diff --git "a/data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Order.Module.Pointwise.sym.json" "b/data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Order.Module.Pointwise.sym.json" new file mode 100644--- /dev/null +++ "b/data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Order.Module.Pointwise.sym.json" @@ -0,0 +1 @@ +[{"isProp":true,"kind":"theorem","name":["lowerBounds_smul_of_neg"],"typeFallback":"forall {α : Type.{u_1}} {β : Type.{u_2}} [inst._@.Mathlib.Algebra.Order.Module.Pointwise.1962474121._hygCtx._hyg.4 : Field.{u_1} α] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.1962474121._hygCtx._hyg.7 : LinearOrder.{u_1} α] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.1962474121._hygCtx._hyg.10 : IsStrictOrderedRing.{u_1} α (DivisionSemiring.toSemiring.{u_1} α (Semifield.toDivisionSemiring.{u_1} α (Field.toSemifield.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.1962474121._hygCtx._hyg.4))) (SemilatticeInf.toPartialOrder.{u_1} α (Lattice.toSemilatticeInf.{u_1} α (DistribLattice.toLattice.{u_1} α (instDistribLatticeOfLinearOrder.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.1962474121._hygCtx._hyg.7))))] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.1962474121._hygCtx._hyg.13 : AddCommGroup.{u_2} β] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.1962474121._hygCtx._hyg.16 : PartialOrder.{u_2} β] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.1962474121._hygCtx._hyg.19 : IsOrderedAddMonoid.{u_2} β (AddCommGroup.toAddCommMonoid.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.1962474121._hygCtx._hyg.13) (PartialOrder.toPreorder.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.1962474121._hygCtx._hyg.16)] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.1962474121._hygCtx._hyg.22 : Module.{u_1, u_2} α β (DivisionSemiring.toSemiring.{u_1} α (Semifield.toDivisionSemiring.{u_1} α (Field.toSemifield.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.1962474121._hygCtx._hyg.4))) (AddCommGroup.toAddCommMonoid.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.1962474121._hygCtx._hyg.13)] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.1962474121._hygCtx._hyg.26 : PosSMulMono.{u_1, u_2} α β (SMulZeroClass.toSMul.{u_1, u_2} α β (AddZero.toZero.{u_2} β (AddZeroClass.toAddZero.{u_2} β (AddMonoid.toAddZeroClass.{u_2} β (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.1962474121._hygCtx._hyg.13)))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} α β (AddMonoid.toAddZeroClass.{u_2} β (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.1962474121._hygCtx._hyg.13)))) (DistribMulAction.toDistribSMul.{u_1, u_2} α β (MonoidWithZero.toMonoid.{u_1} α (Semiring.toMonoidWithZero.{u_1} α (DivisionSemiring.toSemiring.{u_1} α (Semifield.toDivisionSemiring.{u_1} α (Field.toSemifield.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.1962474121._hygCtx._hyg.4))))) (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.1962474121._hygCtx._hyg.13))) (Module.toDistribMulAction.{u_1, u_2} α β (DivisionSemiring.toSemiring.{u_1} α (Semifield.toDivisionSemiring.{u_1} α (Field.toSemifield.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.1962474121._hygCtx._hyg.4))) (AddCommGroup.toAddCommMonoid.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.1962474121._hygCtx._hyg.13) inst._@.Mathlib.Algebra.Order.Module.Pointwise.1962474121._hygCtx._hyg.22)))) (PartialOrder.toPreorder.{u_1} α (SemilatticeInf.toPartialOrder.{u_1} α (Lattice.toSemilatticeInf.{u_1} α (DistribLattice.toLattice.{u_1} α (instDistribLatticeOfLinearOrder.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.1962474121._hygCtx._hyg.7))))) (PartialOrder.toPreorder.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.1962474121._hygCtx._hyg.16) (MulZeroClass.toZero.{u_1} α (NonUnitalNonAssocSemiring.toMulZeroClass.{u_1} α (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_1} α (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_1} α (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_1} α (CommRing.toNonUnitalCommRing.{u_1} α (Field.toCommRing.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.1962474121._hygCtx._hyg.4)))))))] {s : Set.{u_2} β} {a : α}, (LT.lt.{u_1} α (Preorder.toLT.{u_1} α (PartialOrder.toPreorder.{u_1} α (SemilatticeInf.toPartialOrder.{u_1} α (Lattice.toSemilatticeInf.{u_1} α (DistribLattice.toLattice.{u_1} α (instDistribLatticeOfLinearOrder.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.1962474121._hygCtx._hyg.7)))))) a (OfNat.ofNat.{u_1} α 0 (Zero.toOfNat0.{u_1} α (MulZeroClass.toZero.{u_1} α (NonUnitalNonAssocSemiring.toMulZeroClass.{u_1} α (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_1} α (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_1} α (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_1} α (CommRing.toNonUnitalCommRing.{u_1} α (Field.toCommRing.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.1962474121._hygCtx._hyg.4)))))))))) -> (Eq.{succ u_2} (Set.{u_2} β) (lowerBounds.{u_2} β (Preorder.toLE.{u_2} β (PartialOrder.toPreorder.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.1962474121._hygCtx._hyg.16)) (HSMul.hSMul.{u_1, u_2, u_2} α (Set.{u_2} β) (Set.{u_2} β) (instHSMul.{u_1, u_2} α (Set.{u_2} β) (Set.smulSet.{u_1, u_2} α β (SMulZeroClass.toSMul.{u_1, u_2} α β (AddZero.toZero.{u_2} β (AddZeroClass.toAddZero.{u_2} β (AddMonoid.toAddZeroClass.{u_2} β (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.1962474121._hygCtx._hyg.13)))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} α β (AddMonoid.toAddZeroClass.{u_2} β (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.1962474121._hygCtx._hyg.13)))) (DistribMulAction.toDistribSMul.{u_1, u_2} α β (MonoidWithZero.toMonoid.{u_1} α (Semiring.toMonoidWithZero.{u_1} α (DivisionSemiring.toSemiring.{u_1} α (Semifield.toDivisionSemiring.{u_1} α (Field.toSemifield.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.1962474121._hygCtx._hyg.4))))) (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.1962474121._hygCtx._hyg.13))) (Module.toDistribMulAction.{u_1, u_2} α β (DivisionSemiring.toSemiring.{u_1} α (Semifield.toDivisionSemiring.{u_1} α (Field.toSemifield.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.1962474121._hygCtx._hyg.4))) (AddCommGroup.toAddCommMonoid.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.1962474121._hygCtx._hyg.13) inst._@.Mathlib.Algebra.Order.Module.Pointwise.1962474121._hygCtx._hyg.22)))))) a s)) (HSMul.hSMul.{u_1, u_2, u_2} α (Set.{u_2} β) (Set.{u_2} β) (instHSMul.{u_1, u_2} α (Set.{u_2} β) (Set.smulSet.{u_1, u_2} α β (SMulZeroClass.toSMul.{u_1, u_2} α β (AddZero.toZero.{u_2} β (AddZeroClass.toAddZero.{u_2} β (AddMonoid.toAddZeroClass.{u_2} β (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.1962474121._hygCtx._hyg.13)))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} α β (AddMonoid.toAddZeroClass.{u_2} β (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.1962474121._hygCtx._hyg.13)))) (DistribMulAction.toDistribSMul.{u_1, u_2} α β (MonoidWithZero.toMonoid.{u_1} α (Semiring.toMonoidWithZero.{u_1} α (DivisionSemiring.toSemiring.{u_1} α (Semifield.toDivisionSemiring.{u_1} α (Field.toSemifield.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.1962474121._hygCtx._hyg.4))))) (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.1962474121._hygCtx._hyg.13))) (Module.toDistribMulAction.{u_1, u_2} α β (DivisionSemiring.toSemiring.{u_1} α (Semifield.toDivisionSemiring.{u_1} α (Field.toSemifield.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.1962474121._hygCtx._hyg.4))) (AddCommGroup.toAddCommMonoid.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.1962474121._hygCtx._hyg.13) inst._@.Mathlib.Algebra.Order.Module.Pointwise.1962474121._hygCtx._hyg.22)))))) a (upperBounds.{u_2} β (Preorder.toLE.{u_2} β (PartialOrder.toPreorder.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.1962474121._hygCtx._hyg.16)) s)))","typeFull":"∀ {α : Type u_1} {β : Type u_2} [inst : Field α] [inst_1 : LinearOrder α] [IsStrictOrderedRing α]\n [inst_3 : AddCommGroup β] [inst_4 : PartialOrder β] [IsOrderedAddMonoid β] [inst_6 : Module α β] [PosSMulMono α β]\n {s : Set β} {a : α}, a < 0 → lowerBounds (a • s) = a • upperBounds s","typeReadable":"∀ {α : Type u_1} {β : Type u_2} [inst : Field α] [inst_1 : LinearOrder α] [IsStrictOrderedRing α]\n [inst_3 : AddCommGroup β] [inst_4 : PartialOrder β] [IsOrderedAddMonoid β] [inst_6 : Module α β] [PosSMulMono α β]\n {s : Set β} {a : α}, a < 0 → lowerBounds (a • s) = a • upperBounds s","typeReferences":[["PartialOrder","toPreorder"],["Field"],["Module"],["AddCommGroup","toAddGroup"],["Preorder","toLT"],["SMulZeroClass","toSMul"],["CommRing","toNonUnitalCommRing"],["PosSMulMono"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["instDistribLatticeOfLinearOrder"],["IsOrderedAddMonoid"],["PartialOrder"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["MonoidWithZero","toMonoid"],["Set","smulSet"],["instHSMul"],["Zero","toOfNat0"],["Semifield","toDivisionSemiring"],["AddGroup","toSubNegMonoid"],["Preorder","toLE"],["Eq"],["SemilatticeInf","toPartialOrder"],["DistribSMul","toSMulZeroClass"],["Field","toCommRing"],["Lattice","toSemilatticeInf"],["IsStrictOrderedRing"],["Set"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["DistribMulAction","toDistribSMul"],["LinearOrder"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["DivisionSemiring","toSemiring"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["LT","lt"],["Module","toDistribMulAction"],["DistribLattice","toLattice"],["SubNegMonoid","toAddMonoid"],["upperBounds"],["MulZeroClass","toZero"],["HSMul","hSMul"],["lowerBounds"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["Field","toSemifield"],["AddCommGroup","toAddCommMonoid"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["OrderIso","smulRightDual"],["OrderDual","instPreorder"],["OrderDual"],["PartialOrder","toPreorder"],["OrderIso","upperBounds_image"]]},{"isProp":true,"kind":"theorem","name":["BddAbove","smul_of_nonneg"],"typeFallback":"forall {α : Type.{u_1}} {β : Type.{u_2}} [inst._@.Mathlib.Algebra.Order.Module.Pointwise.911681139._hygCtx._hyg.4 : SMul.{u_1, u_2} α β] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.911681139._hygCtx._hyg.8 : Preorder.{u_1} α] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.911681139._hygCtx._hyg.11 : Preorder.{u_2} β] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.911681139._hygCtx._hyg.14 : Zero.{u_1} α] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.911681139._hygCtx._hyg.17 : PosSMulMono.{u_1, u_2} α β inst._@.Mathlib.Algebra.Order.Module.Pointwise.911681139._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Order.Module.Pointwise.911681139._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Order.Module.Pointwise.911681139._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Order.Module.Pointwise.911681139._hygCtx._hyg.14] {a : α} {s : Set.{u_2} β}, (BddAbove.{u_2} β (Preorder.toLE.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.911681139._hygCtx._hyg.11) s) -> (LE.le.{u_1} α (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.911681139._hygCtx._hyg.8) (OfNat.ofNat.{u_1} α 0 (Zero.toOfNat0.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.911681139._hygCtx._hyg.14)) a) -> (BddAbove.{u_2} β (Preorder.toLE.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.911681139._hygCtx._hyg.11) (HSMul.hSMul.{u_1, u_2, u_2} α (Set.{u_2} β) (Set.{u_2} β) (instHSMul.{u_1, u_2} α (Set.{u_2} β) (Set.smulSet.{u_1, u_2} α β inst._@.Mathlib.Algebra.Order.Module.Pointwise.911681139._hygCtx._hyg.4)) a s))","typeFull":"∀ {α : Type u_1} {β : Type u_2} [inst : SMul α β] [inst_1 : Preorder α] [inst_2 : Preorder β] [inst_3 : Zero α]\n [PosSMulMono α β] {a : α} {s : Set β}, BddAbove s → 0 ≤ a → BddAbove (a • s)","typeReadable":"∀ {α : Type u_1} {β : Type u_2} [inst : SMul α β] [inst_1 : Preorder α] [inst_2 : Preorder β] [inst_3 : Zero α]\n [PosSMulMono α β] {a : α} {s : Set β}, BddAbove s → 0 ≤ a → BddAbove (a • s)","typeReferences":[["Set"],["BddAbove"],["OfNat","ofNat"],["PosSMulMono"],["SMul"],["Preorder"],["LE","le"],["HSMul","hSMul"],["Set","smulSet"],["instHSMul"],["Zero","toOfNat0"],["Zero"],["Preorder","toLE"]],"valueReferences":[["HSMul","hSMul"],["instHSMul"],["monotone_smul_left_of_nonneg"],["Monotone","map_bddAbove"]]},{"isProp":true,"kind":"theorem","name":["bddBelow_smul_iff_of_neg","_simp_1"],"typeFallback":"forall {α : Type.{u_1}} {β : Type.{u_2}} [inst._@.Mathlib.Algebra.Order.Module.Pointwise.967861698._hygCtx._hyg.4 : Field.{u_1} α] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.967861698._hygCtx._hyg.7 : LinearOrder.{u_1} α] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.967861698._hygCtx._hyg.10 : IsStrictOrderedRing.{u_1} α (DivisionSemiring.toSemiring.{u_1} α (Semifield.toDivisionSemiring.{u_1} α (Field.toSemifield.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.967861698._hygCtx._hyg.4))) (SemilatticeInf.toPartialOrder.{u_1} α (Lattice.toSemilatticeInf.{u_1} α (DistribLattice.toLattice.{u_1} α (instDistribLatticeOfLinearOrder.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.967861698._hygCtx._hyg.7))))] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.967861698._hygCtx._hyg.13 : AddCommGroup.{u_2} β] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.967861698._hygCtx._hyg.16 : PartialOrder.{u_2} β] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.967861698._hygCtx._hyg.19 : IsOrderedAddMonoid.{u_2} β (AddCommGroup.toAddCommMonoid.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.967861698._hygCtx._hyg.13) (PartialOrder.toPreorder.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.967861698._hygCtx._hyg.16)] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.967861698._hygCtx._hyg.22 : Module.{u_1, u_2} α β (DivisionSemiring.toSemiring.{u_1} α (Semifield.toDivisionSemiring.{u_1} α (Field.toSemifield.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.967861698._hygCtx._hyg.4))) (AddCommGroup.toAddCommMonoid.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.967861698._hygCtx._hyg.13)] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.967861698._hygCtx._hyg.26 : PosSMulMono.{u_1, u_2} α β (SMulZeroClass.toSMul.{u_1, u_2} α β (AddZero.toZero.{u_2} β (AddZeroClass.toAddZero.{u_2} β (AddMonoid.toAddZeroClass.{u_2} β (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.967861698._hygCtx._hyg.13)))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} α β (AddMonoid.toAddZeroClass.{u_2} β (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.967861698._hygCtx._hyg.13)))) (DistribMulAction.toDistribSMul.{u_1, u_2} α β (MonoidWithZero.toMonoid.{u_1} α (Semiring.toMonoidWithZero.{u_1} α (DivisionSemiring.toSemiring.{u_1} α (Semifield.toDivisionSemiring.{u_1} α (Field.toSemifield.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.967861698._hygCtx._hyg.4))))) (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.967861698._hygCtx._hyg.13))) (Module.toDistribMulAction.{u_1, u_2} α β (DivisionSemiring.toSemiring.{u_1} α (Semifield.toDivisionSemiring.{u_1} α (Field.toSemifield.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.967861698._hygCtx._hyg.4))) (AddCommGroup.toAddCommMonoid.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.967861698._hygCtx._hyg.13) inst._@.Mathlib.Algebra.Order.Module.Pointwise.967861698._hygCtx._hyg.22)))) (PartialOrder.toPreorder.{u_1} α (SemilatticeInf.toPartialOrder.{u_1} α (Lattice.toSemilatticeInf.{u_1} α (DistribLattice.toLattice.{u_1} α (instDistribLatticeOfLinearOrder.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.967861698._hygCtx._hyg.7))))) (PartialOrder.toPreorder.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.967861698._hygCtx._hyg.16) (MulZeroClass.toZero.{u_1} α (NonUnitalNonAssocSemiring.toMulZeroClass.