Defs

📁 Source: Mathlib/Order/CompleteLattice/Defs.lean

Statistics

Metric	Count
Definitions toBoundedOrder, toCompleteSemilatticeInf, toCompleteSemilatticeSup, toInfSet, toLattice, toPartialOrder', toSupSet, toBiheytingAlgebra, toCompl, toCompleteLattice, toDecidableEq, toDecidableLE, toDecidableLT, toHImp, toHNot, toLinearOrder, toOrd, toSDiff, CompleteSemilatticeInf, toInfSet, toPartialOrder, CompleteSemilatticeSup, toPartialOrder, toSupSet, infSet, instCompleteLattice, instCompleteLinearOrder, supSet, completeLatticeOfCompleteSemilatticeInf, completeLatticeOfCompleteSemilatticeSup, completeLatticeOfInf, completeLatticeOfSup, instCompleteSemilatticeInfOrderDualOfCompleteSemilatticeSup, instCompleteSemilatticeSupOrderDualOfCompleteSemilatticeInf	34
Theorems le_sInf, le_sSup, sInf_le, sSup_le, compare_eq_compareOfLessAndEq, himp_bot, le_himp_iff, le_total, sdiff_le_iff, top_sdiff, le_sInf, sInf_le, le_sSup, sSup_le, sInf_eq, sSup_eq, biInf_lt_iff, iInf_le_iff, iInf_lt_iff, isGLB_iff_sInf_eq, isGLB_sInf, isLUB_iff_sSup_eq, isLUB_sSup, le_iSup_iff, le_sInf, le_sInf_iff, le_sSup, le_sSup_iff, le_sSup_of_le, lt_biSup_iff, lt_iSup_iff, lt_sSup_iff, ofDual_iInf, ofDual_iSup, ofDual_sInf, ofDual_sSup, sInf_eq_bot, sInf_le, sInf_le_iff, sInf_le_of_le, sInf_le_sInf, sInf_lt_iff, sSup_eq_top, sSup_le, sSup_le_iff, sSup_le_sSup, toDual_iInf, toDual_iSup, toDual_sInf, toDual_sSup	50
Total	84

CompleteLattice

Definitions

Name	Category	Theorems
toBoundedOrder 📖	CompOp	1344 math math: AlgebraicGeometry.Scheme.toSpecΓ_apply, AlgebraicGeometry.Scheme.Opens.ι_image_basicOpen, AlgebraicGeometry.Γ_map_morphismRestrict, AlgebraicGeometry.Scheme.Hom.smoothLocus_eq_top_iff, IsCyclotomicExtension.adjoin_primitive_root_eq_top, Algebra.range_ofId, NonUnitalSubalgebra.center_eq_top, sInf_univ, CategoryTheory.sheafBotEquivalence_functor, Subalgebra.LinearDisjoint.inf_eq_bot, AlgebraicGeometry.Scheme.ΓSpecIso_inv_naturality_assoc, Submodule.rTensorOne_symm_apply, AlgebraicGeometry.Scheme.Hom.toPartialMap_hom, AlgebraicGeometry.Scheme.map_PrimeSpectrum_basicOpen_of_affine, coinduced_bot, AlgebraicGeometry.Scheme.bot_mem_grothendieckTopology, IntermediateField.finInsepDegree_top, OrderIso.map_radical, SSet.Subcomplex.preimage_eq_top_iff, measurable_from_top, AlgebraicGeometry.IsAffineOpen.isoSpec_inv_appTop, AlgebraicGeometry.Scheme.germ_stalkClosedPointTo, CompleteLatticeHom.coe_toBoundedLatticeHom, IntermediateField.adjoin_eq_top_iff_of_isAlgebraic, iSupIndep.injOn_iInf, IntermediateField.LinearDisjoint.iff_inf_eq_bot, AlgebraicGeometry.ΓSpec.locallyRingedSpaceAdjunction_counit_app, MeasurableSpace.comap_const, AlgebraicGeometry.Scheme.IsQuasiAffine.toIsImmersion, AlgebraicGeometry.StructureSheaf.globalSectionsIso_inv, MonoidHom.ker_transferSylow_disjoint, TopologicalSpace.Opens.mem_top, AlgebraicGeometry.iSup_affineOpens_eq_top, AlgebraicGeometry.isAffineOpen_top, AlgebraicGeometry.AffineSpace.isoOfIsAffine_inv, MeasureTheory.Measure.sub_top, eq_sInf_coatoms, Submodule.comm_trans_lTensorOne, IntermediateField.coe_algebraMap_over_bot, RingCon.toCon_eq_bot, NonUnitalStarAlgebra.mem_bot, IsDedekindDomain.HeightOneSpectrum.iInf_localization_eq_bot, Algebra.adjoin_singleton_intCast, Subalgebra.pi_top, AlgebraicGeometry.Scheme.Opens.toSpecΓ_SpecMap_presheaf_map_top, AlgebraicGeometry.LocallyRingedSpace.notMem_prime_iff_unit_in_stalk, sSup_univ, AlgebraicGeometry.Scheme.restrictFunctorΓ_inv_app, MonoidAlgebra.Submodule.exists_isCompl, AlgebraicGeometry.instIsDomainCarrierObjOppositeOpensCarrierCarrierCommRingCatPresheafOpOpensTopOfIsIntegral, MeasurableSpace.comap_bot, MeasureTheory.OuterMeasure.mkMetric_top, GradedAlgebra.exists_finset_adjoin_eq_top_and_homogeneous_ne_zero, SeminormedGroup.disjoint_nhds, AffineBasis.tot, ImplicitFunctionData.isCompl_ker, Algebra.mem_top, AlgebraicGeometry.AffineSpace.toSpecMvPolyIntEquiv_symm_apply, Filter.liminf_const_top, AlgebraicGeometry.Scheme.fromSpecStalk_toSpecΓ_assoc, AlgebraicGeometry.AffineSpace.SpecIso_hom_appTop, IsOpen.exists_subset_affineIndependent_span_eq_top, SSet.iSup_subcomplexOfSimplex_prod_eq_top, IntermediateField.coe_bot, Submodule.mulMap_one_right_eq, OrdinalApprox.gfp_mem_range_gfpApprox, isCompactElement_finsetSup, LieAlgebra.IsKilling.isCompl_ker_weight_span_coroot, MeasureTheory.Measure.mutuallySingular_tfae, Ring.isField_iff_isSimpleOrder_ideal, AlgebraicGeometry.Scheme.zeroLocus_eq_univ_iff_subset_nilradical, AlgebraicGeometry.Scheme.Opens.topIso_inv, Opens.pretopology_toGrothendieck, iSupIndep.disjoint_biSup, CompleteSublattice.isComplemented_iff, CategoryTheory.Sieve.mem_iff_pullback_eq_top, IntermediateField.lift_insepDegree_bot', Filter.disjoint_comap_iff, NonUnitalAlgebra.coe_bot, Equiv.Perm.disjoint_of_disjoint_support, iInf_of_empty, sSup_eq_bot, inseparable_top, TensorAlgebra.adjoin_range_ι, NonUnitalStarAlgebra.coe_bot, NonUnitalStarAlgebra.mem_top, IsCompactlyGenerated.BooleanGenerators.isAtom, AlgHom.equalizer_eq_top, iSup_false, algebraicClosure.algebraicClosure_eq_bot, AffineSubspace.comap_bot, Subgroup.isCoatom_map, iInf_false, AlgebraicGeometry.Scheme.Hom.id_appTop, CategoryTheory.Presheaf.imageSieve_app, bot_lt_iSup, Directed.disjoint_iSup_right, iInf_eq_top, IsGaloisGroup.fixingSubgroup_top, CategoryTheory.Presieve.isSheaf_bot, MeasurableSpace.generateFrom_empty, AddSubgroup.map_eq_range_iff, IsGaloisGroup.fixedPoints_top, RingCon.comap_bot, MeasureTheory.condLExp_bot', AddCommMonoid.primaryComponent.disjoint, Submodule.ClosedComplemented.exists_isClosed_isCompl, AlgebraicGeometry.Scheme.ideal_ker_le_ker_ΓSpecIso_inv_comp, Algebra.toSubsemiring_eq_top, IntermediateField.botEquiv_symm, Subalgebra.op_bot, MeasureTheory.OuterMeasure.trim_top, LieModule.disjoint_genWeightSpaceOf, IsLocalRing.closedPoint_mem_iff, AlgebraicGeometry.Scheme.isoSpec_inv_toSpecΓ, QuotientAddGroup.leftRel_eq_top, LinearMap.BilinForm.restrict_nondegenerate_iff_isCompl_orthogonal, Filter.disjoint_cocompact_left, TopologicalSpace.Opens.isCoatom_iff, disjoint_nhds_cocompact, Algebra.adjoin_singleton_algebraMap, RingCon.matrix_top, SSet.Subcomplex.range_eq_top, AlgHom.fieldRange_eq_map, IntermediateField.relfinrank_bot_right, Field.exists_primitive_element, isGalois_iff_isGalois_top, AlgebraicGeometry.isIso_ΓSpec_adjunction_unit_app_basicOpen, AlgebraicGeometry.isNoetherian_iff_of_finite_affine_openCover, AlgebraicGeometry.Scheme.