diff --git "a/data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Order.Module.Archimedean.sym.json" "b/data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Order.Module.Archimedean.sym.json" new file mode 100644--- /dev/null +++ "b/data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Order.Module.Archimedean.sym.json" @@ -0,0 +1 @@ +[{"isProp":false,"kind":"definition","name":["ArchimedeanClass","ball"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.3 : AddCommGroup.{u_1} M] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.6 : LinearOrder.{u_1} M] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.9 : IsOrderedAddMonoid.{u_1} M (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.3) (PartialOrder.toPreorder.{u_1} M (SemilatticeInf.toPartialOrder.{u_1} M (Lattice.toSemilatticeInf.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.6)))))] (K : Type.{u_2}) [inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.13 : Ring.{u_2} K] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.16 : LinearOrder.{u_2} K] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.19 : IsOrderedRing.{u_2} K (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.13) (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.16))))] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.22 : Archimedean.{u_2} K (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonAssocRing.toNonUnitalNonAssocRing.{u_2} K (Ring.toNonAssocRing.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.13)))) (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.16))))] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.25 : Module.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.3)] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.29 : PosSMulMono.{u_2, u_1} K M (SMulZeroClass.toSMul.{u_2, u_1} K M (AddZero.toZero.{u_1} M (AddZeroClass.toAddZero.{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.Order.Module.Archimedean.224592742._hygCtx._hyg.3)))))) (DistribSMul.toSMulZeroClass.{u_2, u_1} K M (AddMonoid.toAddZeroClass.{u_1} M (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.3)))) (DistribMulAction.toDistribSMul.{u_2, u_1} K M (MonoidWithZero.toMonoid.{u_2} K (Semiring.toMonoidWithZero.{u_2} K (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.13))) (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.3))) (Module.toDistribMulAction.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.25)))) (PartialOrder.toPreorder.{u_2} K (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.16))))) (PartialOrder.toPreorder.{u_1} M (SemilatticeInf.toPartialOrder.{u_1} M (Lattice.toSemilatticeInf.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.6))))) (MulZeroClass.toZero.{u_2} K (NonUnitalNonAssocSemiring.toMulZeroClass.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonAssocRing.toNonUnitalNonAssocRing.{u_2} K (Ring.toNonAssocRing.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.13)))))], (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.9) -> (Submodule.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.25)","typeFull":"{M : Type u_1} →\n [inst : AddCommGroup M] →\n [inst_1 : LinearOrder M] →\n [inst_2 : IsOrderedAddMonoid M] →\n (K : Type u_2) →\n [inst_3 : Ring K] →\n [inst_4 : LinearOrder K] →\n [IsOrderedRing K] →\n [Archimedean K] → [inst_7 : Module K M] → [PosSMulMono K M] → ArchimedeanClass M → Submodule K M","typeReadable":"{M : Type u_1} →\n [inst : AddCommGroup M] →\n [inst_1 : LinearOrder M] →\n [inst_2 : IsOrderedAddMonoid M] →\n (K : Type u_2) →\n [inst_3 : Ring K] →\n [inst_4 : LinearOrder K] →\n [IsOrderedRing K] →\n [Archimedean K] → [inst_7 : Module K M] → [PosSMulMono K M] → ArchimedeanClass M → Submodule K M","typeReferences":[["PartialOrder","toPreorder"],["Ring","toNonAssocRing"],["ArchimedeanClass"],["Module"],["AddCommGroup","toAddGroup"],["SMulZeroClass","toSMul"],["Submodule"],["PosSMulMono"],["instDistribLatticeOfLinearOrder"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["IsOrderedAddMonoid"],["MonoidWithZero","toMonoid"],["AddGroup","toSubNegMonoid"],["NonAssocRing","toNonUnitalNonAssocRing"],["DistribSMul","toSMulZeroClass"],["SemilatticeInf","toPartialOrder"],["Lattice","toSemilatticeInf"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["DistribMulAction","toDistribSMul"],["LinearOrder"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["AddZeroClass","toAddZero"],["Ring","toSemiring"],["IsOrderedRing"],["Module","toDistribMulAction"],["DistribLattice","toLattice"],["SubNegMonoid","toAddMonoid"],["MulZeroClass","toZero"],["Archimedean"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["AddCommGroup","toAddCommMonoid"],["AddZero","toZero"],["Ring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["instDistribLatticeOfLinearOrder"],["DistribLattice","toLattice"],["Lattice","toSemilatticeInf"],["PartialOrder","toPreorder"],["ArchimedeanClass"],["UpperSet","Ioi"],["ArchimedeanClass","instLinearOrder"],["SemilatticeInf","toPartialOrder"],["ArchimedeanClass","submodule"]]},{"isProp":true,"kind":"theorem","name":["ArchimedeanClass","ball_antitone"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Order.Module.Archimedean.306243331._hygCtx._hyg.3 : AddCommGroup.{u_1} M] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.306243331._hygCtx._hyg.6 : LinearOrder.{u_1} M] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.306243331._hygCtx._hyg.9 : IsOrderedAddMonoid.{u_1} M (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.306243331._hygCtx._hyg.3) (PartialOrder.toPreorder.{u_1} M (SemilatticeInf.toPartialOrder.{u_1} M (Lattice.toSemilatticeInf.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.306243331._hygCtx._hyg.6)))))] (K : Type.{u_2}) [inst._@.Mathlib.Algebra.Order.Module.Archimedean.306243331._hygCtx._hyg.13 : Ring.{u_2} K] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.306243331._hygCtx._hyg.16 : LinearOrder.{u_2} K] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.306243331._hygCtx._hyg.19 : IsOrderedRing.{u_2} K (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.306243331._hygCtx._hyg.13) (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.306243331._hygCtx._hyg.16))))] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.306243331._hygCtx._hyg.22 : Archimedean.{u_2} K (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonAssocRing.toNonUnitalNonAssocRing.{u_2} K (Ring.toNonAssocRing.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.306243331._hygCtx._hyg.13)))) (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.306243331._hygCtx._hyg.16))))] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.306243331._hygCtx._hyg.25 : Module.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.306243331._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.306243331._hygCtx._hyg.3)] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.306243331._hygCtx._hyg.29 : PosSMulMono.{u_2, u_1} K M (SMulZeroClass.toSMul.{u_2, u_1} K M (AddZero.toZero.{u_1} M (AddZeroClass.toAddZero.{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.Order.Module.Archimedean.306243331._hygCtx._hyg.3)))))) (DistribSMul.toSMulZeroClass.{u_2, u_1} K M (AddMonoid.toAddZeroClass.{u_1} M (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.306243331._hygCtx._hyg.3)))) (DistribMulAction.toDistribSMul.{u_2, u_1} K M (MonoidWithZero.toMonoid.{u_2} K (Semiring.toMonoidWithZero.{u_2} K (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.306243331._hygCtx._hyg.13))) (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.306243331._hygCtx._hyg.3))) (Module.toDistribMulAction.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.306243331._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.306243331._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.306243331._hygCtx._hyg.25)))) (PartialOrder.toPreorder.{u_2} K (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.306243331._hygCtx._hyg.16))))) (PartialOrder.toPreorder.{u_1} M (SemilatticeInf.toPartialOrder.{u_1} M (Lattice.toSemilatticeInf.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.306243331._hygCtx._hyg.6))))) (MulZeroClass.toZero.{u_2} K (NonUnitalNonAssocSemiring.toMulZeroClass.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonAssocRing.toNonUnitalNonAssocRing.{u_2} K (Ring.toNonAssocRing.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.306243331._hygCtx._hyg.13)))))], Antitone.{u_1, u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.306243331._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.306243331._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.306243331._hygCtx._hyg.9) (Submodule.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.306243331._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.306243331._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.306243331._hygCtx._hyg.25) (PartialOrder.toPreorder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.306243331._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.306243331._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.306243331._hygCtx._hyg.9) (SemilatticeInf.toPartialOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.306243331._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.306243331._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.306243331._hygCtx._hyg.9) (Lattice.toSemilatticeInf.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.306243331._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.306243331._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.306243331._hygCtx._hyg.9) (DistribLattice.toLattice.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.306243331._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.306243331._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.306243331._hygCtx._hyg.9) (instDistribLatticeOfLinearOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.306243331._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.306243331._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.306243331._hygCtx._hyg.9) (ArchimedeanClass.instLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.306243331._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.306243331._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.306243331._hygCtx._hyg.9)))))) (PartialOrder.toPreorder.{u_1} (Submodule.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.306243331._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.306243331._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.306243331._hygCtx._hyg.25) (Submodule.instPartialOrder.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.306243331._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.306243331._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.306243331._hygCtx._hyg.25)) (ArchimedeanClass.ball.{u_1, u_2} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.306243331._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.306243331._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.306243331._hygCtx._hyg.9 K inst._@.Mathlib.Algebra.Order.Module.Archimedean.306243331._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Order.Module.Archimedean.306243331._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Order.Module.Archimedean.306243331._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Order.Module.Archimedean.306243331._hygCtx._hyg.22 inst._@.Mathlib.Algebra.Order.Module.Archimedean.306243331._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Order.Module.Archimedean.306243331._hygCtx._hyg.29)","typeFull":"∀ {M : Type u_1} [inst : AddCommGroup M] [inst_1 : LinearOrder M] [inst_2 : IsOrderedAddMonoid M] (K : Type u_2)\n [inst_3 : Ring K] [inst_4 : LinearOrder K] [inst_5 : IsOrderedRing K] [inst_6 : Archimedean K] [inst_7 : Module K M]\n [inst_8 : PosSMulMono K M], Antitone (ArchimedeanClass.ball K)","typeReadable":"∀ {M : Type u_1} [inst : AddCommGroup M] [inst_1 : LinearOrder M] [inst_2 : IsOrderedAddMonoid M] (K : Type u_2)\n [inst_3 : Ring K] [inst_4 : LinearOrder K] [inst_5 : IsOrderedRing K] [inst_6 : Archimedean K] [inst_7 : Module K M]\n [inst_8 : PosSMulMono K M], Antitone (ArchimedeanClass.ball K)","typeReferences":[["PartialOrder","toPreorder"],["Ring","toNonAssocRing"],["ArchimedeanClass"],["Module"],["AddCommGroup","toAddGroup"],["SMulZeroClass","toSMul"],["Submodule"],["PosSMulMono"],["instDistribLatticeOfLinearOrder"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["IsOrderedAddMonoid"],["ArchimedeanClass","instLinearOrder"],["MonoidWithZero","toMonoid"],["AddGroup","toSubNegMonoid"],["NonAssocRing","toNonUnitalNonAssocRing"],["DistribSMul","toSMulZeroClass"],["SemilatticeInf","toPartialOrder"],["Lattice","toSemilatticeInf"],["Submodule","instPartialOrder"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["DistribMulAction","toDistribSMul"],["LinearOrder"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["AddZeroClass","toAddZero"],["Ring","toSemiring"],["IsOrderedRing"],["Module","toDistribMulAction"],["DistribLattice","toLattice"],["SubNegMonoid","toAddMonoid"],["MulZeroClass","toZero"],["ArchimedeanClass","ball"],["Antitone"],["Archimedean"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["AddCommGroup","toAddCommMonoid"],["AddZero","toZero"],["Ring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["PartialOrder","toPreorder"],["AddCommGroup","toAddGroup"],["Submodule","instPartialOrder"],["Submodule","toAddSubgroup_le"],["Iff","mp"],["AddSubgroup","instPartialOrder"],["Submodule","toAddSubgroup"],["Submodule"],["Ring","toSemiring"],["AddSubgroup"],["ArchimedeanClass","ballAddSubgroup_antitone"],["ArchimedeanClass","ball"],["LE","le"],["AddCommGroup","toAddCommMonoid"],["Preorder","toLE"]]},{"isProp":false,"kind":"definition","name":["ArchimedeanClass","submodule"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.3 : AddCommGroup.{u_1} M] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.6 : LinearOrder.{u_1} M] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.9 : IsOrderedAddMonoid.{u_1} M (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.3) (PartialOrder.toPreorder.{u_1} M (SemilatticeInf.toPartialOrder.{u_1} M (Lattice.toSemilatticeInf.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.6)))))] (K : Type.{u_2}) [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.13 : Ring.{u_2} K] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.16 : LinearOrder.{u_2} K] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.19 : IsOrderedRing.{u_2} K (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.13) (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.16))))] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.22 : Archimedean.{u_2} K (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonAssocRing.toNonUnitalNonAssocRing.{u_2} K (Ring.toNonAssocRing.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.13)))) (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.16))))] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.25 : Module.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.3)] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.29 : PosSMulMono.{u_2, u_1} K M (SMulZeroClass.toSMul.{u_2, u_1} K M (AddZero.toZero.{u_1} M (AddZeroClass.toAddZero.{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.Order.Module.Archimedean.3843450101._hygCtx._hyg.3)))))) (DistribSMul.toSMulZeroClass.{u_2, u_1} K M (AddMonoid.toAddZeroClass.{u_1} M (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.3)))) (DistribMulAction.toDistribSMul.{u_2, u_1} K M (MonoidWithZero.toMonoid.{u_2} K (Semiring.toMonoidWithZero.{u_2} K (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.13))) (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.3))) (Module.toDistribMulAction.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.25)))) (PartialOrder.toPreorder.{u_2} K (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.16))))) (PartialOrder.toPreorder.{u_1} M (SemilatticeInf.toPartialOrder.{u_1} M (Lattice.toSemilatticeInf.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.6))))) (MulZeroClass.toZero.{u_2} K (NonUnitalNonAssocSemiring.toMulZeroClass.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonAssocRing.toNonUnitalNonAssocRing.{u_2} K (Ring.toNonAssocRing.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.13)))))], (UpperSet.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.9) (Preorder.toLE.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.9) (PartialOrder.toPreorder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.9) (SemilatticeInf.toPartialOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.9) (Lattice.toSemilatticeInf.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.9) (DistribLattice.toLattice.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.9) (instDistribLatticeOfLinearOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.9) (ArchimedeanClass.instLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.9)))))))) -> (Submodule.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.25)","typeFull":"{M : Type u_1} →\n [inst : AddCommGroup M] →\n [inst_1 : LinearOrder M] →\n [inst_2 : IsOrderedAddMonoid M] →\n (K : Type u_2) →\n [inst_3 : Ring K] →\n [inst_4 : LinearOrder K] →\n [IsOrderedRing K] →\n [Archimedean K] →\n [inst_7 : Module K M] → [PosSMulMono K M] → UpperSet (ArchimedeanClass M) → Submodule K M","typeReadable":"{M : Type u_1} →\n [inst : AddCommGroup M] →\n [inst_1 : LinearOrder M] →\n [inst_2 : IsOrderedAddMonoid M] →\n (K : Type u_2) →\n [inst_3 : Ring K] →\n [inst_4 : LinearOrder K] →\n [IsOrderedRing K] →\n [Archimedean K] →\n [inst_7 : Module K M] → [PosSMulMono K M] → UpperSet (ArchimedeanClass M) → Submodule K M","typeReferences":[["PartialOrder","toPreorder"],["Ring","toNonAssocRing"],["ArchimedeanClass"],["Module"],["AddCommGroup","toAddGroup"],["SMulZeroClass","toSMul"],["Submodule"],["PosSMulMono"],["instDistribLatticeOfLinearOrder"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["IsOrderedAddMonoid"],["ArchimedeanClass","instLinearOrder"],["MonoidWithZero","toMonoid"],["AddGroup","toSubNegMonoid"],["Preorder","toLE"],["NonAssocRing","toNonUnitalNonAssocRing"],["DistribSMul","toSMulZeroClass"],["SemilatticeInf","toPartialOrder"],["Lattice","toSemilatticeInf"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["DistribMulAction","toDistribSMul"],["LinearOrder"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["AddZeroClass","toAddZero"],["UpperSet"],["Ring","toSemiring"],["IsOrderedRing"],["Module","toDistribMulAction"],["DistribLattice","toLattice"],["SubNegMonoid","toAddMonoid"],["MulZeroClass","toZero"],["Archimedean"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["AddCommGroup","toAddCommMonoid"],["AddZero","toZero"],["Ring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["AddSubgroup","toAddSubmonoid"],["Submodule","mk"],["AddCommGroup","toAddGroup"],["AddCommGroup","toAddCommMonoid"],["ArchimedeanClass","submodule","_proof_2"],["ArchimedeanClass","addSubgroup"],["Ring","toSemiring"]]},{"isProp":true,"kind":"theorem","name":["ArchimedeanClass","toAddSubgroup_ball"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917684._hygCtx._hyg.3 : AddCommGroup.{u_1} M] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917684._hygCtx._hyg.6 : LinearOrder.{u_1} M] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917684._hygCtx._hyg.9 : IsOrderedAddMonoid.{u_1} M (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917684._hygCtx._hyg.3) (PartialOrder.toPreorder.{u_1} M (SemilatticeInf.toPartialOrder.{u_1} M (Lattice.toSemilatticeInf.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917684._hygCtx._hyg.6)))))] (K : Type.{u_2}) [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917684._hygCtx._hyg.13 : Ring.{u_2} K] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917684._hygCtx._hyg.16 : LinearOrder.{u_2} K] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917684._hygCtx._hyg.19 : IsOrderedRing.{u_2} K (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917684._hygCtx._hyg.13) (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917684._hygCtx._hyg.16))))] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917684._hygCtx._hyg.22 : Archimedean.{u_2} K (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonAssocRing.toNonUnitalNonAssocRing.{u_2} K (Ring.toNonAssocRing.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917684._hygCtx._hyg.13)))) (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917684._hygCtx._hyg.16))))] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917684._hygCtx._hyg.25 : Module.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917684._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917684._hygCtx._hyg.3)] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917684._hygCtx._hyg.29 : PosSMulMono.{u_2, u_1} K M (SMulZeroClass.toSMul.{u_2, u_1} K M (AddZero.toZero.{u_1} M (AddZeroClass.toAddZero.{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.Order.Module.Archimedean.3336917684._hygCtx._hyg.3)))))) (DistribSMul.toSMulZeroClass.{u_2, u_1} K M (AddMonoid.toAddZeroClass.{u_1} M (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917684._hygCtx._hyg.3)))) (DistribMulAction.toDistribSMul.{u_2, u_1} K M (MonoidWithZero.toMonoid.{u_2} K (Semiring.toMonoidWithZero.{u_2} K (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917684._hygCtx._hyg.13))) (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917684._hygCtx._hyg.3))) (Module.toDistribMulAction.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917684._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917684._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917684._hygCtx._hyg.25)))) (PartialOrder.toPreorder.{u_2} K (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917684._hygCtx._hyg.16))))) (PartialOrder.toPreorder.{u_1} M (SemilatticeInf.toPartialOrder.{u_1} M (Lattice.toSemilatticeInf.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917684._hygCtx._hyg.6))))) (MulZeroClass.toZero.{u_2} K (NonUnitalNonAssocSemiring.toMulZeroClass.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonAssocRing.toNonUnitalNonAssocRing.{u_2} K (Ring.toNonAssocRing.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917684._hygCtx._hyg.13)))))] (c : ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917684._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917684._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917684._hygCtx._hyg.9), Eq.{succ u_1} (AddSubgroup.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917684._hygCtx._hyg.3)) (Submodule.toAddSubgroup.{u_2, u_1} K M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917684._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917684._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917684._hygCtx._hyg.25 (ArchimedeanClass.ball.{u_1, u_2} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917684._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917684._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917684._hygCtx._hyg.9 K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917684._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917684._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917684._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917684._hygCtx._hyg.22 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917684._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917684._hygCtx._hyg.29 c)) (ArchimedeanClass.ballAddSubgroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917684._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917684._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917684._hygCtx._hyg.9 c)","typeFull":"∀ {M : Type u_1} [inst : AddCommGroup M] [inst_1 : LinearOrder M] [inst_2 : IsOrderedAddMonoid M] (K : Type u_2)\n [inst_3 : Ring K] [inst_4 : LinearOrder K] [inst_5 : IsOrderedRing K] [inst_6 : Archimedean K] [inst_7 : Module K M]\n [inst_8 : PosSMulMono K M] (c : ArchimedeanClass M), (ArchimedeanClass.ball K c).toAddSubgroup = c.ballAddSubgroup","typeReadable":"∀ {M : Type u_1} [inst : AddCommGroup M] [inst_1 : LinearOrder M] [inst_2 : IsOrderedAddMonoid M] (K : Type u_2)\n [inst_3 : Ring K] [inst_4 : LinearOrder K] [inst_5 : IsOrderedRing K] [inst_6 : Archimedean K] [inst_7 : Module K M]\n [inst_8 : PosSMulMono K M] (c : ArchimedeanClass M), (ArchimedeanClass.ball K c).toAddSubgroup = c.ballAddSubgroup","typeReferences":[["PartialOrder","toPreorder"],["Ring","toNonAssocRing"],["ArchimedeanClass"],["Module"],["AddCommGroup","toAddGroup"],["SMulZeroClass","toSMul"],["PosSMulMono"],["instDistribLatticeOfLinearOrder"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["ArchimedeanClass","ballAddSubgroup"],["IsOrderedAddMonoid"],["MonoidWithZero","toMonoid"],["AddGroup","toSubNegMonoid"],["Eq"],["NonAssocRing","toNonUnitalNonAssocRing"],["DistribSMul","toSMulZeroClass"],["SemilatticeInf","toPartialOrder"],["Lattice","toSemilatticeInf"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["DistribMulAction","toDistribSMul"],["LinearOrder"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["Submodule","toAddSubgroup"],["AddZeroClass","toAddZero"],["Ring","toSemiring"],["AddSubgroup"],["IsOrderedRing"],["Module","toDistribMulAction"],["DistribLattice","toLattice"],["SubNegMonoid","toAddMonoid"],["MulZeroClass","toZero"],["ArchimedeanClass","ball"],["Archimedean"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["AddCommGroup","toAddCommMonoid"],["AddZero","toZero"],["Ring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["rfl"],["AddSubgroup"],["ArchimedeanClass","ball"],["AddCommGroup","toAddGroup"],["Submodule","toAddSubgroup"]]},{"isProp":true,"kind":"theorem","name":["FiniteArchimedeanClass","mem_closedBall_iff"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.3 : AddCommGroup.{u_1} M] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.6 : LinearOrder.{u_1} M] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.9 : IsOrderedAddMonoid.{u_1} M (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.3) (PartialOrder.toPreorder.{u_1} M (SemilatticeInf.toPartialOrder.{u_1} M (Lattice.toSemilatticeInf.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.6)))))] (K : Type.{u_2}) [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.13 : Ring.{u_2} K] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.16 : LinearOrder.{u_2} K] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.19 : IsOrderedRing.{u_2} K (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.13) (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.16))))] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.22 : Archimedean.{u_2} K (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonAssocRing.toNonUnitalNonAssocRing.{u_2} K (Ring.toNonAssocRing.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.13)))) (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.16))))] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.25 : Module.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.3)] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.29 : PosSMulMono.{u_2, u_1} K M (SMulZeroClass.toSMul.{u_2, u_1} K M (AddZero.toZero.{u_1} M (AddZeroClass.toAddZero.{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.Order.Module.Archimedean.2781882086._hygCtx._hyg.3)))))) (DistribSMul.toSMulZeroClass.{u_2, u_1} K M (AddMonoid.toAddZeroClass.{u_1} M (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.3)))) (DistribMulAction.toDistribSMul.{u_2, u_1} K M (MonoidWithZero.toMonoid.{u_2} K (Semiring.toMonoidWithZero.{u_2} K (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.13))) (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.3))) (Module.toDistribMulAction.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.25)))) (PartialOrder.toPreorder.{u_2} K (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.16))))) (PartialOrder.toPreorder.{u_1} M (SemilatticeInf.toPartialOrder.{u_1} M (Lattice.toSemilatticeInf.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.6))))) (MulZeroClass.toZero.{u_2} K (NonUnitalNonAssocSemiring.toMulZeroClass.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonAssocRing.toNonUnitalNonAssocRing.{u_2} K (Ring.toNonAssocRing.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.13)))))] {a : M} {c : FiniteArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.9}, Iff (Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.25) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.25) M (Submodule.setLike.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.25)) (FiniteArchimedeanClass.closedBall.{u_1, u_2} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.9 K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.22 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.29 c) a) (forall (h : Ne.{succ u_1} M a (OfNat.ofNat.{u_1} M 0 (Zero.toOfNat0.{u_1} M (NegZeroClass.toZero.{u_1} M (SubNegZeroMonoid.toNegZeroClass.{u_1} M (SubtractionMonoid.toSubNegZeroMonoid.{u_1} M (SubtractionCommMonoid.toSubtractionMonoid.{u_1} M (AddCommGroup.toDivisionAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.3)))))))), LE.le.{u_1} (FiniteArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.9) (Subtype.instLE.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.9) (Preorder.toLE.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.9) (PartialOrder.toPreorder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.9) (SemilatticeInf.toPartialOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.9) (Lattice.toSemilatticeInf.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.9) (DistribLattice.toLattice.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.9) (instDistribLatticeOfLinearOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.9) (ArchimedeanClass.instLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.9))))))) (fun (A : ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.9) => Ne.{succ u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.9) A (Top.top.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.9) (OrderTop.toTop.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.9) (Preorder.toLE.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.9) (PartialOrder.toPreorder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.9) (SemilatticeInf.toPartialOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.9) (Lattice.toSemilatticeInf.