{u_1} α (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_1} α (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_1} α (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_1} α (CommRing.toNonUnitalCommRing.{u_1} α (Field.toCommRing.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.967861698._hygCtx._hyg.4)))))))] {s : Set.{u_2} β} {a : α}, (LT.lt.{u_1} α (Preorder.toLT.{u_1} α (PartialOrder.toPreorder.{u_1} α (SemilatticeInf.toPartialOrder.{u_1} α (Lattice.toSemilatticeInf.{u_1} α (DistribLattice.toLattice.{u_1} α (instDistribLatticeOfLinearOrder.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.967861698._hygCtx._hyg.7)))))) a (OfNat.ofNat.{u_1} α 0 (Zero.toOfNat0.{u_1} α (MulZeroClass.toZero.{u_1} α (NonUnitalNonAssocSemiring.toMulZeroClass.{u_1} α (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_1} α (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_1} α (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_1} α (CommRing.toNonUnitalCommRing.{u_1} α (Field.toCommRing.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.967861698._hygCtx._hyg.4)))))))))) -> (Eq.{1} Prop (BddBelow.{u_2} β (Preorder.toLE.{u_2} β (PartialOrder.toPreorder.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.967861698._hygCtx._hyg.16)) (HSMul.hSMul.{u_1, u_2, u_2} α (Set.{u_2} β) (Set.{u_2} β) (instHSMul.{u_1, u_2} α (Set.{u_2} β) (Set.smulSet.{u_1, u_2} α β (SMulZeroClass.toSMul.{u_1, u_2} α β (AddZero.toZero.{u_2} β (AddZeroClass.toAddZero.{u_2} β (AddMonoid.toAddZeroClass.{u_2} β (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.967861698._hygCtx._hyg.13)))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} α β (AddMonoid.toAddZeroClass.{u_2} β (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.967861698._hygCtx._hyg.13)))) (DistribMulAction.toDistribSMul.{u_1, u_2} α β (MonoidWithZero.toMonoid.{u_1} α (Semiring.toMonoidWithZero.{u_1} α (DivisionSemiring.toSemiring.{u_1} α (Semifield.toDivisionSemiring.{u_1} α (Field.toSemifield.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.967861698._hygCtx._hyg.4))))) (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.967861698._hygCtx._hyg.13))) (Module.toDistribMulAction.{u_1, u_2} α β (DivisionSemiring.toSemiring.{u_1} α (Semifield.toDivisionSemiring.{u_1} α (Field.toSemifield.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.967861698._hygCtx._hyg.4))) (AddCommGroup.toAddCommMonoid.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.967861698._hygCtx._hyg.13) inst._@.Mathlib.Algebra.Order.Module.Pointwise.967861698._hygCtx._hyg.22)))))) a s)) (BddAbove.{u_2} β (Preorder.toLE.{u_2} β (PartialOrder.toPreorder.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.967861698._hygCtx._hyg.16)) s))","typeFull":"∀ {α : Type u_1} {β : Type u_2} [inst : Field α] [inst_1 : LinearOrder α] [IsStrictOrderedRing α]\n [inst_3 : AddCommGroup β] [inst_4 : PartialOrder β] [IsOrderedAddMonoid β] [inst_6 : Module α β] [PosSMulMono α β]\n {s : Set β} {a : α}, a < 0 → BddBelow (a • s) = BddAbove s","typeReadable":"∀ {α : Type u_1} {β : Type u_2} [inst : Field α] [inst_1 : LinearOrder α] [IsStrictOrderedRing α]\n [inst_3 : AddCommGroup β] [inst_4 : PartialOrder β] [IsOrderedAddMonoid β] [inst_6 : Module α β] [PosSMulMono α β]\n {s : Set β} {a : α}, a < 0 → BddBelow (a • s) = BddAbove s","typeReferences":[["PartialOrder","toPreorder"],["Field"],["Module"],["AddCommGroup","toAddGroup"],["Preorder","toLT"],["SMulZeroClass","toSMul"],["CommRing","toNonUnitalCommRing"],["PosSMulMono"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["instDistribLatticeOfLinearOrder"],["IsOrderedAddMonoid"],["PartialOrder"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["MonoidWithZero","toMonoid"],["Set","smulSet"],["instHSMul"],["Zero","toOfNat0"],["Semifield","toDivisionSemiring"],["AddGroup","toSubNegMonoid"],["Preorder","toLE"],["Eq"],["SemilatticeInf","toPartialOrder"],["DistribSMul","toSMulZeroClass"],["Field","toCommRing"],["Lattice","toSemilatticeInf"],["IsStrictOrderedRing"],["Set"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["DistribMulAction","toDistribSMul"],["LinearOrder"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["BddBelow"],["DivisionSemiring","toSemiring"],["AddZeroClass","toAddZero"],["BddAbove"],["OfNat","ofNat"],["LT","lt"],["Module","toDistribMulAction"],["DistribLattice","toLattice"],["SubNegMonoid","toAddMonoid"],["MulZeroClass","toZero"],["HSMul","hSMul"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["Field","toSemifield"],["AddCommGroup","toAddCommMonoid"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["PartialOrder","toPreorder"],["AddCommGroup","toAddGroup"],["SMulZeroClass","toSMul"],["bddBelow_smul_iff_of_neg"],["MonoidWithZero","toMonoid"],["Set","smulSet"],["instHSMul"],["AddGroup","toSubNegMonoid"],["Semifield","toDivisionSemiring"],["Preorder","toLE"],["propext"],["DistribSMul","toSMulZeroClass"],["Set"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["BddBelow"],["DivisionSemiring","toSemiring"],["AddZeroClass","toAddZero"],["BddAbove"],["Module","toDistribMulAction"],["SubNegMonoid","toAddMonoid"],["HSMul","hSMul"],["Field","toSemifield"],["AddCommGroup","toAddCommMonoid"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["BddAbove","smul_of_nonpos"],"typeFallback":"forall {α : Type.{u_1}} {β : Type.{u_2}} [inst._@.Mathlib.Algebra.Order.Module.Pointwise.3286293553._hygCtx._hyg.4 : Ring.{u_1} α] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.3286293553._hygCtx._hyg.7 : PartialOrder.{u_1} α] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.3286293553._hygCtx._hyg.10 : IsOrderedRing.{u_1} α (Ring.toSemiring.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3286293553._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Order.Module.Pointwise.3286293553._hygCtx._hyg.7] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.3286293553._hygCtx._hyg.13 : AddCommGroup.{u_2} β] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.3286293553._hygCtx._hyg.16 : PartialOrder.{u_2} β] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.3286293553._hygCtx._hyg.19 : IsOrderedAddMonoid.{u_2} β (AddCommGroup.toAddCommMonoid.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3286293553._hygCtx._hyg.13) (PartialOrder.toPreorder.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3286293553._hygCtx._hyg.16)] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.3286293553._hygCtx._hyg.22 : Module.{u_1, u_2} α β (Ring.toSemiring.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3286293553._hygCtx._hyg.4) (AddCommGroup.toAddCommMonoid.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3286293553._hygCtx._hyg.13)] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.3286293553._hygCtx._hyg.26 : PosSMulMono.{u_1, u_2} α β (SMulZeroClass.toSMul.{u_1, u_2} α β (AddZero.toZero.{u_2} β (AddZeroClass.toAddZero.{u_2} β (AddMonoid.toAddZeroClass.{u_2} β (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3286293553._hygCtx._hyg.13)))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} α β (AddMonoid.toAddZeroClass.{u_2} β (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3286293553._hygCtx._hyg.13)))) (DistribMulAction.toDistribSMul.{u_1, u_2} α β (MonoidWithZero.toMonoid.{u_1} α (Semiring.toMonoidWithZero.{u_1} α (Ring.toSemiring.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3286293553._hygCtx._hyg.4))) (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3286293553._hygCtx._hyg.13))) (Module.toDistribMulAction.{u_1, u_2} α β (Ring.toSemiring.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3286293553._hygCtx._hyg.4) (AddCommGroup.toAddCommMonoid.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3286293553._hygCtx._hyg.13) inst._@.Mathlib.Algebra.Order.Module.Pointwise.3286293553._hygCtx._hyg.22)))) (PartialOrder.toPreorder.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3286293553._hygCtx._hyg.7) (PartialOrder.toPreorder.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3286293553._hygCtx._hyg.16) (MulZeroClass.toZero.{u_1} α (NonUnitalNonAssocSemiring.toMulZeroClass.{u_1} α (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_1} α (NonAssocRing.toNonUnitalNonAssocRing.{u_1} α (Ring.toNonAssocRing.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3286293553._hygCtx._hyg.4)))))] {s : Set.{u_2} β} {a : α}, (LE.le.{u_1} α (Preorder.toLE.{u_1} α (PartialOrder.toPreorder.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3286293553._hygCtx._hyg.7)) a (OfNat.ofNat.{u_1} α 0 (Zero.toOfNat0.{u_1} α (MulZeroClass.toZero.{u_1} α (NonUnitalNonAssocSemiring.toMulZeroClass.{u_1} α (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_1} α (NonAssocRing.toNonUnitalNonAssocRing.{u_1} α (Ring.toNonAssocRing.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3286293553._hygCtx._hyg.4)))))))) -> (BddAbove.{u_2} β (Preorder.toLE.{u_2} β (PartialOrder.toPreorder.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3286293553._hygCtx._hyg.16)) s) -> (BddBelow.{u_2} β (Preorder.toLE.{u_2} β (PartialOrder.toPreorder.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3286293553._hygCtx._hyg.16)) (HSMul.hSMul.{u_1, u_2, u_2} α (Set.{u_2} β) (Set.{u_2} β) (instHSMul.{u_1, u_2} α (Set.{u_2} β) (Set.smulSet.{u_1, u_2} α β (SMulZeroClass.toSMul.{u_1, u_2} α β (AddZero.toZero.{u_2} β (AddZeroClass.toAddZero.{u_2} β (AddMonoid.toAddZeroClass.{u_2} β (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3286293553._hygCtx._hyg.13)))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} α β (AddMonoid.toAddZeroClass.{u_2} β (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3286293553._hygCtx._hyg.13)))) (DistribMulAction.toDistribSMul.{u_1, u_2} α β (MonoidWithZero.toMonoid.{u_1} α (Semiring.toMonoidWithZero.{u_1} α (Ring.toSemiring.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3286293553._hygCtx._hyg.4))) (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3286293553._hygCtx._hyg.13))) (Module.toDistribMulAction.{u_1, u_2} α β (Ring.toSemiring.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3286293553._hygCtx._hyg.4) (AddCommGroup.toAddCommMonoid.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3286293553._hygCtx._hyg.13) inst._@.Mathlib.Algebra.Order.Module.Pointwise.3286293553._hygCtx._hyg.22)))))) a s))","typeFull":"∀ {α : Type u_1} {β : Type u_2} [inst : Ring α] [inst_1 : PartialOrder α] [IsOrderedRing α] [inst_3 : AddCommGroup β]\n [inst_4 : PartialOrder β] [IsOrderedAddMonoid β] [inst_6 : Module α β] [PosSMulMono α β] {s : Set β} {a : α},\n a ≤ 0 → BddAbove s → BddBelow (a • s)","typeReadable":"∀ {α : Type u_1} {β : Type u_2} [inst : Ring α] [inst_1 : PartialOrder α] [IsOrderedRing α] [inst_3 : AddCommGroup β]\n [inst_4 : PartialOrder β] [IsOrderedAddMonoid β] [inst_6 : Module α β] [PosSMulMono α β] {s : Set β} {a : α},\n a ≤ 0 → BddAbove s → BddBelow (a • s)","typeReferences":[["Ring","toNonAssocRing"],["PartialOrder","toPreorder"],["Module"],["AddCommGroup","toAddGroup"],["SMulZeroClass","toSMul"],["PosSMulMono"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["IsOrderedAddMonoid"],["PartialOrder"],["MonoidWithZero","toMonoid"],["Set","smulSet"],["instHSMul"],["Zero","toOfNat0"],["AddGroup","toSubNegMonoid"],["Preorder","toLE"],["NonAssocRing","toNonUnitalNonAssocRing"],["DistribSMul","toSMulZeroClass"],["Set"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["BddBelow"],["AddZeroClass","toAddZero"],["BddAbove"],["OfNat","ofNat"],["Ring","toSemiring"],["IsOrderedRing"],["Module","toDistribMulAction"],["SubNegMonoid","toAddMonoid"],["MulZeroClass","toZero"],["HSMul","hSMul"],["LE","le"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["AddCommGroup","toAddCommMonoid"],["AddZero","toZero"],["Ring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["PartialOrder","toPreorder"],["Antitone","map_bddAbove"],["AddCommGroup","toAddGroup"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddZeroClass","toAddZero"],["Ring","toSemiring"],["Module","toDistribMulAction"],["SubNegMonoid","toAddMonoid"],["MonoidWithZero","toMonoid"],["HSMul","hSMul"],["AddCommGroup","toAddCommMonoid"],["instHSMul"],["antitone_smul_left"],["AddGroup","toSubNegMonoid"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]]},{"isProp":true,"kind":"theorem","name":["bddAbove_smul_iff_of_pos"],"typeFallback":"forall {α : Type.{u_1}} {β : Type.{u_2}} [inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.4 : Preorder.{u_1} α] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.7 : Preorder.{u_2} β] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.10 : GroupWithZero.{u_1} α] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.13 : Zero.{u_2} β] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.16 : MulActionWithZero.{u_1, u_2} α β (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.13] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.20 : PosSMulMono.{u_1, u_2} α β (SMulZeroClass.toSMul.{u_1, u_2} α β inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.13 (SMulWithZero.toSMulZeroClass.{u_1, u_2} α β (MulZeroClass.toZero.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α (MonoidWithZero.toMulZeroOneClass.{u_1} α (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.10)))) inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.13 (MulActionWithZero.toSMulWithZero.{u_1, u_2} α β (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.16))) inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.7 (MulZeroClass.toZero.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α (MonoidWithZero.toMulZeroOneClass.{u_1} α (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.10))))] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.24 : PosSMulReflectLE.{u_1, u_2} α β (SMulZeroClass.toSMul.{u_1, u_2} α β inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.13 (SMulWithZero.toSMulZeroClass.{u_1, u_2} α β (MulZeroClass.toZero.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α (MonoidWithZero.toMulZeroOneClass.{u_1} α (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.10)))) inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.13 (MulActionWithZero.toSMulWithZero.{u_1, u_2} α β (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.16))) inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.7 (MulZeroClass.toZero.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α (MonoidWithZero.toMulZeroOneClass.{u_1} α (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.10))))] {s : Set.{u_2} β} {a : α}, (LT.lt.{u_1} α (Preorder.toLT.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.4) (OfNat.ofNat.{u_1} α 0 (Zero.toOfNat0.{u_1} α (MulZeroClass.toZero.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α (MonoidWithZero.toMulZeroOneClass.{u_1} α (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.10)))))) a) -> (Iff (BddAbove.{u_2} β (Preorder.toLE.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.7) (HSMul.hSMul.{u_1, u_2, u_2} α (Set.{u_2} β) (Set.{u_2} β) (instHSMul.{u_1, u_2} α (Set.{u_2} β) (Set.smulSet.{u_1, u_2} α β (SMulZeroClass.toSMul.{u_1, u_2} α β inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.13 (SMulWithZero.toSMulZeroClass.{u_1, u_2} α β (MulZeroClass.toZero.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α (MonoidWithZero.toMulZeroOneClass.{u_1} α (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.10)))) inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.13 (MulActionWithZero.toSMulWithZero.{u_1, u_2} α β (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.16))))) a s)) (BddAbove.{u_2} β (Preorder.toLE.