ι_toIso_inv, CategoryTheory.Subfunctor.range_toRange, UnitAddTorus.mFourierSubalgebra_closure_eq_top, InfiniteGalois.mem_bot_iff_fixed, Frm.Iso.mk_hom, Algebra.toSubring_eq_top, AddCon.toSetoid_top, ProbabilityTheory.condIndep_bot_left, NumberField.adjoin_eq_top_of_infinitePlace_lt, LieAlgebra.IsSemisimple.booleanGenerators, Polynomial.IsSplittingField.splits_iff, AlgebraicGeometry.Scheme.IdealSheafData.equivOfIsAffine_apply, Partition.bot_notMem, SSet.Subcomplex.topIso_inv_app_coe, t1Space_iff_disjoint_nhds_pure, bot_lt_affineSpan, polynomialFunctions.starClosure_topologicalClosure, AlgebraicGeometry.Scheme.IdealSheafData.equivOfIsAffine_symm_apply, CategoryTheory.GrothendieckTopology.trivial_covering, AffineEquiv.span_eq_top_iff, AlgebraicGeometry.Scheme.SpecΓIdentity_hom_app, AffineSubspace.direction_eq_top_iff_of_nonempty, Filter.disjoint_principal_left, isSeparatedMap_iff_disjoint_nhds, StarSubalgebra.bot_toSubalgebra, iSupIndep.pairwiseDisjoint, Subfield.splits_bot, induced_const, Algebra.toSubring_bot, affineSpan_singleton_union_vadd_eq_top_of_span_eq_top, integralClosure_eq_top_iff, interior_convexHull_nonempty_iff_affineSpan_eq_top, CompleteSublattice.top_mem, AlgebraicGeometry.Scheme.isoSpec_inv_preimage_zeroLocus, Subalgebra.LinearDisjoint.inf_eq_bot_of_commute, TopologicalSpace.eq_top_iff_forall_inseparable, sSup_empty, LieSubmodule.disjoint_toSubmodule, AlgebraicGeometry.ProjectiveSpectrum.Proj.toOpen_toSpec_val_c_app_assoc, AlgebraicGeometry.PresheafedSpace.Γ_obj_op, IntermediateField.adjoin_zero, IntermediateField.adjoin_univ, Filter.disjoint_comap_iff_map', Finset.inf_id_eq_sInf, AlgebraicGeometry.Scheme.Modules.toOpen_fromTildeΓ_app, TopCat.GlueData.MkCore.V_id, AffineSubspace.linear_topEquiv, Subalgebra.algebra_isAlgebraic_bot_right, UniformSpace.toTopologicalSpace_top, T25Space.t2_5, CategoryTheory.Subfunctor.Subpresheaf.preimage_eq_top_iff, Subalgebra.comm_trans_rTensorBot, AffineSubspace.not_wSameSide_bot, AlgebraicGeometry.Scheme.bot_mem_precoverage, CategoryTheory.Sieve.top_apply, Subalgebra.algebra_isAlgebraic_bot_left_iff, Algebra.EssFiniteType.adjoin_mem_finset, AlgebraicGeometry.Scheme.Hom.preimage_top, Polynomial.IsSplittingField.adjoin_rootSet, AddCon.toSetoid_bot, AlgebraicGeometry.Scheme.isoSpec_hom, AlgebraicGeometry.Scheme.ι_toIso_inv_assoc, Equiv.Perm.disjoint_closure_of_disjoint_support, DirectedOn.disjoint_sSup_left, IsSemisimpleModule.toComplementedLattice, TopologicalSpace.Closeds.coe_eq_empty, AlgebraicGeometry.Proj.basicOpenToSpec_app_top, sSupIndep.pairwiseDisjoint, NonUnitalSubalgebra.prod_top, TopologicalSpace.NoetherianSpace.exists_finset_irreducible, Metric.disjoint_cobounded_nhdsSet, AlgebraicGeometry.Scheme.Hom.image_top_eq_opensRange, sSup_atoms_le_eq, MeasurableSpace.map_const, LieSubmodule.isCompl_toSubmodule, NonUnitalStarAlgebra.coe