) (SMulZeroClass.toSMul.{u_2, u_3} R M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_2, u_3} R M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_2, u_3} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.8) (Module.toDistribMulAction.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.11))))] (a : α) (s : Set.{u_3} M), Eq.{succ u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.11) (Submodule.span.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.11 (HSMul.hSMul.{u_1, u_3, u_3} α (Set.{u_3} M) (Set.{u_3} M) (instHSMul.{u_1, u_3} α (Set.{u_3} M) (Set.smulSet.{u_1, u_3} α M (SMulZeroClass.toSMul.{u_1, u_3} α M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_1, u_3} α M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_1, u_3} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.18))))) a s)) (HSMul.hSMul.{u_1, u_3, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.11) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.11) (instHSMul.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.11) (SMulZeroClass.toSMul.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.11) (AddZero.toZero.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.11) (AddZeroClass.toAddZero.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.11) (AddMonoid.toAddZeroClass.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.11) (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.11))))) (DistribSMul.toSMulZeroClass.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.11) (AddMonoid.toAddZeroClass.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.11) (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.11))) (DistribMulAction.toDistribSMul.{u_1, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.11) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.15 (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.11)) (Submodule.pointwiseDistribMulAction.{u_1, u_2, u_3} α R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.22))))) a (Submodule.span.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.382235205._hygCtx._hyg.11 s))","typeFull":"∀ {α : Type u_1} {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  [inst_3 : Monoid α] [inst_4 : DistribMulAction α M] [inst_5 : SMulCommClass α R M] (a : α) (s : Set M),\n  Submodule.span R (a • s) = a • Submodule.span R s","typeReadable":"∀ {α : Type u_1} {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  [inst_3 : Monoid α] [inst_4 : DistribMulAction α M] [inst_5 : SMulCommClass α R M] (a : α) (s : Set M),\n  Submodule.span R (a • s) = a • Submodule.span R 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{R : Type.{u_2}} {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1158753761._hygCtx._hyg.5 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1158753761._hygCtx._hyg.8 : AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1158753761._hygCtx._hyg.11 : Module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1158753761._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1158753761._hygCtx._hyg.8] {S : Type.{u_4}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1158753761._hygCtx._hyg.16 : Monoid.{u_4} S] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1158753761._hygCtx._hyg.19 : DistribMulAction.{u_4, u_3} S M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1158753761._hygCtx._hyg.16 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1158753761._hygCtx._hyg.8)], CovariantClass.{u_4, u_3} (Set.{u_4} S) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1158753761._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1158753761._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1158753761._hygCtx._hyg.11) (HSMul.hSMul.{u_4, u_3, u_3} (Set.{u_4} S) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1158753761._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1158753761._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1158753761._hygCtx._hyg.11) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1158753761._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1158753761._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1158753761._hygCtx._hyg.11) (instHSMul.{u_4, u_3} (Set.{u_4} S) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1158753761._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1158753761._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1158753761._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_3, u_4} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1158753761._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1158753761._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1158753761._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1158753761._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1158753761._hygCtx._hyg.19))) (LE.le.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1158753761._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1158753761._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1158753761._hygCtx._hyg.11) (Preorder.toLE.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1158753761._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1158753761._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1158753761._hygCtx._hyg.11) (PartialOrder.toPreorder.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1158753761._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1158753761._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1158753761._hygCtx._hyg.11) (Submodule.instPartialOrder.