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.7) s))","typeFull":"∀ {α : Type u_1} {β : Type u_2} [inst : Preorder α] [inst_1 : Preorder β] [inst_2 : GroupWithZero α] [inst_3 : Zero β]\n [inst_4 : MulActionWithZero α β] [PosSMulMono α β] [PosSMulReflectLE α β] {s : Set β} {a : α},\n 0 < a → (BddAbove (a • s) ↔ BddAbove s)","typeReadable":"∀ {α : Type u_1} {β : Type u_2} [inst : Preorder α] [inst_1 : Preorder β] [inst_2 : GroupWithZero α] [inst_3 : Zero β]\n [inst_4 : MulActionWithZero α β] [PosSMulMono α β] [PosSMulReflectLE α β] {s : Set β} {a : α},\n 0 < a → (BddAbove (a • s) ↔ BddAbove s)","typeReferences":[["Set"],["Preorder","toLT"],["SMulZeroClass","toSMul"],["MonoidWithZero","toMulZeroOneClass"],["GroupWithZero"],["SMulWithZero","toSMulZeroClass"],["BddAbove"],["OfNat","ofNat"],["PosSMulMono"],["LT","lt"],["PosSMulReflectLE"],["Preorder"],["MulZeroOneClass","toMulZeroClass"],["MulZeroClass","toZero"],["Iff"],["MulActionWithZero"],["HSMul","hSMul"],["GroupWithZero","toMonoidWithZero"],["Set","smulSet"],["instHSMul"],["Zero","toOfNat0"],["Zero"],["Preorder","toLE"],["MulActionWithZero","toSMulWithZero"]],"valueReferences":[["OrderIso","bddAbove_image"],["GroupWithZero","toMonoidWithZero"],["OrderIso","smulRight"],["MulActionWithZero","toMulAction"]]},{"isProp":true,"kind":"theorem","name":["BddBelow","smul_of_nonpos"],"typeFallback":"forall {α : Type.{u_1}} {β : Type.{u_2}} [inst._@.Mathlib.Algebra.Order.Module.Pointwise.3286293552._hygCtx._hyg.4 : Ring.{u_1} α] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.3286293552._hygCtx._hyg.7 : PartialOrder.{u_1} α] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.3286293552._hygCtx._hyg.10 : IsOrderedRing.{u_1} α (Ring.toSemiring.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3286293552._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Order.Module.Pointwise.3286293552._hygCtx._hyg.7] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.3286293552._hygCtx._hyg.13 : AddCommGroup.{u_2} β] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.3286293552._hygCtx._hyg.16 : PartialOrder.{u_2} β] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.3286293552._hygCtx._hyg.19 : IsOrderedAddMonoid.{u_2} β (AddCommGroup.toAddCommMonoid.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3286293552._hygCtx._hyg.13) (PartialOrder.toPreorder.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3286293552._hygCtx._hyg.16)] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.3286293552._hygCtx._hyg.22 : Module.{u_1, u_2} α β (Ring.toSemiring.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3286293552._hygCtx._hyg.4) (AddCommGroup.toAddCommMonoid.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3286293552._hygCtx._hyg.13)] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.3286293552._hygCtx._hyg.26 : PosSMulMono.{u_1, u_2} α β (SMulZeroClass.toSMul.{u_1, u_2} α β (AddZero.toZero.{u_2} β (AddZeroClass.toAddZero.{u_2} β (AddMonoid.toAddZeroClass.{u_2} β (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3286293552._hygCtx._hyg.13)))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} α β (AddMonoid.toAddZeroClass.{u_2} β (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3286293552._hygCtx._hyg.13)))) (DistribMulAction.toDistribSMul.{u_1, u_2} α β (MonoidWithZero.toMonoid.{u_1} α (Semiring.toMonoidWithZero.{u_1} α (Ring.toSemiring.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3286293552._hygCtx._hyg.4))) (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3286293552._hygCtx._hyg.13))) (Module.toDistribMulAction.{u_1, u_2} α β (Ring.toSemiring.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3286293552._hygCtx._hyg.4) (AddCommGroup.toAddCommMonoid.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3286293552._hygCtx._hyg.13) inst._@.Mathlib.Algebra.Order.Module.Pointwise.3286293552._hygCtx._hyg.22)))) (PartialOrder.toPreorder.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3286293552._hygCtx._hyg.7) (PartialOrder.toPreorder.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3286293552._hygCtx._hyg.16) (MulZeroClass.toZero.{u_1} α (NonUnitalNonAssocSemiring.toMulZeroClass.{u_1} α (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_1} α (NonAssocRing.toNonUnitalNonAssocRing.{u_1} α (Ring.toNonAssocRing.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3286293552._hygCtx._hyg.4)))))] {s : Set.{u_2} β} {a : α}, (LE.le.{u_1} α (Preorder.toLE.{u_1} α (PartialOrder.toPreorder.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3286293552._hygCtx._hyg.7)) a (OfNat.ofNat.{u_1} α 0 (Zero.toOfNat0.{u_1} α (MulZeroClass.toZero.{u_1} α (NonUnitalNonAssocSemiring.toMulZeroClass.{u_1} α (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_1} α (NonAssocRing.toNonUnitalNonAssocRing.{u_1} α (Ring.toNonAssocRing.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3286293552._hygCtx._hyg.4)))))))) -> (BddBelow.{u_2} β (Preorder.toLE.{u_2} β (PartialOrder.toPreorder.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3286293552._hygCtx._hyg.16)) s) -> (BddAbove.{u_2} β (Preorder.toLE.{u_2} β (PartialOrder.toPreorder.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3286293552._hygCtx._hyg.16)) (HSMul.hSMul.{u_1, u_2, u_2} α (Set.{u_2} β) (Set.{u_2} β) (instHSMul.{u_1, u_2} α (Set.{u_2} β) (Set.smulSet.{u_1, u_2} α β (SMulZeroClass.toSMul.{u_1, u_2} α β (AddZero.toZero.{u_2} β (AddZeroClass.toAddZero.{u_2} β (AddMonoid.toAddZeroClass.{u_2} β (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3286293552._hygCtx._hyg.13)))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} α β (AddMonoid.toAddZeroClass.{u_2} β (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3286293552._hygCtx._hyg.13)))) (DistribMulAction.toDistribSMul.{u_1, u_2} α β (MonoidWithZero.toMonoid.{u_1} α (Semiring.toMonoidWithZero.{u_1} α (Ring.toSemiring.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3286293552._hygCtx._hyg.4))) (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3286293552._hygCtx._hyg.13))) (Module.toDistribMulAction.{u_1, u_2} α β (Ring.toSemiring.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3286293552._hygCtx._hyg.4) (AddCommGroup.toAddCommMonoid.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3286293552._hygCtx._hyg.13) inst._@.Mathlib.Algebra.Order.Module.Pointwise.3286293552._hygCtx._hyg.22)))))) a s))","typeFull":"∀ {α : Type u_1} {β : Type u_2} [inst : Ring α] [inst_1 : PartialOrder α] [IsOrderedRing α] [inst_3 : AddCommGroup β]\n [inst_4 : PartialOrder β] [IsOrderedAddMonoid β] [inst_6 : Module α β] [PosSMulMono α β] {s : Set β} {a : α},\n a ≤ 0 → BddBelow s → BddAbove (a • s)","typeReadable":"∀ {α : Type u_1} {β : Type u_2} [inst : Ring α] [inst_1 : PartialOrder α] [IsOrderedRing α] [inst_3 : AddCommGroup β]\n [inst_4 : PartialOrder β] [IsOrderedAddMonoid β] [inst_6 : Module α β] [PosSMulMono α β] {s : Set β} {a : α},\n a ≤ 0 → BddBelow s → BddAbove (a • s)","typeReferences":[["Ring","toNonAssocRing"],["PartialOrder","toPreorder"],["Module"],["AddCommGroup","toAddGroup"],["SMulZeroClass","toSMul"],["PosSMulMono"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["IsOrderedAddMonoid"],["PartialOrder"],["MonoidWithZero","toMonoid"],["Set","smulSet"],["instHSMul"],["Zero","toOfNat0"],["AddGroup","toSubNegMonoid"],["Preorder","toLE"],["NonAssocRing","toNonUnitalNonAssocRing"],["DistribSMul","toSMulZeroClass"],["Set"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["BddBelow"],["AddZeroClass","toAddZero"],["BddAbove"],["OfNat","ofNat"],["Ring","toSemiring"],["IsOrderedRing"],["Module","toDistribMulAction"],["SubNegMonoid","toAddMonoid"],["MulZeroClass","toZero"],["HSMul","hSMul"],["LE","le"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["AddCommGroup","toAddCommMonoid"],["AddZero","toZero"],["Ring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["PartialOrder","toPreorder"],["Antitone","map_bddBelow"],["AddCommGroup","toAddGroup"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddZeroClass","toAddZero"],["Ring","toSemiring"],["Module","toDistribMulAction"],["SubNegMonoid","toAddMonoid"],["MonoidWithZero","toMonoid"],["HSMul","hSMul"],["AddCommGroup","toAddCommMonoid"],["instHSMul"],["antitone_smul_left"],["AddGroup","toSubNegMonoid"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]]},{"isProp":true,"kind":"theorem","name":["smul_lowerBounds_subset_lowerBounds_smul_of_nonneg"],"typeFallback":"forall {α : Type.{u_1}} {β : Type.{u_2}} [inst._@.Mathlib.Algebra.Order.Module.Pointwise.4074436909._hygCtx._hyg.4 : SMul.{u_1, u_2} α β] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.4074436909._hygCtx._hyg.8 : Preorder.{u_1} α] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.4074436909._hygCtx._hyg.11 : Preorder.{u_2} β] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.4074436909._hygCtx._hyg.14 : Zero.{u_1} α] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.4074436909._hygCtx._hyg.17 : PosSMulMono.{u_1, u_2} α β inst._@.Mathlib.Algebra.Order.Module.Pointwise.4074436909._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Order.Module.Pointwise.4074436909._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Order.Module.Pointwise.4074436909._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Order.Module.Pointwise.4074436909._hygCtx._hyg.14] {a : α} {s : Set.{u_2} β}, (LE.le.{u_1} α (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.4074436909._hygCtx._hyg.8) (OfNat.ofNat.{u_1} α 0 (Zero.toOfNat0.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.4074436909._hygCtx._hyg.14)) a) -> (HasSubset.Subset.{u_2} (Set.{u_2} β) (Set.instHasSubset.{u_2} β) (HSMul.hSMul.{u_1, u_2, u_2} α (Set.{u_2} β) (Set.{u_2} β) (instHSMul.{u_1, u_2} α (Set.{u_2} β) (Set.smulSet.{u_1, u_2} α β inst._@.Mathlib.Algebra.Order.Module.Pointwise.4074436909._hygCtx._hyg.4)) a (lowerBounds.{u_2} β (Preorder.toLE.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.4074436909._hygCtx._hyg.11) s)) (lowerBounds.{u_2} β (Preorder.toLE.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.4074436909._hygCtx._hyg.11) (HSMul.hSMul.{u_1, u_2, u_2} α (Set.{u_2} β) (Set.{u_2} β) (instHSMul.{u_1, u_2} α (Set.{u_2} β) (Set.smulSet.{u_1, u_2} α β inst._@.Mathlib.Algebra.Order.Module.Pointwise.4074436909._hygCtx._hyg.4)) a s)))","typeFull":"∀ {α : Type u_1} {β : Type u_2} [inst : SMul α β] [inst_1 : Preorder α] [inst_2 : Preorder β] [inst_3 : Zero α]\n [PosSMulMono α β] {a : α} {s : Set β}, 0 ≤ a → a • lowerBounds s ⊆ lowerBounds (a • s)","typeReadable":"∀ {α : Type u_1} {β : Type u_2} [inst : SMul α β] [inst_1 : Preorder α] [inst_2 : Preorder β] [inst_3 : Zero α]\n [PosSMulMono α β] {a : α} {s : Set β}, 0 ≤ a → a • lowerBounds s ⊆ lowerBounds (a • s)","typeReferences":[["Set"],["OfNat","ofNat"],["PosSMulMono"],["Set","instHasSubset"],["SMul"],["Preorder"],["HasSubset","Subset"],["lowerBounds"],["LE","le"],["HSMul","hSMul"],["Set","smulSet"],["instHSMul"],["Zero","toOfNat0"],["Zero"],["Preorder","toLE"]],"valueReferences":[["HSMul","hSMul"],["Monotone","image_lowerBounds_subset_lowerBounds_image"],["instHSMul"],["monotone_smul_left_of_nonneg"]]},{"isProp":true,"kind":"theorem","name":["smul_upperBounds_subset_lowerBounds_smul"],"typeFallback":"forall {α : Type.{u_1}} {β : Type.{u_2}} [inst._@.Mathlib.Algebra.Order.Module.Pointwise.2655550140._hygCtx._hyg.4 : Ring.{u_1} α] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.2655550140._hygCtx._hyg.7 : PartialOrder.{u_1} α] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.2655550140._hygCtx._hyg.10 : IsOrderedRing.{u_1} α (Ring.toSemiring.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.2655550140._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Order.Module.Pointwise.2655550140._hygCtx._hyg.7] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.2655550140._hygCtx._hyg.13 : AddCommGroup.{u_2} β] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.2655550140._hygCtx._hyg.16 : PartialOrder.{u_2} β] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.2655550140._hygCtx._hyg.19 : IsOrderedAddMonoid.{u_2} β (AddCommGroup.toAddCommMonoid.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.2655550140._hygCtx._hyg.13) (PartialOrder.toPreorder.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.2655550140._hygCtx._hyg.16)] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.2655550140._hygCtx._hyg.22 : Module.{u_1, u_2} α β (Ring.toSemiring.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.2655550140._hygCtx._hyg.4) (AddCommGroup.toAddCommMonoid.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.2655550140._hygCtx._hyg.13)] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.2655550140._hygCtx._hyg.26 : PosSMulMono.{u_1, u_2} α β (SMulZeroClass.toSMul.{u_1, u_2} α β (AddZero.toZero.{u_2} β (AddZeroClass.toAddZero.{u_2} β (AddMonoid.toAddZeroClass.{u_2} β (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.2655550140._hygCtx._hyg.13)))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} α β (AddMonoid.toAddZeroClass.{u_2} β (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.2655550140._hygCtx._hyg.13)))) (DistribMulAction.toDistribSMul.{u_1, u_2} α β (MonoidWithZero.toMonoid.{u_1} α (Semiring.toMonoidWithZero.{u_1} α (Ring.toSemiring.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.2655550140._hygCtx._hyg.4))) (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.2655550140._hygCtx._hyg.13))) (Module.toDistribMulAction.{u_1, u_2} α β (Ring.toSemiring.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.2655550140._hygCtx._hyg.4) (AddCommGroup.toAddCommMonoid.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.2655550140._hygCtx._hyg.13) inst._@.Mathlib.Algebra.Order.Module.Pointwise.2655550140._hygCtx._hyg.22)))) (PartialOrder.toPreorder.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.2655550140._hygCtx._hyg.7) (PartialOrder.toPreorder.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.2655550140._hygCtx._hyg.16) (MulZeroClass.toZero.{u_1} α (NonUnitalNonAssocSemiring.toMulZeroClass.{u_1} α (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_1} α (NonAssocRing.toNonUnitalNonAssocRing.{u_1} α (Ring.toNonAssocRing.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.2655550140._hygCtx._hyg.4)))))] {s : Set.{u_2} β} {a : α}, (LE.le.{u_1} α (Preorder.toLE.{u_1} α (PartialOrder.toPreorder.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.2655550140._hygCtx._hyg.7)) a (OfNat.ofNat.{u_1} α 0 (Zero.toOfNat0.{u_1} α (MulZeroClass.toZero.{u_1} α (NonUnitalNonAssocSemiring.toMulZeroClass.{u_1} α (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_1} α (NonAssocRing.toNonUnitalNonAssocRing.{u_1} α (Ring.toNonAssocRing.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.2655550140._hygCtx._hyg.4)))))))) -> (HasSubset.Subset.{u_2} (Set.{u_2} β) (Set.instHasSubset.{u_2} β) (HSMul.hSMul.{u_1, u_2, u_2} α (Set.{u_2} β) (Set.{u_2} β) (instHSMul.{u_1, u_2} α (Set.{u_2} β) (Set.smulSet.{u_1, u_2} α β (SMulZeroClass.toSMul.{u_1, u_2} α β (AddZero.toZero.{u_2} β (AddZeroClass.toAddZero.{u_2} β (AddMonoid.toAddZeroClass.{u_2} β (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.2655550140._hygCtx._hyg.13)))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} α β (AddMonoid.toAddZeroClass.{u_2} β (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.2655550140._hygCtx._hyg.13)))) (DistribMulAction.toDistribSMul.{u_1, u_2} α β (MonoidWithZero.toMonoid.{u_1} α (Semiring.toMonoidWithZero.{u_1} α (Ring.toSemiring.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.2655550140._hygCtx._hyg.4))) (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.2655550140._hygCtx._hyg.13))) (Module.toDistribMulAction.{u_1, u_2} α β (Ring.toSemiring.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.2655550140._hygCtx._hyg.4) (AddCommGroup.toAddCommMonoid.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.2655550140._hygCtx._hyg.13) inst._@.Mathlib.Algebra.Order.Module.Pointwise.2655550140._hygCtx._hyg.22)))))) a (upperBounds.{u_2} β (Preorder.toLE.{u_2} β (PartialOrder.toPreorder.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.2655550140._hygCtx._hyg.16)) s)) (lowerBounds.{u_2} β (Preorder.toLE.{u_2} β (PartialOrder.toPreorder.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.2655550140._hygCtx._hyg.16)) (HSMul.hSMul.{u_1, u_2, u_2} α (Set.{u_2} β) (Set.{u_2} β) (instHSMul.{u_1, u_2} α (Set.{u_2} β) (Set.smulSet.{u_1, u_2} α β (SMulZeroClass.toSMul.