_top, AlgebraicGeometry.Scheme.toSpecΓ_naturality_assoc, Submodule.coe_isComplEquivProj_symm_apply, sSupHom.coe_bot, Sublocale.top_mem, AlgebraicGeometry.SurjectiveOnStalks.iff_of_isAffine, AlgebraicGeometry.SpecMap_ΓSpecIso_hom, disjoint_rootsOfUnity_of_coprime, Subalgebra.center_eq_top, NumberField.mixedEmbedding.exists_primitive_element_lt_of_isComplex, IsIntegrallyClosedIn.integralClosure_eq_bot, SSet.iSup_skeleton, integralClosure_idem, PrimeSpectrum.iSup_basicOpen_eq_top_iff, IsNormalClosure.adjoin_rootSet, instIsAbelianGaloisSubtypeMemIntermediateFieldBot, AffineBasis.tot', CategoryTheory.Sieve.functorPullback_bot, TopologicalSpace.Opens.isOpenEmbedding_obj_top, AlgebraicGeometry.LocallyRingedSpace.Γ_obj_op, Algebra.mem_bot, AlgebraicGeometry.AffineSpace.isoOfIsAffine_inv_appTop_coord, Module.Dual.isCompl_ker_of_disjoint_of_ne_bot, AlgebraicGeometry.AffineSpace.isoOfIsAffine_inv_over_assoc, CategoryTheory.Sieve.generate_bot, SSet.Subcomplex.eq_top_iff_of_hasDimensionLT, AlgebraicGeometry.Scheme.isoSpec_hom_naturality, Filter.HasBasis.disjoint_iff_right, iInf₂_eq_top, induced_topology_pure, AlgebraicGeometry.Scheme.toSpecΓ_isoSpec_inv_assoc, FirstOrder.Language.Substructure.small_bot, LinearMap.isCompl_range_inl_inr, alternatingGroup.isCoatom_stabilizer_singleton, WellFoundedGT.finite_ne_bot_of_iSupIndep, essInf_measure_zero, Polynomial.SplittingFieldAux.adjoin_rootSet, RingCon.subsingleton_quotient, CategoryTheory.Sieve.generate_top, StarSubalgebra.mem_bot, SSet.Subcomplex.topIso_inv_ι, AlgebraicGeometry.Scheme.toSpecΓ_naturality, AlgebraicGeometry.Scheme.Hom.quasiFiniteLocus_eq_top, IntermediateField.fixingSubgroup_top, AffineSubspace.coe_eq_univ_iff, CategoryTheory.Sieve.equalizer_self, IntermediateField.bot_toSubalgebra, TopologicalSpace.Opens.mk_univ, Algebra.toSubmodule_bot, AlgebraicGeometry.LocallyRingedSpace.Γ_Spec_left_triangle, Algebra.map_bot, Subfield.mem_bot_iff_pow_eq_self, AlgebraicGeometry.RingedSpace.mem_top_basicOpen, Subfield.isTotallyReal_bot, PMF.toOuterMeasure_caratheodory, Metric.disjoint_nhds_cobounded, IntermediateField.relfinrank_top_right, Filter.disjoint_atTop_principal_Iio, IndiscreteTopology.eq_top, Algebra.adjoin_singleton_zero, AlgebraicGeometry.Scheme.Hom.toPartialMap_domain, Algebra.coe_bot, CompleteLinearOrder.top_sdiff, Setoid.injective_iff_ker_bot, Subgroup.IsComplement'.isCompl, Submodule.toConvexCone_top, MeasureTheory.condLExp_bot, LinearMap.BilinForm.isCompl_orthogonal_iff_disjoint, FrameHomClass.toBoundedLatticeHomClass, sSupHom.bot_apply, AlgebraicGeometry.IsAffineOpen.isoSpec_inv, AlgebraicGeometry.Scheme.Γ_obj, CategoryTheory.Presieve.isSheafFor_top_sieve, AlgebraicGeometry.isLocallyNoetherian_iff_of_affine_openCover, SSet.hasDimensionLT_subcomplex_top_iff, IntermediateField.restrictScalars_top, SSet.Subcomplex.instSubsingletonHomToSSetBot, LinearMap.isCompl_iSup_ker_pow_iInf_range_pow, EuclideanGeometry.Sphere.orthRadius_center, TopologicalSpace.isOpen_top_iff, AlgebraicGeometry.IsLocallyArtinian.isArtinianRing_of_isAffine, AlgebraicGeometry.StructureSheaf.toPushforwardStalkAlgHom_apply, AlgebraicGeometry.Scheme.preimage_basicOpen_top, MvPolynomial.adjoin_range_X, AlgebraicGeometry.Scheme.