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1158753761._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1158753761._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.1158753761._hygCtx._hyg.11))))","typeFull":"∀ {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {S : Type u_4}\n  [inst_3 : Monoid S] [inst_4 : DistribMulAction S M], CovariantClass (Set S) (Submodule R M) HSMul.hSMul LE.le","typeReadable":"∀ {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {S : Type u_4}\n  [inst_3 : Monoid S] [inst_4 : DistribMulAction S M], CovariantClass (Set S) (Submodule R M) HSMul.hSMul LE.le","typeReferences":[["PartialOrder","toPreorder"],["Submodule","pointwiseSetSMul"],["Module"],["Set"],["Submodule","instPartialOrder"],["CovariantClass"],["AddCommMonoid","toAddMonoid"],["Submodule"],["AddCommMonoid"],["HSMul","hSMul"],["LE","le"],["Monoid"],["instHSMul"],["Preorder","toLE"],["DistribMulAction"],["Semiring"]],"valueReferences":[["PartialOrder","toPreorder"],["Submodule","pointwiseSetSMul"],["Set"],["Submodule","instPartialOrder"],["LE","le"],["HSMul","hSMul"],["instHSMul"],["Submodule","set_smul_le"],["Preorder","toLE"],["CovariantClass","mk"],["Submodule","mem_set_smul_of_mem_mem"],["Submodule"]]},{"isProp":true,"kind":"theorem","name":["Submodule","neg_eq_self_iff_neg_le"],"typeFallback":"forall {R : Type.{u_2}} {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3210662264._hygCtx._hyg.5 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3210662264._hygCtx._hyg.8 : AddCommGroup.{u_3} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3210662264._hygCtx._hyg.11 : Module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3210662264._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3210662264._hygCtx._hyg.8)] {S : Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3210662264._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3210662264._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3210662264._hygCtx._hyg.11}, Iff (Eq.{succ u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3210662264._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3210662264._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3210662264._hygCtx._hyg.11) (Neg.neg.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3210662264._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3210662264._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3210662264._hygCtx._hyg.11) (Submodule.pointwiseNeg.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3210662264._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3210662264._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3210662264._hygCtx._hyg.11) S) S) (LE.le.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3210662264._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3210662264._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3210662264._hygCtx._hyg.11) (Preorder.toLE.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3210662264._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3210662264._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3210662264._hygCtx._hyg.11) (PartialOrder.toPreorder.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3210662264._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3210662264._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3210662264._hygCtx._hyg.11) (Submodule.instPartialOrder.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3210662264._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3210662264._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3210662264._hygCtx._hyg.11))) (Neg.neg.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3210662264._hygCtx._hyg.5 (AddCommGroup.toAddCommMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3210662264._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3210662264._hygCtx._hyg.11) (Submodule.pointwiseNeg.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3210662264._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3210662264._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3210662264._hygCtx._hyg.11) S) S)","typeFull":"∀ {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommGroup M] [inst_2 : Module R M] {S : Submodule R M},\n  -S = S ↔ -S ≤ S","typeReadable":"∀ {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommGroup M] [inst_2 : Module R M] {S : Submodule R M},\n  -S = S ↔ -S ≤ S","typeReferences":[["PartialOrder","toPreorder"],["Module"],["Neg","neg"],["Submodule","instPartialOrder"],["AddCommGroup"],["Submodule"],["Iff"],["LE","le"],["AddCommGroup","toAddCommMonoid"],["Submodule","pointwiseNeg"],["Eq"],["Preorder","toLE"],["Semiring"]],"valueReferences":[["PartialOrder","toPreorder"],["instAntisymmLe"],["Neg","neg"],["Submodule","instPartialOrder"],["Iff","mp"],["antisymm"],["Iff","intro"],["Submodule"],["Submodule","neg_le"],["LE","le"],["le_of_eq"],["AddCommGroup","toAddCommMonoid"],["Submodule","pointwiseNeg"],["Eq"],["Preorder","toLE"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Algebra","Module","Submodule","Pointwise",0,"Submodule","set_smul_eq_map","_simp_1_2"],"typeFallback":"forall {α : Type.{u}} {ι : Sort.{v}} {x : α} {s : ι -> (Set.{u} α)}, Eq.{1} Prop (Membership.mem.{u, u} α (Set.{u} α) (Set.instMembership.{u} α) (Set.iInter.{u, v} α ι (fun (i : ι) => s i)) x) (forall (i : ι), Membership.mem.{u, u} α (Set.{u} α) (Set.instMembership.