{u_1, u_2} α β (AddZero.toZero.{u_2} β (AddZeroClass.toAddZero.{u_2} β (AddMonoid.toAddZeroClass.{u_2} β (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.2655550140._hygCtx._hyg.13)))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} α β (AddMonoid.toAddZeroClass.{u_2} β (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.2655550140._hygCtx._hyg.13)))) (DistribMulAction.toDistribSMul.{u_1, u_2} α β (MonoidWithZero.toMonoid.{u_1} α (Semiring.toMonoidWithZero.{u_1} α (Ring.toSemiring.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.2655550140._hygCtx._hyg.4))) (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.2655550140._hygCtx._hyg.13))) (Module.toDistribMulAction.{u_1, u_2} α β (Ring.toSemiring.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.2655550140._hygCtx._hyg.4) (AddCommGroup.toAddCommMonoid.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.2655550140._hygCtx._hyg.13) inst._@.Mathlib.Algebra.Order.Module.Pointwise.2655550140._hygCtx._hyg.22)))))) a s)))","typeFull":"∀ {α : Type u_1} {β : Type u_2} [inst : Ring α] [inst_1 : PartialOrder α] [IsOrderedRing α] [inst_3 : AddCommGroup β]\n [inst_4 : PartialOrder β] [IsOrderedAddMonoid β] [inst_6 : Module α β] [PosSMulMono α β] {s : Set β} {a : α},\n a ≤ 0 → a • upperBounds s ⊆ lowerBounds (a • s)","typeReadable":"∀ {α : Type u_1} {β : Type u_2} [inst : Ring α] [inst_1 : PartialOrder α] [IsOrderedRing α] [inst_3 : AddCommGroup β]\n [inst_4 : PartialOrder β] [IsOrderedAddMonoid β] [inst_6 : Module α β] [PosSMulMono α β] {s : Set β} {a : α},\n a ≤ 0 → a • upperBounds s ⊆ lowerBounds (a • s)","typeReferences":[["Ring","toNonAssocRing"],["PartialOrder","toPreorder"],["Module"],["AddCommGroup","toAddGroup"],["SMulZeroClass","toSMul"],["PosSMulMono"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["IsOrderedAddMonoid"],["PartialOrder"],["MonoidWithZero","toMonoid"],["Set","smulSet"],["instHSMul"],["Zero","toOfNat0"],["AddGroup","toSubNegMonoid"],["Preorder","toLE"],["NonAssocRing","toNonUnitalNonAssocRing"],["DistribSMul","toSMulZeroClass"],["Set"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["Ring","toSemiring"],["Set","instHasSubset"],["IsOrderedRing"],["Module","toDistribMulAction"],["HasSubset","Subset"],["SubNegMonoid","toAddMonoid"],["upperBounds"],["MulZeroClass","toZero"],["lowerBounds"],["HSMul","hSMul"],["LE","le"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["AddCommGroup","toAddCommMonoid"],["AddZero","toZero"],["Ring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["Antitone","image_upperBounds_subset_lowerBounds_image"],["PartialOrder","toPreorder"],["AddCommGroup","toAddGroup"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddZeroClass","toAddZero"],["Ring","toSemiring"],["Module","toDistribMulAction"],["SubNegMonoid","toAddMonoid"],["MonoidWithZero","toMonoid"],["HSMul","hSMul"],["AddCommGroup","toAddCommMonoid"],["instHSMul"],["antitone_smul_left"],["AddGroup","toSubNegMonoid"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]]},{"isProp":true,"kind":"theorem","name":["smul_upperBounds_subset_upperBounds_smul_of_nonneg"],"typeFallback":"forall {α : Type.{u_1}} {β : Type.{u_2}} [inst._@.Mathlib.Algebra.Order.Module.Pointwise.2516670960._hygCtx._hyg.4 : SMul.{u_1, u_2} α β] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.2516670960._hygCtx._hyg.8 : Preorder.{u_1} α] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.2516670960._hygCtx._hyg.11 : Preorder.{u_2} β] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.2516670960._hygCtx._hyg.14 : Zero.{u_1} α] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.2516670960._hygCtx._hyg.17 : PosSMulMono.{u_1, u_2} α β inst._@.Mathlib.Algebra.Order.Module.Pointwise.2516670960._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Order.Module.Pointwise.2516670960._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Order.Module.Pointwise.2516670960._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Order.Module.Pointwise.2516670960._hygCtx._hyg.14] {a : α} {s : Set.{u_2} β}, (LE.le.{u_1} α (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.2516670960._hygCtx._hyg.8) (OfNat.ofNat.{u_1} α 0 (Zero.toOfNat0.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.2516670960._hygCtx._hyg.14)) a) -> (HasSubset.Subset.{u_2} (Set.{u_2} β) (Set.instHasSubset.{u_2} β) (HSMul.hSMul.{u_1, u_2, u_2} α (Set.{u_2} β) (Set.{u_2} β) (instHSMul.{u_1, u_2} α (Set.{u_2} β) (Set.smulSet.{u_1, u_2} α β inst._@.Mathlib.Algebra.Order.Module.Pointwise.2516670960._hygCtx._hyg.4)) a (upperBounds.{u_2} β (Preorder.toLE.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.2516670960._hygCtx._hyg.11) s)) (upperBounds.{u_2} β (Preorder.toLE.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.2516670960._hygCtx._hyg.11) (HSMul.hSMul.{u_1, u_2, u_2} α (Set.{u_2} β) (Set.{u_2} β) (instHSMul.{u_1, u_2} α (Set.{u_2} β) (Set.smulSet.{u_1, u_2} α β inst._@.Mathlib.Algebra.Order.Module.Pointwise.2516670960._hygCtx._hyg.4)) a s)))","typeFull":"∀ {α : Type u_1} {β : Type u_2} [inst : SMul α β] [inst_1 : Preorder α] [inst_2 : Preorder β] [inst_3 : Zero α]\n [PosSMulMono α β] {a : α} {s : Set β}, 0 ≤ a → a • upperBounds s ⊆ upperBounds (a • s)","typeReadable":"∀ {α : Type u_1} {β : Type u_2} [inst : SMul α β] [inst_1 : Preorder α] [inst_2 : Preorder β] [inst_3 : Zero α]\n [PosSMulMono α β] {a : α} {s : Set β}, 0 ≤ a → a • upperBounds s ⊆ upperBounds (a • s)","typeReferences":[["Set"],["OfNat","ofNat"],["PosSMulMono"],["Set","instHasSubset"],["SMul"],["Preorder"],["HasSubset","Subset"],["upperBounds"],["LE","le"],["HSMul","hSMul"],["Set","smulSet"],["instHSMul"],["Zero","toOfNat0"],["Zero"],["Preorder","toLE"]],"valueReferences":[["HSMul","hSMul"],["instHSMul"],["Monotone","image_upperBounds_subset_upperBounds_image"],["monotone_smul_left_of_nonneg"]]},{"isProp":true,"kind":"theorem","name":["BddBelow","smul_of_nonneg"],"typeFallback":"forall {α : Type.{u_1}} {β : Type.{u_2}} [inst._@.Mathlib.Algebra.Order.Module.Pointwise.911681138._hygCtx._hyg.4 : SMul.{u_1, u_2} α β] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.911681138._hygCtx._hyg.8 : Preorder.{u_1} α] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.911681138._hygCtx._hyg.11 : Preorder.{u_2} β] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.911681138._hygCtx._hyg.14 : Zero.{u_1} α] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.911681138._hygCtx._hyg.17 : PosSMulMono.{u_1, u_2} α β inst._@.Mathlib.Algebra.Order.Module.Pointwise.911681138._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Order.Module.Pointwise.911681138._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Order.Module.Pointwise.911681138._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Order.Module.Pointwise.911681138._hygCtx._hyg.14] {a : α} {s : Set.{u_2} β}, (BddBelow.{u_2} β (Preorder.toLE.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.911681138._hygCtx._hyg.11) s) -> (LE.le.{u_1} α (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.911681138._hygCtx._hyg.8) (OfNat.ofNat.{u_1} α 0 (Zero.toOfNat0.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.911681138._hygCtx._hyg.14)) a) -> (BddBelow.{u_2} β (Preorder.toLE.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.911681138._hygCtx._hyg.11) (HSMul.hSMul.{u_1, u_2, u_2} α (Set.{u_2} β) (Set.{u_2} β) (instHSMul.{u_1, u_2} α (Set.{u_2} β) (Set.smulSet.{u_1, u_2} α β inst._@.Mathlib.Algebra.Order.Module.Pointwise.911681138._hygCtx._hyg.4)) a s))","typeFull":"∀ {α : Type u_1} {β : Type u_2} [inst : SMul α β] [inst_1 : Preorder α] [inst_2 : Preorder β] [inst_3 : Zero α]\n [PosSMulMono α β] {a : α} {s : Set β}, BddBelow s → 0 ≤ a → BddBelow (a • s)","typeReadable":"∀ {α : Type u_1} {β : Type u_2} [inst : SMul α β] [inst_1 : Preorder α] [inst_2 : Preorder β] [inst_3 : Zero α]\n [PosSMulMono α β] {a : α} {s : Set β}, BddBelow s → 0 ≤ a → BddBelow (a • s)","typeReferences":[["Set"],["BddBelow"],["OfNat","ofNat"],["PosSMulMono"],["SMul"],["Preorder"],["LE","le"],["HSMul","hSMul"],["Set","smulSet"],["instHSMul"],["Zero","toOfNat0"],["Zero"],["Preorder","toLE"]],"valueReferences":[["Monotone","map_bddBelow"],["HSMul","hSMul"],["instHSMul"],["monotone_smul_left_of_nonneg"]]},{"isProp":true,"kind":"theorem","name":["bddBelow_smul_iff_of_neg"],"typeFallback":"forall {α : Type.{u_1}} {β : Type.{u_2}} [inst._@.Mathlib.Algebra.Order.Module.Pointwise.967861698._hygCtx._hyg.4 : Field.{u_1} α] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.967861698._hygCtx._hyg.7 : LinearOrder.{u_1} α] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.967861698._hygCtx._hyg.10 : IsStrictOrderedRing.{u_1} α (DivisionSemiring.toSemiring.{u_1} α (Semifield.toDivisionSemiring.{u_1} α (Field.toSemifield.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.967861698._hygCtx._hyg.4))) (SemilatticeInf.toPartialOrder.{u_1} α (Lattice.toSemilatticeInf.{u_1} α (DistribLattice.toLattice.{u_1} α (instDistribLatticeOfLinearOrder.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.967861698._hygCtx._hyg.7))))] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.967861698._hygCtx._hyg.13 : AddCommGroup.{u_2} β] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.967861698._hygCtx._hyg.16 : PartialOrder.{u_2} β] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.967861698._hygCtx._hyg.19 : IsOrderedAddMonoid.{u_2} β (AddCommGroup.toAddCommMonoid.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.967861698._hygCtx._hyg.13) (PartialOrder.toPreorder.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.967861698._hygCtx._hyg.16)] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.967861698._hygCtx._hyg.22 : Module.{u_1, u_2} α β (DivisionSemiring.toSemiring.{u_1} α (Semifield.toDivisionSemiring.{u_1} α (Field.toSemifield.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.967861698._hygCtx._hyg.4))) (AddCommGroup.toAddCommMonoid.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.967861698._hygCtx._hyg.13)] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.967861698._hygCtx._hyg.26 : PosSMulMono.{u_1, u_2} α β (SMulZeroClass.toSMul.{u_1, u_2} α β (AddZero.toZero.{u_2} β (AddZeroClass.toAddZero.{u_2} β (AddMonoid.toAddZeroClass.{u_2} β (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.967861698._hygCtx._hyg.13)))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} α β (AddMonoid.toAddZeroClass.{u_2} β (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.967861698._hygCtx._hyg.13)))) (DistribMulAction.toDistribSMul.{u_1, u_2} α β (MonoidWithZero.toMonoid.{u_1} α (Semiring.toMonoidWithZero.{u_1} α (DivisionSemiring.toSemiring.{u_1} α (Semifield.toDivisionSemiring.{u_1} α (Field.toSemifield.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.967861698._hygCtx._hyg.4))))) (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.967861698._hygCtx._hyg.13))) (Module.toDistribMulAction.{u_1, u_2} α β (DivisionSemiring.toSemiring.{u_1} α (Semifield.toDivisionSemiring.{u_1} α (Field.toSemifield.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.967861698._hygCtx._hyg.4))) (AddCommGroup.toAddCommMonoid.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.967861698._hygCtx._hyg.13) inst._@.Mathlib.Algebra.Order.Module.Pointwise.967861698._hygCtx._hyg.22)))) (PartialOrder.toPreorder.{u_1} α (SemilatticeInf.toPartialOrder.{u_1} α (Lattice.toSemilatticeInf.{u_1} α (DistribLattice.toLattice.{u_1} α (instDistribLatticeOfLinearOrder.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.967861698._hygCtx._hyg.7))))) (PartialOrder.toPreorder.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.967861698._hygCtx._hyg.16) (MulZeroClass.toZero.{u_1} α (NonUnitalNonAssocSemiring.toMulZeroClass.{u_1} α (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_1} α (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_1} α (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_1} α (CommRing.toNonUnitalCommRing.{u_1} α (Field.toCommRing.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.967861698._hygCtx._hyg.4)))))))] {s : Set.{u_2} β} {a : α}, (LT.lt.{u_1} α (Preorder.toLT.{u_1} α (PartialOrder.toPreorder.{u_1} α (SemilatticeInf.toPartialOrder.{u_1} α (Lattice.toSemilatticeInf.{u_1} α (DistribLattice.toLattice.{u_1} α (instDistribLatticeOfLinearOrder.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.967861698._hygCtx._hyg.7)))))) a (OfNat.ofNat.{u_1} α 0 (Zero.toOfNat0.{u_1} α (MulZeroClass.toZero.{u_1} α (NonUnitalNonAssocSemiring.toMulZeroClass.{u_1} α (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_1} α (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_1} α (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_1} α (CommRing.toNonUnitalCommRing.{u_1} α (Field.toCommRing.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.967861698._hygCtx._hyg.4)))))))))) -> (Iff (BddBelow.{u_2} β (Preorder.toLE.{u_2} β (PartialOrder.toPreorder.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.967861698._hygCtx._hyg.16)) (HSMul.hSMul.{u_1, u_2, u_2} α (Set.{u_2} β) (Set.{u_2} β) (instHSMul.{u_1, u_2} α (Set.{u_2} β) (Set.smulSet.{u_1, u_2} α β (SMulZeroClass.toSMul.{u_1, u_2} α β (AddZero.toZero.{u_2} β (AddZeroClass.toAddZero.{u_2} β (AddMonoid.toAddZeroClass.{u_2} β (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.967861698._hygCtx._hyg.13)))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} α β (AddMonoid.toAddZeroClass.{u_2} β (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.967861698._hygCtx._hyg.13)))) (DistribMulAction.toDistribSMul.{u_1, u_2} α β (MonoidWithZero.toMonoid.{u_1} α (Semiring.toMonoidWithZero.{u_1} α (DivisionSemiring.toSemiring.{u_1} α (Semifield.toDivisionSemiring.{u_1} α (Field.toSemifield.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.967861698._hygCtx._hyg.4))))) (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.967861698._hygCtx._hyg.13))) (Module.toDistribMulAction.{u_1, u_2} α β (DivisionSemiring.toSemiring.{u_1} α (Semifield.toDivisionSemiring.{u_1} α (Field.toSemifield.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.967861698._hygCtx._hyg.4))) (AddCommGroup.toAddCommMonoid.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.967861698._hygCtx._hyg.13) inst._@.Mathlib.Algebra.Order.Module.Pointwise.967861698._hygCtx._hyg.22)))))) a s)) (BddAbove.{u_2} β (Preorder.toLE.{u_2} β (PartialOrder.toPreorder.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.967861698._hygCtx._hyg.16)) s))","typeFull":"∀ {α : Type u_1} {β : Type u_2} [inst : Field α] [inst_1 : LinearOrder α] [IsStrictOrderedRing α]\n [inst_3 : AddCommGroup β] [inst_4 : PartialOrder β] [IsOrderedAddMonoid β] [inst_6 : Module α β] [PosSMulMono α β]\n {s : Set β} {a : α}, a < 0 → (BddBelow (a • s) ↔ BddAbove s)","typeReadable":"∀ {α : Type u_1} {β : Type u_2} [inst : Field α] [inst_1 : LinearOrder α] [IsStrictOrderedRing α]\n [inst_3 : AddCommGroup β] [inst_4 : PartialOrder β] [IsOrderedAddMonoid β] [inst_6 : Module α β] [PosSMulMono α β]\n {s : Set β} {a : α}, a < 0 → (BddBelow (a • s) ↔ BddAbove s)","typeReferences":[["PartialOrder","toPreorder"],["Field"],["Module"],["AddCommGroup","toAddGroup"],["Preorder","toLT"],["SMulZeroClass","toSMul"],["CommRing","toNonUnitalCommRing"],["PosSMulMono"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["instDistribLatticeOfLinearOrder"],["IsOrderedAddMonoid"],["PartialOrder"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["MonoidWithZero","toMonoid"],["Set","smulSet"],["instHSMul"],["Zero","toOfNat0"],["Semifield","toDivisionSemiring"],["AddGroup","toSubNegMonoid"],["Preorder","toLE"],["SemilatticeInf","toPartialOrder"],["DistribSMul","toSMulZeroClass"],["Field","toCommRing"],["Lattice","toSemilatticeInf"],["IsStrictOrderedRing"],["Set"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["DistribMulAction","toDistribSMul"],["LinearOrder"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["BddBelow"],["DivisionSemiring","toSemiring"],["AddZeroClass","toAddZero"],["BddAbove"],["OfNat","ofNat"],["LT","lt"],["Module","toDistribMulAction"],["DistribLattice","toLattice"],["SubNegMonoid","toAddMonoid"],["MulZeroClass","toZero"],["Iff"],["HSMul","hSMul"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["Field","toSemifield"],["AddCommGroup","toAddCommMonoid"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["OrderIso","smulRightDual"],["OrderDual","instPreorder"],["OrderDual"],["PartialOrder","toPreorder"],["OrderIso","bddAbove_image"]]},{"isProp":true,"kind":"theorem","name":["bddAbove_smul_iff_of_neg","_simp_1"],"typeFallback":"forall {α : Type.