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.9) (DistribLattice.toLattice.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.9) (instDistribLatticeOfLinearOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.9) (ArchimedeanClass.instLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.9))))))) (ArchimedeanClass.instOrderTop.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.9))))) c (FiniteArchimedeanClass.mk.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.9 a h))","typeFull":"∀ {M : Type u_1} [inst : AddCommGroup M] [inst_1 : LinearOrder M] [inst_2 : IsOrderedAddMonoid M] (K : Type u_2)\n [inst_3 : Ring K] [inst_4 : LinearOrder K] [inst_5 : IsOrderedRing K] [inst_6 : Archimedean K] [inst_7 : Module K M]\n [inst_8 : PosSMulMono K M] {a : M} {c : FiniteArchimedeanClass M},\n a ∈ FiniteArchimedeanClass.closedBall K c ↔ ∀ (h : a ≠ 0), c ≤ FiniteArchimedeanClass.mk a h","typeReadable":"∀ {M : Type u_1} [inst : AddCommGroup M] [inst_1 : LinearOrder M] [inst_2 : IsOrderedAddMonoid M] (K : Type u_2)\n [inst_3 : Ring K] [inst_4 : LinearOrder K] [inst_5 : IsOrderedRing K] [inst_6 : Archimedean K] [inst_7 : Module K M]\n [inst_8 : PosSMulMono K M] {a : M} {c : FiniteArchimedeanClass M},\n a ∈ FiniteArchimedeanClass.closedBall K c ↔ ∀ (h : a ≠ 0), c ≤ FiniteArchimedeanClass.mk a h","typeReferences":[["SubtractionMonoid","toSubNegZeroMonoid"],["Ring","toNonAssocRing"],["PartialOrder","toPreorder"],["Subtype","instLE"],["AddCommGroup","toAddGroup"],["Membership","mem"],["SMulZeroClass","toSMul"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["IsOrderedAddMonoid"],["ArchimedeanClass","instLinearOrder"],["FiniteArchimedeanClass"],["FiniteArchimedeanClass","mk"],["AddGroup","toSubNegMonoid"],["NonAssocRing","toNonUnitalNonAssocRing"],["SemilatticeInf","toPartialOrder"],["DistribSMul","toSMulZeroClass"],["SetLike","instMembership"],["LinearOrder"],["DistribMulAction","toDistribSMul"],["AddZeroClass","toAddZero"],["ArchimedeanClass","instOrderTop"],["Ring","toSemiring"],["Iff"],["Top","top"],["AddCommGroup","toAddCommMonoid"],["NegZeroClass","toZero"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["FiniteArchimedeanClass","closedBall"],["ArchimedeanClass"],["Module"],["SubtractionCommMonoid","toSubtractionMonoid"],["SubNegZeroMonoid","toNegZeroClass"],["PosSMulMono"],["Submodule"],["instDistribLatticeOfLinearOrder"],["MonoidWithZero","toMonoid"],["Zero","toOfNat0"],["Preorder","toLE"],["Lattice","toSemilatticeInf"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["OfNat","ofNat"],["IsOrderedRing"],["Module","toDistribMulAction"],["DistribLattice","toLattice"],["SubNegMonoid","toAddMonoid"],["AddCommGroup","toDivisionAddCommMonoid"],["MulZeroClass","toZero"],["Submodule","setLike"],["Archimedean"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["LE","le"],["Ne"],["OrderTop","toTop"],["Ring"]],"valueReferences":[["FiniteArchimedeanClass","mem_closedBallAddSubgroup_iff"]]},{"isProp":true,"kind":"theorem","name":["FiniteArchimedeanClass","toAddSubgroup_ball"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917685._hygCtx._hyg.3 : AddCommGroup.{u_1} M] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917685._hygCtx._hyg.6 : LinearOrder.{u_1} M] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917685._hygCtx._hyg.9 : IsOrderedAddMonoid.{u_1} M (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917685._hygCtx._hyg.3) (PartialOrder.toPreorder.{u_1} M (SemilatticeInf.toPartialOrder.{u_1} M (Lattice.toSemilatticeInf.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917685._hygCtx._hyg.6)))))] (K : Type.{u_2}) [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917685._hygCtx._hyg.13 : Ring.{u_2} K] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917685._hygCtx._hyg.16 : LinearOrder.{u_2} K] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917685._hygCtx._hyg.19 : IsOrderedRing.{u_2} K (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917685._hygCtx._hyg.13) (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917685._hygCtx._hyg.16))))] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917685._hygCtx._hyg.22 : Archimedean.{u_2} K (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonAssocRing.toNonUnitalNonAssocRing.{u_2} K (Ring.toNonAssocRing.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917685._hygCtx._hyg.13)))) (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917685._hygCtx._hyg.16))))] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917685._hygCtx._hyg.25 : Module.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917685._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917685._hygCtx._hyg.3)] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917685._hygCtx._hyg.29 : PosSMulMono.{u_2, u_1} K M (SMulZeroClass.toSMul.{u_2, u_1} K M (AddZero.toZero.{u_1} M (AddZeroClass.toAddZero.{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.Order.Module.Archimedean.3336917685._hygCtx._hyg.3)))))) (DistribSMul.toSMulZeroClass.{u_2, u_1} K M (AddMonoid.toAddZeroClass.{u_1} M (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917685._hygCtx._hyg.3)))) (DistribMulAction.toDistribSMul.{u_2, u_1} K M (MonoidWithZero.toMonoid.{u_2} K (Semiring.toMonoidWithZero.{u_2} K (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917685._hygCtx._hyg.13))) (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917685._hygCtx._hyg.3))) (Module.toDistribMulAction.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917685._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917685._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917685._hygCtx._hyg.25)))) (PartialOrder.toPreorder.{u_2} K (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917685._hygCtx._hyg.16))))) (PartialOrder.toPreorder.{u_1} M (SemilatticeInf.toPartialOrder.{u_1} M (Lattice.toSemilatticeInf.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917685._hygCtx._hyg.6))))) (MulZeroClass.toZero.{u_2} K (NonUnitalNonAssocSemiring.toMulZeroClass.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonAssocRing.toNonUnitalNonAssocRing.{u_2} K (Ring.toNonAssocRing.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917685._hygCtx._hyg.13)))))] (c : FiniteArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917685._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917685._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917685._hygCtx._hyg.9), Eq.{succ u_1} (AddSubgroup.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917685._hygCtx._hyg.3)) (Submodule.toAddSubgroup.{u_2, u_1} K M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917685._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917685._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917685._hygCtx._hyg.25 (FiniteArchimedeanClass.ball.{u_1, u_2} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917685._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917685._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917685._hygCtx._hyg.9 K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917685._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917685._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917685._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917685._hygCtx._hyg.22 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917685._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917685._hygCtx._hyg.29 c)) (FiniteArchimedeanClass.ballAddSubgroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917685._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917685._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3336917685._hygCtx._hyg.9 c)","typeFull":"∀ {M : Type u_1} [inst : AddCommGroup M] [inst_1 : LinearOrder M] [inst_2 : IsOrderedAddMonoid M] (K : Type u_2)\n [inst_3 : Ring K] [inst_4 : LinearOrder K] [inst_5 : IsOrderedRing K] [inst_6 : Archimedean K] [inst_7 : Module K M]\n [inst_8 : PosSMulMono K M] (c : FiniteArchimedeanClass M),\n (FiniteArchimedeanClass.ball K c).toAddSubgroup = c.ballAddSubgroup","typeReadable":"∀ {M : Type u_1} [inst : AddCommGroup M] [inst_1 : LinearOrder M] [inst_2 : IsOrderedAddMonoid M] (K : Type u_2)\n [inst_3 : Ring K] [inst_4 : LinearOrder K] [inst_5 : IsOrderedRing K] [inst_6 : Archimedean K] [inst_7 : Module K M]\n [inst_8 : PosSMulMono K M] (c : FiniteArchimedeanClass M),\n (FiniteArchimedeanClass.ball K c).toAddSubgroup = c.ballAddSubgroup","typeReferences":[["PartialOrder","toPreorder"],["Ring","toNonAssocRing"],["Module"],["AddCommGroup","toAddGroup"],["SMulZeroClass","toSMul"],["PosSMulMono"],["instDistribLatticeOfLinearOrder"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["IsOrderedAddMonoid"],["MonoidWithZero","toMonoid"],["FiniteArchimedeanClass"],["AddGroup","toSubNegMonoid"],["Eq"],["FiniteArchimedeanClass","ballAddSubgroup"],["NonAssocRing","toNonUnitalNonAssocRing"],["DistribSMul","toSMulZeroClass"],["SemilatticeInf","toPartialOrder"],["Lattice","toSemilatticeInf"],["FiniteArchimedeanClass","ball"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["DistribMulAction","toDistribSMul"],["LinearOrder"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["Submodule","toAddSubgroup"],["AddZeroClass","toAddZero"],["Ring","toSemiring"],["AddSubgroup"],["IsOrderedRing"],["Module","toDistribMulAction"],["DistribLattice","toLattice"],["SubNegMonoid","toAddMonoid"],["MulZeroClass","toZero"],["Archimedean"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["AddCommGroup","toAddCommMonoid"],["AddZero","toZero"],["Ring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["rfl"],["AddSubgroup"],["AddCommGroup","toAddGroup"],["FiniteArchimedeanClass","ball"],["Submodule","toAddSubgroup"]]},{"isProp":true,"kind":"theorem","name":["ArchimedeanClass","mem_ball_iff","_simp_1"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.3 : AddCommGroup.{u_1} M] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.6 : LinearOrder.{u_1} M] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.9 : IsOrderedAddMonoid.{u_1} M (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.3) (PartialOrder.toPreorder.{u_1} M (SemilatticeInf.toPartialOrder.{u_1} M (Lattice.toSemilatticeInf.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.6)))))] (K : Type.{u_2}) [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.13 : Ring.{u_2} K] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.16 : LinearOrder.{u_2} K] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.19 : IsOrderedRing.{u_2} K (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.13) (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.16))))] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.22 : Archimedean.{u_2} K (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonAssocRing.toNonUnitalNonAssocRing.{u_2} K (Ring.toNonAssocRing.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.13)))) (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.16))))] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.25 : Module.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.3)] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.29 : PosSMulMono.{u_2, u_1} K M (SMulZeroClass.toSMul.{u_2, u_1} K M (AddZero.toZero.{u_1} M (AddZeroClass.toAddZero.{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.Order.Module.Archimedean.2496125325._hygCtx._hyg.3)))))) (DistribSMul.toSMulZeroClass.{u_2, u_1} K M (AddMonoid.toAddZeroClass.{u_1} M (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.3)))) (DistribMulAction.toDistribSMul.{u_2, u_1} K M (MonoidWithZero.toMonoid.{u_2} K (Semiring.toMonoidWithZero.{u_2} K (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.13))) (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.3))) (Module.toDistribMulAction.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.25)))) (PartialOrder.toPreorder.{u_2} K (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.16))))) (PartialOrder.toPreorder.{u_1} M (SemilatticeInf.toPartialOrder.{u_1} M (Lattice.toSemilatticeInf.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.6))))) (MulZeroClass.toZero.{u_2} K (NonUnitalNonAssocSemiring.toMulZeroClass.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonAssocRing.toNonUnitalNonAssocRing.{u_2} K (Ring.toNonAssocRing.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.13)))))] {a : M} {c : ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.9}, (Ne.{succ u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.9) c (Top.top.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.9) (OrderTop.toTop.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.9) (Preorder.toLE.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.9) (PartialOrder.toPreorder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.9) (SemilatticeInf.toPartialOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.9) (Lattice.toSemilatticeInf.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.9) (DistribLattice.toLattice.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.9) (instDistribLatticeOfLinearOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.9) (ArchimedeanClass.instLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.9))))))) (ArchimedeanClass.instOrderTop.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.9)))) -> (Eq.{1} Prop (Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.25) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.25) M (Submodule.setLike.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.25)) (ArchimedeanClass.ball.{u_1, u_2} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.9 K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.22 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.29 c) a) (LT.lt.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.9) (Preorder.toLT.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.9) (PartialOrder.toPreorder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.9) (SemilatticeInf.toPartialOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.9) (Lattice.toSemilatticeInf.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.9) (DistribLattice.toLattice.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.9) (instDistribLatticeOfLinearOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.9) (ArchimedeanClass.instLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.9))))))) c (ArchimedeanClass.mk.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.9 a)))","typeFull":"∀ {M : Type u_1} [inst : AddCommGroup M] [inst_1 : LinearOrder M] [inst_2 : IsOrderedAddMonoid M] (K : Type u_2)\n [inst_3 : Ring K] [inst_4 : LinearOrder K] [inst_5 : IsOrderedRing K] [inst_6 : Archimedean K] [inst_7 : Module K M]\n [inst_8 : PosSMulMono K M] {a : M} {c : ArchimedeanClass M},\n c ≠ ⊤ → (a ∈ ArchimedeanClass.ball K c) = (c < ArchimedeanClass.mk a)","typeReadable":"∀ {M : Type u_1} [inst : AddCommGroup M] [inst_1 : LinearOrder M] [inst_2 : IsOrderedAddMonoid M] (K : Type u_2)\n [inst_3 : Ring K] [inst_4 : LinearOrder K] [inst_5 : IsOrderedRing K] [inst_6 : Archimedean K] [inst_7 : Module K M]\n [inst_8 : PosSMulMono K M] {a : M} {c : ArchimedeanClass M},\n c ≠ ⊤ → (a ∈ ArchimedeanClass.ball K c) = (c < ArchimedeanClass.mk a)","typeReferences":[["Ring","toNonAssocRing"],["PartialOrder","toPreorder"],["AddCommGroup","toAddGroup"],["Membership","mem"],["Preorder","toLT"],["SMulZeroClass","toSMul"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["IsOrderedAddMonoid"],["ArchimedeanClass","instLinearOrder"],["AddGroup","toSubNegMonoid"],["NonAssocRing","toNonUnitalNonAssocRing"],["SemilatticeInf","toPartialOrder"],["DistribSMul","toSMulZeroClass"],["SetLike","instMembership"],["LinearOrder"],["DistribMulAction","toDistribSMul"],["AddZeroClass","toAddZero"],["Ring","toSemiring"],["ArchimedeanClass","instOrderTop"],["ArchimedeanClass","ball"],["AddCommGroup","toAddCommMonoid"],["Top","top"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Module"],["ArchimedeanClass"],["PosSMulMono"],["Submodule"],["instDistribLatticeOfLinearOrder"],["MonoidWithZero","toMonoid"],["Preorder","toLE"],["Eq"],["Lattice","toSemilatticeInf"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["ArchimedeanClass","mk"],["LT","lt"],["IsOrderedRing"],["Module","toDistribMulAction"],["DistribLattice","toLattice"],["SubNegMonoid","toAddMonoid"],["MulZeroClass","toZero"],["Submodule","setLike"],["Archimedean"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["Ne"],["OrderTop","toTop"],["Ring"]],"valueReferences":[["Lattice","toSemilatticeInf"],["SetLike","instMembership"],["PartialOrder","toPreorder"],["ArchimedeanClass"],["ArchimedeanClass","mem_ball_iff"],["Membership","mem"],["Preorder","toLT"],["ArchimedeanClass","mk"],["Submodule"],["Ring","toSemiring"],["LT","lt"],["instDistribLatticeOfLinearOrder"],["DistribLattice","toLattice"],["Submodule","setLike"],["ArchimedeanClass","ball"],["ArchimedeanClass","instLinearOrder"],["AddCommGroup","toAddCommMonoid"],["propext"],["SemilatticeInf","toPartialOrder"]]},{"isProp":true,"kind":"theorem","name":["ArchimedeanClass","congr_simp"],"typeFallback":"forall (M : Type.{u_1}) [inst._@.Mathlib.Algebra.Order.Archimedean.Class.290876463._hygCtx._hyg.3 : AddCommGroup.{u_1} M] [inst._@.Mathlib.Algebra.Order.Archimedean.Class.290876463._hygCtx._hyg.6 : LinearOrder.{u_1} M] [inst._@.Mathlib.Algebra.Order.Archimedean.Class.290876463._hygCtx._hyg.9 : IsOrderedAddMonoid.{u_1} M (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Archimedean.Class.290876463._hygCtx._hyg.3) (PartialOrder.toPreorder.{u_1} M (SemilatticeInf.toPartialOrder.{u_1} M (Lattice.toSemilatticeInf.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Archimedean.Class.290876463._hygCtx._hyg.6)))))], Eq.{succ (succ u_1)} Type.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Archimedean.Class.290876463._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Archimedean.Class.290876463._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Archimedean.Class.290876463._hygCtx._hyg.9) (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Archimedean.Class.290876463._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Archimedean.Class.290876463._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Archimedean.Class.290876463._hygCtx._hyg.9)","typeFull":"∀ (M : Type u_1) [inst : AddCommGroup M] [inst_1 : LinearOrder M] [inst_2 : IsOrderedAddMonoid M],\n ArchimedeanClass M = ArchimedeanClass M","typeReadable":"∀ (M : Type u_1) [inst : AddCommGroup M] [inst_1 : LinearOrder M] [inst_2 : IsOrderedAddMonoid M],\n ArchimedeanClass M = ArchimedeanClass M","typeReferences":[["instDistribLatticeOfLinearOrder"],["DistribLattice","toLattice"],["Lattice","toSemilatticeInf"],["PartialOrder","toPreorder"],["ArchimedeanClass"],["IsOrderedAddMonoid"],["AddCommGroup","toAddCommMonoid"],["LinearOrder"],["AddCommGroup"],["Eq"],["SemilatticeInf","toPartialOrder"]],"valueReferences":[["ArchimedeanClass"],["Eq","refl"]]},{"isProp":true,"kind":"theorem","name":["FiniteArchimedeanClass","submodule","_proof_2"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 : AddCommGroup.{u_1} M] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 : LinearOrder.{u_1} M] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9 : IsOrderedAddMonoid.{u_1} M (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3) (PartialOrder.toPreorder.{u_1} M (SemilatticeInf.toPartialOrder.{u_1} M (Lattice.toSemilatticeInf.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6)))))] (K : Type.{u_2}) [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.13 : Ring.{u_2} K] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.16 : LinearOrder.{u_2} K] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.19 : IsOrderedRing.{u_2} K (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.13) (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.16))))] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.22 : Archimedean.{u_2} K (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonAssocRing.toNonUnitalNonAssocRing.{u_2} K (Ring.toNonAssocRing.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.13)))) (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.16))))] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.25 : Module.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3)] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.29 : PosSMulMono.{u_2, u_1} K M (SMulZeroClass.toSMul.{u_2, u_1} K M (AddZero.toZero.{u_1} M (AddZeroClass.toAddZero.{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.Order.Module.Archimedean.3843450102._hygCtx._hyg.3)))))) (DistribSMul.toSMulZeroClass.{u_2, u_1} K M (AddMonoid.toAddZeroClass.{u_1} M (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3)))) (DistribMulAction.toDistribSMul.{u_2, u_1} K M (MonoidWithZero.toMonoid.{u_2} K (Semiring.toMonoidWithZero.{u_2} K (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.13))) (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3))) (Module.toDistribMulAction.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.25)))) (PartialOrder.toPreorder.{u_2} K (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.16))))) (PartialOrder.toPreorder.{u_1} M (SemilatticeInf.toPartialOrder.{u_1} M (Lattice.toSemilatticeInf.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6))))) (MulZeroClass.toZero.{u_2} K (NonUnitalNonAssocSemiring.toMulZeroClass.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonAssocRing.toNonUnitalNonAssocRing.{u_2} K (Ring.toNonAssocRing.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.13)))))] (s : UpperSet.{u_1} (FiniteArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (Subtype.instLE.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (Preorder.toLE.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (PartialOrder.toPreorder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (SemilatticeInf.toPartialOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (Lattice.toSemilatticeInf.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (DistribLattice.toLattice.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (instDistribLatticeOfLinearOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (ArchimedeanClass.instLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9))))))) (fun (A : ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) => Ne.{succ u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) A (Top.top.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (OrderTop.toTop.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (Preorder.toLE.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (PartialOrder.toPreorder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (SemilatticeInf.toPartialOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (Lattice.toSemilatticeInf.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (DistribLattice.toLattice.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (instDistribLatticeOfLinearOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (ArchimedeanClass.instLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9))))))) (ArchimedeanClass.instOrderTop.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9)))))) (k : K) {a : 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 (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3)))))) (AddSubmonoid.toAddSubsemigroup.{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.Order.Module.Archimedean.3843450102._hygCtx._hyg.3)))) (AddSubgroup.toAddSubmonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3) (FiniteArchimedeanClass.addSubgroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9 s)))) a) -> (forall (ne : Ne.{succ u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (ArchimedeanClass.mk.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9 (HSMul.hSMul.{u_2, u_1, u_1} K M M (instHSMul.{u_2, u_1} K M (SMulZeroClass.toSMul.{u_2, u_1} K 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.Order.Module.Archimedean.3843450102._hygCtx._hyg.3))))) (DistribSMul.toSMulZeroClass.{u_2, u_1} K M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3))) (DistribMulAction.toDistribSMul.{u_2, u_1} K M (MonoidWithZero.toMonoid.{u_2} K (Semiring.toMonoidWithZero.{u_2} K (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.13))) (AddCommMonoid.toAddMonoid.{u_1} M (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3)) (Module.toDistribMulAction.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.25))))) k a)) (Top.top.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (OrderTop.toTop.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (Preorder.toLE.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (PartialOrder.toPreorder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (SemilatticeInf.toPartialOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (Lattice.toSemilatticeInf.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (DistribLattice.toLattice.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (instDistribLatticeOfLinearOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (ArchimedeanClass.instLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9))))))) (ArchimedeanClass.instOrderTop.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9)))), Membership.mem.{u_1, u_1} (FiniteArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (Set.{u_1} (FiniteArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9)) (Set.instMembership.{u_1} (FiniteArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9)) (SetLike.coe.{u_1, u_1} (UpperSet.{u_1} (FiniteArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (Subtype.instLE.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (Preorder.toLE.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (PartialOrder.toPreorder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (SemilatticeInf.toPartialOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (Lattice.toSemilatticeInf.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (DistribLattice.toLattice.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (instDistribLatticeOfLinearOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (ArchimedeanClass.instLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9))))))) (fun (A : ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) => Ne.{succ u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) A (Top.top.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (OrderTop.toTop.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (Preorder.toLE.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (PartialOrder.toPreorder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (SemilatticeInf.toPartialOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (Lattice.toSemilatticeInf.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (DistribLattice.toLattice.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (instDistribLatticeOfLinearOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (ArchimedeanClass.instLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9))))))) (ArchimedeanClass.instOrderTop.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9)))))) (FiniteArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (UpperSet.instSetLike.{u_1} (FiniteArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (Subtype.instLE.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (Preorder.toLE.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (PartialOrder.toPreorder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (SemilatticeInf.toPartialOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (Lattice.toSemilatticeInf.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (DistribLattice.toLattice.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (instDistribLatticeOfLinearOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (ArchimedeanClass.instLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9))))))) (fun (A : ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) => Ne.{succ u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) A (Top.top.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (OrderTop.toTop.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (Preorder.toLE.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (PartialOrder.toPreorder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (SemilatticeInf.toPartialOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (Lattice.toSemilatticeInf.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (DistribLattice.toLattice.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (instDistribLatticeOfLinearOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (ArchimedeanClass.instLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9))))))) (ArchimedeanClass.instOrderTop.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9)))))) s) (Subtype.mk.{succ u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (fun (A : ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) => Ne.{succ u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) A (Top.top.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (OrderTop.toTop.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (Preorder.toLE.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (PartialOrder.toPreorder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (SemilatticeInf.toPartialOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (Lattice.toSemilatticeInf.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (DistribLattice.toLattice.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (instDistribLatticeOfLinearOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (ArchimedeanClass.instLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9))))))) (ArchimedeanClass.instOrderTop.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9)))) (ArchimedeanClass.mk.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9 (HSMul.hSMul.{u_2, u_1, u_1} K M M (instHSMul.{u_2, u_1} K M (SMulZeroClass.toSMul.{u_2, u_1} K 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.Order.Module.Archimedean.3843450102._hygCtx._hyg.3))))) (DistribSMul.toSMulZeroClass.{u_2, u_1} K M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3))) (DistribMulAction.toDistribSMul.{u_2, u_1} K M (MonoidWithZero.toMonoid.