{u} α) (s i) x)","typeFull":"∀ {α : Type u} {ι : Sort v} {x : α} {s : ι → Set α}, (x ∈ ⋂ i, s i) = ∀ (i : ι), x ∈ s i","typeReadable":"∀ {α : Type u} {ι : Sort v} {x : α} {s : ι → Set α}, (x ∈ ⋂ i, s i) = ∀ (i : ι), x ∈ s i","typeReferences":[["Set"],["Membership","mem"],["Set","iInter"],["Eq"],["Set","instMembership"]],"valueReferences":[["Set"],["Membership","mem"],["Set","mem_iInter"],["Set","iInter"],["propext"],["Set","instMembership"]]},{"isProp":true,"kind":"theorem","name":["Submodule","pointwiseAddCommMonoid","_proof_3"],"typeFallback":"forall {R : Type.{u_1}} {M : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11 : Module.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8] (a : Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11) (b : Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11) (c : Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11), Eq.{succ u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11) (Max.max.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11) (SemilatticeSup.toMax.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11) (Lattice.toSemilatticeSup.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11) (ConditionallyCompleteLattice.toLattice.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11) (CompleteLattice.toConditionallyCompleteLattice.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11) (Submodule.completeLattice.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11))))) (Max.max.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11) (SemilatticeSup.toMax.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11) (Lattice.toSemilatticeSup.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11) (ConditionallyCompleteLattice.toLattice.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11) (CompleteLattice.toConditionallyCompleteLattice.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11) (Submodule.completeLattice.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11))))) a b) c) (Max.max.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11) (SemilatticeSup.toMax.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11) (Lattice.toSemilatticeSup.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11) (ConditionallyCompleteLattice.toLattice.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11) (CompleteLattice.toConditionallyCompleteLattice.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11) (Submodule.completeLattice.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11))))) a (Max.max.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11) (SemilatticeSup.toMax.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11) (Lattice.toSemilatticeSup.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11) (ConditionallyCompleteLattice.toLattice.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11) (CompleteLattice.toConditionallyCompleteLattice.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11) (Submodule.completeLattice.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11))))) b c))","typeFull":"∀ {R : Type u_1} {M : Type u_2} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  (a b c : Submodule R M), a ⊔ b ⊔ c = a ⊔ (b ⊔ c)","typeReadable":"∀ {R : Type u_1} {M : Type u_2} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  (a b c : Submodule R M), a ⊔ b ⊔ c = a ⊔ (b ⊔ c)","typeReferences":[["Lattice","toSemilatticeSup"],["Max","max"],["AddCommMonoid"],["Module"],["SemilatticeSup","toMax"],["Submodule","completeLattice"],["ConditionallyCompleteLattice","toLattice"],["Eq"],["CompleteLattice","toConditionallyCompleteLattice"],["Submodule"],["Semiring"]],"valueReferences":[["Lattice","toSemilatticeSup"],["sup_assoc"],["Submodule","completeLattice"],["ConditionallyCompleteLattice","toLattice"],["CompleteLattice","toConditionallyCompleteLattice"],["Submodule"]]},{"isProp":false,"kind":"definition","name":["Submodule","pointwiseAdd"],"typeFallback":"forall {R : Type.{u_2}} {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.634769551._hygCtx._hyg.5 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.634769551._hygCtx._hyg.8 : AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.634769551._hygCtx._hyg.11 : Module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.634769551._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.634769551._hygCtx._hyg.8], Add.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.634769551._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.634769551._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.634769551._hygCtx._hyg.11)","typeFull":"{R : Type u_2} →\n  {M : Type u_3} → [inst : Semiring R] → [inst_1 : AddCommMonoid M] → [inst_2 : Module R M] → Add (Submodule R M)","typeReadable":"{R : Type u_2} →\n  {M : Type u_3} → [inst : Semiring R] → [inst_1 : AddCommMonoid M] → [inst_2 : Module R M] → Add (Submodule R M)","typeReferences":[["AddCommMonoid"],["Add"],["Module"],["Submodule"],["Semiring"]],"valueReferences":[["Lattice","toSemilatticeSup"],["Max","max"],["SemilatticeSup","toMax"],["Submodule","completeLattice"],["Add","mk"],["ConditionallyCompleteLattice","toLattice"],["CompleteLattice","toConditionallyCompleteLattice"],["Submodule"]]},{"isProp":true,"kind":"theorem","name":["Submodule","pointwiseSetMulAction","_proof_5"],"typeFallback":"forall {R : Type.{u_1}} {M : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11 : Module.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8] 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inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11)))) (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 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8) (Module.toDistribMulAction.