{u_1}} {β : Type.{u_2}} [inst._@.Mathlib.Algebra.Order.Module.Pointwise.3332345626._hygCtx._hyg.4 : Field.{u_1} α] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.3332345626._hygCtx._hyg.7 : LinearOrder.{u_1} α] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.3332345626._hygCtx._hyg.10 : IsStrictOrderedRing.{u_1} α (DivisionSemiring.toSemiring.{u_1} α (Semifield.toDivisionSemiring.{u_1} α (Field.toSemifield.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3332345626._hygCtx._hyg.4))) (SemilatticeInf.toPartialOrder.{u_1} α (Lattice.toSemilatticeInf.{u_1} α (DistribLattice.toLattice.{u_1} α (instDistribLatticeOfLinearOrder.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3332345626._hygCtx._hyg.7))))] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.3332345626._hygCtx._hyg.13 : AddCommGroup.{u_2} β] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.3332345626._hygCtx._hyg.16 : PartialOrder.{u_2} β] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.3332345626._hygCtx._hyg.19 : IsOrderedAddMonoid.{u_2} β (AddCommGroup.toAddCommMonoid.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3332345626._hygCtx._hyg.13) (PartialOrder.toPreorder.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3332345626._hygCtx._hyg.16)] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.3332345626._hygCtx._hyg.22 : Module.{u_1, u_2} α β (DivisionSemiring.toSemiring.{u_1} α (Semifield.toDivisionSemiring.{u_1} α (Field.toSemifield.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3332345626._hygCtx._hyg.4))) (AddCommGroup.toAddCommMonoid.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3332345626._hygCtx._hyg.13)] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.3332345626._hygCtx._hyg.26 : PosSMulMono.{u_1, u_2} α β (SMulZeroClass.toSMul.{u_1, u_2} α β (AddZero.toZero.{u_2} β (AddZeroClass.toAddZero.{u_2} β (AddMonoid.toAddZeroClass.{u_2} β (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3332345626._hygCtx._hyg.13)))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} α β (AddMonoid.toAddZeroClass.{u_2} β (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3332345626._hygCtx._hyg.13)))) (DistribMulAction.toDistribSMul.{u_1, u_2} α β (MonoidWithZero.toMonoid.{u_1} α (Semiring.toMonoidWithZero.{u_1} α (DivisionSemiring.toSemiring.{u_1} α (Semifield.toDivisionSemiring.{u_1} α (Field.toSemifield.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3332345626._hygCtx._hyg.4))))) (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3332345626._hygCtx._hyg.13))) (Module.toDistribMulAction.{u_1, u_2} α β (DivisionSemiring.toSemiring.{u_1} α (Semifield.toDivisionSemiring.{u_1} α (Field.toSemifield.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3332345626._hygCtx._hyg.4))) (AddCommGroup.toAddCommMonoid.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3332345626._hygCtx._hyg.13) inst._@.Mathlib.Algebra.Order.Module.Pointwise.3332345626._hygCtx._hyg.22)))) (PartialOrder.toPreorder.{u_1} α (SemilatticeInf.toPartialOrder.{u_1} α (Lattice.toSemilatticeInf.{u_1} α (DistribLattice.toLattice.{u_1} α (instDistribLatticeOfLinearOrder.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3332345626._hygCtx._hyg.7))))) (PartialOrder.toPreorder.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3332345626._hygCtx._hyg.16) (MulZeroClass.toZero.{u_1} α (NonUnitalNonAssocSemiring.toMulZeroClass.{u_1} α (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_1} α (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_1} α (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_1} α (CommRing.toNonUnitalCommRing.{u_1} α (Field.toCommRing.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3332345626._hygCtx._hyg.4)))))))] {s : Set.{u_2} β} {a : α}, (LT.lt.{u_1} α (Preorder.toLT.{u_1} α (PartialOrder.toPreorder.{u_1} α (SemilatticeInf.toPartialOrder.{u_1} α (Lattice.toSemilatticeInf.{u_1} α (DistribLattice.toLattice.{u_1} α (instDistribLatticeOfLinearOrder.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3332345626._hygCtx._hyg.7)))))) a (OfNat.ofNat.{u_1} α 0 (Zero.toOfNat0.{u_1} α (MulZeroClass.toZero.{u_1} α (NonUnitalNonAssocSemiring.toMulZeroClass.{u_1} α (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_1} α (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_1} α (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_1} α (CommRing.toNonUnitalCommRing.{u_1} α (Field.toCommRing.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3332345626._hygCtx._hyg.4)))))))))) -> (Eq.{1} Prop (BddAbove.{u_2} β (Preorder.toLE.{u_2} β (PartialOrder.toPreorder.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3332345626._hygCtx._hyg.16)) (HSMul.hSMul.{u_1, u_2, u_2} α (Set.{u_2} β) (Set.{u_2} β) (instHSMul.{u_1, u_2} α (Set.{u_2} β) (Set.smulSet.{u_1, u_2} α β (SMulZeroClass.toSMul.{u_1, u_2} α β (AddZero.toZero.{u_2} β (AddZeroClass.toAddZero.{u_2} β (AddMonoid.toAddZeroClass.{u_2} β (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3332345626._hygCtx._hyg.13)))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} α β (AddMonoid.toAddZeroClass.{u_2} β (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3332345626._hygCtx._hyg.13)))) (DistribMulAction.toDistribSMul.{u_1, u_2} α β (MonoidWithZero.toMonoid.{u_1} α (Semiring.toMonoidWithZero.{u_1} α (DivisionSemiring.toSemiring.{u_1} α (Semifield.toDivisionSemiring.{u_1} α (Field.toSemifield.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3332345626._hygCtx._hyg.4))))) (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3332345626._hygCtx._hyg.13))) (Module.toDistribMulAction.{u_1, u_2} α β (DivisionSemiring.toSemiring.{u_1} α (Semifield.toDivisionSemiring.{u_1} α (Field.toSemifield.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3332345626._hygCtx._hyg.4))) (AddCommGroup.toAddCommMonoid.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3332345626._hygCtx._hyg.13) inst._@.Mathlib.Algebra.Order.Module.Pointwise.3332345626._hygCtx._hyg.22)))))) a s)) (BddBelow.{u_2} β (Preorder.toLE.{u_2} β (PartialOrder.toPreorder.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3332345626._hygCtx._hyg.16)) s))","typeFull":"∀ {α : Type u_1} {β : Type u_2} [inst : Field α] [inst_1 : LinearOrder α] [IsStrictOrderedRing α]\n [inst_3 : AddCommGroup β] [inst_4 : PartialOrder β] [IsOrderedAddMonoid β] [inst_6 : Module α β] [PosSMulMono α β]\n {s : Set β} {a : α}, a < 0 → BddAbove (a • s) = BddBelow s","typeReadable":"∀ {α : Type u_1} {β : Type u_2} [inst : Field α] [inst_1 : LinearOrder α] [IsStrictOrderedRing α]\n [inst_3 : AddCommGroup β] [inst_4 : PartialOrder β] [IsOrderedAddMonoid β] [inst_6 : Module α β] [PosSMulMono α β]\n {s : Set β} {a : α}, a < 0 → BddAbove (a • s) = BddBelow s","typeReferences":[["PartialOrder","toPreorder"],["Field"],["Module"],["AddCommGroup","toAddGroup"],["Preorder","toLT"],["SMulZeroClass","toSMul"],["CommRing","toNonUnitalCommRing"],["PosSMulMono"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["instDistribLatticeOfLinearOrder"],["IsOrderedAddMonoid"],["PartialOrder"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["MonoidWithZero","toMonoid"],["Set","smulSet"],["instHSMul"],["Zero","toOfNat0"],["Semifield","toDivisionSemiring"],["AddGroup","toSubNegMonoid"],["Preorder","toLE"],["Eq"],["SemilatticeInf","toPartialOrder"],["DistribSMul","toSMulZeroClass"],["Field","toCommRing"],["Lattice","toSemilatticeInf"],["IsStrictOrderedRing"],["Set"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["DistribMulAction","toDistribSMul"],["LinearOrder"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["BddBelow"],["DivisionSemiring","toSemiring"],["AddZeroClass","toAddZero"],["BddAbove"],["OfNat","ofNat"],["LT","lt"],["Module","toDistribMulAction"],["DistribLattice","toLattice"],["SubNegMonoid","toAddMonoid"],["MulZeroClass","toZero"],["HSMul","hSMul"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["Field","toSemifield"],["AddCommGroup","toAddCommMonoid"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["PartialOrder","toPreorder"],["AddCommGroup","toAddGroup"],["SMulZeroClass","toSMul"],["MonoidWithZero","toMonoid"],["Set","smulSet"],["bddAbove_smul_iff_of_neg"],["instHSMul"],["AddGroup","toSubNegMonoid"],["Semifield","toDivisionSemiring"],["Preorder","toLE"],["propext"],["DistribSMul","toSMulZeroClass"],["Set"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["DivisionSemiring","toSemiring"],["BddBelow"],["AddZeroClass","toAddZero"],["BddAbove"],["Module","toDistribMulAction"],["SubNegMonoid","toAddMonoid"],["HSMul","hSMul"],["Field","toSemifield"],["AddCommGroup","toAddCommMonoid"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["smul_lowerBounds_subset_upperBounds_smul"],"typeFallback":"forall {α : Type.{u_1}} {β : Type.{u_2}} [inst._@.Mathlib.Algebra.Order.Module.Pointwise.1776562198._hygCtx._hyg.4 : Ring.{u_1} α] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.1776562198._hygCtx._hyg.7 : PartialOrder.{u_1} α] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.1776562198._hygCtx._hyg.10 : IsOrderedRing.{u_1} α (Ring.toSemiring.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.1776562198._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Order.Module.Pointwise.1776562198._hygCtx._hyg.7] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.1776562198._hygCtx._hyg.13 : AddCommGroup.{u_2} β] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.1776562198._hygCtx._hyg.16 : PartialOrder.{u_2} β] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.1776562198._hygCtx._hyg.19 : IsOrderedAddMonoid.{u_2} β (AddCommGroup.toAddCommMonoid.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.1776562198._hygCtx._hyg.13) (PartialOrder.toPreorder.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.1776562198._hygCtx._hyg.16)] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.1776562198._hygCtx._hyg.22 : Module.{u_1, u_2} α β (Ring.toSemiring.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.1776562198._hygCtx._hyg.4) (AddCommGroup.toAddCommMonoid.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.1776562198._hygCtx._hyg.13)] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.1776562198._hygCtx._hyg.26 : PosSMulMono.{u_1, u_2} α β (SMulZeroClass.toSMul.{u_1, u_2} α β (AddZero.toZero.{u_2} β (AddZeroClass.toAddZero.{u_2} β (AddMonoid.toAddZeroClass.{u_2} β (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.1776562198._hygCtx._hyg.13)))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} α β (AddMonoid.toAddZeroClass.{u_2} β (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.1776562198._hygCtx._hyg.13)))) (DistribMulAction.toDistribSMul.{u_1, u_2} α β (MonoidWithZero.toMonoid.{u_1} α (Semiring.toMonoidWithZero.{u_1} α (Ring.toSemiring.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.1776562198._hygCtx._hyg.4))) (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.1776562198._hygCtx._hyg.13))) (Module.toDistribMulAction.{u_1, u_2} α β (Ring.toSemiring.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.1776562198._hygCtx._hyg.4) (AddCommGroup.toAddCommMonoid.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.1776562198._hygCtx._hyg.13) inst._@.Mathlib.Algebra.Order.Module.Pointwise.1776562198._hygCtx._hyg.22)))) (PartialOrder.toPreorder.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.1776562198._hygCtx._hyg.7) (PartialOrder.toPreorder.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.1776562198._hygCtx._hyg.16) (MulZeroClass.toZero.{u_1} α (NonUnitalNonAssocSemiring.toMulZeroClass.{u_1} α (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_1} α (NonAssocRing.toNonUnitalNonAssocRing.{u_1} α (Ring.toNonAssocRing.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.1776562198._hygCtx._hyg.4)))))] {s : Set.{u_2} β} {a : α}, (LE.le.{u_1} α (Preorder.toLE.{u_1} α (PartialOrder.toPreorder.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.1776562198._hygCtx._hyg.7)) a (OfNat.ofNat.{u_1} α 0 (Zero.toOfNat0.{u_1} α (MulZeroClass.toZero.{u_1} α (NonUnitalNonAssocSemiring.toMulZeroClass.{u_1} α (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_1} α (NonAssocRing.toNonUnitalNonAssocRing.{u_1} α (Ring.toNonAssocRing.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.1776562198._hygCtx._hyg.4)))))))) -> (HasSubset.Subset.{u_2} (Set.{u_2} β) (Set.instHasSubset.{u_2} β) (HSMul.hSMul.{u_1, u_2, u_2} α (Set.{u_2} β) (Set.{u_2} β) (instHSMul.{u_1, u_2} α (Set.{u_2} β) (Set.smulSet.{u_1, u_2} α β (SMulZeroClass.toSMul.{u_1, u_2} α β (AddZero.toZero.{u_2} β (AddZeroClass.toAddZero.{u_2} β (AddMonoid.toAddZeroClass.{u_2} β (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.1776562198._hygCtx._hyg.13)))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} α β (AddMonoid.toAddZeroClass.{u_2} β (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.1776562198._hygCtx._hyg.13)))) (DistribMulAction.toDistribSMul.{u_1, u_2} α β (MonoidWithZero.toMonoid.{u_1} α (Semiring.toMonoidWithZero.{u_1} α (Ring.toSemiring.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.1776562198._hygCtx._hyg.4))) (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.1776562198._hygCtx._hyg.13))) (Module.toDistribMulAction.{u_1, u_2} α β (Ring.toSemiring.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.1776562198._hygCtx._hyg.4) (AddCommGroup.toAddCommMonoid.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.1776562198._hygCtx._hyg.13) inst._@.Mathlib.Algebra.Order.Module.Pointwise.1776562198._hygCtx._hyg.22)))))) a (lowerBounds.{u_2} β (Preorder.toLE.{u_2} β (PartialOrder.toPreorder.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.1776562198._hygCtx._hyg.16)) s)) (upperBounds.{u_2} β (Preorder.toLE.{u_2} β (PartialOrder.toPreorder.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.1776562198._hygCtx._hyg.16)) (HSMul.hSMul.{u_1, u_2, u_2} α (Set.{u_2} β) (Set.{u_2} β) (instHSMul.{u_1, u_2} α (Set.{u_2} β) (Set.smulSet.{u_1, u_2} α β (SMulZeroClass.toSMul.{u_1, u_2} α β (AddZero.toZero.{u_2} β (AddZeroClass.toAddZero.{u_2} β (AddMonoid.toAddZeroClass.{u_2} β (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.1776562198._hygCtx._hyg.13)))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} α β (AddMonoid.toAddZeroClass.{u_2} β (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.1776562198._hygCtx._hyg.13)))) (DistribMulAction.toDistribSMul.{u_1, u_2} α β (MonoidWithZero.toMonoid.{u_1} α (Semiring.toMonoidWithZero.{u_1} α (Ring.toSemiring.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.1776562198._hygCtx._hyg.4))) (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.1776562198._hygCtx._hyg.13))) (Module.toDistribMulAction.{u_1, u_2} α β (Ring.toSemiring.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.1776562198._hygCtx._hyg.4) (AddCommGroup.toAddCommMonoid.