{u_2} K (Semiring.toMonoidWithZero.{u_2} K (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.13))) (AddCommMonoid.toAddMonoid.{u_1} M (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3)) (Module.toDistribMulAction.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.25))))) k a)) ne))","typeFull":"∀ {M : Type u_1} [inst : AddCommGroup M] [inst_1 : LinearOrder M] [inst_2 : IsOrderedAddMonoid M] (K : Type u_2)\n [inst_3 : Ring K] [inst_4 : LinearOrder K] [IsOrderedRing K] [Archimedean K] [inst_7 : Module K M] [PosSMulMono K M]\n (s : UpperSet (FiniteArchimedeanClass M)) (k : K) {a : M},\n a ∈ (FiniteArchimedeanClass.addSubgroup s).carrier →\n ∀ (ne : ArchimedeanClass.mk (k • a) ≠ ⊤), ⟨ArchimedeanClass.mk (k • a), ne⟩ ∈ ↑s","typeReadable":"∀ {M : Type u_1} [inst : AddCommGroup M] [inst_1 : LinearOrder M] [inst_2 : IsOrderedAddMonoid M] (K : Type u_2)\n [inst_3 : Ring K] [inst_4 : LinearOrder K] [IsOrderedRing K] [Archimedean K] [inst_7 : Module K M] [PosSMulMono K M]\n (s : UpperSet (FiniteArchimedeanClass M)) (k : K) {a : M},\n a ∈ (FiniteArchimedeanClass.addSubgroup s).carrier →\n ∀ (ne : ArchimedeanClass.mk (k • a) ≠ ⊤), ⟨ArchimedeanClass.mk (k • a), ne⟩ ∈ ↑s","typeReferences":[["Ring","toNonAssocRing"],["PartialOrder","toPreorder"],["Subtype","instLE"],["AddCommGroup","toAddGroup"],["Membership","mem"],["SMulZeroClass","toSMul"],["FiniteArchimedeanClass","addSubgroup"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["AddSubgroup","toAddSubmonoid"],["IsOrderedAddMonoid"],["ArchimedeanClass","instLinearOrder"],["FiniteArchimedeanClass"],["AddGroup","toSubNegMonoid"],["NonAssocRing","toNonUnitalNonAssocRing"],["SemilatticeInf","toPartialOrder"],["DistribSMul","toSMulZeroClass"],["LinearOrder"],["DistribMulAction","toDistribSMul"],["AddZeroClass","toAddZero"],["Ring","toSemiring"],["ArchimedeanClass","instOrderTop"],["Set","instMembership"],["SetLike","coe"],["HSMul","hSMul"],["AddCommGroup","toAddCommMonoid"],["Top","top"],["AddSubmonoid","toAddSubsemigroup"],["UpperSet","instSetLike"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Module"],["ArchimedeanClass"],["AddCommMonoid","toAddMonoid"],["PosSMulMono"],["AddSubsemigroup","carrier"],["instDistribLatticeOfLinearOrder"],["MonoidWithZero","toMonoid"],["instHSMul"],["Preorder","toLE"],["Lattice","toSemilatticeInf"],["Set"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["AddZero","toAdd"],["UpperSet"],["ArchimedeanClass","mk"],["IsOrderedRing"],["Module","toDistribMulAction"],["DistribLattice","toLattice"],["SubNegMonoid","toAddMonoid"],["MulZeroClass","toZero"],["Archimedean"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["Ne"],["Subtype","mk"],["OrderTop","toTop"],["Ring"]],"valueReferences":[["UpperSet","upper"],["PartialOrder","toPreorder"],["ArchimedeanClass"],["Subtype","instLE"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["instDistribLatticeOfLinearOrder"],["MonoidWithZero","toMonoid"],["ArchimedeanClass","instLinearOrder"],["FiniteArchimedeanClass"],["instHSMul"],["ArchimedeanClass","mk_le_mk_smul"],["Preorder","toLE"],["DistribSMul","toSMulZeroClass"],["SemilatticeInf","toPartialOrder"],["Lattice","toSemilatticeInf"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["FiniteArchimedeanClass","submodule","_proof_1"],["AddZeroClass","toAddZero"],["ArchimedeanClass","mk"],["Ring","toSemiring"],["ArchimedeanClass","instOrderTop"],["Module","toDistribMulAction"],["DistribLattice","toLattice"],["HSMul","hSMul"],["AddCommGroup","toAddCommMonoid"],["Top","top"],["Ne"],["Subtype","mk"],["OrderTop","toTop"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]]},{"isProp":false,"kind":"definition","name":["ArchimedeanClass","closedBall"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.3 : AddCommGroup.{u_1} M] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.6 : LinearOrder.{u_1} M] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.9 : IsOrderedAddMonoid.{u_1} M (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.3) (PartialOrder.toPreorder.{u_1} M (SemilatticeInf.toPartialOrder.{u_1} M (Lattice.toSemilatticeInf.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.6)))))] (K : Type.{u_2}) [inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.13 : Ring.{u_2} K] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.16 : LinearOrder.{u_2} K] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.19 : IsOrderedRing.{u_2} K (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.13) (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.16))))] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.22 : Archimedean.{u_2} K (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonAssocRing.toNonUnitalNonAssocRing.{u_2} K (Ring.toNonAssocRing.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.13)))) (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.16))))] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.25 : Module.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.3)] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.29 : PosSMulMono.{u_2, u_1} K M (SMulZeroClass.toSMul.{u_2, u_1} K M (AddZero.toZero.{u_1} M (AddZeroClass.toAddZero.{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.Order.Module.Archimedean.113252810._hygCtx._hyg.3)))))) (DistribSMul.toSMulZeroClass.{u_2, u_1} K M (AddMonoid.toAddZeroClass.{u_1} M (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.3)))) (DistribMulAction.toDistribSMul.{u_2, u_1} K M (MonoidWithZero.toMonoid.{u_2} K (Semiring.toMonoidWithZero.{u_2} K (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.13))) (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.3))) (Module.toDistribMulAction.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.25)))) (PartialOrder.toPreorder.{u_2} K (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.16))))) (PartialOrder.toPreorder.{u_1} M (SemilatticeInf.toPartialOrder.{u_1} M (Lattice.toSemilatticeInf.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.6))))) (MulZeroClass.toZero.{u_2} K (NonUnitalNonAssocSemiring.toMulZeroClass.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonAssocRing.toNonUnitalNonAssocRing.{u_2} K (Ring.toNonAssocRing.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.13)))))], (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.9) -> (Submodule.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.25)","typeFull":"{M : Type u_1} →\n [inst : AddCommGroup M] →\n [inst_1 : LinearOrder M] →\n [inst_2 : IsOrderedAddMonoid M] →\n (K : Type u_2) →\n [inst_3 : Ring K] →\n [inst_4 : LinearOrder K] →\n [IsOrderedRing K] →\n [Archimedean K] → [inst_7 : Module K M] → [PosSMulMono K M] → ArchimedeanClass M → Submodule K M","typeReadable":"{M : Type u_1} →\n [inst : AddCommGroup M] →\n [inst_1 : LinearOrder M] →\n [inst_2 : IsOrderedAddMonoid M] →\n (K : Type u_2) →\n [inst_3 : Ring K] →\n [inst_4 : LinearOrder K] →\n [IsOrderedRing K] →\n [Archimedean K] → [inst_7 : Module K M] → [PosSMulMono K M] → ArchimedeanClass M → Submodule K M","typeReferences":[["PartialOrder","toPreorder"],["Ring","toNonAssocRing"],["ArchimedeanClass"],["Module"],["AddCommGroup","toAddGroup"],["SMulZeroClass","toSMul"],["Submodule"],["PosSMulMono"],["instDistribLatticeOfLinearOrder"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["IsOrderedAddMonoid"],["MonoidWithZero","toMonoid"],["AddGroup","toSubNegMonoid"],["NonAssocRing","toNonUnitalNonAssocRing"],["DistribSMul","toSMulZeroClass"],["SemilatticeInf","toPartialOrder"],["Lattice","toSemilatticeInf"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["DistribMulAction","toDistribSMul"],["LinearOrder"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["AddZeroClass","toAddZero"],["Ring","toSemiring"],["IsOrderedRing"],["Module","toDistribMulAction"],["DistribLattice","toLattice"],["SubNegMonoid","toAddMonoid"],["MulZeroClass","toZero"],["Archimedean"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["AddCommGroup","toAddCommMonoid"],["AddZero","toZero"],["Ring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["instDistribLatticeOfLinearOrder"],["DistribLattice","toLattice"],["Lattice","toSemilatticeInf"],["PartialOrder","toPreorder"],["ArchimedeanClass"],["ArchimedeanClass","instLinearOrder"],["UpperSet","Ici"],["SemilatticeInf","toPartialOrder"],["ArchimedeanClass","submodule"]]},{"isProp":true,"kind":"theorem","name":["ArchimedeanClass","closedBall_top"],"typeFallback":"forall (M : Type.{u_1}) [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.3 : AddCommGroup.{u_1} M] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.6 : LinearOrder.{u_1} M] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.9 : IsOrderedAddMonoid.{u_1} M (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.3) (PartialOrder.toPreorder.{u_1} M (SemilatticeInf.toPartialOrder.{u_1} M (Lattice.toSemilatticeInf.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.6)))))] (K : Type.{u_2}) [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.13 : Ring.{u_2} K] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.16 : LinearOrder.{u_2} K] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.19 : IsOrderedRing.{u_2} K (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.13) (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.16))))] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.22 : Archimedean.{u_2} K (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonAssocRing.toNonUnitalNonAssocRing.{u_2} K (Ring.toNonAssocRing.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.13)))) (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.16))))] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.25 : Module.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.3)] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.29 : PosSMulMono.{u_2, u_1} K M (SMulZeroClass.toSMul.{u_2, u_1} K M (AddZero.toZero.{u_1} M (AddZeroClass.toAddZero.{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.Order.Module.Archimedean.3789009670._hygCtx._hyg.3)))))) (DistribSMul.toSMulZeroClass.{u_2, u_1} K M (AddMonoid.toAddZeroClass.{u_1} M (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.3)))) (DistribMulAction.toDistribSMul.{u_2, u_1} K M (MonoidWithZero.toMonoid.{u_2} K (Semiring.toMonoidWithZero.{u_2} K (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.13))) (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.3))) (Module.toDistribMulAction.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.25)))) (PartialOrder.toPreorder.{u_2} K (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.16))))) (PartialOrder.toPreorder.{u_1} M (SemilatticeInf.toPartialOrder.{u_1} M (Lattice.toSemilatticeInf.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.6))))) (MulZeroClass.toZero.{u_2} K (NonUnitalNonAssocSemiring.toMulZeroClass.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonAssocRing.toNonUnitalNonAssocRing.{u_2} K (Ring.toNonAssocRing.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.13)))))], Eq.{succ u_1} (Submodule.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.25) (ArchimedeanClass.closedBall.{u_1, u_2} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.9 K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.22 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.29 (Top.top.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.9) (OrderTop.toTop.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.9) (Preorder.toLE.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.9) (PartialOrder.toPreorder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.9) (SemilatticeInf.toPartialOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.9) (Lattice.toSemilatticeInf.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.9) (DistribLattice.toLattice.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.9) (instDistribLatticeOfLinearOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.9) (ArchimedeanClass.instLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.9))))))) (ArchimedeanClass.instOrderTop.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.9)))) (Bot.bot.{u_1} (Submodule.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.25) (Submodule.instBot.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.3789009670._hygCtx._hyg.25))","typeFull":"∀ (M : Type u_1) [inst : AddCommGroup M] [inst_1 : LinearOrder M] [inst_2 : IsOrderedAddMonoid M] (K : Type u_2)\n [inst_3 : Ring K] [inst_4 : LinearOrder K] [inst_5 : IsOrderedRing K] [inst_6 : Archimedean K] [inst_7 : Module K M]\n [inst_8 : PosSMulMono K M], ArchimedeanClass.closedBall K ⊤ = ⊥","typeReadable":"∀ (M : Type u_1) [inst : AddCommGroup M] [inst_1 : LinearOrder M] [inst_2 : IsOrderedAddMonoid M] (K : Type u_2)\n [inst_3 : Ring K] [inst_4 : LinearOrder K] [inst_5 : IsOrderedRing K] [inst_6 : Archimedean K] [inst_7 : Module K M]\n [inst_8 : PosSMulMono K M], ArchimedeanClass.closedBall K ⊤ = ⊥","typeReferences":[["PartialOrder","toPreorder"],["Ring","toNonAssocRing"],["ArchimedeanClass"],["Module"],["AddCommGroup","toAddGroup"],["SMulZeroClass","toSMul"],["Submodule"],["PosSMulMono"],["instDistribLatticeOfLinearOrder"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["IsOrderedAddMonoid"],["ArchimedeanClass","instLinearOrder"],["MonoidWithZero","toMonoid"],["AddGroup","toSubNegMonoid"],["Preorder","toLE"],["Eq"],["NonAssocRing","toNonUnitalNonAssocRing"],["DistribSMul","toSMulZeroClass"],["SemilatticeInf","toPartialOrder"],["Lattice","toSemilatticeInf"],["ArchimedeanClass","closedBall"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["DistribMulAction","toDistribSMul"],["LinearOrder"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["AddZeroClass","toAddZero"],["Bot","bot"],["ArchimedeanClass","instOrderTop"],["Ring","toSemiring"],["IsOrderedRing"],["Module","toDistribMulAction"],["DistribLattice","toLattice"],["SubNegMonoid","toAddMonoid"],["MulZeroClass","toZero"],["Archimedean"],["Submodule","instBot"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["Top","top"],["AddCommGroup","toAddCommMonoid"],["OrderTop","toTop"],["AddZero","toZero"],["Ring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["PartialOrder","toPreorder"],["Lattice","toSemilatticeInf"],["Submodule","toAddSubgroup_inj"],["ArchimedeanClass","closedBall"],["ArchimedeanClass"],["AddCommGroup","toAddGroup"],["Iff","mp"],["Submodule","toAddSubgroup"],["Bot","bot"],["Ring","toSemiring"],["Submodule"],["ArchimedeanClass","instOrderTop"],["instDistribLatticeOfLinearOrder"],["AddSubgroup"],["ArchimedeanClass","closedBallAddSubgroup_top"],["DistribLattice","toLattice"],["Submodule","instBot"],["ArchimedeanClass","instLinearOrder"],["AddCommGroup","toAddCommMonoid"],["Top","top"],["Eq"],["Preorder","toLE"],["OrderTop","toTop"],["SemilatticeInf","toPartialOrder"]]},{"isProp":true,"kind":"theorem","name":["FiniteArchimedeanClass","ball_strictAnti"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.3 : AddCommGroup.{u_1} M] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.6 : LinearOrder.{u_1} M] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.9 : IsOrderedAddMonoid.{u_1} M (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.3) (PartialOrder.toPreorder.{u_1} M (SemilatticeInf.toPartialOrder.{u_1} M (Lattice.toSemilatticeInf.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.6)))))] (K : Type.{u_2}) [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.13 : Ring.{u_2} K] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.16 : LinearOrder.{u_2} K] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.19 : IsOrderedRing.{u_2} K (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.13) (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.16))))] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.22 : Archimedean.{u_2} K (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonAssocRing.toNonUnitalNonAssocRing.{u_2} K (Ring.toNonAssocRing.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.13)))) (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.16))))] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.25 : Module.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.3)] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.29 : PosSMulMono.{u_2, u_1} K M (SMulZeroClass.toSMul.{u_2, u_1} K M (AddZero.toZero.{u_1} M (AddZeroClass.toAddZero.{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.Order.Module.Archimedean.2180991096._hygCtx._hyg.3)))))) (DistribSMul.toSMulZeroClass.{u_2, u_1} K M (AddMonoid.toAddZeroClass.{u_1} M (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.3)))) (DistribMulAction.toDistribSMul.{u_2, u_1} K M (MonoidWithZero.toMonoid.{u_2} K (Semiring.toMonoidWithZero.{u_2} K (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.13))) (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.3))) (Module.toDistribMulAction.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.25)))) (PartialOrder.toPreorder.{u_2} K (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.16))))) (PartialOrder.toPreorder.{u_1} M (SemilatticeInf.toPartialOrder.{u_1} M (Lattice.toSemilatticeInf.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.6))))) (MulZeroClass.toZero.{u_2} K (NonUnitalNonAssocSemiring.toMulZeroClass.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonAssocRing.toNonUnitalNonAssocRing.{u_2} K (Ring.toNonAssocRing.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.13)))))], StrictAnti.{u_1, u_1} (FiniteArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.9) (Submodule.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.25) (Subtype.preorder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.9) (PartialOrder.toPreorder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.9) (SemilatticeInf.toPartialOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.9) (Lattice.toSemilatticeInf.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.9) (DistribLattice.toLattice.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.9) (instDistribLatticeOfLinearOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.9) (ArchimedeanClass.instLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.9)))))) (fun (A : ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.9) => Ne.{succ u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.9) A (Top.top.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.9) (OrderTop.toTop.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.9) (Preorder.toLE.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.9) (PartialOrder.toPreorder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.9) (SemilatticeInf.toPartialOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.9) (Lattice.toSemilatticeInf.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.9) (DistribLattice.toLattice.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.9) (instDistribLatticeOfLinearOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.9) (ArchimedeanClass.instLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.9))))))) (ArchimedeanClass.instOrderTop.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.9))))) (PartialOrder.toPreorder.{u_1} (Submodule.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.25) (Submodule.instPartialOrder.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.25)) (FiniteArchimedeanClass.ball.{u_1, u_2} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.9 K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.22 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2180991096._hygCtx._hyg.29)","typeFull":"∀ {M : Type u_1} [inst : AddCommGroup M] [inst_1 : LinearOrder M] [inst_2 : IsOrderedAddMonoid M] (K : Type u_2)\n [inst_3 : Ring K] [inst_4 : LinearOrder K] [inst_5 : IsOrderedRing K] [inst_6 : Archimedean K] [inst_7 : Module K M]\n [inst_8 : PosSMulMono K M], StrictAnti (FiniteArchimedeanClass.ball K)","typeReadable":"∀ {M : Type u_1} [inst : AddCommGroup M] [inst_1 : LinearOrder M] [inst_2 : IsOrderedAddMonoid M] (K : Type u_2)\n [inst_3 : Ring K] [inst_4 : LinearOrder K] [inst_5 : IsOrderedRing K] [inst_6 : Archimedean K] [inst_7 : Module K M]\n [inst_8 : PosSMulMono K M], StrictAnti (FiniteArchimedeanClass.ball K)","typeReferences":[["PartialOrder","toPreorder"],["Ring","toNonAssocRing"],["ArchimedeanClass"],["Module"],["AddCommGroup","toAddGroup"],["Subtype","preorder"],["SMulZeroClass","toSMul"],["Submodule"],["PosSMulMono"],["instDistribLatticeOfLinearOrder"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["IsOrderedAddMonoid"],["ArchimedeanClass","instLinearOrder"],["MonoidWithZero","toMonoid"],["FiniteArchimedeanClass"],["AddGroup","toSubNegMonoid"],["Preorder","toLE"],["NonAssocRing","toNonUnitalNonAssocRing"],["DistribSMul","toSMulZeroClass"],["SemilatticeInf","toPartialOrder"],["Lattice","toSemilatticeInf"],["FiniteArchimedeanClass","ball"],["Submodule","instPartialOrder"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["DistribMulAction","toDistribSMul"],["LinearOrder"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["AddZeroClass","toAddZero"],["ArchimedeanClass","instOrderTop"],["Ring","toSemiring"],["StrictAnti"],["IsOrderedRing"],["Module","toDistribMulAction"],["DistribLattice","toLattice"],["SubNegMonoid","toAddMonoid"],["MulZeroClass","toZero"],["Archimedean"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["Top","top"],["AddCommGroup","toAddCommMonoid"],["Ne"],["OrderTop","toTop"],["AddZero","toZero"],["Ring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["FiniteArchimedeanClass","ballAddSubgroup_strictAnti"]]},{"isProp":true,"kind":"theorem","name":["FiniteArchimedeanClass","ball_lt_closedBall"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3426652509._hygCtx._hyg.3 : AddCommGroup.{u_1} M] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3426652509._hygCtx._hyg.6 : LinearOrder.{u_1} M] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3426652509._hygCtx._hyg.9 : IsOrderedAddMonoid.{u_1} M (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3426652509._hygCtx._hyg.3) (PartialOrder.toPreorder.{u_1} M (SemilatticeInf.toPartialOrder.{u_1} M (Lattice.toSemilatticeInf.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3426652509._hygCtx._hyg.6)))))] (K : Type.{u_2}) [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3426652509._hygCtx._hyg.13 : Ring.{u_2} K] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3426652509._hygCtx._hyg.16 : LinearOrder.{u_2} K] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3426652509._hygCtx._hyg.19 : IsOrderedRing.{u_2} K (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3426652509._hygCtx._hyg.13) (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3426652509._hygCtx._hyg.16))))] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3426652509._hygCtx._hyg.22 : Archimedean.{u_2} K (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonAssocRing.toNonUnitalNonAssocRing.{u_2} K (Ring.toNonAssocRing.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3426652509._hygCtx._hyg.13)))) (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3426652509._hygCtx._hyg.16))))] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3426652509._hygCtx._hyg.25 : Module.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3426652509._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3426652509._hygCtx._hyg.3)] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3426652509._hygCtx._hyg.29 : PosSMulMono.{u_2, u_1} K M (SMulZeroClass.toSMul.{u_2, u_1} K M (AddZero.toZero.{u_1} M (AddZeroClass.toAddZero.{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.Order.Module.Archimedean.3426652509._hygCtx._hyg.3)))))) (DistribSMul.toSMulZeroClass.{u_2, u_1} K M (AddMonoid.toAddZeroClass.{u_1} M (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3426652509._hygCtx._hyg.3)))) (DistribMulAction.toDistribSMul.{u_2, u_1} K M (MonoidWithZero.toMonoid.{u_2} K (Semiring.toMonoidWithZero.{u_2} K (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3426652509._hygCtx._hyg.13))) (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3426652509._hygCtx._hyg.3))) (Module.toDistribMulAction.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3426652509._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3426652509._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.3426652509._hygCtx._hyg.25)))) (PartialOrder.toPreorder.{u_2} K (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3426652509._hygCtx._hyg.16))))) (PartialOrder.toPreorder.{u_1} M (SemilatticeInf.toPartialOrder.{u_1} M (Lattice.toSemilatticeInf.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3426652509._hygCtx._hyg.6))))) (MulZeroClass.toZero.{u_2} K (NonUnitalNonAssocSemiring.toMulZeroClass.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonAssocRing.toNonUnitalNonAssocRing.{u_2} K (Ring.toNonAssocRing.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3426652509._hygCtx._hyg.13)))))] {c : FiniteArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3426652509._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3426652509._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3426652509._hygCtx._hyg.9}, LT.lt.{u_1} (Submodule.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3426652509._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3426652509._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.3426652509._hygCtx._hyg.25) (Preorder.toLT.{u_1} (Submodule.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3426652509._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3426652509._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.3426652509._hygCtx._hyg.25) (PartialOrder.toPreorder.{u_1} (Submodule.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3426652509._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3426652509._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.3426652509._hygCtx._hyg.25) (Submodule.instPartialOrder.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3426652509._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3426652509._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.3426652509._hygCtx._hyg.25))) (FiniteArchimedeanClass.ball.{u_1, u_2} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3426652509._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3426652509._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3426652509._hygCtx._hyg.9 K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3426652509._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3426652509._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3426652509._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3426652509._hygCtx._hyg.22 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3426652509._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3426652509._hygCtx._hyg.29 c) (FiniteArchimedeanClass.closedBall.{u_1, u_2} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3426652509._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3426652509._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3426652509._hygCtx._hyg.9 K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3426652509._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3426652509._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3426652509._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3426652509._hygCtx._hyg.22 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3426652509._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3426652509._hygCtx._hyg.29 c)","typeFull":"∀ {M : Type u_1} [inst : AddCommGroup M] [inst_1 : LinearOrder M] [inst_2 : IsOrderedAddMonoid M] (K : Type u_2)\n [inst_3 : Ring K] [inst_4 : LinearOrder K] [inst_5 : IsOrderedRing K] [inst_6 : Archimedean K] [inst_7 : Module K M]\n [inst_8 : PosSMulMono K M] {c : FiniteArchimedeanClass M},\n FiniteArchimedeanClass.ball K c < FiniteArchimedeanClass.closedBall K c","typeReadable":"∀ {M : Type u_1} [inst : AddCommGroup M] [inst_1 : LinearOrder M] [inst_2 : IsOrderedAddMonoid M] (K : Type u_2)\n [inst_3 : Ring K] [inst_4 : LinearOrder K] [inst_5 : IsOrderedRing K] [inst_6 : Archimedean K] [inst_7 : Module K M]\n [inst_8 : PosSMulMono K M] {c : FiniteArchimedeanClass M},\n FiniteArchimedeanClass.ball K c < FiniteArchimedeanClass.closedBall K c","typeReferences":[["FiniteArchimedeanClass","closedBall"],["PartialOrder","toPreorder"],["Ring","toNonAssocRing"],["Module"],["AddCommGroup","toAddGroup"],["Preorder","toLT"],["SMulZeroClass","toSMul"],["Submodule"],["PosSMulMono"],["instDistribLatticeOfLinearOrder"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["IsOrderedAddMonoid"],["MonoidWithZero","toMonoid"],["FiniteArchimedeanClass"],["AddGroup","toSubNegMonoid"],["NonAssocRing","toNonUnitalNonAssocRing"],["DistribSMul","toSMulZeroClass"],["SemilatticeInf","toPartialOrder"],["Lattice","toSemilatticeInf"],["FiniteArchimedeanClass","ball"],["Submodule","instPartialOrder"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["DistribMulAction","toDistribSMul"],["LinearOrder"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["AddZeroClass","toAddZero"],["Ring","toSemiring"],["LT","lt"],["IsOrderedRing"],["Module","toDistribMulAction"],["DistribLattice","toLattice"],["SubNegMonoid","toAddMonoid"],["MulZeroClass","toZero"],["Archimedean"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["AddCommGroup","toAddCommMonoid"],["AddZero","toZero"],["Ring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["PartialOrder","toPreorder"],["Lattice","toSemilatticeInf"],["ArchimedeanClass"],["Subtype","preorder"],["Set","Ioi_ssubset_Ici_self"],["ArchimedeanClass","instOrderTop"],["instDistribLatticeOfLinearOrder"],["FiniteArchimedeanClass","submodule_strictAnti"],["DistribLattice","toLattice"],["UpperSet","Ioi"],["ArchimedeanClass","instLinearOrder"],["FiniteArchimedeanClass"],["Top","top"],["Ne"],["Preorder","toLE"],["OrderTop","toTop"],["UpperSet","Ici"],["SemilatticeInf","toPartialOrder"]]},{"isProp":true,"kind":"theorem","name":["ArchimedeanClass","ball_top"],"typeFallback":"forall (M : Type.{u_1}) [inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.3 : AddCommGroup.{u_1} M] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.6 : LinearOrder.{u_1} M] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.9 : IsOrderedAddMonoid.{u_1} M (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.3) (PartialOrder.toPreorder.