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11))))] (x : Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11), Eq.{succ u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11) (HSMul.hSMul.{u_1, u_2, u_2} (Set.{u_1} R) (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11) (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11) (instHSMul.{u_1, u_2} (Set.{u_1} R) (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_1, u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11 R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5)) (Module.toDistribMulAction.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.11))) (Singleton.singleton.{u_1, u_1} R (Set.{u_1} R) (Set.instSingletonSet.{u_1} R) (OfNat.ofNat.{u_1} R 1 (One.toOfNat1.{u_1} R (AddMonoidWithOne.toOne.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2393484299._hygCtx._hyg.5))))))) x) x","typeFull":"∀ {R : Type u_1} {M : Type u_2} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  [SMulCommClass R R M] (x : Submodule R M), {1} • x = x","typeReadable":"∀ {R : Type u_1} {M : Type u_2} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  [SMulCommClass R R M] (x : Submodule R M), {1} • x = x","typeReferences":[["Module"],["Singleton","singleton"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Submodule"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["instHSMul"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Eq"],["NonAssocSemiring","toAddCommMonoidWithOne"],["DistribSMul","toSMulZeroClass"],["Submodule","pointwiseSetSMul"],["Set"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["Set","instSingletonSet"],["OfNat","ofNat"],["SMulCommClass"],["Module","toDistribMulAction"],["One","toOfNat1"],["HSMul","hSMul"],["AddMonoidWithOne","toOne"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["MulOneClass","toMulOne"],["Singleton","singleton"],["Membership","mem"],["SemigroupAction","toSMul"],["Exists","intro"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Iff","intro"],["Submodule"],["And","intro"],["Submodule","mem_singleton_set_smul"],["Semiring","toNonAssocSemiring"],["Monoid","toMulOneClass"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["Eq","symm"],["instHSMul"],["Monoid","toSemigroup"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Eq"],["NonAssocSemiring","toAddCommMonoidWithOne"],["DistribSMul","toSMulZeroClass"],["Iff","trans"],["Submodule","pointwiseSetSMul"],["SetLike","instMembership"],["Exists"],["MulOne","toOne"],["Set"],["DistribMulAction","toDistribSMul"],["And"],["Semiring","toMonoidWithZero"],["Submodule","pointwiseSetMulAction","_proof_4"],["AddZeroClass","toAddZero"],["Set","instSingletonSet"],["OfNat","ofNat"],["Module","toDistribMulAction"],["One","toOfNat1"],["Submodule","setLike"],["one_smul"],["HSMul","hSMul"],["AddMonoidWithOne","toOne"],["SetLike","ext"],["MulAction","toSemigroupAction"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Submodule","pointwiseSetDistribMulAction","_proof_1"],"typeFallback":"forall {R : Type.{u_1}} {M : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11 : Module.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.30 : SMulCommClass.{u_1, u_1, u_2} R R M (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 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8) (Module.toDistribMulAction.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11)))) (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 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8) (Module.toDistribMulAction.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11))))] (s : Set.{u_1} R) (x : Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11) (y : Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11) {{r : R}} {{n : M}}, (Membership.mem.{u_1, u_1} R (Set.{u_1} R) (Set.instMembership.{u_1} R) s r) -> (Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11) M (Submodule.setLike.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11)) (HAdd.hAdd.{u_2, u_2, u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11) (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11) (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11) (instHAdd.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11) (AddSemigroup.toAdd.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11) (AddMonoid.toAddSemigroup.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11) (AddCommMonoid.toAddMonoid.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11))))) x y) n) -> (Membership.mem.{u_2, u_2} M (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11) (SetLike.instMembership.{u_2, u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11) M (Submodule.setLike.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11)) (HAdd.hAdd.{u_2, u_2, u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11) (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11) (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11) (instHAdd.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11) (AddSemigroup.toAdd.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11) (AddMonoid.toAddSemigroup.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11) (AddCommMonoid.toAddMonoid.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11))))) (HSMul.hSMul.{u_1, u_2, u_2} (Set.{u_1} R) (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11) (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11) (instHSMul.{u_1, u_2} (Set.{u_1} R) (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11) (SemigroupAction.toSMul.