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.1776562198._hygCtx._hyg.13) inst._@.Mathlib.Algebra.Order.Module.Pointwise.1776562198._hygCtx._hyg.22)))))) a s)))","typeFull":"∀ {α : Type u_1} {β : Type u_2} [inst : Ring α] [inst_1 : PartialOrder α] [IsOrderedRing α] [inst_3 : AddCommGroup β]\n [inst_4 : PartialOrder β] [IsOrderedAddMonoid β] [inst_6 : Module α β] [PosSMulMono α β] {s : Set β} {a : α},\n a ≤ 0 → a • lowerBounds s ⊆ upperBounds (a • s)","typeReadable":"∀ {α : Type u_1} {β : Type u_2} [inst : Ring α] [inst_1 : PartialOrder α] [IsOrderedRing α] [inst_3 : AddCommGroup β]\n [inst_4 : PartialOrder β] [IsOrderedAddMonoid β] [inst_6 : Module α β] [PosSMulMono α β] {s : Set β} {a : α},\n a ≤ 0 → a • lowerBounds s ⊆ upperBounds (a • s)","typeReferences":[["Ring","toNonAssocRing"],["PartialOrder","toPreorder"],["Module"],["AddCommGroup","toAddGroup"],["SMulZeroClass","toSMul"],["PosSMulMono"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["IsOrderedAddMonoid"],["PartialOrder"],["MonoidWithZero","toMonoid"],["Set","smulSet"],["instHSMul"],["Zero","toOfNat0"],["AddGroup","toSubNegMonoid"],["Preorder","toLE"],["NonAssocRing","toNonUnitalNonAssocRing"],["DistribSMul","toSMulZeroClass"],["Set"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["Ring","toSemiring"],["Set","instHasSubset"],["IsOrderedRing"],["Module","toDistribMulAction"],["HasSubset","Subset"],["SubNegMonoid","toAddMonoid"],["upperBounds"],["MulZeroClass","toZero"],["lowerBounds"],["HSMul","hSMul"],["LE","le"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["AddCommGroup","toAddCommMonoid"],["AddZero","toZero"],["Ring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["PartialOrder","toPreorder"],["AddCommGroup","toAddGroup"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddZeroClass","toAddZero"],["Ring","toSemiring"],["Antitone","image_lowerBounds_subset_upperBounds_image"],["Module","toDistribMulAction"],["SubNegMonoid","toAddMonoid"],["MonoidWithZero","toMonoid"],["HSMul","hSMul"],["AddCommGroup","toAddCommMonoid"],["instHSMul"],["antitone_smul_left"],["AddGroup","toSubNegMonoid"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]]},{"isProp":true,"kind":"theorem","name":["upperBounds_smul_of_neg"],"typeFallback":"forall {α : Type.{u_1}} {β : Type.{u_2}} [inst._@.Mathlib.Algebra.Order.Module.Pointwise.3647333820._hygCtx._hyg.4 : Field.{u_1} α] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.3647333820._hygCtx._hyg.7 : LinearOrder.{u_1} α] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.3647333820._hygCtx._hyg.10 : IsStrictOrderedRing.{u_1} α (DivisionSemiring.toSemiring.{u_1} α (Semifield.toDivisionSemiring.{u_1} α (Field.toSemifield.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3647333820._hygCtx._hyg.4))) (SemilatticeInf.toPartialOrder.{u_1} α (Lattice.toSemilatticeInf.{u_1} α (DistribLattice.toLattice.{u_1} α (instDistribLatticeOfLinearOrder.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3647333820._hygCtx._hyg.7))))] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.3647333820._hygCtx._hyg.13 : AddCommGroup.{u_2} β] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.3647333820._hygCtx._hyg.16 : PartialOrder.{u_2} β] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.3647333820._hygCtx._hyg.19 : IsOrderedAddMonoid.{u_2} β (AddCommGroup.toAddCommMonoid.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3647333820._hygCtx._hyg.13) (PartialOrder.toPreorder.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3647333820._hygCtx._hyg.16)] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.3647333820._hygCtx._hyg.22 : Module.{u_1, u_2} α β (DivisionSemiring.toSemiring.{u_1} α (Semifield.toDivisionSemiring.{u_1} α (Field.toSemifield.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3647333820._hygCtx._hyg.4))) (AddCommGroup.toAddCommMonoid.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3647333820._hygCtx._hyg.13)] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.3647333820._hygCtx._hyg.26 : PosSMulMono.{u_1, u_2} α β (SMulZeroClass.toSMul.{u_1, u_2} α β (AddZero.toZero.{u_2} β (AddZeroClass.toAddZero.{u_2} β (AddMonoid.toAddZeroClass.{u_2} β (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3647333820._hygCtx._hyg.13)))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} α β (AddMonoid.toAddZeroClass.{u_2} β (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3647333820._hygCtx._hyg.13)))) (DistribMulAction.toDistribSMul.{u_1, u_2} α β (MonoidWithZero.toMonoid.{u_1} α (Semiring.toMonoidWithZero.{u_1} α (DivisionSemiring.toSemiring.{u_1} α (Semifield.toDivisionSemiring.{u_1} α (Field.toSemifield.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3647333820._hygCtx._hyg.4))))) (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3647333820._hygCtx._hyg.13))) (Module.toDistribMulAction.{u_1, u_2} α β (DivisionSemiring.toSemiring.{u_1} α (Semifield.toDivisionSemiring.{u_1} α (Field.toSemifield.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3647333820._hygCtx._hyg.4))) (AddCommGroup.toAddCommMonoid.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3647333820._hygCtx._hyg.13) inst._@.Mathlib.Algebra.Order.Module.Pointwise.3647333820._hygCtx._hyg.22)))) (PartialOrder.toPreorder.{u_1} α (SemilatticeInf.toPartialOrder.{u_1} α (Lattice.toSemilatticeInf.{u_1} α (DistribLattice.toLattice.{u_1} α (instDistribLatticeOfLinearOrder.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3647333820._hygCtx._hyg.7))))) (PartialOrder.toPreorder.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3647333820._hygCtx._hyg.16) (MulZeroClass.toZero.{u_1} α (NonUnitalNonAssocSemiring.toMulZeroClass.{u_1} α (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_1} α (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_1} α (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_1} α (CommRing.toNonUnitalCommRing.{u_1} α (Field.toCommRing.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3647333820._hygCtx._hyg.4)))))))] {s : Set.{u_2} β} {a : α}, (LT.lt.{u_1} α (Preorder.toLT.{u_1} α (PartialOrder.toPreorder.{u_1} α (SemilatticeInf.toPartialOrder.{u_1} α (Lattice.toSemilatticeInf.{u_1} α (DistribLattice.toLattice.{u_1} α (instDistribLatticeOfLinearOrder.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3647333820._hygCtx._hyg.7)))))) a (OfNat.ofNat.{u_1} α 0 (Zero.toOfNat0.{u_1} α (MulZeroClass.toZero.{u_1} α (NonUnitalNonAssocSemiring.toMulZeroClass.{u_1} α (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_1} α (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_1} α (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_1} α (CommRing.toNonUnitalCommRing.{u_1} α (Field.toCommRing.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3647333820._hygCtx._hyg.4)))))))))) -> (Eq.{succ u_2} (Set.{u_2} β) (upperBounds.{u_2} β (Preorder.toLE.{u_2} β (PartialOrder.toPreorder.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3647333820._hygCtx._hyg.16)) (HSMul.hSMul.{u_1, u_2, u_2} α (Set.{u_2} β) (Set.{u_2} β) (instHSMul.{u_1, u_2} α (Set.{u_2} β) (Set.smulSet.{u_1, u_2} α β (SMulZeroClass.toSMul.{u_1, u_2} α β (AddZero.toZero.{u_2} β (AddZeroClass.toAddZero.{u_2} β (AddMonoid.toAddZeroClass.{u_2} β (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3647333820._hygCtx._hyg.13)))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} α β (AddMonoid.toAddZeroClass.{u_2} β (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3647333820._hygCtx._hyg.13)))) (DistribMulAction.toDistribSMul.{u_1, u_2} α β (MonoidWithZero.toMonoid.{u_1} α (Semiring.toMonoidWithZero.{u_1} α (DivisionSemiring.toSemiring.{u_1} α (Semifield.toDivisionSemiring.{u_1} α (Field.toSemifield.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3647333820._hygCtx._hyg.4))))) (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3647333820._hygCtx._hyg.13))) (Module.toDistribMulAction.{u_1, u_2} α β (DivisionSemiring.toSemiring.{u_1} α (Semifield.toDivisionSemiring.{u_1} α (Field.toSemifield.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3647333820._hygCtx._hyg.4))) (AddCommGroup.toAddCommMonoid.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3647333820._hygCtx._hyg.13) inst._@.Mathlib.Algebra.Order.Module.Pointwise.3647333820._hygCtx._hyg.22)))))) a s)) (HSMul.hSMul.{u_1, u_2, u_2} α (Set.{u_2} β) (Set.{u_2} β) (instHSMul.{u_1, u_2} α (Set.{u_2} β) (Set.smulSet.{u_1, u_2} α β (SMulZeroClass.toSMul.{u_1, u_2} α β (AddZero.toZero.{u_2} β (AddZeroClass.toAddZero.{u_2} β (AddMonoid.toAddZeroClass.{u_2} β (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3647333820._hygCtx._hyg.13)))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} α β (AddMonoid.toAddZeroClass.{u_2} β (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3647333820._hygCtx._hyg.13)))) (DistribMulAction.toDistribSMul.{u_1, u_2} α β (MonoidWithZero.toMonoid.{u_1} α (Semiring.toMonoidWithZero.{u_1} α (DivisionSemiring.toSemiring.{u_1} α (Semifield.toDivisionSemiring.{u_1} α (Field.toSemifield.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3647333820._hygCtx._hyg.4))))) (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3647333820._hygCtx._hyg.13))) (Module.toDistribMulAction.{u_1, u_2} α β (DivisionSemiring.toSemiring.{u_1} α (Semifield.toDivisionSemiring.{u_1} α (Field.toSemifield.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3647333820._hygCtx._hyg.4))) (AddCommGroup.toAddCommMonoid.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3647333820._hygCtx._hyg.13) inst._@.Mathlib.Algebra.Order.Module.Pointwise.3647333820._hygCtx._hyg.22)))))) a (lowerBounds.{u_2} β (Preorder.toLE.{u_2} β (PartialOrder.toPreorder.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3647333820._hygCtx._hyg.16)) s)))","typeFull":"∀ {α : Type u_1} {β : Type u_2} [inst : Field α] [inst_1 : LinearOrder α] [IsStrictOrderedRing α]\n [inst_3 : AddCommGroup β] [inst_4 : PartialOrder β] [IsOrderedAddMonoid β] [inst_6 : Module α β] [PosSMulMono α β]\n {s : Set β} {a : α}, a < 0 → upperBounds (a • s) = a • lowerBounds s","typeReadable":"∀ {α : Type u_1} {β : Type u_2} [inst : Field α] [inst_1 : LinearOrder α] [IsStrictOrderedRing α]\n [inst_3 : AddCommGroup β] [inst_4 : PartialOrder β] [IsOrderedAddMonoid β] [inst_6 : Module α β] [PosSMulMono α β]\n {s : Set β} {a : α}, a < 0 → upperBounds (a • s) = a • lowerBounds s","typeReferences":[["PartialOrder","toPreorder"],["Field"],["Module"],["AddCommGroup","toAddGroup"],["Preorder","toLT"],["SMulZeroClass","toSMul"],["CommRing","toNonUnitalCommRing"],["PosSMulMono"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["instDistribLatticeOfLinearOrder"],["IsOrderedAddMonoid"],["PartialOrder"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["MonoidWithZero","toMonoid"],["Set","smulSet"],["instHSMul"],["Zero","toOfNat0"],["Semifield","toDivisionSemiring"],["AddGroup","toSubNegMonoid"],["Preorder","toLE"],["Eq"],["SemilatticeInf","toPartialOrder"],["DistribSMul","toSMulZeroClass"],["Field","toCommRing"],["Lattice","toSemilatticeInf"],["IsStrictOrderedRing"],["Set"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["DistribMulAction","toDistribSMul"],["LinearOrder"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["DivisionSemiring","toSemiring"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["LT","lt"],["Module","toDistribMulAction"],["DistribLattice","toLattice"],["SubNegMonoid","toAddMonoid"],["upperBounds"],["MulZeroClass","toZero"],["lowerBounds"],["HSMul","hSMul"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["Field","toSemifield"],["AddCommGroup","toAddCommMonoid"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["OrderIso","smulRightDual"],["OrderIso","lowerBounds_image"],["OrderDual","instPreorder"],["OrderDual"],["PartialOrder","toPreorder"]]},{"isProp":true,"kind":"theorem","name":["lowerBounds_smul_of_pos"],"typeFallback":"forall {α : Type.{u_1}} {β : Type.{u_2}} [inst._@.Mathlib.Algebra.Order.Module.Pointwise.1740256422._hygCtx._hyg.4 : Preorder.{u_1} α] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.1740256422._hygCtx._hyg.7 : Preorder.{u_2} β] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.1740256422._hygCtx._hyg.10 : GroupWithZero.{u_1} α] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.1740256422._hygCtx._hyg.13 : Zero.{u_2} β] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.1740256422._hygCtx._hyg.16 : MulActionWithZero.{u_1, u_2} α β (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.1740256422._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Order.Module.Pointwise.1740256422._hygCtx._hyg.13] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.1740256422._hygCtx._hyg.20 : PosSMulMono.{u_1, u_2} α β (SMulZeroClass.toSMul.{u_1, u_2} α β inst._@.Mathlib.Algebra.Order.Module.Pointwise.1740256422._hygCtx._hyg.13 (SMulWithZero.toSMulZeroClass.{u_1, u_2} α β (MulZeroClass.toZero.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α (MonoidWithZero.toMulZeroOneClass.{u_1} α (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.1740256422._hygCtx._hyg.10)))) inst._@.Mathlib.Algebra.Order.Module.Pointwise.1740256422._hygCtx._hyg.13 (MulActionWithZero.toSMulWithZero.{u_1, u_2} α β (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.1740256422._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Order.Module.Pointwise.1740256422._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Order.Module.Pointwise.1740256422._hygCtx._hyg.16))) inst._@.Mathlib.Algebra.Order.Module.Pointwise.1740256422._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Order.Module.Pointwise.1740256422._hygCtx._hyg.7 (MulZeroClass.toZero.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α (MonoidWithZero.toMulZeroOneClass.{u_1} α (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.1740256422._hygCtx._hyg.10))))] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.1740256422._hygCtx._hyg.24 : PosSMulReflectLE.{u_1, u_2} α β (SMulZeroClass.toSMul.{u_1, u_2} α β inst._@.Mathlib.Algebra.Order.Module.Pointwise.1740256422._hygCtx._hyg.13 (SMulWithZero.toSMulZeroClass.{u_1, u_2} α β (MulZeroClass.toZero.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α (MonoidWithZero.toMulZeroOneClass.{u_1} α (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.1740256422._hygCtx._hyg.10)))) inst._@.Mathlib.Algebra.Order.Module.Pointwise.1740256422._hygCtx._hyg.13 (MulActionWithZero.toSMulWithZero.{u_1, u_2} α β (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.1740256422._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Order.Module.Pointwise.1740256422._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Order.Module.Pointwise.1740256422._hygCtx._hyg.16))) inst._@.Mathlib.Algebra.Order.Module.Pointwise.1740256422._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Order.Module.Pointwise.1740256422._hygCtx._hyg.7 (MulZeroClass.toZero.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α (MonoidWithZero.toMulZeroOneClass.{u_1} α (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.1740256422._hygCtx._hyg.10))))] {s : Set.{u_2} β} {a : α}, (LT.lt.{u_1} α (Preorder.toLT.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.1740256422._hygCtx._hyg.4) (OfNat.ofNat.{u_1} α 0 (Zero.toOfNat0.{u_1} α (MulZeroClass.toZero.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α (MonoidWithZero.toMulZeroOneClass.{u_1} α (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.1740256422._hygCtx._hyg.10)))))) a) -> (Eq.{succ u_2} (Set.{u_2} β) (lowerBounds.{u_2} β (Preorder.toLE.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.1740256422._hygCtx._hyg.7) (HSMul.hSMul.{u_1, u_2, u_2} α (Set.{u_2} β) (Set.{u_2} β) (instHSMul.{u_1, u_2} α (Set.{u_2} β) (Set.smulSet.{u_1, u_2} α β (SMulZeroClass.toSMul.{u_1, u_2} α β inst._@.Mathlib.Algebra.Order.Module.Pointwise.1740256422._hygCtx._hyg.13 (SMulWithZero.toSMulZeroClass.{u_1, u_2} α β (MulZeroClass.toZero.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α (MonoidWithZero.toMulZeroOneClass.{u_1} α (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.1740256422._hygCtx._hyg.10)))) inst._@.Mathlib.Algebra.Order.Module.Pointwise.1740256422._hygCtx._hyg.13 (MulActionWithZero.toSMulWithZero.{u_1, u_2} α β (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.1740256422._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Order.Module.Pointwise.1740256422._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Order.Module.Pointwise.1740256422._hygCtx._hyg.16))))) a s)) (HSMul.hSMul.{u_1, u_2, u_2} α (Set.{u_2} β) (Set.{u_2} β) (instHSMul.