{u_1} M (SemilatticeInf.toPartialOrder.{u_1} M (Lattice.toSemilatticeInf.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.6)))))] (K : Type.{u_2}) [inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.13 : Ring.{u_2} K] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.16 : LinearOrder.{u_2} K] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.19 : IsOrderedRing.{u_2} K (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.13) (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.16))))] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.22 : Archimedean.{u_2} K (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonAssocRing.toNonUnitalNonAssocRing.{u_2} K (Ring.toNonAssocRing.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.13)))) (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.16))))] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.25 : Module.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.3)] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.29 : PosSMulMono.{u_2, u_1} K M (SMulZeroClass.toSMul.{u_2, u_1} K M (AddZero.toZero.{u_1} M (AddZeroClass.toAddZero.{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.Order.Module.Archimedean.1900385171._hygCtx._hyg.3)))))) (DistribSMul.toSMulZeroClass.{u_2, u_1} K M (AddMonoid.toAddZeroClass.{u_1} M (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.3)))) (DistribMulAction.toDistribSMul.{u_2, u_1} K M (MonoidWithZero.toMonoid.{u_2} K (Semiring.toMonoidWithZero.{u_2} K (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.13))) (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.3))) (Module.toDistribMulAction.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.25)))) (PartialOrder.toPreorder.{u_2} K (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.16))))) (PartialOrder.toPreorder.{u_1} M (SemilatticeInf.toPartialOrder.{u_1} M (Lattice.toSemilatticeInf.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.6))))) (MulZeroClass.toZero.{u_2} K (NonUnitalNonAssocSemiring.toMulZeroClass.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonAssocRing.toNonUnitalNonAssocRing.{u_2} K (Ring.toNonAssocRing.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.13)))))], Eq.{succ u_1} (Submodule.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.25) (ArchimedeanClass.ball.{u_1, u_2} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.9 K inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.22 inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.29 (Top.top.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.9) (OrderTop.toTop.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.9) (Preorder.toLE.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.9) (PartialOrder.toPreorder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.9) (SemilatticeInf.toPartialOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.9) (Lattice.toSemilatticeInf.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.9) (DistribLattice.toLattice.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.9) (instDistribLatticeOfLinearOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.9) (ArchimedeanClass.instLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.9))))))) (ArchimedeanClass.instOrderTop.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.9)))) (Bot.bot.{u_1} (Submodule.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.25) (Submodule.instBot.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.1900385171._hygCtx._hyg.25))","typeFull":"∀ (M : Type u_1) [inst : AddCommGroup M] [inst_1 : LinearOrder M] [inst_2 : IsOrderedAddMonoid M] (K : Type u_2)\n [inst_3 : Ring K] [inst_4 : LinearOrder K] [inst_5 : IsOrderedRing K] [inst_6 : Archimedean K] [inst_7 : Module K M]\n [inst_8 : PosSMulMono K M], ArchimedeanClass.ball K ⊤ = ⊥","typeReadable":"∀ (M : Type u_1) [inst : AddCommGroup M] [inst_1 : LinearOrder M] [inst_2 : IsOrderedAddMonoid M] (K : Type u_2)\n [inst_3 : Ring K] [inst_4 : LinearOrder K] [inst_5 : IsOrderedRing K] [inst_6 : Archimedean K] [inst_7 : Module K M]\n [inst_8 : PosSMulMono K M], ArchimedeanClass.ball K ⊤ = ⊥","typeReferences":[["PartialOrder","toPreorder"],["Ring","toNonAssocRing"],["ArchimedeanClass"],["Module"],["AddCommGroup","toAddGroup"],["SMulZeroClass","toSMul"],["Submodule"],["PosSMulMono"],["instDistribLatticeOfLinearOrder"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["IsOrderedAddMonoid"],["ArchimedeanClass","instLinearOrder"],["MonoidWithZero","toMonoid"],["AddGroup","toSubNegMonoid"],["Preorder","toLE"],["Eq"],["NonAssocRing","toNonUnitalNonAssocRing"],["DistribSMul","toSMulZeroClass"],["SemilatticeInf","toPartialOrder"],["Lattice","toSemilatticeInf"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["DistribMulAction","toDistribSMul"],["LinearOrder"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["AddZeroClass","toAddZero"],["Bot","bot"],["ArchimedeanClass","instOrderTop"],["Ring","toSemiring"],["IsOrderedRing"],["Module","toDistribMulAction"],["DistribLattice","toLattice"],["SubNegMonoid","toAddMonoid"],["MulZeroClass","toZero"],["ArchimedeanClass","ball"],["Archimedean"],["Submodule","instBot"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["Top","top"],["AddCommGroup","toAddCommMonoid"],["OrderTop","toTop"],["AddZero","toZero"],["Ring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["PartialOrder","toPreorder"],["Lattice","toSemilatticeInf"],["Submodule","toAddSubgroup_inj"],["ArchimedeanClass"],["AddCommGroup","toAddGroup"],["Iff","mp"],["ArchimedeanClass","ballAddSubgroup_top"],["Submodule","toAddSubgroup"],["Bot","bot"],["Ring","toSemiring"],["Submodule"],["ArchimedeanClass","instOrderTop"],["instDistribLatticeOfLinearOrder"],["AddSubgroup"],["DistribLattice","toLattice"],["ArchimedeanClass","ball"],["Submodule","instBot"],["ArchimedeanClass","instLinearOrder"],["AddCommGroup","toAddCommMonoid"],["Top","top"],["Eq"],["Preorder","toLE"],["OrderTop","toTop"],["SemilatticeInf","toPartialOrder"]]},{"isProp":true,"kind":"theorem","name":["ArchimedeanClass","mk_smul"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2672574235._hygCtx._hyg.3 : AddCommGroup.{u_1} M] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2672574235._hygCtx._hyg.6 : LinearOrder.{u_1} M] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2672574235._hygCtx._hyg.9 : IsOrderedAddMonoid.{u_1} M (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2672574235._hygCtx._hyg.3) (PartialOrder.toPreorder.{u_1} M (SemilatticeInf.toPartialOrder.{u_1} M (Lattice.toSemilatticeInf.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2672574235._hygCtx._hyg.6)))))] {K : Type.{u_2}} [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2672574235._hygCtx._hyg.13 : Ring.{u_2} K] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2672574235._hygCtx._hyg.16 : LinearOrder.{u_2} K] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2672574235._hygCtx._hyg.19 : IsOrderedRing.{u_2} K (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2672574235._hygCtx._hyg.13) (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2672574235._hygCtx._hyg.16))))] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2672574235._hygCtx._hyg.22 : Archimedean.{u_2} K (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonAssocRing.toNonUnitalNonAssocRing.{u_2} K (Ring.toNonAssocRing.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2672574235._hygCtx._hyg.13)))) (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2672574235._hygCtx._hyg.16))))] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2672574235._hygCtx._hyg.25 : Module.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2672574235._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2672574235._hygCtx._hyg.3)] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2672574235._hygCtx._hyg.29 : PosSMulMono.{u_2, u_1} K M (SMulZeroClass.toSMul.{u_2, u_1} K M (AddZero.toZero.{u_1} M (AddZeroClass.toAddZero.{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.Order.Module.Archimedean.2672574235._hygCtx._hyg.3)))))) (DistribSMul.toSMulZeroClass.{u_2, u_1} K M (AddMonoid.toAddZeroClass.{u_1} M (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2672574235._hygCtx._hyg.3)))) (DistribMulAction.toDistribSMul.{u_2, u_1} K M (MonoidWithZero.toMonoid.{u_2} K (Semiring.toMonoidWithZero.{u_2} K (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2672574235._hygCtx._hyg.13))) (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2672574235._hygCtx._hyg.3))) (Module.toDistribMulAction.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2672574235._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2672574235._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.2672574235._hygCtx._hyg.25)))) (PartialOrder.toPreorder.{u_2} K (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2672574235._hygCtx._hyg.16))))) (PartialOrder.toPreorder.{u_1} M (SemilatticeInf.toPartialOrder.{u_1} M (Lattice.toSemilatticeInf.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2672574235._hygCtx._hyg.6))))) (MulZeroClass.toZero.{u_2} K (NonUnitalNonAssocSemiring.toMulZeroClass.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonAssocRing.toNonUnitalNonAssocRing.{u_2} K (Ring.toNonAssocRing.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2672574235._hygCtx._hyg.13)))))] (a : M) {k : K}, (Ne.{succ u_2} K k (OfNat.ofNat.{u_2} K 0 (Zero.toOfNat0.{u_2} K (MulZeroClass.toZero.{u_2} K (NonUnitalNonAssocSemiring.toMulZeroClass.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonAssocRing.toNonUnitalNonAssocRing.{u_2} K (Ring.toNonAssocRing.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2672574235._hygCtx._hyg.13)))))))) -> (Eq.{succ u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2672574235._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2672574235._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2672574235._hygCtx._hyg.9) (ArchimedeanClass.mk.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2672574235._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2672574235._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2672574235._hygCtx._hyg.9 (HSMul.hSMul.{u_2, u_1, u_1} K M M (instHSMul.{u_2, u_1} K M (SMulZeroClass.toSMul.{u_2, u_1} K M (AddZero.toZero.{u_1} M (AddZeroClass.toAddZero.{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.Order.Module.Archimedean.2672574235._hygCtx._hyg.3)))))) (DistribSMul.toSMulZeroClass.{u_2, u_1} K M (AddMonoid.toAddZeroClass.{u_1} M (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2672574235._hygCtx._hyg.3)))) (DistribMulAction.toDistribSMul.{u_2, u_1} K M (MonoidWithZero.toMonoid.{u_2} K (Semiring.toMonoidWithZero.{u_2} K (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2672574235._hygCtx._hyg.13))) (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2672574235._hygCtx._hyg.3))) (Module.toDistribMulAction.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2672574235._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2672574235._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.2672574235._hygCtx._hyg.25))))) k a)) (ArchimedeanClass.mk.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2672574235._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2672574235._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2672574235._hygCtx._hyg.9 a))","typeFull":"∀ {M : Type u_1} [inst : AddCommGroup M] [inst_1 : LinearOrder M] [inst_2 : IsOrderedAddMonoid M] {K : Type u_2}\n [inst_3 : Ring K] [inst_4 : LinearOrder K] [IsOrderedRing K] [Archimedean K] [inst_7 : Module K M] [PosSMulMono K M]\n (a : M) {k : K}, k ≠ 0 → ArchimedeanClass.mk (k • a) = ArchimedeanClass.mk a","typeReadable":"∀ {M : Type u_1} [inst : AddCommGroup M] [inst_1 : LinearOrder M] [inst_2 : IsOrderedAddMonoid M] {K : Type u_2}\n [inst_3 : Ring K] [inst_4 : LinearOrder K] [IsOrderedRing K] [Archimedean K] [inst_7 : Module K M] [PosSMulMono K M]\n (a : M) {k : K}, k ≠ 0 → ArchimedeanClass.mk (k • a) = ArchimedeanClass.mk a","typeReferences":[["PartialOrder","toPreorder"],["Ring","toNonAssocRing"],["ArchimedeanClass"],["Module"],["AddCommGroup","toAddGroup"],["SMulZeroClass","toSMul"],["PosSMulMono"],["instDistribLatticeOfLinearOrder"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["IsOrderedAddMonoid"],["MonoidWithZero","toMonoid"],["instHSMul"],["Zero","toOfNat0"],["AddGroup","toSubNegMonoid"],["Eq"],["NonAssocRing","toNonUnitalNonAssocRing"],["DistribSMul","toSMulZeroClass"],["SemilatticeInf","toPartialOrder"],["Lattice","toSemilatticeInf"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["DistribMulAction","toDistribSMul"],["LinearOrder"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["AddZeroClass","toAddZero"],["ArchimedeanClass","mk"],["OfNat","ofNat"],["Ring","toSemiring"],["IsOrderedRing"],["Module","toDistribMulAction"],["DistribLattice","toLattice"],["SubNegMonoid","toAddMonoid"],["MulZeroClass","toZero"],["Archimedean"],["HSMul","hSMul"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["AddCommGroup","toAddCommMonoid"],["Ne"],["AddZero","toZero"],["Ring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["SubtractionMonoid","toSubNegZeroMonoid"],["Ring","toNonAssocRing"],["AddCommGroup","toAddGroup"],["Exists","intro"],["AddGroupWithOne","toAddMonoidWithOne"],["SemigroupAction","toSMul"],["SMulZeroClass","toSMul"],["nontrivial_iff"],["AddGroup","toSubtractionMonoid"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["Eq","symm"],["Monoid","toSemigroup"],["abs"],["IsOrderedRing","toZeroLEOneClass"],["Exists"],["PosSMulMono","toSMulPosMono"],["Ring","toSemiring"],["abs_nonneg","_simp_1"],["Eq","refl"],["AddMonoidWithOne","toOne"],["one_smul"],["AddCommGroup","toAddCommMonoid"],["Eq","mpr"],["covariant_swap_add_of_covariant_add"],["AddMonoid","toAddZeroClass"],["MulOneClass","toMulOne"],["abs_smul"],["SubtractionCommMonoid","toSubtractionMonoid"],["AddCommMonoid","toAddMonoid"],["IsOrderedRing","toIsOrderedAddMonoid"],["congr"],["Eq"],["Preorder","toLE"],["zero_lt_one","_simp_1"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["OfNat","ofNat"],["Module","toDistribMulAction"],["AddGroupWithOne","toAddGroup"],["AddCommGroup","toDivisionAddCommMonoid"],["MulZeroClass","toZero"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["Ne"],["PartialOrder","toPreorder"],["abs_pos","_simp_1"],["Preorder","toLT"],["AddMonoidWithOne","toAddMonoid"],["And","intro"],["Semiring","toNonAssocSemiring"],["Ring","toAddGroupWithOne"],["funext"],["DistribMulAction","toMulAction"],["AddGroup","toSubNegMonoid"],["NonAssocRing","toNonUnitalNonAssocRing"],["NonAssocSemiring","toMulZeroOneClass"],["DistribSMul","toSMulZeroClass"],["SemilatticeInf","toPartialOrder"],["MulOne","toOne"],["DistribMulAction","toDistribSMul"],["And"],["IsOrderedAddMonoid","toAddLeftMono"],["AddZeroClass","toAddZero"],["Archimedean","arch"],["smul_le_smul_of_nonneg_right"],["Exists","casesOn"],["Nat"],["AddMonoid","toNSMul"],["Iff","mpr"],["HSMul","hSMul"],["Nontrivial"],["NegZeroClass","toZero"],["id"],["AddCommMonoid","nat_isScalarTower"],["AddZero","toZero"],["le_smul_of_one_le_left"],["ArchimedeanClass"],["NeZero","one"],["SubNegZeroMonoid","toNegZeroClass"],["_private","Mathlib","Algebra","Order","Module","Archimedean",0,"ArchimedeanClass","mk_smul","_simp_1_1"],["congrArg"],["instDistribLatticeOfLinearOrder"],["MonoidWithZero","toMonoid"],["Monoid","toMulOneClass"],["instHSMul"],["Zero","toOfNat0"],["congrFun'"],["Not"],["Lattice","toSemilatticeInf"],["Semiring","toMonoidWithZero"],["ArchimedeanClass","mk"],["LT","lt"],["DistribLattice","toLattice"],["SubNegMonoid","toAddMonoid"],["One","toOfNat1"],["of_eq_true"],["AddCommMonoid","toAddCommSemigroup"],["LE","le"],["smul_assoc"],["MulAction","toSemigroupAction"]]},{"isProp":true,"kind":"theorem","name":["FiniteArchimedeanClass","mem_closedBall_iff","_simp_1"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.3 : AddCommGroup.{u_1} M] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.6 : LinearOrder.{u_1} M] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.9 : IsOrderedAddMonoid.{u_1} M (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.3) (PartialOrder.toPreorder.{u_1} M (SemilatticeInf.toPartialOrder.{u_1} M (Lattice.toSemilatticeInf.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.6)))))] (K : Type.{u_2}) [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.13 : Ring.{u_2} K] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.16 : LinearOrder.{u_2} K] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.19 : IsOrderedRing.{u_2} K (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.13) (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.16))))] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.22 : Archimedean.{u_2} K (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonAssocRing.toNonUnitalNonAssocRing.{u_2} K (Ring.toNonAssocRing.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.13)))) (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.16))))] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.25 : Module.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.3)] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.29 : PosSMulMono.{u_2, u_1} K M (SMulZeroClass.toSMul.{u_2, u_1} K M (AddZero.toZero.{u_1} M (AddZeroClass.toAddZero.{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.Order.Module.Archimedean.2781882086._hygCtx._hyg.3)))))) (DistribSMul.toSMulZeroClass.{u_2, u_1} K M (AddMonoid.toAddZeroClass.{u_1} M (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.3)))) (DistribMulAction.toDistribSMul.{u_2, u_1} K M (MonoidWithZero.toMonoid.{u_2} K (Semiring.toMonoidWithZero.{u_2} K (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.13))) (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.3))) (Module.toDistribMulAction.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.25)))) (PartialOrder.toPreorder.{u_2} K (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.16))))) (PartialOrder.toPreorder.{u_1} M (SemilatticeInf.toPartialOrder.{u_1} M (Lattice.toSemilatticeInf.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.6))))) (MulZeroClass.toZero.{u_2} K (NonUnitalNonAssocSemiring.toMulZeroClass.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonAssocRing.toNonUnitalNonAssocRing.{u_2} K (Ring.toNonAssocRing.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.13)))))] {a : M} {c : FiniteArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.9}, Eq.{1} Prop (Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.25) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.25) M (Submodule.setLike.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.25)) (FiniteArchimedeanClass.closedBall.{u_1, u_2} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.9 K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.22 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.29 c) a) (forall (h : Ne.{succ u_1} M a (OfNat.ofNat.{u_1} M 0 (Zero.toOfNat0.{u_1} M (NegZeroClass.toZero.{u_1} M (SubNegZeroMonoid.toNegZeroClass.{u_1} M (SubtractionMonoid.toSubNegZeroMonoid.{u_1} M (SubtractionCommMonoid.toSubtractionMonoid.{u_1} M (AddCommGroup.toDivisionAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.3)))))))), LE.le.{u_1} (FiniteArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.9) (Subtype.instLE.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.9) (Preorder.toLE.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.9) (PartialOrder.toPreorder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.9) (SemilatticeInf.toPartialOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.9) (Lattice.toSemilatticeInf.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.9) (DistribLattice.toLattice.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.9) (instDistribLatticeOfLinearOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.9) (ArchimedeanClass.instLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.9))))))) (fun (A : ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.9) => Ne.{succ u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.9) A (Top.top.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.9) (OrderTop.toTop.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.9) (Preorder.toLE.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.9) (PartialOrder.toPreorder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.9) (SemilatticeInf.toPartialOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.9) (Lattice.toSemilatticeInf.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.9) (DistribLattice.toLattice.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.9) (instDistribLatticeOfLinearOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.9) (ArchimedeanClass.instLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.9))))))) (ArchimedeanClass.instOrderTop.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.9))))) c (FiniteArchimedeanClass.mk.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882086._hygCtx._hyg.9 a h))","typeFull":"∀ {M : Type u_1} [inst : AddCommGroup M] [inst_1 : LinearOrder M] [inst_2 : IsOrderedAddMonoid M] (K : Type u_2)\n [inst_3 : Ring K] [inst_4 : LinearOrder K] [inst_5 : IsOrderedRing K] [inst_6 : Archimedean K] [inst_7 : Module K M]\n [inst_8 : PosSMulMono K M] {a : M} {c : FiniteArchimedeanClass M},\n (a ∈ FiniteArchimedeanClass.closedBall K c) = ∀ (h : a ≠ 0), c ≤ FiniteArchimedeanClass.mk a h","typeReadable":"∀ {M : Type u_1} [inst : AddCommGroup M] [inst_1 : LinearOrder M] [inst_2 : IsOrderedAddMonoid M] (K : Type u_2)\n [inst_3 : Ring K] [inst_4 : LinearOrder K] [inst_5 : IsOrderedRing K] [inst_6 : Archimedean K] [inst_7 : Module K M]\n [inst_8 : PosSMulMono K M] {a : M} {c : FiniteArchimedeanClass M},\n (a ∈ FiniteArchimedeanClass.closedBall K c) = ∀ (h : a ≠ 0), c ≤ FiniteArchimedeanClass.mk a h","typeReferences":[["SubtractionMonoid","toSubNegZeroMonoid"],["Ring","toNonAssocRing"],["PartialOrder","toPreorder"],["Subtype","instLE"],["AddCommGroup","toAddGroup"],["Membership","mem"],["SMulZeroClass","toSMul"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["IsOrderedAddMonoid"],["ArchimedeanClass","instLinearOrder"],["FiniteArchimedeanClass"],["FiniteArchimedeanClass","mk"],["AddGroup","toSubNegMonoid"],["NonAssocRing","toNonUnitalNonAssocRing"],["SemilatticeInf","toPartialOrder"],["DistribSMul","toSMulZeroClass"],["SetLike","instMembership"],["LinearOrder"],["DistribMulAction","toDistribSMul"],["AddZeroClass","toAddZero"],["ArchimedeanClass","instOrderTop"],["Ring","toSemiring"],["Top","top"],["AddCommGroup","toAddCommMonoid"],["NegZeroClass","toZero"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["FiniteArchimedeanClass","closedBall"],["ArchimedeanClass"],["Module"],["SubtractionCommMonoid","toSubtractionMonoid"],["SubNegZeroMonoid","toNegZeroClass"],["PosSMulMono"],["Submodule"],["instDistribLatticeOfLinearOrder"],["MonoidWithZero","toMonoid"],["Zero","toOfNat0"],["Preorder","toLE"],["Eq"],["Lattice","toSemilatticeInf"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["OfNat","ofNat"],["IsOrderedRing"],["Module","toDistribMulAction"],["DistribLattice","toLattice"],["SubNegMonoid","toAddMonoid"],["AddCommGroup","toDivisionAddCommMonoid"],["MulZeroClass","toZero"],["Submodule","setLike"],["Archimedean"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["LE","le"],["Ne"],["OrderTop","toTop"],["Ring"]],"valueReferences":[["FiniteArchimedeanClass","closedBall"],["SubtractionMonoid","toSubNegZeroMonoid"],["PartialOrder","toPreorder"],["Subtype","instLE"],["ArchimedeanClass"],["Membership","mem"],["SubtractionCommMonoid","toSubtractionMonoid"],["SubNegZeroMonoid","toNegZeroClass"],["Submodule"],["instDistribLatticeOfLinearOrder"],["ArchimedeanClass","instLinearOrder"],["FiniteArchimedeanClass"],["FiniteArchimedeanClass","mk"],["Zero","toOfNat0"],["Preorder","toLE"],["propext"],["SemilatticeInf","toPartialOrder"],["SetLike","instMembership"],["Lattice","toSemilatticeInf"],["FiniteArchimedeanClass","mem_closedBall_iff"],["OfNat","ofNat"],["ArchimedeanClass","instOrderTop"],["Ring","toSemiring"],["DistribLattice","toLattice"],["AddCommGroup","toDivisionAddCommMonoid"],["Submodule","setLike"],["LE","le"],["Top","top"],["NegZeroClass","toZero"],["AddCommGroup","toAddCommMonoid"],["Ne"],["OrderTop","toTop"]]},{"isProp":true,"kind":"theorem","name":["FiniteArchimedeanClass","mem_ball_iff","_simp_1"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.3 : AddCommGroup.{u_1} M] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.6 : LinearOrder.{u_1} M] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.9 : IsOrderedAddMonoid.{u_1} M (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.3) (PartialOrder.toPreorder.{u_1} M (SemilatticeInf.toPartialOrder.{u_1} M (Lattice.toSemilatticeInf.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.6)))))] (K : Type.{u_2}) [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.13 : Ring.{u_2} K] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.16 : LinearOrder.{u_2} K] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.19 : IsOrderedRing.{u_2} K (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.13) (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.16))))] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.22 : Archimedean.{u_2} K (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonAssocRing.toNonUnitalNonAssocRing.{u_2} K (Ring.toNonAssocRing.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.13)))) (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.16))))] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.25 : Module.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.3)] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.29 : PosSMulMono.{u_2, u_1} K M (SMulZeroClass.toSMul.{u_2, u_1} K M (AddZero.toZero.{u_1} M (AddZeroClass.toAddZero.{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.Order.Module.Archimedean.2496125326._hygCtx._hyg.3)))))) (DistribSMul.toSMulZeroClass.{u_2, u_1} K M (AddMonoid.toAddZeroClass.{u_1} M (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.3)))) (DistribMulAction.toDistribSMul.{u_2, u_1} K M (MonoidWithZero.toMonoid.{u_2} K (Semiring.toMonoidWithZero.{u_2} K (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.13))) (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.3))) (Module.toDistribMulAction.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.25)))) (PartialOrder.toPreorder.{u_2} K (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.16))))) (PartialOrder.toPreorder.{u_1} M (SemilatticeInf.toPartialOrder.{u_1} M (Lattice.toSemilatticeInf.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.6))))) (MulZeroClass.toZero.{u_2} K (NonUnitalNonAssocSemiring.toMulZeroClass.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonAssocRing.toNonUnitalNonAssocRing.{u_2} K (Ring.toNonAssocRing.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.13)))))] {a : M} {c : FiniteArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.9}, Eq.{1} Prop (Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.25) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.25) M (Submodule.setLike.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.25)) (FiniteArchimedeanClass.ball.{u_1, u_2} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.9 K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.22 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.29 c) a) (forall (h : Ne.{succ u_1} M a (OfNat.ofNat.{u_1} M 0 (Zero.toOfNat0.{u_1} M (NegZeroClass.toZero.{u_1} M (SubNegZeroMonoid.toNegZeroClass.{u_1} M (SubtractionMonoid.toSubNegZeroMonoid.{u_1} M (SubtractionCommMonoid.toSubtractionMonoid.{u_1} M (AddCommGroup.toDivisionAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.3)))))))), LT.lt.{u_1} (FiniteArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.9) (Subtype.instLT.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.9) (Preorder.toLT.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.9) (PartialOrder.toPreorder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.9) (SemilatticeInf.toPartialOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.9) (Lattice.toSemilatticeInf.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.9) (DistribLattice.toLattice.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.9) (instDistribLatticeOfLinearOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.9) (ArchimedeanClass.instLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.9))))))) (fun (A : ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.9) => Ne.{succ u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.9) A (Top.top.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.9) (OrderTop.toTop.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.9) (Preorder.toLE.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.9) (PartialOrder.toPreorder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.9) (SemilatticeInf.toPartialOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.9) (Lattice.toSemilatticeInf.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.9) (DistribLattice.toLattice.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.9) (instDistribLatticeOfLinearOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.9) (ArchimedeanClass.instLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.9))))))) (ArchimedeanClass.instOrderTop.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.9))))) c (FiniteArchimedeanClass.mk.