{u_1, u_2} (Set.{u_1} R) (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11) (Monoid.toSemigroup.{u_1} (Set.{u_1} R) (Set.monoid.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5)))) (MulAction.toSemigroupAction.{u_1, u_2} (Set.{u_1} R) (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11) (Set.monoid.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5))) (Submodule.pointwiseSetMulAction.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.30)))) s x) (HSMul.hSMul.{u_1, u_2, u_2} (Set.{u_1} R) (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11) (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11) (instHSMul.{u_1, u_2} (Set.{u_1} R) (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11) (SemigroupAction.toSMul.{u_1, u_2} (Set.{u_1} R) (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11) (Monoid.toSemigroup.{u_1} (Set.{u_1} R) (Set.monoid.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5)))) (MulAction.toSemigroupAction.{u_1, u_2} (Set.{u_1} R) (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11) (Set.monoid.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5))) (Submodule.pointwiseSetMulAction.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.30)))) s y)) (HSMul.hSMul.{u_1, u_2, u_2} R M M (instHSMul.{u_1, u_2} R M (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 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_1, u_2} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8) (Module.toDistribMulAction.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3724804579._hygCtx._hyg.11))))) r n))","typeFull":"∀ {R : Type u_1} {M : Type u_2} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  [inst_3 : SMulCommClass R R M] (s : Set R) (x y : Submodule R M) ⦃r : R⦄ ⦃n : M⦄,\n  r ∈ s → n ∈ x + y → r • n ∈ s • x + s • y","typeReadable":"∀ {R : Type u_1} {M : Type u_2} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  [inst_3 : SMulCommClass R R M] (s : Set R) (x y : Submodule R M) ⦃r : R⦄ ⦃n : M⦄,\n  r ∈ s → n ∈ x + y → r • n ∈ s • x + s • y","typeReferences":[["Module"],["Membership","mem"],["SemigroupAction","toSMul"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Submodule"],["AddCommMonoid"],["MonoidWithZero","toMonoid"],["instHSMul"],["Monoid","toSemigroup"],["DistribSMul","toSMulZeroClass"],["AddSemigroup","toAdd"],["SetLike","instMembership"],["Set"],["instHAdd"],["DistribMulAction","toDistribSMul"],["Set","monoid"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["Set","instMembership"],["HAdd","hAdd"],["Submodule","pointwiseSetMulAction"],["SMulCommClass"],["Module","toDistribMulAction"],["Submodule","setLike"],["AddMonoid","toAddSemigroup"],["HSMul","hSMul"],["MulAction","toSemigroupAction"],["AddZero","toZero"],["Submodule","pointwiseAddCommMonoid"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["Lattice","toSemilatticeSup"],["Membership","mem"],["Exists","intro"],["SemigroupAction","toSMul"],["SMulZeroClass","toSMul"],["And","intro"],["Monoid","toSemigroup"],["Eq","ndrec"],["AddSemigroup","toAdd"],["DistribSMul","toSMulZeroClass"],["rfl"],["SetLike","instMembership"],["Exists"],["And"],["DistribMulAction","toDistribSMul"],["Set","monoid"],["AddZeroClass","toAddZero"],["Exists","casesOn"],["Submodule","pointwiseAdd"],["AddMonoid","toAddSemigroup"],["HSMul","hSMul"],["id"],["Eq","mpr"],["ConditionallyCompleteLattice","toLattice"],["AddZero","toZero"],["Submodule","pointwiseAddCommMonoid"],["AddMonoid","toAddZeroClass"],["Eq","mp"],["AddCommMonoid","toAddMonoid"],["Submodule"],["congrArg"],["Submodule","mem_sup"],["MonoidWithZero","toMonoid"],["instHSMul"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["Eq"],["CompleteLattice","toConditionallyCompleteLattice"],["propext"],["Set"],["instHAdd"],["Submodule","add_eq_sup"],["Semiring","toMonoidWithZero"],["AddZero","toAdd"],["HAdd","hAdd"],["Submodule","pointwiseSetMulAction"],["Module","toDistribMulAction"],["Max","max"],["AddCommMonoid","toAddCommSemigroup"],["Submodule","setLike"],["SemilatticeSup","toMax"],["Submodule","completeLattice"],["smul_add"],["MulAction","toSemigroupAction"],["Submodule","mem_set_smul_of_mem_mem"],["And","casesOn"]]},{"isProp":true,"kind":"theorem","name":["Submodule","pointwiseAddCommMonoid","_proof_5"],"typeFallback":"forall {R : Type.{u_1}} {M : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11 : Module.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8] (a : Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11), Eq.{succ u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11) (Max.max.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11) (SemilatticeSup.toMax.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11) (Lattice.toSemilatticeSup.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11) (ConditionallyCompleteLattice.toLattice.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11) (CompleteLattice.toConditionallyCompleteLattice.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11) (Submodule.completeLattice.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11))))) a (Bot.bot.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11) (OrderBot.toBot.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11) (Preorder.toLE.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11) (PartialOrder.toPreorder.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11) (SemilatticeSup.toPartialOrder.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11) (Lattice.toSemilatticeSup.