{u_1, u_2} α (Set.{u_2} β) (Set.smulSet.{u_1, u_2} α β (SMulZeroClass.toSMul.{u_1, u_2} α β inst._@.Mathlib.Algebra.Order.Module.Pointwise.1740256422._hygCtx._hyg.13 (SMulWithZero.toSMulZeroClass.{u_1, u_2} α β (MulZeroClass.toZero.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α (MonoidWithZero.toMulZeroOneClass.{u_1} α (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.1740256422._hygCtx._hyg.10)))) inst._@.Mathlib.Algebra.Order.Module.Pointwise.1740256422._hygCtx._hyg.13 (MulActionWithZero.toSMulWithZero.{u_1, u_2} α β (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.1740256422._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Order.Module.Pointwise.1740256422._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Order.Module.Pointwise.1740256422._hygCtx._hyg.16))))) a (lowerBounds.{u_2} β (Preorder.toLE.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.1740256422._hygCtx._hyg.7) s)))","typeFull":"∀ {α : Type u_1} {β : Type u_2} [inst : Preorder α] [inst_1 : Preorder β] [inst_2 : GroupWithZero α] [inst_3 : Zero β]\n [inst_4 : MulActionWithZero α β] [PosSMulMono α β] [PosSMulReflectLE α β] {s : Set β} {a : α},\n 0 < a → lowerBounds (a • s) = a • lowerBounds s","typeReadable":"∀ {α : Type u_1} {β : Type u_2} [inst : Preorder α] [inst_1 : Preorder β] [inst_2 : GroupWithZero α] [inst_3 : Zero β]\n [inst_4 : MulActionWithZero α β] [PosSMulMono α β] [PosSMulReflectLE α β] {s : Set β} {a : α},\n 0 < a → lowerBounds (a • s) = a • lowerBounds s","typeReferences":[["Set"],["Preorder","toLT"],["SMulZeroClass","toSMul"],["MonoidWithZero","toMulZeroOneClass"],["GroupWithZero"],["SMulWithZero","toSMulZeroClass"],["OfNat","ofNat"],["PosSMulMono"],["LT","lt"],["PosSMulReflectLE"],["Preorder"],["MulZeroOneClass","toMulZeroClass"],["MulZeroClass","toZero"],["MulActionWithZero"],["HSMul","hSMul"],["lowerBounds"],["GroupWithZero","toMonoidWithZero"],["Set","smulSet"],["instHSMul"],["Zero","toOfNat0"],["Zero"],["Preorder","toLE"],["Eq"],["MulActionWithZero","toSMulWithZero"]],"valueReferences":[["OrderIso","lowerBounds_image"],["GroupWithZero","toMonoidWithZero"],["OrderIso","smulRight"],["MulActionWithZero","toMulAction"]]},{"isProp":true,"kind":"theorem","name":["bddBelow_smul_iff_of_pos","_simp_1"],"typeFallback":"forall {α : Type.{u_1}} {β : Type.{u_2}} [inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.4 : Preorder.{u_1} α] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.7 : Preorder.{u_2} β] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.10 : GroupWithZero.{u_1} α] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.13 : Zero.{u_2} β] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.16 : MulActionWithZero.{u_1, u_2} α β (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.13] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.20 : PosSMulMono.{u_1, u_2} α β (SMulZeroClass.toSMul.{u_1, u_2} α β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.13 (SMulWithZero.toSMulZeroClass.{u_1, u_2} α β (MulZeroClass.toZero.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α (MonoidWithZero.toMulZeroOneClass.{u_1} α (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.10)))) inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.13 (MulActionWithZero.toSMulWithZero.{u_1, u_2} α β (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.16))) inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.7 (MulZeroClass.toZero.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α (MonoidWithZero.toMulZeroOneClass.{u_1} α (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.10))))] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.24 : PosSMulReflectLE.{u_1, u_2} α β (SMulZeroClass.toSMul.{u_1, u_2} α β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.13 (SMulWithZero.toSMulZeroClass.{u_1, u_2} α β (MulZeroClass.toZero.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α (MonoidWithZero.toMulZeroOneClass.{u_1} α (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.10)))) inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.13 (MulActionWithZero.toSMulWithZero.{u_1, u_2} α β (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.16))) inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.7 (MulZeroClass.toZero.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α (MonoidWithZero.toMulZeroOneClass.{u_1} α (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.10))))] {s : Set.{u_2} β} {a : α}, (LT.lt.{u_1} α (Preorder.toLT.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.4) (OfNat.ofNat.{u_1} α 0 (Zero.toOfNat0.{u_1} α (MulZeroClass.toZero.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α (MonoidWithZero.toMulZeroOneClass.{u_1} α (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.10)))))) a) -> (Eq.{1} Prop (BddBelow.{u_2} β (Preorder.toLE.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.7) (HSMul.hSMul.{u_1, u_2, u_2} α (Set.{u_2} β) (Set.{u_2} β) (instHSMul.{u_1, u_2} α (Set.{u_2} β) (Set.smulSet.{u_1, u_2} α β (SMulZeroClass.toSMul.{u_1, u_2} α β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.13 (SMulWithZero.toSMulZeroClass.{u_1, u_2} α β (MulZeroClass.toZero.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α (MonoidWithZero.toMulZeroOneClass.{u_1} α (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.10)))) inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.13 (MulActionWithZero.toSMulWithZero.{u_1, u_2} α β (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.16))))) a s)) (BddBelow.{u_2} β (Preorder.toLE.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.7) s))","typeFull":"∀ {α : Type u_1} {β : Type u_2} [inst : Preorder α] [inst_1 : Preorder β] [inst_2 : GroupWithZero α] [inst_3 : Zero β]\n [inst_4 : MulActionWithZero α β] [PosSMulMono α β] [PosSMulReflectLE α β] {s : Set β} {a : α},\n 0 < a → BddBelow (a • s) = BddBelow s","typeReadable":"∀ {α : Type u_1} {β : Type u_2} [inst : Preorder α] [inst_1 : Preorder β] [inst_2 : GroupWithZero α] [inst_3 : Zero β]\n [inst_4 : MulActionWithZero α β] [PosSMulMono α β] [PosSMulReflectLE α β] {s : Set β} {a : α},\n 0 < a → BddBelow (a • s) = BddBelow s","typeReferences":[["Set"],["Preorder","toLT"],["SMulZeroClass","toSMul"],["MonoidWithZero","toMulZeroOneClass"],["BddBelow"],["GroupWithZero"],["SMulWithZero","toSMulZeroClass"],["OfNat","ofNat"],["PosSMulMono"],["LT","lt"],["PosSMulReflectLE"],["Preorder"],["MulZeroOneClass","toMulZeroClass"],["MulZeroClass","toZero"],["MulActionWithZero"],["HSMul","hSMul"],["GroupWithZero","toMonoidWithZero"],["Set","smulSet"],["instHSMul"],["Zero","toOfNat0"],["Zero"],["Preorder","toLE"],["Eq"],["MulActionWithZero","toSMulWithZero"]],"valueReferences":[["Set"],["SMulZeroClass","toSMul"],["MonoidWithZero","toMulZeroOneClass"],["BddBelow"],["SMulWithZero","toSMulZeroClass"],["MulZeroOneClass","toMulZeroClass"],["MulZeroClass","toZero"],["HSMul","hSMul"],["GroupWithZero","toMonoidWithZero"],["Set","smulSet"],["instHSMul"],["Preorder","toLE"],["bddBelow_smul_iff_of_pos"],["MulActionWithZero","toSMulWithZero"],["propext"]]},{"isProp":true,"kind":"theorem","name":["bddAbove_smul_iff_of_pos","_simp_1"],"typeFallback":"forall {α : Type.{u_1}} {β : Type.{u_2}} [inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.4 : Preorder.{u_1} α] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.7 : Preorder.{u_2} β] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.10 : GroupWithZero.{u_1} α] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.13 : Zero.{u_2} β] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.16 : MulActionWithZero.{u_1, u_2} α β (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.13] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.20 : PosSMulMono.{u_1, u_2} α β (SMulZeroClass.toSMul.{u_1, u_2} α β inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.13 (SMulWithZero.toSMulZeroClass.{u_1, u_2} α β (MulZeroClass.toZero.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α (MonoidWithZero.toMulZeroOneClass.{u_1} α (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.10)))) inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.13 (MulActionWithZero.toSMulWithZero.{u_1, u_2} α β (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.16))) inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.7 (MulZeroClass.toZero.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α (MonoidWithZero.toMulZeroOneClass.{u_1} α (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.10))))] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.24 : PosSMulReflectLE.{u_1, u_2} α β (SMulZeroClass.toSMul.{u_1, u_2} α β inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.13 (SMulWithZero.toSMulZeroClass.{u_1, u_2} α β (MulZeroClass.toZero.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α (MonoidWithZero.toMulZeroOneClass.{u_1} α (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.10)))) inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.13 (MulActionWithZero.toSMulWithZero.{u_1, u_2} α β (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.16))) inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.7 (MulZeroClass.toZero.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α (MonoidWithZero.toMulZeroOneClass.{u_1} α (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.10))))] {s : Set.{u_2} β} {a : α}, (LT.lt.{u_1} α (Preorder.toLT.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.4) (OfNat.ofNat.{u_1} α 0 (Zero.toOfNat0.{u_1} α (MulZeroClass.toZero.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α (MonoidWithZero.toMulZeroOneClass.{u_1} α (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.10)))))) a) -> (Eq.{1} Prop (BddAbove.{u_2} β (Preorder.toLE.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.7) (HSMul.hSMul.{u_1, u_2, u_2} α (Set.{u_2} β) (Set.{u_2} β) (instHSMul.{u_1, u_2} α (Set.{u_2} β) (Set.smulSet.{u_1, u_2} α β (SMulZeroClass.toSMul.{u_1, u_2} α β inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.13 (SMulWithZero.toSMulZeroClass.{u_1, u_2} α β (MulZeroClass.toZero.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α (MonoidWithZero.toMulZeroOneClass.{u_1} α (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.10)))) inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.13 (MulActionWithZero.toSMulWithZero.{u_1, u_2} α β (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.16))))) a s)) (BddAbove.{u_2} β (Preorder.toLE.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.585061352._hygCtx._hyg.7) s))","typeFull":"∀ {α : Type u_1} {β : Type u_2} [inst : Preorder α] [inst_1 : Preorder β] [inst_2 : GroupWithZero α] [inst_3 : Zero β]\n [inst_4 : MulActionWithZero α β] [PosSMulMono α β] [PosSMulReflectLE α β] {s : Set β} {a : α},\n 0 < a → BddAbove (a • s) = BddAbove s","typeReadable":"∀ {α : Type u_1} {β : Type u_2} [inst : Preorder α] [inst_1 : Preorder β] [inst_2 : GroupWithZero α] [inst_3 : Zero β]\n [inst_4 : MulActionWithZero α β] [PosSMulMono α β] [PosSMulReflectLE α β] {s : Set β} {a : α},\n 0 < a → BddAbove (a • s) = BddAbove s","typeReferences":[["Set"],["Preorder","toLT"],["SMulZeroClass","toSMul"],["MonoidWithZero","toMulZeroOneClass"],["GroupWithZero"],["SMulWithZero","toSMulZeroClass"],["BddAbove"],["OfNat","ofNat"],["PosSMulMono"],["LT","lt"],["PosSMulReflectLE"],["Preorder"],["MulZeroOneClass","toMulZeroClass"],["MulZeroClass","toZero"],["MulActionWithZero"],["HSMul","hSMul"],["GroupWithZero","toMonoidWithZero"],["Set","smulSet"],["instHSMul"],["Zero","toOfNat0"],["Zero"],["Preorder","toLE"],["Eq"],["MulActionWithZero","toSMulWithZero"]],"valueReferences":[["bddAbove_smul_iff_of_pos"],["Set"],["SMulZeroClass","toSMul"],["MonoidWithZero","toMulZeroOneClass"],["SMulWithZero","toSMulZeroClass"],["BddAbove"],["MulZeroOneClass","toMulZeroClass"],["MulZeroClass","toZero"],["HSMul","hSMul"],["GroupWithZero","toMonoidWithZero"],["Set","smulSet"],["instHSMul"],["Preorder","toLE"],["MulActionWithZero","toSMulWithZero"],["propext"]]},{"isProp":true,"kind":"theorem","name":["bddAbove_smul_iff_of_neg"],"typeFallback":"forall {α : Type.{u_1}} {β : Type.{u_2}} [inst._@.Mathlib.Algebra.Order.Module.Pointwise.3332345626._hygCtx._hyg.4 : Field.{u_1} α] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.3332345626._hygCtx._hyg.7 : LinearOrder.{u_1} α] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.3332345626._hygCtx._hyg.10 : IsStrictOrderedRing.{u_1} α (DivisionSemiring.toSemiring.{u_1} α (Semifield.toDivisionSemiring.{u_1} α (Field.toSemifield.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3332345626._hygCtx._hyg.4))) (SemilatticeInf.toPartialOrder.{u_1} α (Lattice.toSemilatticeInf.{u_1} α (DistribLattice.toLattice.{u_1} α (instDistribLatticeOfLinearOrder.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3332345626._hygCtx._hyg.7))))] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.3332345626._hygCtx._hyg.13 : AddCommGroup.{u_2} β] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.3332345626._hygCtx._hyg.16 : PartialOrder.{u_2} β] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.3332345626._hygCtx._hyg.19 : IsOrderedAddMonoid.{u_2} β (AddCommGroup.toAddCommMonoid.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3332345626._hygCtx._hyg.13) (PartialOrder.toPreorder.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3332345626._hygCtx._hyg.16)] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.3332345626._hygCtx._hyg.22 : Module.{u_1, u_2} α β (DivisionSemiring.toSemiring.{u_1} α (Semifield.toDivisionSemiring.{u_1} α (Field.toSemifield.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3332345626._hygCtx._hyg.4))) (AddCommGroup.toAddCommMonoid.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3332345626._hygCtx._hyg.13)] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.3332345626._hygCtx._hyg.26 : PosSMulMono.{u_1, u_2} α β (SMulZeroClass.toSMul.{u_1, u_2} α β (AddZero.toZero.{u_2} β (AddZeroClass.toAddZero.{u_2} β (AddMonoid.toAddZeroClass.{u_2} β (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3332345626._hygCtx._hyg.13)))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} α β (AddMonoid.toAddZeroClass.{u_2} β (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3332345626._hygCtx._hyg.13)))) (DistribMulAction.toDistribSMul.{u_1, u_2} α β (MonoidWithZero.toMonoid.{u_1} α (Semiring.toMonoidWithZero.{u_1} α (DivisionSemiring.toSemiring.{u_1} α (Semifield.toDivisionSemiring.{u_1} α (Field.toSemifield.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3332345626._hygCtx._hyg.4))))) (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3332345626._hygCtx._hyg.13))) (Module.toDistribMulAction.{u_1, u_2} α β (DivisionSemiring.toSemiring.{u_1} α (Semifield.toDivisionSemiring.{u_1} α (Field.toSemifield.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3332345626._hygCtx._hyg.4))) (AddCommGroup.toAddCommMonoid.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3332345626._hygCtx._hyg.13) inst._@.Mathlib.Algebra.Order.Module.Pointwise.3332345626._hygCtx._hyg.22)))) (PartialOrder.toPreorder.{u_1} α (SemilatticeInf.toPartialOrder.{u_1} α (Lattice.toSemilatticeInf.{u_1} α (DistribLattice.toLattice.{u_1} α (instDistribLatticeOfLinearOrder.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3332345626._hygCtx._hyg.7))))) (PartialOrder.toPreorder.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3332345626._hygCtx._hyg.16) (MulZeroClass.toZero.{u_1} α (NonUnitalNonAssocSemiring.toMulZeroClass.{u_1} α (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_1} α (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_1} α (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_1} α (CommRing.toNonUnitalCommRing.{u_1} α (Field.toCommRing.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3332345626._hygCtx._hyg.4)))))))] {s : Set.{u_2} β} {a : α}, (LT.lt.{u_1} α (Preorder.toLT.{u_1} α (PartialOrder.toPreorder.{u_1} α (SemilatticeInf.toPartialOrder.{u_1} α (Lattice.toSemilatticeInf.