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.9 a h))","typeFull":"∀ {M : Type u_1} [inst : AddCommGroup M] [inst_1 : LinearOrder M] [inst_2 : IsOrderedAddMonoid M] (K : Type u_2)\n [inst_3 : Ring K] [inst_4 : LinearOrder K] [inst_5 : IsOrderedRing K] [inst_6 : Archimedean K] [inst_7 : Module K M]\n [inst_8 : PosSMulMono K M] {a : M} {c : FiniteArchimedeanClass M},\n (a ∈ FiniteArchimedeanClass.ball K c) = ∀ (h : a ≠ 0), c < FiniteArchimedeanClass.mk a h","typeReadable":"∀ {M : Type u_1} [inst : AddCommGroup M] [inst_1 : LinearOrder M] [inst_2 : IsOrderedAddMonoid M] (K : Type u_2)\n [inst_3 : Ring K] [inst_4 : LinearOrder K] [inst_5 : IsOrderedRing K] [inst_6 : Archimedean K] [inst_7 : Module K M]\n [inst_8 : PosSMulMono K M] {a : M} {c : FiniteArchimedeanClass M},\n (a ∈ FiniteArchimedeanClass.ball K c) = ∀ (h : a ≠ 0), c < FiniteArchimedeanClass.mk a h","typeReferences":[["SubtractionMonoid","toSubNegZeroMonoid"],["Ring","toNonAssocRing"],["PartialOrder","toPreorder"],["AddCommGroup","toAddGroup"],["Membership","mem"],["Preorder","toLT"],["SMulZeroClass","toSMul"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["IsOrderedAddMonoid"],["ArchimedeanClass","instLinearOrder"],["FiniteArchimedeanClass"],["FiniteArchimedeanClass","mk"],["AddGroup","toSubNegMonoid"],["NonAssocRing","toNonUnitalNonAssocRing"],["SemilatticeInf","toPartialOrder"],["DistribSMul","toSMulZeroClass"],["Subtype","instLT"],["SetLike","instMembership"],["FiniteArchimedeanClass","ball"],["LinearOrder"],["DistribMulAction","toDistribSMul"],["AddZeroClass","toAddZero"],["ArchimedeanClass","instOrderTop"],["Ring","toSemiring"],["Top","top"],["AddCommGroup","toAddCommMonoid"],["NegZeroClass","toZero"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["ArchimedeanClass"],["Module"],["SubtractionCommMonoid","toSubtractionMonoid"],["SubNegZeroMonoid","toNegZeroClass"],["PosSMulMono"],["Submodule"],["instDistribLatticeOfLinearOrder"],["MonoidWithZero","toMonoid"],["Zero","toOfNat0"],["Preorder","toLE"],["Eq"],["Lattice","toSemilatticeInf"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["OfNat","ofNat"],["LT","lt"],["IsOrderedRing"],["Module","toDistribMulAction"],["DistribLattice","toLattice"],["SubNegMonoid","toAddMonoid"],["AddCommGroup","toDivisionAddCommMonoid"],["MulZeroClass","toZero"],["Submodule","setLike"],["Archimedean"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["Ne"],["OrderTop","toTop"],["Ring"]],"valueReferences":[["SubtractionMonoid","toSubNegZeroMonoid"],["PartialOrder","toPreorder"],["ArchimedeanClass"],["Membership","mem"],["Preorder","toLT"],["SubtractionCommMonoid","toSubtractionMonoid"],["SubNegZeroMonoid","toNegZeroClass"],["Submodule"],["instDistribLatticeOfLinearOrder"],["ArchimedeanClass","instLinearOrder"],["FiniteArchimedeanClass"],["FiniteArchimedeanClass","mk"],["Zero","toOfNat0"],["Preorder","toLE"],["propext"],["SemilatticeInf","toPartialOrder"],["Subtype","instLT"],["SetLike","instMembership"],["Lattice","toSemilatticeInf"],["FiniteArchimedeanClass","ball"],["FiniteArchimedeanClass","mem_ball_iff"],["OfNat","ofNat"],["ArchimedeanClass","instOrderTop"],["Ring","toSemiring"],["LT","lt"],["DistribLattice","toLattice"],["AddCommGroup","toDivisionAddCommMonoid"],["Submodule","setLike"],["Top","top"],["NegZeroClass","toZero"],["AddCommGroup","toAddCommMonoid"],["Ne"],["OrderTop","toTop"]]},{"isProp":true,"kind":"theorem","name":["ArchimedeanClass","mem_closedBall_iff"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.3 : AddCommGroup.{u_1} M] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.6 : LinearOrder.{u_1} M] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.9 : IsOrderedAddMonoid.{u_1} M (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.3) (PartialOrder.toPreorder.{u_1} M (SemilatticeInf.toPartialOrder.{u_1} M (Lattice.toSemilatticeInf.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.6)))))] (K : Type.{u_2}) [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.13 : Ring.{u_2} K] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.16 : LinearOrder.{u_2} K] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.19 : IsOrderedRing.{u_2} K (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.13) (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.16))))] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.22 : Archimedean.{u_2} K (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonAssocRing.toNonUnitalNonAssocRing.{u_2} K (Ring.toNonAssocRing.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.13)))) (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.16))))] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.25 : Module.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.3)] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.29 : PosSMulMono.{u_2, u_1} K M (SMulZeroClass.toSMul.{u_2, u_1} K M (AddZero.toZero.{u_1} M (AddZeroClass.toAddZero.{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.Order.Module.Archimedean.2781882085._hygCtx._hyg.3)))))) (DistribSMul.toSMulZeroClass.{u_2, u_1} K M (AddMonoid.toAddZeroClass.{u_1} M (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.3)))) (DistribMulAction.toDistribSMul.{u_2, u_1} K M (MonoidWithZero.toMonoid.{u_2} K (Semiring.toMonoidWithZero.{u_2} K (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.13))) (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.3))) (Module.toDistribMulAction.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.25)))) (PartialOrder.toPreorder.{u_2} K (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.16))))) (PartialOrder.toPreorder.{u_1} M (SemilatticeInf.toPartialOrder.{u_1} M (Lattice.toSemilatticeInf.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.6))))) (MulZeroClass.toZero.{u_2} K (NonUnitalNonAssocSemiring.toMulZeroClass.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonAssocRing.toNonUnitalNonAssocRing.{u_2} K (Ring.toNonAssocRing.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.13)))))] {a : M} {c : ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.9}, Iff (Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.25) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.25) M (Submodule.setLike.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.25)) (ArchimedeanClass.closedBall.{u_1, u_2} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.9 K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.22 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.29 c) a) (LE.le.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.9) (Preorder.toLE.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.9) (PartialOrder.toPreorder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.9) (SemilatticeInf.toPartialOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.9) (Lattice.toSemilatticeInf.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.9) (DistribLattice.toLattice.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.9) (instDistribLatticeOfLinearOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.9) (ArchimedeanClass.instLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.9))))))) c (ArchimedeanClass.mk.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.9 a))","typeFull":"∀ {M : Type u_1} [inst : AddCommGroup M] [inst_1 : LinearOrder M] [inst_2 : IsOrderedAddMonoid M] (K : Type u_2)\n [inst_3 : Ring K] [inst_4 : LinearOrder K] [inst_5 : IsOrderedRing K] [inst_6 : Archimedean K] [inst_7 : Module K M]\n [inst_8 : PosSMulMono K M] {a : M} {c : ArchimedeanClass M},\n a ∈ ArchimedeanClass.closedBall K c ↔ c ≤ ArchimedeanClass.mk a","typeReadable":"∀ {M : Type u_1} [inst : AddCommGroup M] [inst_1 : LinearOrder M] [inst_2 : IsOrderedAddMonoid M] (K : Type u_2)\n [inst_3 : Ring K] [inst_4 : LinearOrder K] [inst_5 : IsOrderedRing K] [inst_6 : Archimedean K] [inst_7 : Module K M]\n [inst_8 : PosSMulMono K M] {a : M} {c : ArchimedeanClass M},\n a ∈ ArchimedeanClass.closedBall K c ↔ c ≤ ArchimedeanClass.mk a","typeReferences":[["PartialOrder","toPreorder"],["Ring","toNonAssocRing"],["ArchimedeanClass"],["Module"],["AddCommGroup","toAddGroup"],["Membership","mem"],["SMulZeroClass","toSMul"],["Submodule"],["PosSMulMono"],["instDistribLatticeOfLinearOrder"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["IsOrderedAddMonoid"],["ArchimedeanClass","instLinearOrder"],["MonoidWithZero","toMonoid"],["AddGroup","toSubNegMonoid"],["Preorder","toLE"],["NonAssocRing","toNonUnitalNonAssocRing"],["DistribSMul","toSMulZeroClass"],["SemilatticeInf","toPartialOrder"],["SetLike","instMembership"],["Lattice","toSemilatticeInf"],["ArchimedeanClass","closedBall"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["DistribMulAction","toDistribSMul"],["LinearOrder"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["AddZeroClass","toAddZero"],["ArchimedeanClass","mk"],["Ring","toSemiring"],["IsOrderedRing"],["Module","toDistribMulAction"],["DistribLattice","toLattice"],["SubNegMonoid","toAddMonoid"],["Submodule","setLike"],["MulZeroClass","toZero"],["Iff"],["Archimedean"],["LE","le"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["AddCommGroup","toAddCommMonoid"],["AddZero","toZero"],["Ring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["ArchimedeanClass","mem_closedBallAddSubgroup_iff"]]},{"isProp":true,"kind":"theorem","name":["ArchimedeanClass","ball","congr_simp"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.3 : AddCommGroup.{u_1} M] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.6 : LinearOrder.{u_1} M] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.9 : IsOrderedAddMonoid.{u_1} M (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.3) (PartialOrder.toPreorder.{u_1} M (SemilatticeInf.toPartialOrder.{u_1} M (Lattice.toSemilatticeInf.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.6)))))] (K : Type.{u_2}) [inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.13 : Ring.{u_2} K] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.16 : LinearOrder.{u_2} K] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.19 : IsOrderedRing.{u_2} K (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.13) (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.16))))] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.22 : Archimedean.{u_2} K (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonAssocRing.toNonUnitalNonAssocRing.{u_2} K (Ring.toNonAssocRing.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.13)))) (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.16))))] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.25 : Module.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.3)] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.29 : PosSMulMono.{u_2, u_1} K M (SMulZeroClass.toSMul.{u_2, u_1} K M (AddZero.toZero.{u_1} M (AddZeroClass.toAddZero.{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.Order.Module.Archimedean.224592742._hygCtx._hyg.3)))))) (DistribSMul.toSMulZeroClass.{u_2, u_1} K M (AddMonoid.toAddZeroClass.{u_1} M (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.3)))) (DistribMulAction.toDistribSMul.{u_2, u_1} K M (MonoidWithZero.toMonoid.{u_2} K (Semiring.toMonoidWithZero.{u_2} K (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.13))) (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.3))) (Module.toDistribMulAction.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.25)))) (PartialOrder.toPreorder.{u_2} K (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.16))))) (PartialOrder.toPreorder.{u_1} M (SemilatticeInf.toPartialOrder.{u_1} M (Lattice.toSemilatticeInf.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.6))))) (MulZeroClass.toZero.{u_2} K (NonUnitalNonAssocSemiring.toMulZeroClass.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonAssocRing.toNonUnitalNonAssocRing.{u_2} K (Ring.toNonAssocRing.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.13)))))] (c : ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.9) (c_1 : ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.9), (Eq.{succ u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.9) c c_1) -> (Eq.{succ u_1} (Submodule.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.25) (ArchimedeanClass.ball.{u_1, u_2} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.9 K inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.22 inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.29 c) (ArchimedeanClass.ball.{u_1, u_2} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.9 K inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.22 inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592742._hygCtx._hyg.29 c_1))","typeFull":"∀ {M : Type u_1} [inst : AddCommGroup M] [inst_1 : LinearOrder M] [inst_2 : IsOrderedAddMonoid M] (K : Type u_2)\n [inst_3 : Ring K] [inst_4 : LinearOrder K] [inst_5 : IsOrderedRing K] [inst_6 : Archimedean K] [inst_7 : Module K M]\n [inst_8 : PosSMulMono K M] (c c_1 : ArchimedeanClass M),\n c = c_1 → ArchimedeanClass.ball K c = ArchimedeanClass.ball K c_1","typeReadable":"∀ {M : Type u_1} [inst : AddCommGroup M] [inst_1 : LinearOrder M] [inst_2 : IsOrderedAddMonoid M] (K : Type u_2)\n [inst_3 : Ring K] [inst_4 : LinearOrder K] [inst_5 : IsOrderedRing K] [inst_6 : Archimedean K] [inst_7 : Module K M]\n [inst_8 : PosSMulMono K M] (c c_1 : ArchimedeanClass M),\n c = c_1 → ArchimedeanClass.ball K c = ArchimedeanClass.ball K c_1","typeReferences":[["PartialOrder","toPreorder"],["Ring","toNonAssocRing"],["ArchimedeanClass"],["Module"],["AddCommGroup","toAddGroup"],["SMulZeroClass","toSMul"],["Submodule"],["PosSMulMono"],["instDistribLatticeOfLinearOrder"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["IsOrderedAddMonoid"],["MonoidWithZero","toMonoid"],["AddGroup","toSubNegMonoid"],["Eq"],["NonAssocRing","toNonUnitalNonAssocRing"],["DistribSMul","toSMulZeroClass"],["SemilatticeInf","toPartialOrder"],["Lattice","toSemilatticeInf"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["DistribMulAction","toDistribSMul"],["LinearOrder"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["AddZeroClass","toAddZero"],["Ring","toSemiring"],["IsOrderedRing"],["Module","toDistribMulAction"],["DistribLattice","toLattice"],["SubNegMonoid","toAddMonoid"],["MulZeroClass","toZero"],["ArchimedeanClass","ball"],["Archimedean"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["AddCommGroup","toAddCommMonoid"],["AddZero","toZero"],["Ring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["ArchimedeanClass"],["Eq","refl"],["ArchimedeanClass","ball"],["AddCommGroup","toAddCommMonoid"],["Eq"],["Eq","rec"],["Ring","toSemiring"],["Submodule"]]},{"isProp":true,"kind":"theorem","name":["ArchimedeanClass","mk_le_mk_smul"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Order.Module.Archimedean.1818556423._hygCtx._hyg.3 : AddCommGroup.{u_1} M] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.1818556423._hygCtx._hyg.6 : LinearOrder.{u_1} M] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.1818556423._hygCtx._hyg.9 : IsOrderedAddMonoid.{u_1} M (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.1818556423._hygCtx._hyg.3) (PartialOrder.toPreorder.{u_1} M (SemilatticeInf.toPartialOrder.{u_1} M (Lattice.toSemilatticeInf.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.1818556423._hygCtx._hyg.6)))))] {K : Type.{u_2}} [inst._@.Mathlib.Algebra.Order.Module.Archimedean.1818556423._hygCtx._hyg.13 : Ring.{u_2} K] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.1818556423._hygCtx._hyg.16 : LinearOrder.{u_2} K] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.1818556423._hygCtx._hyg.19 : IsOrderedRing.{u_2} K (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.1818556423._hygCtx._hyg.13) (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.1818556423._hygCtx._hyg.16))))] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.1818556423._hygCtx._hyg.22 : Archimedean.{u_2} K (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonAssocRing.toNonUnitalNonAssocRing.{u_2} K (Ring.toNonAssocRing.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.1818556423._hygCtx._hyg.13)))) (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.1818556423._hygCtx._hyg.16))))] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.1818556423._hygCtx._hyg.25 : Module.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.1818556423._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.1818556423._hygCtx._hyg.3)] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.1818556423._hygCtx._hyg.29 : PosSMulMono.{u_2, u_1} K M (SMulZeroClass.toSMul.{u_2, u_1} K M (AddZero.toZero.{u_1} M (AddZeroClass.toAddZero.{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.Order.Module.Archimedean.1818556423._hygCtx._hyg.3)))))) (DistribSMul.toSMulZeroClass.{u_2, u_1} K M (AddMonoid.toAddZeroClass.{u_1} M (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.1818556423._hygCtx._hyg.3)))) (DistribMulAction.toDistribSMul.{u_2, u_1} K M (MonoidWithZero.toMonoid.{u_2} K (Semiring.toMonoidWithZero.{u_2} K (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.1818556423._hygCtx._hyg.13))) (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.1818556423._hygCtx._hyg.3))) (Module.toDistribMulAction.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.1818556423._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.1818556423._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.1818556423._hygCtx._hyg.25)))) (PartialOrder.toPreorder.{u_2} K (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.1818556423._hygCtx._hyg.16))))) (PartialOrder.toPreorder.{u_1} M (SemilatticeInf.toPartialOrder.{u_1} M (Lattice.toSemilatticeInf.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.1818556423._hygCtx._hyg.6))))) (MulZeroClass.toZero.{u_2} K (NonUnitalNonAssocSemiring.toMulZeroClass.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonAssocRing.toNonUnitalNonAssocRing.{u_2} K (Ring.toNonAssocRing.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.1818556423._hygCtx._hyg.13)))))] (a : M) (k : K), LE.le.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.1818556423._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.1818556423._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.1818556423._hygCtx._hyg.9) (Preorder.toLE.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.1818556423._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.1818556423._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.1818556423._hygCtx._hyg.9) (PartialOrder.toPreorder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.1818556423._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.1818556423._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.1818556423._hygCtx._hyg.9) (SemilatticeInf.toPartialOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.1818556423._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.1818556423._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.1818556423._hygCtx._hyg.9) (Lattice.toSemilatticeInf.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.1818556423._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.1818556423._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.1818556423._hygCtx._hyg.9) (DistribLattice.toLattice.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.1818556423._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.1818556423._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.1818556423._hygCtx._hyg.9) (instDistribLatticeOfLinearOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.1818556423._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.1818556423._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.1818556423._hygCtx._hyg.9) (ArchimedeanClass.instLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.1818556423._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.1818556423._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.1818556423._hygCtx._hyg.9))))))) (ArchimedeanClass.mk.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.1818556423._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.1818556423._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.1818556423._hygCtx._hyg.9 a) (ArchimedeanClass.mk.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.1818556423._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.1818556423._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.1818556423._hygCtx._hyg.9 (HSMul.hSMul.{u_2, u_1, u_1} K M M (instHSMul.{u_2, u_1} K M (SMulZeroClass.toSMul.{u_2, u_1} K M (AddZero.toZero.{u_1} M (AddZeroClass.toAddZero.{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.Order.Module.Archimedean.1818556423._hygCtx._hyg.3)))))) (DistribSMul.toSMulZeroClass.{u_2, u_1} K M (AddMonoid.toAddZeroClass.{u_1} M (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.1818556423._hygCtx._hyg.3)))) (DistribMulAction.toDistribSMul.{u_2, u_1} K M (MonoidWithZero.toMonoid.{u_2} K (Semiring.toMonoidWithZero.{u_2} K (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.1818556423._hygCtx._hyg.13))) (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.1818556423._hygCtx._hyg.3))) (Module.toDistribMulAction.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.1818556423._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.1818556423._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.1818556423._hygCtx._hyg.25))))) k a))","typeFull":"∀ {M : Type u_1} [inst : AddCommGroup M] [inst_1 : LinearOrder M] [inst_2 : IsOrderedAddMonoid M] {K : Type u_2}\n [inst_3 : Ring K] [inst_4 : LinearOrder K] [IsOrderedRing K] [Archimedean K] [inst_7 : Module K M] [PosSMulMono K M]\n (a : M) (k : K), ArchimedeanClass.mk a ≤ ArchimedeanClass.mk (k • a)","typeReadable":"∀ {M : Type u_1} [inst : AddCommGroup M] [inst_1 : LinearOrder M] [inst_2 : IsOrderedAddMonoid M] {K : Type u_2}\n [inst_3 : Ring K] [inst_4 : LinearOrder K] [IsOrderedRing K] [Archimedean K] [inst_7 : Module K M] [PosSMulMono K M]\n (a : M) (k : K), ArchimedeanClass.mk a ≤ ArchimedeanClass.mk (k • a)","typeReferences":[["PartialOrder","toPreorder"],["Ring","toNonAssocRing"],["ArchimedeanClass"],["Module"],["AddCommGroup","toAddGroup"],["SMulZeroClass","toSMul"],["PosSMulMono"],["instDistribLatticeOfLinearOrder"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["IsOrderedAddMonoid"],["ArchimedeanClass","instLinearOrder"],["MonoidWithZero","toMonoid"],["instHSMul"],["AddGroup","toSubNegMonoid"],["Preorder","toLE"],["NonAssocRing","toNonUnitalNonAssocRing"],["DistribSMul","toSMulZeroClass"],["SemilatticeInf","toPartialOrder"],["Lattice","toSemilatticeInf"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["DistribMulAction","toDistribSMul"],["LinearOrder"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["AddZeroClass","toAddZero"],["ArchimedeanClass","mk"],["Ring","toSemiring"],["IsOrderedRing"],["Module","toDistribMulAction"],["DistribLattice","toLattice"],["SubNegMonoid","toAddMonoid"],["MulZeroClass","toZero"],["Archimedean"],["HSMul","hSMul"],["LE","le"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["AddCommGroup","toAddCommMonoid"],["AddZero","toZero"],["Ring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["SubtractionMonoid","toSubNegZeroMonoid"],["PartialOrder","toPreorder"],["Ring","toNonAssocRing"],["Eq","trans"],["AddCommGroup","toAddGroup"],["SMulZeroClass","toSMul"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["not_false_eq_true"],["ArchimedeanClass","instLinearOrder"],["Eq","symm"],["AddGroup","toSubNegMonoid"],["Eq","ndrec"],["NonAssocRing","toNonUnitalNonAssocRing"],["SemilatticeInf","toPartialOrder"],["DistribSMul","toSMulZeroClass"],["DistribMulAction","toDistribSMul"],["AddZeroClass","toAddZero"],["Ring","toSemiring"],["ArchimedeanClass","instOrderTop"],["eq_false"],["HSMul","hSMul"],["AddCommGroup","toAddCommMonoid"],["Top","top"],["NegZeroClass","toZero"],["zero_smul"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["ArchimedeanClass","mk_smul"],["ArchimedeanClass"],["SubtractionCommMonoid","toSubtractionMonoid"],["eq_or_ne"],["SubNegZeroMonoid","toNegZeroClass"],["congrArg"],["instDistribLatticeOfLinearOrder"],["le_top","_simp_2"],["MonoidWithZero","toMonoid"],["instHSMul"],["Std","le_refl","_simp_1"],["Zero","toOfNat0"],["Eq"],["Preorder","toLE"],["MulActionWithZero","toSMulWithZero"],["Not"],["Lattice","toSemilatticeInf"],["True"],["Semiring","toMonoidWithZero"],["Module","toMulActionWithZero"],["instReflLe"],["ArchimedeanClass","mk"],["OfNat","ofNat"],["Or","casesOn"],["Module","toDistribMulAction"],["DistribLattice","toLattice"],["AddCommGroup","toDivisionAddCommMonoid"],["SubNegMonoid","toAddMonoid"],["of_eq_true"],["MulZeroClass","toZero"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["LE","le"],["False"],["Ne"],["OrderTop","toTop"]]},{"isProp":false,"kind":"definition","name":["FiniteArchimedeanClass","submodule"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 : AddCommGroup.{u_1} M] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 : LinearOrder.{u_1} M] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9 : IsOrderedAddMonoid.{u_1} M (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3) (PartialOrder.toPreorder.{u_1} M (SemilatticeInf.toPartialOrder.{u_1} M (Lattice.toSemilatticeInf.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6)))))] (K : Type.{u_2}) [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.13 : Ring.{u_2} K] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.16 : LinearOrder.{u_2} K] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.19 : IsOrderedRing.{u_2} K (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.13) (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.16))))] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.22 : Archimedean.{u_2} K (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonAssocRing.toNonUnitalNonAssocRing.{u_2} K (Ring.toNonAssocRing.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.13)))) (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.16))))] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.25 : Module.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3)] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.29 : PosSMulMono.{u_2, u_1} K M (SMulZeroClass.toSMul.{u_2, u_1} K M (AddZero.toZero.{u_1} M (AddZeroClass.toAddZero.{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.Order.Module.Archimedean.3843450102._hygCtx._hyg.3)))))) (DistribSMul.toSMulZeroClass.{u_2, u_1} K M (AddMonoid.toAddZeroClass.{u_1} M (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3)))) (DistribMulAction.toDistribSMul.{u_2, u_1} K M (MonoidWithZero.toMonoid.{u_2} K (Semiring.toMonoidWithZero.{u_2} K (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.13))) (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3))) (Module.toDistribMulAction.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.25)))) (PartialOrder.toPreorder.{u_2} K (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.16))))) (PartialOrder.toPreorder.{u_1} M (SemilatticeInf.toPartialOrder.{u_1} M (Lattice.toSemilatticeInf.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6))))) (MulZeroClass.toZero.{u_2} K (NonUnitalNonAssocSemiring.toMulZeroClass.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonAssocRing.toNonUnitalNonAssocRing.{u_2} K (Ring.toNonAssocRing.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.13)))))], (UpperSet.{u_1} (FiniteArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (Subtype.instLE.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (Preorder.toLE.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (PartialOrder.toPreorder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (SemilatticeInf.toPartialOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (Lattice.toSemilatticeInf.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (DistribLattice.toLattice.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (instDistribLatticeOfLinearOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (ArchimedeanClass.instLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9))))))) (fun (A : ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) => Ne.{succ u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) A (Top.top.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (OrderTop.toTop.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (Preorder.toLE.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (PartialOrder.toPreorder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (SemilatticeInf.toPartialOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (Lattice.toSemilatticeInf.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (DistribLattice.toLattice.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (instDistribLatticeOfLinearOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (ArchimedeanClass.instLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9))))))) (ArchimedeanClass.instOrderTop.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9)))))) -> (Submodule.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.25)","typeFull":"{M : Type u_1} →\n [inst : AddCommGroup M] →\n [inst_1 : LinearOrder M] →\n [inst_2 : IsOrderedAddMonoid M] →\n (K : Type u_2) →\n [inst_3 : Ring K] →\n [inst_4 : LinearOrder K] →\n [IsOrderedRing K] →\n [Archimedean K] →\n [inst_7 : Module K M] → [PosSMulMono K M] → UpperSet (FiniteArchimedeanClass M) → Submodule K M","typeReadable":"{M : Type u_1} →\n [inst : AddCommGroup M] →\n [inst_1 : LinearOrder M] →\n [inst_2 : IsOrderedAddMonoid M] →\n (K : Type u_2) →\n [inst_3 : Ring K] →\n [inst_4 : LinearOrder K] →\n [IsOrderedRing K] →\n [Archimedean K] →\n [inst_7 : Module K M] → [PosSMulMono K M] → UpperSet (FiniteArchimedeanClass M) → Submodule K M","typeReferences":[["PartialOrder","toPreorder"],["Ring","toNonAssocRing"],["ArchimedeanClass"],["Subtype","instLE"],["Module"],["AddCommGroup","toAddGroup"],["SMulZeroClass","toSMul"],["Submodule"],["PosSMulMono"],["instDistribLatticeOfLinearOrder"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["IsOrderedAddMonoid"],["ArchimedeanClass","instLinearOrder"],["MonoidWithZero","toMonoid"],["FiniteArchimedeanClass"],["AddGroup","toSubNegMonoid"],["Preorder","toLE"],["NonAssocRing","toNonUnitalNonAssocRing"],["DistribSMul","toSMulZeroClass"],["SemilatticeInf","toPartialOrder"],["Lattice","toSemilatticeInf"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["DistribMulAction","toDistribSMul"],["LinearOrder"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["AddZeroClass","toAddZero"],["UpperSet"],["ArchimedeanClass","instOrderTop"],["Ring","toSemiring"],["IsOrderedRing"],["Module","toDistribMulAction"],["DistribLattice","toLattice"],["SubNegMonoid","toAddMonoid"],["MulZeroClass","toZero"],["Archimedean"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["Top","top"],["AddCommGroup","toAddCommMonoid"],["Ne"],["OrderTop","toTop"],["AddZero","toZero"],["Ring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["AddSubgroup","toAddSubmonoid"],["FiniteArchimedeanClass","submodule","_proof_2"],["Submodule","mk"],["AddCommGroup","toAddGroup"],["AddCommGroup","toAddCommMonoid"],["FiniteArchimedeanClass","addSubgroup"],["Ring","toSemiring"]]},{"isProp":true,"kind":"theorem","name":["ArchimedeanClass","mem_ball_iff"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.3 : AddCommGroup.