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11) (ConditionallyCompleteLattice.toLattice.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11) (CompleteLattice.toConditionallyCompleteLattice.{u_2} (Submodule.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11) (Submodule.completeLattice.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11))))))) (Submodule.instOrderBot.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2019018475._hygCtx._hyg.11)))) a","typeFull":"∀ {R : Type u_1} {M : Type u_2} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  (a : Submodule R M), a ⊔ ⊥ = a","typeReadable":"∀ {R : Type u_1} {M : Type u_2} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  (a : Submodule R M), a ⊔ ⊥ = a","typeReferences":[["Lattice","toSemilatticeSup"],["PartialOrder","toPreorder"],["Module"],["SemilatticeSup","toPartialOrder"],["Bot","bot"],["Submodule"],["AddCommMonoid"],["Max","max"],["SemilatticeSup","toMax"],["Submodule","completeLattice"],["ConditionallyCompleteLattice","toLattice"],["OrderBot","toBot"],["Submodule","instOrderBot"],["Preorder","toLE"],["Eq"],["CompleteLattice","toConditionallyCompleteLattice"],["Semiring"]],"valueReferences":[["Lattice","toSemilatticeSup"],["sup_bot_eq"],["Submodule","completeLattice"],["ConditionallyCompleteLattice","toLattice"],["Submodule","instOrderBot"],["CompleteLattice","toConditionallyCompleteLattice"],["Submodule"]]},{"isProp":true,"kind":"theorem","name":["Submodule","set_smul_eq_iSup"],"typeFallback":"forall {R : Type.{u_2}} {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.5 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.8 : AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.11 : Module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.8] {S : Type.{u_4}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.16 : Monoid.{u_4} S] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.19 : DistribMulAction.{u_4, u_3} S M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.16 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.8)] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.30 : SMulCommClass.{u_4, u_2, u_3} S R M (SMulZeroClass.toSMul.{u_4, u_3} S M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_4, u_3} S M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_4, u_3} S M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.16 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.19))) (SMulZeroClass.toSMul.{u_2, u_3} R M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_2, u_3} R M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_2, u_3} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.8) (Module.toDistribMulAction.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.11))))] (s : Set.{u_4} S) (N : Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.11), Eq.{succ u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.11) (HSMul.hSMul.{u_4, u_3, u_3} (Set.{u_4} S) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.11) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.11) (instHSMul.{u_4, u_3} (Set.{u_4} S) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.11) (Submodule.pointwiseSetSMul.{u_2, u_3, u_4} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.11 S inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.19)) s N) (iSup.{u_3, succ u_4} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.11) S (ConditionallyCompletePartialOrderSup.toSupSet.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.11) (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.11) (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.11) (CompleteLattice.toConditionallyCompleteLattice.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.11) (Submodule.completeLattice.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.11))))) (fun (a : S) => iSup.{u_3, 0} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.11) (Membership.mem.{u_4, u_4} S (Set.{u_4} S) (Set.instMembership.{u_4} S) s a) (ConditionallyCompletePartialOrderSup.toSupSet.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.11) (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.11) (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.11) (CompleteLattice.toConditionallyCompleteLattice.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.11) (Submodule.completeLattice.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.11))))) (fun (h._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.47 : Membership.mem.{u_4, u_4} S (Set.{u_4} S) (Set.instMembership.{u_4} S) s a) => HSMul.hSMul.{u_4, u_3, u_3} S (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.11) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.11) (instHSMul.{u_4, u_3} S (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.11) (SMulZeroClass.toSMul.{u_4, u_3} S (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.11) (AddZero.toZero.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.11) (AddZeroClass.toAddZero.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.11) (AddMonoid.toAddZeroClass.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.11) (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.11))))) (DistribSMul.toSMulZeroClass.{u_4, u_3} S (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.11) (AddMonoid.toAddZeroClass.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.11) (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.11))) (DistribMulAction.toDistribSMul.