{u_1} α (DistribLattice.toLattice.{u_1} α (instDistribLatticeOfLinearOrder.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3332345626._hygCtx._hyg.7)))))) a (OfNat.ofNat.{u_1} α 0 (Zero.toOfNat0.{u_1} α (MulZeroClass.toZero.{u_1} α (NonUnitalNonAssocSemiring.toMulZeroClass.{u_1} α (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_1} α (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_1} α (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_1} α (CommRing.toNonUnitalCommRing.{u_1} α (Field.toCommRing.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3332345626._hygCtx._hyg.4)))))))))) -> (Iff (BddAbove.{u_2} β (Preorder.toLE.{u_2} β (PartialOrder.toPreorder.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3332345626._hygCtx._hyg.16)) (HSMul.hSMul.{u_1, u_2, u_2} α (Set.{u_2} β) (Set.{u_2} β) (instHSMul.{u_1, u_2} α (Set.{u_2} β) (Set.smulSet.{u_1, u_2} α β (SMulZeroClass.toSMul.{u_1, u_2} α β (AddZero.toZero.{u_2} β (AddZeroClass.toAddZero.{u_2} β (AddMonoid.toAddZeroClass.{u_2} β (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3332345626._hygCtx._hyg.13)))))) (DistribSMul.toSMulZeroClass.{u_1, u_2} α β (AddMonoid.toAddZeroClass.{u_2} β (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3332345626._hygCtx._hyg.13)))) (DistribMulAction.toDistribSMul.{u_1, u_2} α β (MonoidWithZero.toMonoid.{u_1} α (Semiring.toMonoidWithZero.{u_1} α (DivisionSemiring.toSemiring.{u_1} α (Semifield.toDivisionSemiring.{u_1} α (Field.toSemifield.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3332345626._hygCtx._hyg.4))))) (SubNegMonoid.toAddMonoid.{u_2} β (AddGroup.toSubNegMonoid.{u_2} β (AddCommGroup.toAddGroup.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3332345626._hygCtx._hyg.13))) (Module.toDistribMulAction.{u_1, u_2} α β (DivisionSemiring.toSemiring.{u_1} α (Semifield.toDivisionSemiring.{u_1} α (Field.toSemifield.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3332345626._hygCtx._hyg.4))) (AddCommGroup.toAddCommMonoid.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3332345626._hygCtx._hyg.13) inst._@.Mathlib.Algebra.Order.Module.Pointwise.3332345626._hygCtx._hyg.22)))))) a s)) (BddBelow.{u_2} β (Preorder.toLE.{u_2} β (PartialOrder.toPreorder.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3332345626._hygCtx._hyg.16)) s))","typeFull":"∀ {α : Type u_1} {β : Type u_2} [inst : Field α] [inst_1 : LinearOrder α] [IsStrictOrderedRing α]\n [inst_3 : AddCommGroup β] [inst_4 : PartialOrder β] [IsOrderedAddMonoid β] [inst_6 : Module α β] [PosSMulMono α β]\n {s : Set β} {a : α}, a < 0 → (BddAbove (a • s) ↔ BddBelow s)","typeReadable":"∀ {α : Type u_1} {β : Type u_2} [inst : Field α] [inst_1 : LinearOrder α] [IsStrictOrderedRing α]\n [inst_3 : AddCommGroup β] [inst_4 : PartialOrder β] [IsOrderedAddMonoid β] [inst_6 : Module α β] [PosSMulMono α β]\n {s : Set β} {a : α}, a < 0 → (BddAbove (a • s) ↔ BddBelow s)","typeReferences":[["PartialOrder","toPreorder"],["Field"],["Module"],["AddCommGroup","toAddGroup"],["Preorder","toLT"],["SMulZeroClass","toSMul"],["CommRing","toNonUnitalCommRing"],["PosSMulMono"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["instDistribLatticeOfLinearOrder"],["IsOrderedAddMonoid"],["PartialOrder"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["MonoidWithZero","toMonoid"],["Set","smulSet"],["instHSMul"],["Zero","toOfNat0"],["Semifield","toDivisionSemiring"],["AddGroup","toSubNegMonoid"],["Preorder","toLE"],["SemilatticeInf","toPartialOrder"],["DistribSMul","toSMulZeroClass"],["Field","toCommRing"],["Lattice","toSemilatticeInf"],["IsStrictOrderedRing"],["Set"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["DistribMulAction","toDistribSMul"],["LinearOrder"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["BddBelow"],["DivisionSemiring","toSemiring"],["AddZeroClass","toAddZero"],["BddAbove"],["OfNat","ofNat"],["LT","lt"],["Module","toDistribMulAction"],["DistribLattice","toLattice"],["SubNegMonoid","toAddMonoid"],["MulZeroClass","toZero"],["Iff"],["HSMul","hSMul"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["Field","toSemifield"],["AddCommGroup","toAddCommMonoid"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["OrderIso","smulRightDual"],["OrderDual","instPreorder"],["OrderDual"],["PartialOrder","toPreorder"],["OrderIso","bddBelow_image"]]},{"isProp":true,"kind":"theorem","name":["upperBounds_smul_of_pos"],"typeFallback":"forall {α : Type.{u_1}} {β : Type.{u_2}} [inst._@.Mathlib.Algebra.Order.Module.Pointwise.1826223390._hygCtx._hyg.4 : Preorder.{u_1} α] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.1826223390._hygCtx._hyg.7 : Preorder.{u_2} β] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.1826223390._hygCtx._hyg.10 : GroupWithZero.{u_1} α] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.1826223390._hygCtx._hyg.13 : Zero.{u_2} β] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.1826223390._hygCtx._hyg.16 : MulActionWithZero.{u_1, u_2} α β (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.1826223390._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Order.Module.Pointwise.1826223390._hygCtx._hyg.13] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.1826223390._hygCtx._hyg.20 : PosSMulMono.{u_1, u_2} α β (SMulZeroClass.toSMul.{u_1, u_2} α β inst._@.Mathlib.Algebra.Order.Module.Pointwise.1826223390._hygCtx._hyg.13 (SMulWithZero.toSMulZeroClass.{u_1, u_2} α β (MulZeroClass.toZero.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α (MonoidWithZero.toMulZeroOneClass.{u_1} α (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.1826223390._hygCtx._hyg.10)))) inst._@.Mathlib.Algebra.Order.Module.Pointwise.1826223390._hygCtx._hyg.13 (MulActionWithZero.toSMulWithZero.{u_1, u_2} α β (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.1826223390._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Order.Module.Pointwise.1826223390._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Order.Module.Pointwise.1826223390._hygCtx._hyg.16))) inst._@.Mathlib.Algebra.Order.Module.Pointwise.1826223390._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Order.Module.Pointwise.1826223390._hygCtx._hyg.7 (MulZeroClass.toZero.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α (MonoidWithZero.toMulZeroOneClass.{u_1} α (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.1826223390._hygCtx._hyg.10))))] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.1826223390._hygCtx._hyg.24 : PosSMulReflectLE.{u_1, u_2} α β (SMulZeroClass.toSMul.{u_1, u_2} α β inst._@.Mathlib.Algebra.Order.Module.Pointwise.1826223390._hygCtx._hyg.13 (SMulWithZero.toSMulZeroClass.{u_1, u_2} α β (MulZeroClass.toZero.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α (MonoidWithZero.toMulZeroOneClass.{u_1} α (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.1826223390._hygCtx._hyg.10)))) inst._@.Mathlib.Algebra.Order.Module.Pointwise.1826223390._hygCtx._hyg.13 (MulActionWithZero.toSMulWithZero.{u_1, u_2} α β (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.1826223390._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Order.Module.Pointwise.1826223390._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Order.Module.Pointwise.1826223390._hygCtx._hyg.16))) inst._@.Mathlib.Algebra.Order.Module.Pointwise.1826223390._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Order.Module.Pointwise.1826223390._hygCtx._hyg.7 (MulZeroClass.toZero.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α (MonoidWithZero.toMulZeroOneClass.{u_1} α (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.1826223390._hygCtx._hyg.10))))] {s : Set.{u_2} β} {a : α}, (LT.lt.{u_1} α (Preorder.toLT.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.1826223390._hygCtx._hyg.4) (OfNat.ofNat.{u_1} α 0 (Zero.toOfNat0.{u_1} α (MulZeroClass.toZero.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α (MonoidWithZero.toMulZeroOneClass.{u_1} α (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.1826223390._hygCtx._hyg.10)))))) a) -> (Eq.{succ u_2} (Set.{u_2} β) (upperBounds.{u_2} β (Preorder.toLE.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.1826223390._hygCtx._hyg.7) (HSMul.hSMul.{u_1, u_2, u_2} α (Set.{u_2} β) (Set.{u_2} β) (instHSMul.{u_1, u_2} α (Set.{u_2} β) (Set.smulSet.{u_1, u_2} α β (SMulZeroClass.toSMul.{u_1, u_2} α β inst._@.Mathlib.Algebra.Order.Module.Pointwise.1826223390._hygCtx._hyg.13 (SMulWithZero.toSMulZeroClass.{u_1, u_2} α β (MulZeroClass.toZero.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α (MonoidWithZero.toMulZeroOneClass.{u_1} α (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.1826223390._hygCtx._hyg.10)))) inst._@.Mathlib.Algebra.Order.Module.Pointwise.1826223390._hygCtx._hyg.13 (MulActionWithZero.toSMulWithZero.{u_1, u_2} α β (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.1826223390._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Order.Module.Pointwise.1826223390._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Order.Module.Pointwise.1826223390._hygCtx._hyg.16))))) a s)) (HSMul.hSMul.{u_1, u_2, u_2} α (Set.{u_2} β) (Set.{u_2} β) (instHSMul.{u_1, u_2} α (Set.{u_2} β) (Set.smulSet.{u_1, u_2} α β (SMulZeroClass.toSMul.{u_1, u_2} α β inst._@.Mathlib.Algebra.Order.Module.Pointwise.1826223390._hygCtx._hyg.13 (SMulWithZero.toSMulZeroClass.{u_1, u_2} α β (MulZeroClass.toZero.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α (MonoidWithZero.toMulZeroOneClass.{u_1} α (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.1826223390._hygCtx._hyg.10)))) inst._@.Mathlib.Algebra.Order.Module.Pointwise.1826223390._hygCtx._hyg.13 (MulActionWithZero.toSMulWithZero.{u_1, u_2} α β (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.1826223390._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Order.Module.Pointwise.1826223390._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Order.Module.Pointwise.1826223390._hygCtx._hyg.16))))) a (upperBounds.{u_2} β (Preorder.toLE.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.1826223390._hygCtx._hyg.7) s)))","typeFull":"∀ {α : Type u_1} {β : Type u_2} [inst : Preorder α] [inst_1 : Preorder β] [inst_2 : GroupWithZero α] [inst_3 : Zero β]\n [inst_4 : MulActionWithZero α β] [PosSMulMono α β] [PosSMulReflectLE α β] {s : Set β} {a : α},\n 0 < a → upperBounds (a • s) = a • upperBounds s","typeReadable":"∀ {α : Type u_1} {β : Type u_2} [inst : Preorder α] [inst_1 : Preorder β] [inst_2 : GroupWithZero α] [inst_3 : Zero β]\n [inst_4 : MulActionWithZero α β] [PosSMulMono α β] [PosSMulReflectLE α β] {s : Set β} {a : α},\n 0 < a → upperBounds (a • s) = a • upperBounds s","typeReferences":[["Set"],["Preorder","toLT"],["SMulZeroClass","toSMul"],["MonoidWithZero","toMulZeroOneClass"],["GroupWithZero"],["SMulWithZero","toSMulZeroClass"],["OfNat","ofNat"],["PosSMulMono"],["LT","lt"],["PosSMulReflectLE"],["Preorder"],["MulZeroOneClass","toMulZeroClass"],["upperBounds"],["MulZeroClass","toZero"],["MulActionWithZero"],["HSMul","hSMul"],["GroupWithZero","toMonoidWithZero"],["Set","smulSet"],["instHSMul"],["Zero","toOfNat0"],["Zero"],["Preorder","toLE"],["Eq"],["MulActionWithZero","toSMulWithZero"]],"valueReferences":[["OrderIso","upperBounds_image"],["GroupWithZero","toMonoidWithZero"],["OrderIso","smulRight"],["MulActionWithZero","toMulAction"]]},{"isProp":true,"kind":"theorem","name":["bddBelow_smul_iff_of_pos"],"typeFallback":"forall {α : Type.{u_1}} {β : Type.{u_2}} [inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.4 : Preorder.{u_1} α] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.7 : Preorder.{u_2} β] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.10 : GroupWithZero.{u_1} α] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.13 : Zero.{u_2} β] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.16 : MulActionWithZero.{u_1, u_2} α β (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.13] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.20 : PosSMulMono.{u_1, u_2} α β (SMulZeroClass.toSMul.{u_1, u_2} α β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.13 (SMulWithZero.toSMulZeroClass.{u_1, u_2} α β (MulZeroClass.toZero.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α (MonoidWithZero.toMulZeroOneClass.{u_1} α (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.10)))) inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.13 (MulActionWithZero.toSMulWithZero.{u_1, u_2} α β (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.16))) inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.7 (MulZeroClass.toZero.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α (MonoidWithZero.toMulZeroOneClass.{u_1} α (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.10))))] [inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.24 : PosSMulReflectLE.{u_1, u_2} α β (SMulZeroClass.toSMul.{u_1, u_2} α β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.13 (SMulWithZero.toSMulZeroClass.{u_1, u_2} α β (MulZeroClass.toZero.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α (MonoidWithZero.toMulZeroOneClass.{u_1} α (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.10)))) inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.13 (MulActionWithZero.toSMulWithZero.{u_1, u_2} α β (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.16))) inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.7 (MulZeroClass.toZero.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α (MonoidWithZero.toMulZeroOneClass.{u_1} α (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.10))))] {s : Set.{u_2} β} {a : α}, (LT.lt.{u_1} α (Preorder.toLT.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.4) (OfNat.ofNat.{u_1} α 0 (Zero.toOfNat0.{u_1} α (MulZeroClass.toZero.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α (MonoidWithZero.toMulZeroOneClass.{u_1} α (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.10)))))) a) -> (Iff (BddBelow.{u_2} β (Preorder.toLE.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.7) (HSMul.hSMul.{u_1, u_2, u_2} α (Set.{u_2} β) (Set.{u_2} β) (instHSMul.{u_1, u_2} α (Set.{u_2} β) (Set.smulSet.{u_1, u_2} α β (SMulZeroClass.toSMul.{u_1, u_2} α β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.13 (SMulWithZero.toSMulZeroClass.{u_1, u_2} α β (MulZeroClass.toZero.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α (MonoidWithZero.toMulZeroOneClass.{u_1} α (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.10)))) inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.13 (MulActionWithZero.toSMulWithZero.{u_1, u_2} α β (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.10) inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.16))))) a s)) (BddBelow.{u_2} β (Preorder.toLE.{u_2} β inst._@.Mathlib.Algebra.Order.Module.Pointwise.3904177163._hygCtx._hyg.7) s))","typeFull":"∀ {α : Type u_1} {β : Type u_2} [inst : Preorder α] [inst_1 : Preorder β] [inst_2 : GroupWithZero α] [inst_3 : Zero β]\n [inst_4 : MulActionWithZero α β] [PosSMulMono α β] [PosSMulReflectLE α β] {s : Set β} {a : α},\n 0 < a → (BddBelow (a • s) ↔ BddBelow s)","typeReadable":"∀ {α : Type u_1} {β : Type u_2} [inst : Preorder α] [inst_1 : Preorder β] [inst_2 : GroupWithZero α] [inst_3 : Zero β]\n [inst_4 : MulActionWithZero α β] [PosSMulMono α β] [PosSMulReflectLE α β] {s : Set β} {a : α},\n 0 < a → (BddBelow (a • s) ↔ BddBelow s)","typeReferences":[["Set"],["Preorder","toLT"],["SMulZeroClass","toSMul"],["MonoidWithZero","toMulZeroOneClass"],["BddBelow"],["GroupWithZero"],["SMulWithZero","toSMulZeroClass"],["OfNat","ofNat"],["PosSMulMono"],["LT","lt"],["PosSMulReflectLE"],["Preorder"],["MulZeroOneClass","toMulZeroClass"],["MulZeroClass","toZero"],["Iff"],["MulActionWithZero"],["HSMul","hSMul"],["GroupWithZero","toMonoidWithZero"],["Set","smulSet"],["instHSMul"],["Zero","toOfNat0"],["Zero"],["Preorder","toLE"],["MulActionWithZero","toSMulWithZero"]],"valueReferences":[["OrderIso","bddBelow_image"],["GroupWithZero","toMonoidWithZero"],["OrderIso","smulRight"],["MulActionWithZero","toMulAction"]]}]