{u_1} M] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.6 : LinearOrder.{u_1} M] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.9 : IsOrderedAddMonoid.{u_1} M (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.3) (PartialOrder.toPreorder.{u_1} M (SemilatticeInf.toPartialOrder.{u_1} M (Lattice.toSemilatticeInf.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.6)))))] (K : Type.{u_2}) [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.13 : Ring.{u_2} K] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.16 : LinearOrder.{u_2} K] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.19 : IsOrderedRing.{u_2} K (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.13) (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.16))))] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.22 : Archimedean.{u_2} K (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonAssocRing.toNonUnitalNonAssocRing.{u_2} K (Ring.toNonAssocRing.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.13)))) (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.16))))] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.25 : Module.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.3)] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.29 : PosSMulMono.{u_2, u_1} K M (SMulZeroClass.toSMul.{u_2, u_1} K M (AddZero.toZero.{u_1} M (AddZeroClass.toAddZero.{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.Order.Module.Archimedean.2496125325._hygCtx._hyg.3)))))) (DistribSMul.toSMulZeroClass.{u_2, u_1} K M (AddMonoid.toAddZeroClass.{u_1} M (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.3)))) (DistribMulAction.toDistribSMul.{u_2, u_1} K M (MonoidWithZero.toMonoid.{u_2} K (Semiring.toMonoidWithZero.{u_2} K (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.13))) (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.3))) (Module.toDistribMulAction.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.25)))) (PartialOrder.toPreorder.{u_2} K (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.16))))) (PartialOrder.toPreorder.{u_1} M (SemilatticeInf.toPartialOrder.{u_1} M (Lattice.toSemilatticeInf.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.6))))) (MulZeroClass.toZero.{u_2} K (NonUnitalNonAssocSemiring.toMulZeroClass.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonAssocRing.toNonUnitalNonAssocRing.{u_2} K (Ring.toNonAssocRing.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.13)))))] {a : M} {c : ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.9}, (Ne.{succ u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.9) c (Top.top.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.9) (OrderTop.toTop.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.9) (Preorder.toLE.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.9) (PartialOrder.toPreorder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.9) (SemilatticeInf.toPartialOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.9) (Lattice.toSemilatticeInf.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.9) (DistribLattice.toLattice.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.9) (instDistribLatticeOfLinearOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.9) (ArchimedeanClass.instLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.9))))))) (ArchimedeanClass.instOrderTop.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.9)))) -> (Iff (Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.25) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.25) M (Submodule.setLike.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.25)) (ArchimedeanClass.ball.{u_1, u_2} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.9 K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.22 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.29 c) a) (LT.lt.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.9) (Preorder.toLT.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.9) (PartialOrder.toPreorder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.9) (SemilatticeInf.toPartialOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.9) (Lattice.toSemilatticeInf.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.9) (DistribLattice.toLattice.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.9) (instDistribLatticeOfLinearOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.9) (ArchimedeanClass.instLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.9))))))) c (ArchimedeanClass.mk.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125325._hygCtx._hyg.9 a)))","typeFull":"∀ {M : Type u_1} [inst : AddCommGroup M] [inst_1 : LinearOrder M] [inst_2 : IsOrderedAddMonoid M] (K : Type u_2)\n [inst_3 : Ring K] [inst_4 : LinearOrder K] [inst_5 : IsOrderedRing K] [inst_6 : Archimedean K] [inst_7 : Module K M]\n [inst_8 : PosSMulMono K M] {a : M} {c : ArchimedeanClass M},\n c ≠ ⊤ → (a ∈ ArchimedeanClass.ball K c ↔ c < ArchimedeanClass.mk a)","typeReadable":"∀ {M : Type u_1} [inst : AddCommGroup M] [inst_1 : LinearOrder M] [inst_2 : IsOrderedAddMonoid M] (K : Type u_2)\n [inst_3 : Ring K] [inst_4 : LinearOrder K] [inst_5 : IsOrderedRing K] [inst_6 : Archimedean K] [inst_7 : Module K M]\n [inst_8 : PosSMulMono K M] {a : M} {c : ArchimedeanClass M},\n c ≠ ⊤ → (a ∈ ArchimedeanClass.ball K c ↔ c < ArchimedeanClass.mk a)","typeReferences":[["Ring","toNonAssocRing"],["PartialOrder","toPreorder"],["AddCommGroup","toAddGroup"],["Membership","mem"],["Preorder","toLT"],["SMulZeroClass","toSMul"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["IsOrderedAddMonoid"],["ArchimedeanClass","instLinearOrder"],["AddGroup","toSubNegMonoid"],["NonAssocRing","toNonUnitalNonAssocRing"],["SemilatticeInf","toPartialOrder"],["DistribSMul","toSMulZeroClass"],["SetLike","instMembership"],["LinearOrder"],["DistribMulAction","toDistribSMul"],["AddZeroClass","toAddZero"],["Ring","toSemiring"],["ArchimedeanClass","instOrderTop"],["Iff"],["ArchimedeanClass","ball"],["AddCommGroup","toAddCommMonoid"],["Top","top"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Module"],["ArchimedeanClass"],["PosSMulMono"],["Submodule"],["instDistribLatticeOfLinearOrder"],["MonoidWithZero","toMonoid"],["Preorder","toLE"],["Lattice","toSemilatticeInf"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["ArchimedeanClass","mk"],["LT","lt"],["IsOrderedRing"],["Module","toDistribMulAction"],["DistribLattice","toLattice"],["SubNegMonoid","toAddMonoid"],["MulZeroClass","toZero"],["Submodule","setLike"],["Archimedean"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["Ne"],["OrderTop","toTop"],["Ring"]],"valueReferences":[["ArchimedeanClass","mem_ballAddSubgroup_iff"]]},{"isProp":true,"kind":"theorem","name":["ArchimedeanClass","toAddSubgroup_closedBall"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019055._hygCtx._hyg.3 : AddCommGroup.{u_1} M] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019055._hygCtx._hyg.6 : LinearOrder.{u_1} M] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019055._hygCtx._hyg.9 : IsOrderedAddMonoid.{u_1} M (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019055._hygCtx._hyg.3) (PartialOrder.toPreorder.{u_1} M (SemilatticeInf.toPartialOrder.{u_1} M (Lattice.toSemilatticeInf.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019055._hygCtx._hyg.6)))))] (K : Type.{u_2}) [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019055._hygCtx._hyg.13 : Ring.{u_2} K] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019055._hygCtx._hyg.16 : LinearOrder.{u_2} K] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019055._hygCtx._hyg.19 : IsOrderedRing.{u_2} K (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019055._hygCtx._hyg.13) (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019055._hygCtx._hyg.16))))] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019055._hygCtx._hyg.22 : Archimedean.{u_2} K (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonAssocRing.toNonUnitalNonAssocRing.{u_2} K (Ring.toNonAssocRing.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019055._hygCtx._hyg.13)))) (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019055._hygCtx._hyg.16))))] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019055._hygCtx._hyg.25 : Module.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019055._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019055._hygCtx._hyg.3)] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019055._hygCtx._hyg.29 : PosSMulMono.{u_2, u_1} K M (SMulZeroClass.toSMul.{u_2, u_1} K M (AddZero.toZero.{u_1} M (AddZeroClass.toAddZero.{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.Order.Module.Archimedean.2590019055._hygCtx._hyg.3)))))) (DistribSMul.toSMulZeroClass.{u_2, u_1} K M (AddMonoid.toAddZeroClass.{u_1} M (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019055._hygCtx._hyg.3)))) (DistribMulAction.toDistribSMul.{u_2, u_1} K M (MonoidWithZero.toMonoid.{u_2} K (Semiring.toMonoidWithZero.{u_2} K (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019055._hygCtx._hyg.13))) (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019055._hygCtx._hyg.3))) (Module.toDistribMulAction.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019055._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019055._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019055._hygCtx._hyg.25)))) (PartialOrder.toPreorder.{u_2} K (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019055._hygCtx._hyg.16))))) (PartialOrder.toPreorder.{u_1} M (SemilatticeInf.toPartialOrder.{u_1} M (Lattice.toSemilatticeInf.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019055._hygCtx._hyg.6))))) (MulZeroClass.toZero.{u_2} K (NonUnitalNonAssocSemiring.toMulZeroClass.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonAssocRing.toNonUnitalNonAssocRing.{u_2} K (Ring.toNonAssocRing.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019055._hygCtx._hyg.13)))))] (c : ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019055._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019055._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019055._hygCtx._hyg.9), Eq.{succ u_1} (AddSubgroup.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019055._hygCtx._hyg.3)) (Submodule.toAddSubgroup.{u_2, u_1} K M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019055._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019055._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019055._hygCtx._hyg.25 (ArchimedeanClass.closedBall.{u_1, u_2} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019055._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019055._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019055._hygCtx._hyg.9 K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019055._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019055._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019055._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019055._hygCtx._hyg.22 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019055._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019055._hygCtx._hyg.29 c)) (ArchimedeanClass.closedBallAddSubgroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019055._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019055._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019055._hygCtx._hyg.9 c)","typeFull":"∀ {M : Type u_1} [inst : AddCommGroup M] [inst_1 : LinearOrder M] [inst_2 : IsOrderedAddMonoid M] (K : Type u_2)\n [inst_3 : Ring K] [inst_4 : LinearOrder K] [inst_5 : IsOrderedRing K] [inst_6 : Archimedean K] [inst_7 : Module K M]\n [inst_8 : PosSMulMono K M] (c : ArchimedeanClass M),\n (ArchimedeanClass.closedBall K c).toAddSubgroup = c.closedBallAddSubgroup","typeReadable":"∀ {M : Type u_1} [inst : AddCommGroup M] [inst_1 : LinearOrder M] [inst_2 : IsOrderedAddMonoid M] (K : Type u_2)\n [inst_3 : Ring K] [inst_4 : LinearOrder K] [inst_5 : IsOrderedRing K] [inst_6 : Archimedean K] [inst_7 : Module K M]\n [inst_8 : PosSMulMono K M] (c : ArchimedeanClass M),\n (ArchimedeanClass.closedBall K c).toAddSubgroup = c.closedBallAddSubgroup","typeReferences":[["PartialOrder","toPreorder"],["Ring","toNonAssocRing"],["ArchimedeanClass"],["Module"],["AddCommGroup","toAddGroup"],["SMulZeroClass","toSMul"],["PosSMulMono"],["instDistribLatticeOfLinearOrder"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["IsOrderedAddMonoid"],["MonoidWithZero","toMonoid"],["AddGroup","toSubNegMonoid"],["Eq"],["NonAssocRing","toNonUnitalNonAssocRing"],["DistribSMul","toSMulZeroClass"],["SemilatticeInf","toPartialOrder"],["Lattice","toSemilatticeInf"],["ArchimedeanClass","closedBall"],["ArchimedeanClass","closedBallAddSubgroup"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["DistribMulAction","toDistribSMul"],["LinearOrder"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["Submodule","toAddSubgroup"],["AddZeroClass","toAddZero"],["Ring","toSemiring"],["AddSubgroup"],["IsOrderedRing"],["Module","toDistribMulAction"],["DistribLattice","toLattice"],["SubNegMonoid","toAddMonoid"],["MulZeroClass","toZero"],["Archimedean"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["AddCommGroup","toAddCommMonoid"],["AddZero","toZero"],["Ring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["rfl"],["AddSubgroup"],["ArchimedeanClass","closedBall"],["AddCommGroup","toAddGroup"],["Submodule","toAddSubgroup"]]},{"isProp":true,"kind":"theorem","name":["ArchimedeanClass","submodule","_proof_2"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.3 : AddCommGroup.{u_1} M] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.6 : LinearOrder.{u_1} M] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.9 : IsOrderedAddMonoid.{u_1} M (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.3) (PartialOrder.toPreorder.{u_1} M (SemilatticeInf.toPartialOrder.{u_1} M (Lattice.toSemilatticeInf.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.6)))))] (K : Type.{u_2}) [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.13 : Ring.{u_2} K] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.16 : LinearOrder.{u_2} K] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.19 : IsOrderedRing.{u_2} K (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.13) (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.16))))] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.22 : Archimedean.{u_2} K (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonAssocRing.toNonUnitalNonAssocRing.{u_2} K (Ring.toNonAssocRing.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.13)))) (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.16))))] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.25 : Module.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.3)] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.29 : PosSMulMono.{u_2, u_1} K M (SMulZeroClass.toSMul.{u_2, u_1} K M (AddZero.toZero.{u_1} M (AddZeroClass.toAddZero.{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.Order.Module.Archimedean.3843450101._hygCtx._hyg.3)))))) (DistribSMul.toSMulZeroClass.{u_2, u_1} K M (AddMonoid.toAddZeroClass.{u_1} M (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.3)))) (DistribMulAction.toDistribSMul.{u_2, u_1} K M (MonoidWithZero.toMonoid.{u_2} K (Semiring.toMonoidWithZero.{u_2} K (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.13))) (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.3))) (Module.toDistribMulAction.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.25)))) (PartialOrder.toPreorder.{u_2} K (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.16))))) (PartialOrder.toPreorder.{u_1} M (SemilatticeInf.toPartialOrder.{u_1} M (Lattice.toSemilatticeInf.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.6))))) (MulZeroClass.toZero.{u_2} K (NonUnitalNonAssocSemiring.toMulZeroClass.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonAssocRing.toNonUnitalNonAssocRing.{u_2} K (Ring.toNonAssocRing.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.13)))))] (s : UpperSet.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.9) (Preorder.toLE.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.9) (PartialOrder.toPreorder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.9) (SemilatticeInf.toPartialOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.9) (Lattice.toSemilatticeInf.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.9) (DistribLattice.toLattice.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.9) (instDistribLatticeOfLinearOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.9) (ArchimedeanClass.instLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.9)))))))) (k : K) {a : 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 (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.3)))))) (AddSubmonoid.toAddSubsemigroup.{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.Order.Module.Archimedean.3843450101._hygCtx._hyg.3)))) (AddSubgroup.toAddSubmonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.3) (ArchimedeanClass.addSubgroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.9 s)))) a) -> (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 (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.3)))))) (AddSubmonoid.toAddSubsemigroup.{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.Order.Module.Archimedean.3843450101._hygCtx._hyg.3)))) (AddSubgroup.toAddSubmonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.3) (ArchimedeanClass.addSubgroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.9 s)))) (HSMul.hSMul.{u_2, u_1, u_1} K M M (instHSMul.{u_2, u_1} K M (SMulZeroClass.toSMul.{u_2, u_1} K 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.Order.Module.Archimedean.3843450101._hygCtx._hyg.3))))) (DistribSMul.toSMulZeroClass.{u_2, u_1} K M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.3))) (DistribMulAction.toDistribSMul.{u_2, u_1} K M (MonoidWithZero.toMonoid.{u_2} K (Semiring.toMonoidWithZero.{u_2} K (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.13))) (AddCommMonoid.toAddMonoid.{u_1} M (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.3)) (Module.toDistribMulAction.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450101._hygCtx._hyg.25))))) k a))","typeFull":"∀ {M : Type u_1} [inst : AddCommGroup M] [inst_1 : LinearOrder M] [inst_2 : IsOrderedAddMonoid M] (K : Type u_2)\n [inst_3 : Ring K] [inst_4 : LinearOrder K] [IsOrderedRing K] [Archimedean K] [inst_7 : Module K M] [PosSMulMono K M]\n (s : UpperSet (ArchimedeanClass M)) (k : K) {a : M},\n a ∈ (ArchimedeanClass.addSubgroup s).carrier → k • a ∈ (ArchimedeanClass.addSubgroup s).carrier","typeReadable":"∀ {M : Type u_1} [inst : AddCommGroup M] [inst_1 : LinearOrder M] [inst_2 : IsOrderedAddMonoid M] (K : Type u_2)\n [inst_3 : Ring K] [inst_4 : LinearOrder K] [IsOrderedRing K] [Archimedean K] [inst_7 : Module K M] [PosSMulMono K M]\n (s : UpperSet (ArchimedeanClass M)) (k : K) {a : M},\n a ∈ (ArchimedeanClass.addSubgroup s).carrier → k • a ∈ (ArchimedeanClass.addSubgroup s).carrier","typeReferences":[["PartialOrder","toPreorder"],["Ring","toNonAssocRing"],["ArchimedeanClass"],["Module"],["AddCommGroup","toAddGroup"],["Membership","mem"],["ArchimedeanClass","addSubgroup"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["PosSMulMono"],["AddSubsemigroup","carrier"],["instDistribLatticeOfLinearOrder"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["AddSubgroup","toAddSubmonoid"],["IsOrderedAddMonoid"],["ArchimedeanClass","instLinearOrder"],["MonoidWithZero","toMonoid"],["instHSMul"],["AddGroup","toSubNegMonoid"],["Preorder","toLE"],["NonAssocRing","toNonUnitalNonAssocRing"],["DistribSMul","toSMulZeroClass"],["SemilatticeInf","toPartialOrder"],["Lattice","toSemilatticeInf"],["Set"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["DistribMulAction","toDistribSMul"],["LinearOrder"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["AddZero","toAdd"],["AddZeroClass","toAddZero"],["UpperSet"],["Set","instMembership"],["Ring","toSemiring"],["IsOrderedRing"],["Module","toDistribMulAction"],["DistribLattice","toLattice"],["SubNegMonoid","toAddMonoid"],["MulZeroClass","toZero"],["Archimedean"],["HSMul","hSMul"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["AddCommGroup","toAddCommMonoid"],["AddSubmonoid","toAddSubsemigroup"],["AddZero","toZero"],["Ring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["UpperSet","upper"],["implies_congr"],["PartialOrder","toPreorder"],["Eq","trans"],["AddCommGroup","toAddGroup"],["Membership","mem"],["ArchimedeanClass","addSubgroup"],["SMulZeroClass","toSMul"],["AddSubgroup","toAddSubmonoid"],["AddSubmonoidClass","toZeroMemClass"],["ArchimedeanClass","instLinearOrder"],["AddSubsemigroup"],["Eq","symm"],["AddGroup","toSubNegMonoid"],["Eq","ndrec"],["SemilatticeInf","toPartialOrder"],["DistribSMul","toSMulZeroClass"],["SetLike","instMembership"],["AddSubgroup","instSetLike"],["AddSubmonoid","instSetLike"],["DistribMulAction","toDistribSMul"],["AddZeroClass","toAddZero"],["Bot","bot"],["Set","instMembership"],["Ring","toSemiring"],["AddSubmonoid","mem_toSubsemigroup","_simp_1"],["AddSubsemigroup","mem_carrier","_simp_1"],["AddSubgroup"],["AddSubgroup","mem_toAddSubmonoid","_simp_1"],["SetLike","coe"],["ArchimedeanClass","mem_addSubgroup_iff"],["HSMul","hSMul"],["id"],["Top","top"],["AddCommGroup","toAddCommMonoid"],["Eq","mpr"],["AddSubmonoid","toAddSubsemigroup"],["UpperSet","instSetLike"],["UpperSet","instTop"],["AddZero","toZero"],["AddSubmonoid","instAddSubmonoidClass"],["AddMonoid","toAddZeroClass"],["smul_zero"],["Eq","mp"],["ArchimedeanClass"],["AddCommMonoid","toAddMonoid"],["eq_or_ne"],["congrArg"],["AddSubsemigroup","carrier"],["instDistribLatticeOfLinearOrder"],["congr"],["MonoidWithZero","toMonoid"],["instHSMul"],["congrFun'"],["Zero","toOfNat0"],["ArchimedeanClass","mk_le_mk_smul"],["Eq"],["Preorder","toLE"],["propext"],["Lattice","toSemilatticeInf"],["True"],["Set"],["Semiring","toMonoidWithZero"],["AddSubsemigroup","instSetLike"],["AddZero","toAdd"],["UpperSet"],["ArchimedeanClass","mk"],["OfNat","ofNat"],["Or","casesOn"],["ZeroMemClass","zero_mem","_simp_1"],["Module","toDistribMulAction"],["DistribLattice","toLattice"],["SetLike","mem_coe","_simp_1"],["SubNegMonoid","toAddMonoid"],["of_eq_true"],["AddSubmonoid"],["AddSubmonoid","mem_bot","_simp_1"],["ArchimedeanClass","addSubgroup_eq_bot"],["AddSubmonoid","instBot"],["Ne"],["AddSubgroup","instBot"]]},{"isProp":false,"kind":"definition","name":["FiniteArchimedeanClass","closedBall"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252811._hygCtx._hyg.3 : AddCommGroup.{u_1} M] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252811._hygCtx._hyg.6 : LinearOrder.{u_1} M] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252811._hygCtx._hyg.9 : IsOrderedAddMonoid.{u_1} M (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252811._hygCtx._hyg.3) (PartialOrder.toPreorder.{u_1} M (SemilatticeInf.toPartialOrder.{u_1} M (Lattice.toSemilatticeInf.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252811._hygCtx._hyg.6)))))] (K : Type.{u_2}) [inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252811._hygCtx._hyg.13 : Ring.{u_2} K] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252811._hygCtx._hyg.16 : LinearOrder.{u_2} K] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252811._hygCtx._hyg.19 : IsOrderedRing.{u_2} K (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252811._hygCtx._hyg.13) (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252811._hygCtx._hyg.16))))] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252811._hygCtx._hyg.22 : Archimedean.{u_2} K (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonAssocRing.toNonUnitalNonAssocRing.{u_2} K (Ring.toNonAssocRing.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252811._hygCtx._hyg.13)))) (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252811._hygCtx._hyg.16))))] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252811._hygCtx._hyg.25 : Module.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252811._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252811._hygCtx._hyg.3)] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252811._hygCtx._hyg.29 : PosSMulMono.{u_2, u_1} K M (SMulZeroClass.toSMul.{u_2, u_1} K M (AddZero.toZero.{u_1} M (AddZeroClass.toAddZero.{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.Order.Module.Archimedean.113252811._hygCtx._hyg.3)))))) (DistribSMul.toSMulZeroClass.{u_2, u_1} K M (AddMonoid.toAddZeroClass.{u_1} M (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252811._hygCtx._hyg.3)))) (DistribMulAction.toDistribSMul.{u_2, u_1} K M (MonoidWithZero.toMonoid.{u_2} K (Semiring.toMonoidWithZero.{u_2} K (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252811._hygCtx._hyg.13))) (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252811._hygCtx._hyg.3))) (Module.toDistribMulAction.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252811._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252811._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252811._hygCtx._hyg.25)))) (PartialOrder.toPreorder.{u_2} K (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252811._hygCtx._hyg.16))))) (PartialOrder.toPreorder.{u_1} M (SemilatticeInf.toPartialOrder.{u_1} M (Lattice.toSemilatticeInf.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252811._hygCtx._hyg.6))))) (MulZeroClass.toZero.{u_2} K (NonUnitalNonAssocSemiring.toMulZeroClass.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonAssocRing.toNonUnitalNonAssocRing.{u_2} K (Ring.toNonAssocRing.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252811._hygCtx._hyg.13)))))], (FiniteArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252811._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252811._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252811._hygCtx._hyg.9) -> (Submodule.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252811._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252811._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252811._hygCtx._hyg.25)","typeFull":"{M : Type u_1} →\n [inst : AddCommGroup M] →\n [inst_1 : LinearOrder M] →\n [inst_2 : IsOrderedAddMonoid M] →\n (K : Type u_2) →\n [inst_3 : Ring K] →\n [inst_4 : LinearOrder K] →\n [IsOrderedRing K] →\n [Archimedean K] → [inst_7 : Module K M] → [PosSMulMono K M] → FiniteArchimedeanClass M → Submodule K M","typeReadable":"{M : Type u_1} →\n [inst : AddCommGroup M] →\n [inst_1 : LinearOrder M] →\n [inst_2 : IsOrderedAddMonoid M] →\n (K : Type u_2) →\n [inst_3 : Ring K] →\n [inst_4 : LinearOrder K] →\n [IsOrderedRing K] →\n [Archimedean K] → [inst_7 : Module K M] → [PosSMulMono K M] → FiniteArchimedeanClass M → Submodule K M","typeReferences":[["PartialOrder","toPreorder"],["Ring","toNonAssocRing"],["Module"],["AddCommGroup","toAddGroup"],["SMulZeroClass","toSMul"],["Submodule"],["PosSMulMono"],["instDistribLatticeOfLinearOrder"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["IsOrderedAddMonoid"],["MonoidWithZero","toMonoid"],["FiniteArchimedeanClass"],["AddGroup","toSubNegMonoid"],["NonAssocRing","toNonUnitalNonAssocRing"],["DistribSMul","toSMulZeroClass"],["SemilatticeInf","toPartialOrder"],["Lattice","toSemilatticeInf"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["DistribMulAction","toDistribSMul"],["LinearOrder"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["AddZeroClass","toAddZero"],["Ring","toSemiring"],["IsOrderedRing"],["Module","toDistribMulAction"],["DistribLattice","toLattice"],["SubNegMonoid","toAddMonoid"],["MulZeroClass","toZero"],["Archimedean"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["AddCommGroup","toAddCommMonoid"],["AddZero","toZero"],["Ring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["PartialOrder","toPreorder"],["Lattice","toSemilatticeInf"],["ArchimedeanClass"],["Subtype","preorder"],["FiniteArchimedeanClass","submodule"],["ArchimedeanClass","instOrderTop"],["instDistribLatticeOfLinearOrder"],["DistribLattice","toLattice"],["ArchimedeanClass","instLinearOrder"],["FiniteArchimedeanClass"],["Top","top"],["Ne"],["Preorder","toLE"],["OrderTop","toTop"],["UpperSet","Ici"],["SemilatticeInf","toPartialOrder"]]},{"isProp":false,"kind":"definition","name":["FiniteArchimedeanClass","ball"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592743._hygCtx._hyg.3 : AddCommGroup.{u_1} M] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592743._hygCtx._hyg.6 : LinearOrder.{u_1} M] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592743._hygCtx._hyg.9 : IsOrderedAddMonoid.{u_1} M (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592743._hygCtx._hyg.3) (PartialOrder.toPreorder.{u_1} M (SemilatticeInf.toPartialOrder.{u_1} M (Lattice.toSemilatticeInf.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592743._hygCtx._hyg.6)))))] (K : Type.{u_2}) [inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592743._hygCtx._hyg.13 : Ring.{u_2} K] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592743._hygCtx._hyg.16 : LinearOrder.{u_2} K] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592743._hygCtx._hyg.19 : IsOrderedRing.{u_2} K (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592743._hygCtx._hyg.13) (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592743._hygCtx._hyg.16))))] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592743._hygCtx._hyg.22 : Archimedean.{u_2} K (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonAssocRing.toNonUnitalNonAssocRing.{u_2} K (Ring.toNonAssocRing.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592743._hygCtx._hyg.13)))) (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592743._hygCtx._hyg.16))))] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592743._hygCtx._hyg.25 : Module.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592743._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592743._hygCtx._hyg.3)] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592743._hygCtx._hyg.29 : PosSMulMono.