{u_4, u_3} S (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.11) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.16 (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.11)) (Submodule.pointwiseDistribMulAction.{u_4, u_2, u_3} S R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.3944854703._hygCtx._hyg.30))))) a N)))","typeFull":"∀ {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {S : Type u_4}\n  [inst_3 : Monoid S] [inst_4 : DistribMulAction S M] [inst_5 : SMulCommClass S R M] (s : Set S) (N : Submodule R M),\n  s • N = ⨆ a ∈ s, a • N","typeReadable":"∀ {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {S : Type u_4}\n  [inst_3 : Monoid S] [inst_4 : DistribMulAction S M] [inst_5 : SMulCommClass S R M] (s : Set S) (N : Submodule R M),\n  s • N = ⨆ a ∈ s, a • N","typeReferences":[["Module"],["Membership","mem"],["ConditionallyCompleteLattice","toConditionallyCompletePartialOrder"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Submodule"],["AddCommMonoid"],["ConditionallyCompletePartialOrder","toConditionallyCompletePartialOrderSup"],["iSup"],["ConditionallyCompletePartialOrderSup","toSupSet"],["MonoidWithZero","toMonoid"],["instHSMul"],["Eq"],["CompleteLattice","toConditionallyCompleteLattice"],["DistribMulAction"],["DistribSMul","toSMulZeroClass"],["Submodule","pointwiseSetSMul"],["Set"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["Set","instMembership"],["SMulCommClass"],["Module","toDistribMulAction"],["HSMul","hSMul"],["Submodule","completeLattice"],["Monoid"],["AddZero","toZero"],["Submodule","pointwiseDistribMulAction"],["Submodule","pointwiseAddCommMonoid"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["PartialOrder","toPreorder"],["csInf_Ici"],["Eq","trans"],["Membership","mem"],["SMulZeroClass","toSMul"],["DistribSMul","toLinearMap"],["ConditionallyCompletePartialOrder","toConditionallyCompletePartialOrderSup"],["Semiring","toNonAssocSemiring"],["iSup"],["RingHom","id"],["ConditionallyCompletePartialOrderSup","toSupSet"],["forall_comm"],["CompleteSemilatticeInf","toPartialOrder"],["funext"],["forall_congr"],["ConditionallyCompletePartialOrderInf","toPartialOrder"],["DistribSMul","toSMulZeroClass"],["Submodule","pointwiseSetSMul"],["SetLike","instMembership"],["DistribMulAction","toDistribSMul"],["forall₂_congr"],["AddZeroClass","toAddZero"],["Set","instMembership"],["HSMul","hSMul"],["id"],["Set","ext"],["Eq","mpr"],["AddZero","toZero"],["Submodule","pointwiseAddCommMonoid"],["Submodule","pointwiseDistribMulAction"],["AddMonoid","toAddZeroClass"],["setOf"],["Submodule","instInfSet"],["ConditionallyCompleteLattice","toConditionallyCompletePartialOrder"],["AddCommMonoid","toAddMonoid"],["CompleteSemilatticeSup","toSupSet"],["CompleteLattice","toCompleteSemilatticeSup"],["Submodule"],["congrArg"],["_private","Mathlib","Algebra","Module","Submodule","Pointwise",0,"Submodule","set_smul_eq_iSup","_simp_1_2"],["InfSet","sInf"],["instHSMul"],["ConditionallyCompletePartialOrder","toConditionallyCompletePartialOrderInf"],["congrFun'"],["Eq"],["Preorder","toLE"],["Submodule","map_le_iff_le_comap"],["CompleteLattice","toConditionallyCompleteLattice"],["propext"],["Set"],["Iff","symm"],["RingHomSurjective","ids"],["Set","Ici"],["Submodule","setLike"],["LE","le"],["Submodule","completeLattice"],["CompleteLattice","toCompleteSemilatticeInf"]]},{"isProp":true,"kind":"theorem","name":["Submodule","pointwiseMulActionWithZero","_proof_1"],"typeFallback":"forall {α : Type.{u_3}} {R : Type.{u_2}} {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.5 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.8 : AddCommMonoid.{u_1} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.11 : Module.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.8] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.15 : Semiring.{u_3} α] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.18 : Module.{u_3, u_1} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.8] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.22 : SMulCommClass.{u_3, u_2, u_1} α R M 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inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.11))))] (a : α), Eq.{succ u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.11) (HSMul.hSMul.{u_3, u_1, u_1} α (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.11) (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.11) 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(Submodule.pointwiseAddCommMonoid.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.11)) (Submodule.pointwiseDistribMulAction.{u_3, u_2, u_1} α R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.11 (MonoidWithZero.toMonoid.{u_3} α (Semiring.toMonoidWithZero.{u_3} α inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.15)) (Module.toDistribMulAction.{u_3, u_1} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.18) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.22))))) a (OfNat.ofNat.{u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.11) 0 (Zero.toOfNat0.{u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.11) (AddZero.toZero.{u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.11) (AddZeroClass.toAddZero.{u_1} (Submodule.{u_2, u_1} R M 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inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.11)))))))) (OfNat.ofNat.{u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.11) 0 (Zero.toOfNat0.{u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.11) (AddZero.toZero.{u_1} (Submodule.