{u_2, u_1} K M (SMulZeroClass.toSMul.{u_2, u_1} K M (AddZero.toZero.{u_1} M (AddZeroClass.toAddZero.{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.Order.Module.Archimedean.224592743._hygCtx._hyg.3)))))) (DistribSMul.toSMulZeroClass.{u_2, u_1} K M (AddMonoid.toAddZeroClass.{u_1} M (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592743._hygCtx._hyg.3)))) (DistribMulAction.toDistribSMul.{u_2, u_1} K M (MonoidWithZero.toMonoid.{u_2} K (Semiring.toMonoidWithZero.{u_2} K (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592743._hygCtx._hyg.13))) (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592743._hygCtx._hyg.3))) (Module.toDistribMulAction.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592743._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592743._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592743._hygCtx._hyg.25)))) (PartialOrder.toPreorder.{u_2} K (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592743._hygCtx._hyg.16))))) (PartialOrder.toPreorder.{u_1} M (SemilatticeInf.toPartialOrder.{u_1} M (Lattice.toSemilatticeInf.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592743._hygCtx._hyg.6))))) (MulZeroClass.toZero.{u_2} K (NonUnitalNonAssocSemiring.toMulZeroClass.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonAssocRing.toNonUnitalNonAssocRing.{u_2} K (Ring.toNonAssocRing.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592743._hygCtx._hyg.13)))))], (FiniteArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592743._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592743._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592743._hygCtx._hyg.9) -> (Submodule.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592743._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592743._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.224592743._hygCtx._hyg.25)","typeFull":"{M : Type u_1} →\n [inst : AddCommGroup M] →\n [inst_1 : LinearOrder M] →\n [inst_2 : IsOrderedAddMonoid M] →\n (K : Type u_2) →\n [inst_3 : Ring K] →\n [inst_4 : LinearOrder K] →\n [IsOrderedRing K] →\n [Archimedean K] → [inst_7 : Module K M] → [PosSMulMono K M] → FiniteArchimedeanClass M → Submodule K M","typeReadable":"{M : Type u_1} →\n [inst : AddCommGroup M] →\n [inst_1 : LinearOrder M] →\n [inst_2 : IsOrderedAddMonoid M] →\n (K : Type u_2) →\n [inst_3 : Ring K] →\n [inst_4 : LinearOrder K] →\n [IsOrderedRing K] →\n [Archimedean K] → [inst_7 : Module K M] → [PosSMulMono K M] → FiniteArchimedeanClass M → Submodule K M","typeReferences":[["PartialOrder","toPreorder"],["Ring","toNonAssocRing"],["Module"],["AddCommGroup","toAddGroup"],["SMulZeroClass","toSMul"],["Submodule"],["PosSMulMono"],["instDistribLatticeOfLinearOrder"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["IsOrderedAddMonoid"],["MonoidWithZero","toMonoid"],["FiniteArchimedeanClass"],["AddGroup","toSubNegMonoid"],["NonAssocRing","toNonUnitalNonAssocRing"],["DistribSMul","toSMulZeroClass"],["SemilatticeInf","toPartialOrder"],["Lattice","toSemilatticeInf"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["DistribMulAction","toDistribSMul"],["LinearOrder"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["AddZeroClass","toAddZero"],["Ring","toSemiring"],["IsOrderedRing"],["Module","toDistribMulAction"],["DistribLattice","toLattice"],["SubNegMonoid","toAddMonoid"],["MulZeroClass","toZero"],["Archimedean"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["AddCommGroup","toAddCommMonoid"],["AddZero","toZero"],["Ring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["PartialOrder","toPreorder"],["Lattice","toSemilatticeInf"],["ArchimedeanClass"],["Subtype","preorder"],["FiniteArchimedeanClass","submodule"],["ArchimedeanClass","instOrderTop"],["instDistribLatticeOfLinearOrder"],["DistribLattice","toLattice"],["UpperSet","Ioi"],["ArchimedeanClass","instLinearOrder"],["FiniteArchimedeanClass"],["Top","top"],["Ne"],["Preorder","toLE"],["OrderTop","toTop"],["SemilatticeInf","toPartialOrder"]]},{"isProp":true,"kind":"theorem","name":["ArchimedeanClass","mem_closedBall_iff","_simp_1"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.3 : AddCommGroup.{u_1} M] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.6 : LinearOrder.{u_1} M] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.9 : IsOrderedAddMonoid.{u_1} M (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.3) (PartialOrder.toPreorder.{u_1} M (SemilatticeInf.toPartialOrder.{u_1} M (Lattice.toSemilatticeInf.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.6)))))] (K : Type.{u_2}) [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.13 : Ring.{u_2} K] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.16 : LinearOrder.{u_2} K] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.19 : IsOrderedRing.{u_2} K (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.13) (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.16))))] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.22 : Archimedean.{u_2} K (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonAssocRing.toNonUnitalNonAssocRing.{u_2} K (Ring.toNonAssocRing.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.13)))) (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.16))))] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.25 : Module.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.3)] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.29 : PosSMulMono.{u_2, u_1} K M (SMulZeroClass.toSMul.{u_2, u_1} K M (AddZero.toZero.{u_1} M (AddZeroClass.toAddZero.{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.Order.Module.Archimedean.2781882085._hygCtx._hyg.3)))))) (DistribSMul.toSMulZeroClass.{u_2, u_1} K M (AddMonoid.toAddZeroClass.{u_1} M (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.3)))) (DistribMulAction.toDistribSMul.{u_2, u_1} K M (MonoidWithZero.toMonoid.{u_2} K (Semiring.toMonoidWithZero.{u_2} K (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.13))) (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.3))) (Module.toDistribMulAction.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.25)))) (PartialOrder.toPreorder.{u_2} K (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.16))))) (PartialOrder.toPreorder.{u_1} M (SemilatticeInf.toPartialOrder.{u_1} M (Lattice.toSemilatticeInf.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.6))))) (MulZeroClass.toZero.{u_2} K (NonUnitalNonAssocSemiring.toMulZeroClass.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonAssocRing.toNonUnitalNonAssocRing.{u_2} K (Ring.toNonAssocRing.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.13)))))] {a : M} {c : ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.9}, Eq.{1} Prop (Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.25) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.25) M (Submodule.setLike.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.25)) (ArchimedeanClass.closedBall.{u_1, u_2} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.9 K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.22 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.29 c) a) (LE.le.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.9) (Preorder.toLE.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.9) (PartialOrder.toPreorder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.9) (SemilatticeInf.toPartialOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.9) (Lattice.toSemilatticeInf.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.9) (DistribLattice.toLattice.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.9) (instDistribLatticeOfLinearOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.9) (ArchimedeanClass.instLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.9))))))) c (ArchimedeanClass.mk.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2781882085._hygCtx._hyg.9 a))","typeFull":"∀ {M : Type u_1} [inst : AddCommGroup M] [inst_1 : LinearOrder M] [inst_2 : IsOrderedAddMonoid M] (K : Type u_2)\n [inst_3 : Ring K] [inst_4 : LinearOrder K] [inst_5 : IsOrderedRing K] [inst_6 : Archimedean K] [inst_7 : Module K M]\n [inst_8 : PosSMulMono K M] {a : M} {c : ArchimedeanClass M},\n (a ∈ ArchimedeanClass.closedBall K c) = (c ≤ ArchimedeanClass.mk a)","typeReadable":"∀ {M : Type u_1} [inst : AddCommGroup M] [inst_1 : LinearOrder M] [inst_2 : IsOrderedAddMonoid M] (K : Type u_2)\n [inst_3 : Ring K] [inst_4 : LinearOrder K] [inst_5 : IsOrderedRing K] [inst_6 : Archimedean K] [inst_7 : Module K M]\n [inst_8 : PosSMulMono K M] {a : M} {c : ArchimedeanClass M},\n (a ∈ ArchimedeanClass.closedBall K c) = (c ≤ ArchimedeanClass.mk a)","typeReferences":[["PartialOrder","toPreorder"],["Ring","toNonAssocRing"],["ArchimedeanClass"],["Module"],["AddCommGroup","toAddGroup"],["Membership","mem"],["SMulZeroClass","toSMul"],["Submodule"],["PosSMulMono"],["instDistribLatticeOfLinearOrder"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["IsOrderedAddMonoid"],["ArchimedeanClass","instLinearOrder"],["MonoidWithZero","toMonoid"],["AddGroup","toSubNegMonoid"],["Preorder","toLE"],["Eq"],["NonAssocRing","toNonUnitalNonAssocRing"],["DistribSMul","toSMulZeroClass"],["SemilatticeInf","toPartialOrder"],["SetLike","instMembership"],["Lattice","toSemilatticeInf"],["ArchimedeanClass","closedBall"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["DistribMulAction","toDistribSMul"],["LinearOrder"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["AddZeroClass","toAddZero"],["ArchimedeanClass","mk"],["Ring","toSemiring"],["IsOrderedRing"],["Module","toDistribMulAction"],["DistribLattice","toLattice"],["SubNegMonoid","toAddMonoid"],["Submodule","setLike"],["MulZeroClass","toZero"],["Archimedean"],["LE","le"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["AddCommGroup","toAddCommMonoid"],["AddZero","toZero"],["Ring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["Lattice","toSemilatticeInf"],["SetLike","instMembership"],["PartialOrder","toPreorder"],["ArchimedeanClass","closedBall"],["ArchimedeanClass"],["Membership","mem"],["ArchimedeanClass","mem_closedBall_iff"],["ArchimedeanClass","mk"],["Submodule"],["Ring","toSemiring"],["instDistribLatticeOfLinearOrder"],["DistribLattice","toLattice"],["Submodule","setLike"],["ArchimedeanClass","instLinearOrder"],["LE","le"],["AddCommGroup","toAddCommMonoid"],["Preorder","toLE"],["propext"],["SemilatticeInf","toPartialOrder"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Algebra","Order","Module","Archimedean",0,"ArchimedeanClass","mk_smul","_simp_1_1"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Order.Archimedean.Class.2806003247._hygCtx._hyg.3 : AddCommGroup.{u_1} M] [inst._@.Mathlib.Algebra.Order.Archimedean.Class.2806003247._hygCtx._hyg.6 : LinearOrder.{u_1} M] [inst._@.Mathlib.Algebra.Order.Archimedean.Class.2806003247._hygCtx._hyg.9 : IsOrderedAddMonoid.{u_1} M (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Archimedean.Class.2806003247._hygCtx._hyg.3) (PartialOrder.toPreorder.{u_1} M (SemilatticeInf.toPartialOrder.{u_1} M (Lattice.toSemilatticeInf.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Archimedean.Class.2806003247._hygCtx._hyg.6)))))] {a : M} {b : M}, Eq.{1} Prop (Eq.{succ u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Archimedean.Class.2806003247._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Archimedean.Class.2806003247._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Archimedean.Class.2806003247._hygCtx._hyg.9) (ArchimedeanClass.mk.{u_1} M inst._@.Mathlib.Algebra.Order.Archimedean.Class.2806003247._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Archimedean.Class.2806003247._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Archimedean.Class.2806003247._hygCtx._hyg.9 a) (ArchimedeanClass.mk.{u_1} M inst._@.Mathlib.Algebra.Order.Archimedean.Class.2806003247._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Archimedean.Class.2806003247._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Archimedean.Class.2806003247._hygCtx._hyg.9 b)) (And (Exists.{1} Nat (fun (m : Nat) => LE.le.{u_1} M (Preorder.toLE.{u_1} M (PartialOrder.toPreorder.{u_1} M (SemilatticeInf.toPartialOrder.{u_1} M (Lattice.toSemilatticeInf.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Archimedean.Class.2806003247._hygCtx._hyg.6)))))) (abs.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Archimedean.Class.2806003247._hygCtx._hyg.6)) (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Archimedean.Class.2806003247._hygCtx._hyg.3) b) (HSMul.hSMul.{0, u_1, u_1} Nat M M (instHSMul.{0, u_1} Nat M (AddMonoid.toNSMul.{u_1} M (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Archimedean.Class.2806003247._hygCtx._hyg.3))))) m (abs.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Archimedean.Class.2806003247._hygCtx._hyg.6)) (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Archimedean.Class.2806003247._hygCtx._hyg.3) a)))) (Exists.{1} Nat (fun (n : Nat) => LE.le.{u_1} M (Preorder.toLE.{u_1} M (PartialOrder.toPreorder.{u_1} M (SemilatticeInf.toPartialOrder.{u_1} M (Lattice.toSemilatticeInf.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Archimedean.Class.2806003247._hygCtx._hyg.6)))))) (abs.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Archimedean.Class.2806003247._hygCtx._hyg.6)) (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Archimedean.Class.2806003247._hygCtx._hyg.3) a) (HSMul.hSMul.{0, u_1, u_1} Nat M M (instHSMul.{0, u_1} Nat M (AddMonoid.toNSMul.{u_1} M (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Archimedean.Class.2806003247._hygCtx._hyg.3))))) n (abs.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Archimedean.Class.2806003247._hygCtx._hyg.6)) (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Archimedean.Class.2806003247._hygCtx._hyg.3) b)))))","typeFull":"∀ {M : Type u_1} [inst : AddCommGroup M] [inst_1 : LinearOrder M] [inst_2 : IsOrderedAddMonoid M] {a b : M},\n (ArchimedeanClass.mk a = ArchimedeanClass.mk b) = ((∃ m, |b| ≤ m • |a|) ∧ ∃ n, |a| ≤ n • |b|)","typeReadable":"∀ {M : Type u_1} [inst : AddCommGroup M] [inst_1 : LinearOrder M] [inst_2 : IsOrderedAddMonoid M] {a b : M},\n (ArchimedeanClass.mk a = ArchimedeanClass.mk b) = ((∃ m, |b| ≤ m • |a|) ∧ ∃ n, |a| ≤ n • |b|)","typeReferences":[["Exists"],["PartialOrder","toPreorder"],["Lattice","toSemilatticeInf"],["ArchimedeanClass"],["AddCommGroup","toAddGroup"],["LinearOrder"],["And"],["AddCommGroup"],["ArchimedeanClass","mk"],["instDistribLatticeOfLinearOrder"],["Nat"],["DistribLattice","toLattice"],["SubNegMonoid","toAddMonoid"],["AddMonoid","toNSMul"],["IsOrderedAddMonoid"],["HSMul","hSMul"],["LE","le"],["AddCommGroup","toAddCommMonoid"],["instHSMul"],["AddGroup","toSubNegMonoid"],["abs"],["Preorder","toLE"],["Eq"],["SemilatticeInf","toPartialOrder"]],"valueReferences":[["Exists"],["PartialOrder","toPreorder"],["Lattice","toSemilatticeInf"],["ArchimedeanClass"],["AddCommGroup","toAddGroup"],["And"],["ArchimedeanClass","mk_eq_mk"],["ArchimedeanClass","mk"],["instDistribLatticeOfLinearOrder"],["Nat"],["DistribLattice","toLattice"],["SubNegMonoid","toAddMonoid"],["AddMonoid","toNSMul"],["HSMul","hSMul"],["LE","le"],["instHSMul"],["AddGroup","toSubNegMonoid"],["abs"],["Eq"],["Preorder","toLE"],["propext"],["SemilatticeInf","toPartialOrder"]]},{"isProp":true,"kind":"theorem","name":["FiniteArchimedeanClass","submodule","_proof_1"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 : AddCommGroup.{u_1} M] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 : LinearOrder.{u_1} M] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9 : IsOrderedAddMonoid.{u_1} M (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3) (PartialOrder.toPreorder.{u_1} M (SemilatticeInf.toPartialOrder.{u_1} M (Lattice.toSemilatticeInf.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6)))))] (K : Type.{u_2}) [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.13 : Ring.{u_2} K] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.25 : Module.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3)] (k : K) {a : M}, (Eq.{succ u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (ArchimedeanClass.mk.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9 a) (Top.top.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (OrderTop.toTop.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (Preorder.toLE.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (PartialOrder.toPreorder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (SemilatticeInf.toPartialOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (Lattice.toSemilatticeInf.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (DistribLattice.toLattice.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (instDistribLatticeOfLinearOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (ArchimedeanClass.instLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9))))))) (ArchimedeanClass.instOrderTop.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9)))) -> (Eq.{succ u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (ArchimedeanClass.mk.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9 (HSMul.hSMul.{u_2, u_1, u_1} K M M (instHSMul.{u_2, u_1} K M (SMulZeroClass.toSMul.{u_2, u_1} K 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.Order.Module.Archimedean.3843450102._hygCtx._hyg.3))))) (DistribSMul.toSMulZeroClass.{u_2, u_1} K M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3))) (DistribMulAction.toDistribSMul.{u_2, u_1} K M (MonoidWithZero.toMonoid.{u_2} K (Semiring.toMonoidWithZero.{u_2} K (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.13))) (AddCommMonoid.toAddMonoid.{u_1} M (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3)) (Module.toDistribMulAction.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.25))))) k a)) (Top.top.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (OrderTop.toTop.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (Preorder.toLE.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (PartialOrder.toPreorder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (SemilatticeInf.toPartialOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (Lattice.toSemilatticeInf.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (DistribLattice.toLattice.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (instDistribLatticeOfLinearOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9) (ArchimedeanClass.instLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9))))))) (ArchimedeanClass.instOrderTop.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3843450102._hygCtx._hyg.9))))","typeFull":"∀ {M : Type u_1} [inst : AddCommGroup M] [inst_1 : LinearOrder M] [inst_2 : IsOrderedAddMonoid M] (K : Type u_2)\n [inst_3 : Ring K] [inst_4 : Module K M] (k : K) {a : M}, ArchimedeanClass.mk a = ⊤ → ArchimedeanClass.mk (k • a) = ⊤","typeReadable":"∀ {M : Type u_1} [inst : AddCommGroup M] [inst_1 : LinearOrder M] [inst_2 : IsOrderedAddMonoid M] (K : Type u_2)\n [inst_3 : Ring K] [inst_4 : Module K M] (k : K) {a : M}, ArchimedeanClass.mk a = ⊤ → ArchimedeanClass.mk (k • a) = ⊤","typeReferences":[["PartialOrder","toPreorder"],["Module"],["ArchimedeanClass"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["instDistribLatticeOfLinearOrder"],["IsOrderedAddMonoid"],["MonoidWithZero","toMonoid"],["ArchimedeanClass","instLinearOrder"],["instHSMul"],["Eq"],["Preorder","toLE"],["DistribSMul","toSMulZeroClass"],["SemilatticeInf","toPartialOrder"],["Lattice","toSemilatticeInf"],["DistribMulAction","toDistribSMul"],["LinearOrder"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["AddZeroClass","toAddZero"],["ArchimedeanClass","mk"],["Ring","toSemiring"],["ArchimedeanClass","instOrderTop"],["Module","toDistribMulAction"],["DistribLattice","toLattice"],["HSMul","hSMul"],["AddCommGroup","toAddCommMonoid"],["Top","top"],["OrderTop","toTop"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Ring"]],"valueReferences":[["smul_zero"],["SubtractionMonoid","toSubNegZeroMonoid"],["PartialOrder","toPreorder"],["Eq","trans"],["ArchimedeanClass","mk_eq_top_iff"],["ArchimedeanClass"],["AddCommGroup","toAddGroup"],["Iff","mp"],["SubtractionCommMonoid","toSubtractionMonoid"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["SubNegZeroMonoid","toNegZeroClass"],["congrArg"],["instDistribLatticeOfLinearOrder"],["ArchimedeanClass","instLinearOrder"],["MonoidWithZero","toMonoid"],["instHSMul"],["Zero","toOfNat0"],["congrFun'"],["AddGroup","toSubNegMonoid"],["Eq"],["Preorder","toLE"],["DistribSMul","toSMulZeroClass"],["SemilatticeInf","toPartialOrder"],["Lattice","toSemilatticeInf"],["True"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["ArchimedeanClass","mk"],["ArchimedeanClass","instOrderTop"],["Ring","toSemiring"],["eq_self"],["Module","toDistribMulAction"],["DistribLattice","toLattice"],["SubNegMonoid","toAddMonoid"],["AddCommGroup","toDivisionAddCommMonoid"],["of_eq_true"],["HSMul","hSMul"],["NegZeroClass","toZero"],["AddCommGroup","toAddCommMonoid"],["Top","top"],["AddZero","toZero"],["OrderTop","toTop"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["FiniteArchimedeanClass","mem_ball_iff"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.3 : AddCommGroup.{u_1} M] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.6 : LinearOrder.{u_1} M] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.9 : IsOrderedAddMonoid.{u_1} M (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.3) (PartialOrder.toPreorder.{u_1} M (SemilatticeInf.toPartialOrder.{u_1} M (Lattice.toSemilatticeInf.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.6)))))] (K : Type.{u_2}) [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.13 : Ring.{u_2} K] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.16 : LinearOrder.{u_2} K] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.19 : IsOrderedRing.{u_2} K (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.13) (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.16))))] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.22 : Archimedean.{u_2} K (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonAssocRing.toNonUnitalNonAssocRing.{u_2} K (Ring.toNonAssocRing.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.13)))) (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.16))))] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.25 : Module.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.3)] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.29 : PosSMulMono.{u_2, u_1} K M (SMulZeroClass.toSMul.{u_2, u_1} K M (AddZero.toZero.{u_1} M (AddZeroClass.toAddZero.{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.Order.Module.Archimedean.2496125326._hygCtx._hyg.3)))))) (DistribSMul.toSMulZeroClass.{u_2, u_1} K M (AddMonoid.toAddZeroClass.{u_1} M (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.3)))) (DistribMulAction.toDistribSMul.{u_2, u_1} K M (MonoidWithZero.toMonoid.{u_2} K (Semiring.toMonoidWithZero.{u_2} K (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.13))) (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.3))) (Module.toDistribMulAction.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.25)))) (PartialOrder.toPreorder.{u_2} K (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.16))))) (PartialOrder.toPreorder.{u_1} M (SemilatticeInf.toPartialOrder.{u_1} M (Lattice.toSemilatticeInf.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.6))))) (MulZeroClass.toZero.{u_2} K (NonUnitalNonAssocSemiring.toMulZeroClass.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonAssocRing.toNonUnitalNonAssocRing.{u_2} K (Ring.toNonAssocRing.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.13)))))] {a : M} {c : FiniteArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.9}, Iff (Membership.mem.{u_1, u_1} M (Submodule.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.25) (SetLike.instMembership.{u_1, u_1} (Submodule.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.25) M (Submodule.setLike.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.25)) (FiniteArchimedeanClass.ball.{u_1, u_2} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.9 K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.22 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.29 c) a) (forall (h : Ne.{succ u_1} M a (OfNat.ofNat.{u_1} M 0 (Zero.toOfNat0.{u_1} M (NegZeroClass.toZero.{u_1} M (SubNegZeroMonoid.toNegZeroClass.{u_1} M (SubtractionMonoid.toSubNegZeroMonoid.{u_1} M (SubtractionCommMonoid.toSubtractionMonoid.{u_1} M (AddCommGroup.toDivisionAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.3)))))))), LT.lt.{u_1} (FiniteArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.9) (Subtype.instLT.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.9) (Preorder.toLT.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.9) (PartialOrder.toPreorder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.9) (SemilatticeInf.toPartialOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.9) (Lattice.toSemilatticeInf.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.9) (DistribLattice.toLattice.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.9) (instDistribLatticeOfLinearOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.9) (ArchimedeanClass.instLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.9))))))) (fun (A : ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.9) => Ne.{succ u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.9) A (Top.top.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.9) (OrderTop.toTop.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.9) (Preorder.toLE.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.9) (PartialOrder.toPreorder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.9) (SemilatticeInf.toPartialOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.9) (Lattice.toSemilatticeInf.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.9) (DistribLattice.toLattice.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.9) (instDistribLatticeOfLinearOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.9) (ArchimedeanClass.instLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.9))))))) (ArchimedeanClass.instOrderTop.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.9))))) c (FiniteArchimedeanClass.mk.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2496125326._hygCtx._hyg.9 a h))","typeFull":"∀ {M : Type u_1} [inst : AddCommGroup M] [inst_1 : LinearOrder M] [inst_2 : IsOrderedAddMonoid M] (K : Type u_2)\n [inst_3 : Ring K] [inst_4 : LinearOrder K] [inst_5 : IsOrderedRing K] [inst_6 : Archimedean K] [inst_7 : Module K M]\n [inst_8 : PosSMulMono K M] {a : M} {c : FiniteArchimedeanClass M},\n a ∈ FiniteArchimedeanClass.ball K c ↔ ∀ (h : a ≠ 0), c < FiniteArchimedeanClass.mk a h","typeReadable":"∀ {M : Type u_1} [inst : AddCommGroup M] [inst_1 : LinearOrder M] [inst_2 : IsOrderedAddMonoid M] (K : Type u_2)\n [inst_3 : Ring K] [inst_4 : LinearOrder K] [inst_5 : IsOrderedRing K] [inst_6 : Archimedean K] [inst_7 : Module K M]\n [inst_8 : PosSMulMono K M] {a : M} {c : FiniteArchimedeanClass M},\n a ∈ FiniteArchimedeanClass.ball K c ↔ ∀ (h : a ≠ 0), c < FiniteArchimedeanClass.mk a h","typeReferences":[["SubtractionMonoid","toSubNegZeroMonoid"],["Ring","toNonAssocRing"],["PartialOrder","toPreorder"],["AddCommGroup","toAddGroup"],["Membership","mem"],["Preorder","toLT"],["SMulZeroClass","toSMul"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["IsOrderedAddMonoid"],["ArchimedeanClass","instLinearOrder"],["FiniteArchimedeanClass"],["FiniteArchimedeanClass","mk"],["AddGroup","toSubNegMonoid"],["NonAssocRing","toNonUnitalNonAssocRing"],["SemilatticeInf","toPartialOrder"],["DistribSMul","toSMulZeroClass"],["Subtype","instLT"],["SetLike","instMembership"],["FiniteArchimedeanClass","ball"],["LinearOrder"],["DistribMulAction","toDistribSMul"],["AddZeroClass","toAddZero"],["ArchimedeanClass","instOrderTop"],["Ring","toSemiring"],["Iff"],["Top","top"],["AddCommGroup","toAddCommMonoid"],["NegZeroClass","toZero"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["ArchimedeanClass"],["Module"],["SubtractionCommMonoid","toSubtractionMonoid"],["SubNegZeroMonoid","toNegZeroClass"],["PosSMulMono"],["Submodule"],["instDistribLatticeOfLinearOrder"],["MonoidWithZero","toMonoid"],["Zero","toOfNat0"],["Preorder","toLE"],["Lattice","toSemilatticeInf"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["OfNat","ofNat"],["LT","lt"],["IsOrderedRing"],["Module","toDistribMulAction"],["DistribLattice","toLattice"],["SubNegMonoid","toAddMonoid"],["AddCommGroup","toDivisionAddCommMonoid"],["MulZeroClass","toZero"],["Submodule","setLike"],["Archimedean"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["Ne"],["OrderTop","toTop"],["Ring"]],"valueReferences":[["FiniteArchimedeanClass","mem_ballAddSubgroup_iff"]]},{"isProp":true,"kind":"theorem","name":["ArchimedeanClass","closedBall","congr_simp"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.3 : AddCommGroup.{u_1} M] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.6 : LinearOrder.{u_1} M] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.9 : IsOrderedAddMonoid.{u_1} M (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.3) (PartialOrder.toPreorder.{u_1} M (SemilatticeInf.toPartialOrder.{u_1} M (Lattice.toSemilatticeInf.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.6)))))] (K : Type.{u_2}) [inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.13 : Ring.{u_2} K] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.16 : LinearOrder.{u_2} K] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.19 : IsOrderedRing.{u_2} K (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.13) (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.16))))] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.22 : Archimedean.{u_2} K (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonAssocRing.toNonUnitalNonAssocRing.{u_2} K (Ring.toNonAssocRing.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.13)))) (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.16))))] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.25 : Module.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.3)] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.29 : PosSMulMono.{u_2, u_1} K M (SMulZeroClass.toSMul.{u_2, u_1} K M (AddZero.toZero.{u_1} M (AddZeroClass.toAddZero.{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.Order.Module.Archimedean.113252810._hygCtx._hyg.3)))))) (DistribSMul.toSMulZeroClass.{u_2, u_1} K M (AddMonoid.toAddZeroClass.{u_1} M (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.3)))) (DistribMulAction.toDistribSMul.{u_2, u_1} K M (MonoidWithZero.toMonoid.{u_2} K (Semiring.toMonoidWithZero.{u_2} K (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.13))) (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.3))) (Module.toDistribMulAction.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.25)))) (PartialOrder.toPreorder.{u_2} K (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.16))))) (PartialOrder.toPreorder.{u_1} M (SemilatticeInf.toPartialOrder.{u_1} M (Lattice.toSemilatticeInf.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.6))))) (MulZeroClass.toZero.{u_2} K (NonUnitalNonAssocSemiring.toMulZeroClass.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonAssocRing.toNonUnitalNonAssocRing.{u_2} K (Ring.toNonAssocRing.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.13)))))] (c : ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.9) (c_1 : ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.9), (Eq.{succ u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.9) c c_1) -> (Eq.{succ u_1} (Submodule.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.25) (ArchimedeanClass.closedBall.{u_1, u_2} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.9 K inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.22 inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.29 c) (ArchimedeanClass.closedBall.{u_1, u_2} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.9 K inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.22 inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Order.Module.Archimedean.113252810._hygCtx._hyg.29 c_1))","typeFull":"∀ {M : Type u_1} [inst : AddCommGroup M] [inst_1 : LinearOrder M] [inst_2 : IsOrderedAddMonoid M] (K : Type u_2)\n [inst_3 : Ring K] [inst_4 : LinearOrder K] [inst_5 : IsOrderedRing K] [inst_6 : Archimedean K] [inst_7 : Module K M]\n [inst_8 : PosSMulMono K M] (c c_1 : ArchimedeanClass M),\n c = c_1 → ArchimedeanClass.closedBall K c = ArchimedeanClass.closedBall K c_1","typeReadable":"∀ {M : Type u_1} [inst : AddCommGroup M] [inst_1 : LinearOrder M] [inst_2 : IsOrderedAddMonoid M] (K : Type u_2)\n [inst_3 : Ring K] [inst_4 : LinearOrder K] [inst_5 : IsOrderedRing K] [inst_6 : Archimedean K] [inst_7 : Module K M]\n [inst_8 : PosSMulMono K M] (c c_1 : ArchimedeanClass M),\n c = c_1 → ArchimedeanClass.closedBall K c = ArchimedeanClass.closedBall K c_1","typeReferences":[["PartialOrder","toPreorder"],["Ring","toNonAssocRing"],["ArchimedeanClass"],["Module"],["AddCommGroup","toAddGroup"],["SMulZeroClass","toSMul"],["Submodule"],["PosSMulMono"],["instDistribLatticeOfLinearOrder"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["IsOrderedAddMonoid"],["MonoidWithZero","toMonoid"],["AddGroup","toSubNegMonoid"],["Eq"],["NonAssocRing","toNonUnitalNonAssocRing"],["DistribSMul","toSMulZeroClass"],["SemilatticeInf","toPartialOrder"],["Lattice","toSemilatticeInf"],["ArchimedeanClass","closedBall"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["DistribMulAction","toDistribSMul"],["LinearOrder"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["AddZeroClass","toAddZero"],["Ring","toSemiring"],["IsOrderedRing"],["Module","toDistribMulAction"],["DistribLattice","toLattice"],["SubNegMonoid","toAddMonoid"],["MulZeroClass","toZero"],["Archimedean"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["AddCommGroup","toAddCommMonoid"],["AddZero","toZero"],["Ring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["ArchimedeanClass","closedBall"],["ArchimedeanClass"],["Eq","refl"],["AddCommGroup","toAddCommMonoid"],["Eq"],["Eq","rec"],["Ring","toSemiring"],["Submodule"]]},{"isProp":true,"kind":"theorem","name":["FiniteArchimedeanClass","toAddSubgroup_closedBall"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019056._hygCtx._hyg.3 : AddCommGroup.{u_1} M] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019056._hygCtx._hyg.6 : LinearOrder.{u_1} M] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019056._hygCtx._hyg.9 : IsOrderedAddMonoid.{u_1} M (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019056._hygCtx._hyg.3) (PartialOrder.toPreorder.{u_1} M (SemilatticeInf.toPartialOrder.{u_1} M (Lattice.toSemilatticeInf.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019056._hygCtx._hyg.6)))))] (K : Type.{u_2}) [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019056._hygCtx._hyg.13 : Ring.{u_2} K] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019056._hygCtx._hyg.16 : LinearOrder.{u_2} K] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019056._hygCtx._hyg.19 : IsOrderedRing.{u_2} K (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019056._hygCtx._hyg.13) (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019056._hygCtx._hyg.16))))] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019056._hygCtx._hyg.22 : Archimedean.{u_2} K (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonAssocRing.toNonUnitalNonAssocRing.{u_2} K (Ring.toNonAssocRing.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019056._hygCtx._hyg.13)))) (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019056._hygCtx._hyg.16))))] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019056._hygCtx._hyg.25 : Module.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019056._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019056._hygCtx._hyg.3)] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019056._hygCtx._hyg.29 : PosSMulMono.{u_2, u_1} K M (SMulZeroClass.toSMul.{u_2, u_1} K M (AddZero.toZero.{u_1} M (AddZeroClass.toAddZero.{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.Order.Module.Archimedean.2590019056._hygCtx._hyg.3)))))) (DistribSMul.toSMulZeroClass.{u_2, u_1} K M (AddMonoid.toAddZeroClass.{u_1} M (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019056._hygCtx._hyg.3)))) (DistribMulAction.toDistribSMul.{u_2, u_1} K M (MonoidWithZero.toMonoid.{u_2} K (Semiring.toMonoidWithZero.{u_2} K (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019056._hygCtx._hyg.13))) (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019056._hygCtx._hyg.3))) (Module.toDistribMulAction.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019056._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019056._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019056._hygCtx._hyg.25)))) (PartialOrder.toPreorder.{u_2} K (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019056._hygCtx._hyg.16))))) (PartialOrder.toPreorder.{u_1} M (SemilatticeInf.toPartialOrder.{u_1} M (Lattice.toSemilatticeInf.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019056._hygCtx._hyg.6))))) (MulZeroClass.toZero.{u_2} K (NonUnitalNonAssocSemiring.toMulZeroClass.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonAssocRing.toNonUnitalNonAssocRing.{u_2} K (Ring.toNonAssocRing.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019056._hygCtx._hyg.13)))))] (c : FiniteArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019056._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019056._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019056._hygCtx._hyg.9), Eq.{succ u_1} (AddSubgroup.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019056._hygCtx._hyg.3)) (Submodule.toAddSubgroup.{u_2, u_1} K M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019056._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019056._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019056._hygCtx._hyg.25 (FiniteArchimedeanClass.closedBall.{u_1, u_2} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019056._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019056._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019056._hygCtx._hyg.9 K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019056._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019056._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019056._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019056._hygCtx._hyg.22 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019056._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019056._hygCtx._hyg.29 c)) (FiniteArchimedeanClass.closedBallAddSubgroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019056._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019056._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2590019056._hygCtx._hyg.9 c)","typeFull":"∀ {M : Type u_1} [inst : AddCommGroup M] [inst_1 : LinearOrder M] [inst_2 : IsOrderedAddMonoid M] (K : Type u_2)\n [inst_3 : Ring K] [inst_4 : LinearOrder K] [inst_5 : IsOrderedRing K] [inst_6 : Archimedean K] [inst_7 : Module K M]\n [inst_8 : PosSMulMono K M] (c : FiniteArchimedeanClass M),\n (FiniteArchimedeanClass.closedBall K c).toAddSubgroup = c.closedBallAddSubgroup","typeReadable":"∀ {M : Type u_1} [inst : AddCommGroup M] [inst_1 : LinearOrder M] [inst_2 : IsOrderedAddMonoid M] (K : Type u_2)\n [inst_3 : Ring K] [inst_4 : LinearOrder K] [inst_5 : IsOrderedRing K] [inst_6 : Archimedean K] [inst_7 : Module K M]\n [inst_8 : PosSMulMono K M] (c : FiniteArchimedeanClass M),\n (FiniteArchimedeanClass.closedBall K c).toAddSubgroup = c.closedBallAddSubgroup","typeReferences":[["FiniteArchimedeanClass","closedBall"],["PartialOrder","toPreorder"],["Ring","toNonAssocRing"],["Module"],["AddCommGroup","toAddGroup"],["SMulZeroClass","toSMul"],["PosSMulMono"],["instDistribLatticeOfLinearOrder"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["IsOrderedAddMonoid"],["MonoidWithZero","toMonoid"],["FiniteArchimedeanClass"],["AddGroup","toSubNegMonoid"],["Eq"],["NonAssocRing","toNonUnitalNonAssocRing"],["DistribSMul","toSMulZeroClass"],["SemilatticeInf","toPartialOrder"],["Lattice","toSemilatticeInf"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["DistribMulAction","toDistribSMul"],["LinearOrder"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["Submodule","toAddSubgroup"],["AddZeroClass","toAddZero"],["Ring","toSemiring"],["AddSubgroup"],["IsOrderedRing"],["Module","toDistribMulAction"],["DistribLattice","toLattice"],["SubNegMonoid","toAddMonoid"],["MulZeroClass","toZero"],["Archimedean"],["FiniteArchimedeanClass","closedBallAddSubgroup"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["AddCommGroup","toAddCommMonoid"],["AddZero","toZero"],["Ring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["rfl"],["FiniteArchimedeanClass","closedBall"],["AddSubgroup"],["AddCommGroup","toAddGroup"],["Submodule","toAddSubgroup"]]},{"isProp":true,"kind":"theorem","name":["ArchimedeanClass","ball_le_closedBall"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3616962213._hygCtx._hyg.3 : AddCommGroup.{u_1} M] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3616962213._hygCtx._hyg.6 : LinearOrder.{u_1} M] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3616962213._hygCtx._hyg.9 : IsOrderedAddMonoid.{u_1} M (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3616962213._hygCtx._hyg.3) (PartialOrder.toPreorder.{u_1} M (SemilatticeInf.toPartialOrder.{u_1} M (Lattice.toSemilatticeInf.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3616962213._hygCtx._hyg.6)))))] (K : Type.{u_2}) [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3616962213._hygCtx._hyg.13 : Ring.{u_2} K] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3616962213._hygCtx._hyg.16 : LinearOrder.{u_2} K] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3616962213._hygCtx._hyg.19 : IsOrderedRing.{u_2} K (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3616962213._hygCtx._hyg.13) (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3616962213._hygCtx._hyg.16))))] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3616962213._hygCtx._hyg.22 : Archimedean.{u_2} K (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonAssocRing.toNonUnitalNonAssocRing.{u_2} K (Ring.toNonAssocRing.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3616962213._hygCtx._hyg.13)))) (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3616962213._hygCtx._hyg.16))))] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3616962213._hygCtx._hyg.25 : Module.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3616962213._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3616962213._hygCtx._hyg.3)] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.3616962213._hygCtx._hyg.29 : PosSMulMono.{u_2, u_1} K M (SMulZeroClass.toSMul.{u_2, u_1} K M (AddZero.toZero.{u_1} M (AddZeroClass.toAddZero.{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.Order.Module.Archimedean.3616962213._hygCtx._hyg.3)))))) (DistribSMul.toSMulZeroClass.{u_2, u_1} K M (AddMonoid.toAddZeroClass.{u_1} M (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3616962213._hygCtx._hyg.3)))) (DistribMulAction.toDistribSMul.{u_2, u_1} K M (MonoidWithZero.toMonoid.{u_2} K (Semiring.toMonoidWithZero.{u_2} K (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3616962213._hygCtx._hyg.13))) (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3616962213._hygCtx._hyg.3))) (Module.toDistribMulAction.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3616962213._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3616962213._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.3616962213._hygCtx._hyg.25)))) (PartialOrder.toPreorder.{u_2} K (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3616962213._hygCtx._hyg.16))))) (PartialOrder.toPreorder.{u_1} M (SemilatticeInf.toPartialOrder.{u_1} M (Lattice.toSemilatticeInf.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3616962213._hygCtx._hyg.6))))) (MulZeroClass.toZero.{u_2} K (NonUnitalNonAssocSemiring.toMulZeroClass.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonAssocRing.toNonUnitalNonAssocRing.{u_2} K (Ring.toNonAssocRing.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3616962213._hygCtx._hyg.13)))))] {c : ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3616962213._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3616962213._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3616962213._hygCtx._hyg.9}, LE.le.{u_1} (Submodule.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3616962213._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3616962213._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.3616962213._hygCtx._hyg.25) (Preorder.toLE.{u_1} (Submodule.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3616962213._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3616962213._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.3616962213._hygCtx._hyg.25) (PartialOrder.toPreorder.{u_1} (Submodule.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3616962213._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3616962213._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.3616962213._hygCtx._hyg.25) (Submodule.instPartialOrder.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3616962213._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3616962213._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.3616962213._hygCtx._hyg.25))) (ArchimedeanClass.ball.{u_1, u_2} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3616962213._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3616962213._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3616962213._hygCtx._hyg.9 K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3616962213._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3616962213._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3616962213._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3616962213._hygCtx._hyg.22 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3616962213._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3616962213._hygCtx._hyg.29 c) (ArchimedeanClass.closedBall.{u_1, u_2} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.3616962213._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3616962213._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3616962213._hygCtx._hyg.9 K inst._@.Mathlib.Algebra.Order.Module.Archimedean.3616962213._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3616962213._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3616962213._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3616962213._hygCtx._hyg.22 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3616962213._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Order.Module.Archimedean.3616962213._hygCtx._hyg.29 c)","typeFull":"∀ {M : Type u_1} [inst : AddCommGroup M] [inst_1 : LinearOrder M] [inst_2 : IsOrderedAddMonoid M] (K : Type u_2)\n [inst_3 : Ring K] [inst_4 : LinearOrder K] [inst_5 : IsOrderedRing K] [inst_6 : Archimedean K] [inst_7 : Module K M]\n [inst_8 : PosSMulMono K M] {c : ArchimedeanClass M}, ArchimedeanClass.ball K c ≤ ArchimedeanClass.closedBall K c","typeReadable":"∀ {M : Type u_1} [inst : AddCommGroup M] [inst_1 : LinearOrder M] [inst_2 : IsOrderedAddMonoid M] (K : Type u_2)\n [inst_3 : Ring K] [inst_4 : LinearOrder K] [inst_5 : IsOrderedRing K] [inst_6 : Archimedean K] [inst_7 : Module K M]\n [inst_8 : PosSMulMono K M] {c : ArchimedeanClass M}, ArchimedeanClass.ball K c ≤ ArchimedeanClass.closedBall K c","typeReferences":[["PartialOrder","toPreorder"],["Ring","toNonAssocRing"],["ArchimedeanClass"],["Module"],["AddCommGroup","toAddGroup"],["SMulZeroClass","toSMul"],["Submodule"],["PosSMulMono"],["instDistribLatticeOfLinearOrder"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["IsOrderedAddMonoid"],["MonoidWithZero","toMonoid"],["AddGroup","toSubNegMonoid"],["Preorder","toLE"],["NonAssocRing","toNonUnitalNonAssocRing"],["DistribSMul","toSMulZeroClass"],["SemilatticeInf","toPartialOrder"],["Lattice","toSemilatticeInf"],["ArchimedeanClass","closedBall"],["Submodule","instPartialOrder"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["DistribMulAction","toDistribSMul"],["LinearOrder"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["AddZeroClass","toAddZero"],["Ring","toSemiring"],["IsOrderedRing"],["Module","toDistribMulAction"],["DistribLattice","toLattice"],["SubNegMonoid","toAddMonoid"],["MulZeroClass","toZero"],["ArchimedeanClass","ball"],["Archimedean"],["LE","le"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["AddCommGroup","toAddCommMonoid"],["AddZero","toZero"],["Ring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["PartialOrder","toPreorder"],["Eq","trans"],["ArchimedeanClass"],["Membership","mem"],["Preorder","toLT"],["Iff","mp"],["ArchimedeanClass","mem_closedBall_iff","_simp_1"],["eq_or_ne"],["congrArg"],["Submodule"],["instDistribLatticeOfLinearOrder"],["congr"],["ArchimedeanClass","instLinearOrder"],["Eq","symm"],["Std","le_refl","_simp_1"],["ArchimedeanClass","closedBall_top"],["Preorder","toLE"],["Eq"],["Eq","ndrec"],["LT","lt","le"],["SemilatticeInf","toPartialOrder"],["SetLike","instMembership"],["Lattice","toSemilatticeInf"],["True"],["ArchimedeanClass","closedBall"],["ArchimedeanClass","mem_ball_iff"],["Submodule","instPartialOrder"],["ArchimedeanClass","ball_top"],["instReflLe"],["Bot","bot"],["ArchimedeanClass","mk"],["ArchimedeanClass","instOrderTop"],["Ring","toSemiring"],["LT","lt"],["Or","casesOn"],["DistribLattice","toLattice"],["of_eq_true"],["Submodule","setLike"],["ArchimedeanClass","ball"],["Submodule","instBot"],["LE","le"],["id"],["Top","top"],["AddCommGroup","toAddCommMonoid"],["Eq","mpr"],["Ne"],["OrderTop","toTop"]]},{"isProp":true,"kind":"theorem","name":["FiniteArchimedeanClass","submodule_strictAnti"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3 : AddCommGroup.{u_1} M] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.6 : LinearOrder.{u_1} M] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.9 : IsOrderedAddMonoid.{u_1} M (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3) (PartialOrder.toPreorder.{u_1} M (SemilatticeInf.toPartialOrder.{u_1} M (Lattice.toSemilatticeInf.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.6)))))] (K : Type.{u_2}) [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.13 : Ring.{u_2} K] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.16 : LinearOrder.{u_2} K] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.19 : IsOrderedRing.{u_2} K (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.13) (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.16))))] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.22 : Archimedean.{u_2} K (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonAssocRing.toNonUnitalNonAssocRing.{u_2} K (Ring.toNonAssocRing.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.13)))) (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.16))))] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.25 : Module.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3)] [inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.29 : PosSMulMono.{u_2, u_1} K M (SMulZeroClass.toSMul.{u_2, u_1} K M (AddZero.toZero.{u_1} M (AddZeroClass.toAddZero.{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.Order.Module.Archimedean.2863435983._hygCtx._hyg.3)))))) (DistribSMul.toSMulZeroClass.{u_2, u_1} K M (AddMonoid.toAddZeroClass.{u_1} M (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3)))) (DistribMulAction.toDistribSMul.{u_2, u_1} K M (MonoidWithZero.toMonoid.{u_2} K (Semiring.toMonoidWithZero.{u_2} K (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.13))) (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3))) (Module.toDistribMulAction.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.25)))) (PartialOrder.toPreorder.{u_2} K (SemilatticeInf.toPartialOrder.{u_2} K (Lattice.toSemilatticeInf.{u_2} K (DistribLattice.toLattice.{u_2} K (instDistribLatticeOfLinearOrder.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.16))))) (PartialOrder.toPreorder.{u_1} M (SemilatticeInf.toPartialOrder.{u_1} M (Lattice.toSemilatticeInf.{u_1} M (DistribLattice.toLattice.{u_1} M (instDistribLatticeOfLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.6))))) (MulZeroClass.toZero.{u_2} K (NonUnitalNonAssocSemiring.toMulZeroClass.{u_2} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} K (NonAssocRing.toNonUnitalNonAssocRing.{u_2} K (Ring.toNonAssocRing.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.13)))))], StrictAnti.{u_1, u_1} (UpperSet.{u_1} (FiniteArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.9) (Subtype.instLE.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.9) (Preorder.toLE.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.9) (PartialOrder.toPreorder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.9) (SemilatticeInf.toPartialOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.9) (Lattice.toSemilatticeInf.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.9) (DistribLattice.toLattice.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.9) (instDistribLatticeOfLinearOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.9) (ArchimedeanClass.instLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.9))))))) (fun (A : ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.9) => Ne.{succ u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.9) A (Top.top.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.9) (OrderTop.toTop.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.9) (Preorder.toLE.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.9) (PartialOrder.toPreorder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.9) (SemilatticeInf.toPartialOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.9) (Lattice.toSemilatticeInf.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.9) (DistribLattice.toLattice.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.9) (instDistribLatticeOfLinearOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.9) (ArchimedeanClass.instLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.9))))))) (ArchimedeanClass.instOrderTop.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.9)))))) (Submodule.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.25) (PartialOrder.toPreorder.{u_1} (UpperSet.{u_1} (FiniteArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.9) (Subtype.instLE.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.9) (Preorder.toLE.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.9) (PartialOrder.toPreorder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.9) (SemilatticeInf.toPartialOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.9) (Lattice.toSemilatticeInf.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.9) (DistribLattice.toLattice.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.9) (instDistribLatticeOfLinearOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.9) (ArchimedeanClass.instLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.9))))))) (fun (A : ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.9) => Ne.{succ u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.9) A (Top.top.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.9) (OrderTop.toTop.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.9) (Preorder.toLE.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.9) (PartialOrder.toPreorder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.9) (SemilatticeInf.toPartialOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.9) (Lattice.toSemilatticeInf.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.9) (DistribLattice.toLattice.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.9) (instDistribLatticeOfLinearOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.9) (ArchimedeanClass.instLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.9))))))) (ArchimedeanClass.instOrderTop.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.9)))))) (UpperSet.instPartialOrder.{u_1} (FiniteArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.9) (Subtype.instLE.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.9) (Preorder.toLE.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.9) (PartialOrder.toPreorder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.9) (SemilatticeInf.toPartialOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.9) (Lattice.toSemilatticeInf.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.9) (DistribLattice.toLattice.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.9) (instDistribLatticeOfLinearOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.9) (ArchimedeanClass.instLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.9))))))) (fun (A : ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.9) => Ne.{succ u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.9) A (Top.top.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.9) (OrderTop.toTop.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.9) (Preorder.toLE.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.9) (PartialOrder.toPreorder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.9) (SemilatticeInf.toPartialOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.9) (Lattice.toSemilatticeInf.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.9) (DistribLattice.toLattice.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.9) (instDistribLatticeOfLinearOrder.{u_1} (ArchimedeanClass.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.9) (ArchimedeanClass.instLinearOrder.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.9))))))) (ArchimedeanClass.instOrderTop.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.9))))))) (PartialOrder.toPreorder.{u_1} (Submodule.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.25) (Submodule.instPartialOrder.{u_2, u_1} K M (Ring.toSemiring.{u_2} K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.13) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.25)) (FiniteArchimedeanClass.submodule.{u_1, u_2} M inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.9 K inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.22 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.25 inst._@.Mathlib.Algebra.Order.Module.Archimedean.2863435983._hygCtx._hyg.29)","typeFull":"∀ {M : Type u_1} [inst : AddCommGroup M] [inst_1 : LinearOrder M] [inst_2 : IsOrderedAddMonoid M] (K : Type u_2)\n [inst_3 : Ring K] [inst_4 : LinearOrder K] [inst_5 : IsOrderedRing K] [inst_6 : Archimedean K] [inst_7 : Module K M]\n [inst_8 : PosSMulMono K M], StrictAnti (FiniteArchimedeanClass.submodule K)","typeReadable":"∀ {M : Type u_1} [inst : AddCommGroup M] [inst_1 : LinearOrder M] [inst_2 : IsOrderedAddMonoid M] (K : Type u_2)\n [inst_3 : Ring K] [inst_4 : LinearOrder K] [inst_5 : IsOrderedRing K] [inst_6 : Archimedean K] [inst_7 : Module K M]\n [inst_8 : PosSMulMono K M], StrictAnti (FiniteArchimedeanClass.submodule K)","typeReferences":[["PartialOrder","toPreorder"],["Ring","toNonAssocRing"],["ArchimedeanClass"],["Subtype","instLE"],["Module"],["AddCommGroup","toAddGroup"],["SMulZeroClass","toSMul"],["FiniteArchimedeanClass","submodule"],["Submodule"],["PosSMulMono"],["instDistribLatticeOfLinearOrder"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["IsOrderedAddMonoid"],["ArchimedeanClass","instLinearOrder"],["MonoidWithZero","toMonoid"],["FiniteArchimedeanClass"],["AddGroup","toSubNegMonoid"],["Preorder","toLE"],["NonAssocRing","toNonUnitalNonAssocRing"],["DistribSMul","toSMulZeroClass"],["SemilatticeInf","toPartialOrder"],["Lattice","toSemilatticeInf"],["Submodule","instPartialOrder"],["UpperSet","instPartialOrder"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["DistribMulAction","toDistribSMul"],["LinearOrder"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["AddZeroClass","toAddZero"],["UpperSet"],["ArchimedeanClass","instOrderTop"],["Ring","toSemiring"],["StrictAnti"],["IsOrderedRing"],["Module","toDistribMulAction"],["DistribLattice","toLattice"],["SubNegMonoid","toAddMonoid"],["MulZeroClass","toZero"],["Archimedean"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["Top","top"],["AddCommGroup","toAddCommMonoid"],["Ne"],["OrderTop","toTop"],["AddZero","toZero"],["Ring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["FiniteArchimedeanClass","addSubgroup_strictAnti"]]}]