{u_2, u_1} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.8 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inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.2208775318._hygCtx._hyg.11)))))))","typeFull":"∀ {α : Type u_3} {R : Type u_2} {M : Type u_1} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  [inst_3 : Semiring α] [inst_4 : Module α M] [inst_5 : SMulCommClass α R M] (a : α), a • 0 = 0","typeReadable":"∀ {α : Type u_3} {R : Type u_2} {M : Type u_1} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]\n  [inst_3 : Semiring α] [inst_4 : Module α M] [inst_5 : SMulCommClass α R M] (a : α), a • 0 = 0","typeReferences":[["Module"],["SemigroupAction","toSMul"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Submodule"],["AddCommMonoid"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["instHSMul"],["Zero","toOfNat0"],["Monoid","toSemigroup"],["Eq"],["DistribSMul","toSMulZeroClass"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["SMulCommClass"],["Module","toDistribMulAction"],["HSMul","hSMul"],["AddZero","toZero"],["MulAction","toSemigroupAction"],["Submodule","pointwiseAddCommMonoid"],["Submodule","pointwiseDistribMulAction"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["Module","toDistribMulAction"],["MonoidWithZero","toMonoid"],["DistribMulAction","smul_zero"],["Semiring","toMonoidWithZero"],["AddCommMonoid","toAddMonoid"],["Submodule","pointwiseDistribMulAction"],["Submodule","pointwiseAddCommMonoid"],["Submodule"]]},{"isProp":true,"kind":"theorem","name":["Submodule","smul_le_self_of_tower"],"typeFallback":"forall {R : Type.{u_2}} {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.5 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.8 : AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.11 : Module.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.8] {α : Type.{u_4}} [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.28 : Monoid.{u_4} α] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.31 : SMul.{u_4, u_2} α R] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.35 : DistribMulAction.{u_4, u_3} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.28 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.8)] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.39 : SMulCommClass.{u_4, u_2, u_3} α R M (SMulZeroClass.toSMul.{u_4, u_3} α M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_4, u_3} α M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_4, u_3} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.28 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.35))) (SMulZeroClass.toSMul.{u_2, u_3} R M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_2, u_3} R M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_2, u_3} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.8) (Module.toDistribMulAction.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.11))))] [inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.44 : IsScalarTower.{u_4, u_2, u_3} α R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.31 (SMulZeroClass.toSMul.{u_2, u_3} R M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_2, u_3} R M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_2, u_3} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.8) (Module.toDistribMulAction.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.11)))) (SMulZeroClass.toSMul.{u_4, u_3} α M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_4, u_3} α M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_4, u_3} α M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.28 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.35)))] (a : α) (S : Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.11), LE.le.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.11) (Preorder.toLE.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.11) (PartialOrder.toPreorder.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.11) (Submodule.instPartialOrder.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.11))) (HSMul.hSMul.{u_4, u_3, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.11) (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.11) (instHSMul.{u_4, u_3} α 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inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.11))))) (DistribSMul.toSMulZeroClass.{u_4, u_3} α (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.11) (AddMonoid.toAddZeroClass.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.11) (AddCommMonoid.toAddMonoid.{u_3} (Submodule.{u_2, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.11) (Submodule.pointwiseAddCommMonoid.{u_2, 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inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.11)) (Submodule.pointwiseDistribMulAction.{u_4, u_2, u_3} α R M inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.28 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.35 inst._@.Mathlib.Algebra.Module.Submodule.Pointwise.314143019._hygCtx._hyg.39))))) a S) S","typeFull":"∀ {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {α : Type u_4}\n  [inst_3 : Monoid α] [inst_4 : SMul α R] [inst_5 : DistribMulAction α M] [inst_6 : SMulCommClass α R M]\n  [IsScalarTower α R M] (a : α) (S : Submodule R M), a • S ≤ S","typeReadable":"∀ {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {α : Type u_4}\n  [inst_3 : Monoid α] [inst_4 : SMul α R] [inst_5 : DistribMulAction α M] [inst_6 : SMulCommClass α R M]\n  [IsScalarTower α R M] (a : α) (S : Submodule R M), a • S ≤ 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