ΓSpecIso_inv, isOpen_setOf_disjoint_nhds_nhds, NonUnitalAlgebra.range_id, LieModule.disjoint_lowerCentralSeries_maxTrivSubmodule_iff, Partition.pairwiseDisjoint, IntermediateField.mem_top, AlgebraicGeometry.Scheme.isoSpec_hom_naturality_assoc, AlgebraicGeometry.Scheme.isoSpec_inv_image_zeroLocus, Field.primitive_element_iff_algHom_eq_of_eval', CategoryTheory.GrothendieckTopology.top_mem, IsDedekindDomain.integer_empty, AlgebraicGeometry.ΓSpec.locallyRingedSpaceAdjunction_counit_app', AffineSubspace.not_sSameSide_bot, Subgroup.FG.finset_sup, LinearMap.IsProj.isCompl, adjoin_eq_top_of_conductor_eq_top, Subalgebra.eq_bot_of_finrank_one, AlgebraicGeometry.Scheme.Opens.ι_image_basicOpen_topIso_inv, IntermediateField.relrank_top_right, AlgebraicGeometry.SpecMap_ΓSpecIso_inv_toSpecΓ, disjoint_nhds_nhds_iff_not_specializes, AffineBasis.affineSpan_eq_top_of_toMatrix_left_inv, iSupIndep_pair, mem_codiscrete, AlgebraicGeometry.Scheme.toSpecΓ_preimage_zeroLocus, continuous_bot, Algebra.coe_top, IntermediateField.LinearDisjoint.bot_left, TopologicalSpace.Opens.inclusion'_top_functor, FirstOrder.Language.Substructure.cg_bot, IntermediateField.isTotallyReal_bot, MeasureTheory.Measure.mkMetric_top, SeparatedNhds.disjoint_nhdsSet, SSet.Subcomplex.eq_top_iff_contains_nonDegenerate, AlgebraicGeometry.Scheme.fromSpecStalk_appTop, Subalgebra.unop_top, disjoint_nhdsSet_principal, Disjoint.filter_principal, LinearMap.IsPerfectCompl.isCompl_right, AlgebraicGeometry.PresheafedSpace.Γ_map_op, Subalgebra.lTensorBot_one_tmul, Subalgebra.isSimpleOrder_of_finrank, Projectivization.Subspace.span_empty, AlgebraicGeometry.Scheme.id_appTop, sSup_compact_eq_top, iSupIndep.supIndep', Subalgebra.mulMap_bot_left_eq, AlgebraicGeometry.Scheme.Opens.toSpecΓ_appTop_assoc, iSupIndep.supIndep, AffineSubspace.bot_coe, Convex.interior_nonempty_iff_affineSpan_eq_top, IntermediateField.instFiniteSubtypeMemBot, Polynomial.adjoin_rootSet_eq_range, AffineSubspace.parallel_bot_iff_eq_bot, CategoryTheory.Presheaf.equalizerSieve_self_eq_top, Filter.disjoint_iff, essSup_const_bot, Module.End.invtSubmodule.isCompl_iff, SeminormedAddGroup.disjoint_nhds_zero, AlgebraicGeometry.Scheme.fromSpecStalk_toSpecΓ, Filter.not_nonneg_sub_iff, AlgebraicGeometry.Scheme.ker_toSpecΓ, TopCat.Presheaf.map_germ_eq_Γgerm, Algebra.top_toSubmodule, SSet.Subcomplex.preimage_range, AlgebraicGeometry.Scheme.ker_of_isAffine, Subalgebra.fg_of_submodule_fg, AlgebraicGeometry.IsAffineOpen.algebraMap_Spec_obj, AlgebraicGeometry.isBasis_basicOpen, CompleteSublattice.bot_mem, isClosed_and_discrete_iff, AlgebraicGeometry.Scheme.Γevaluation_naturality_apply, r1Space_iff_inseparable_or_disjoint_nhds, CliffordAlgebra.adjoin_range_ι, AlgebraicGeometry.Flat.flat_and_surjective_iff_faithfullyFlat_of_isAffine, isCompactElement_iff_exists_le_iSup_of_le_iSup, AlgebraicGeometry.Scheme.Opens.toSpecΓ_SpecMap_presheaf_map_top_assoc, AlgebraicGeometry.LocallyRingedSpace.coe_toΓSpecSheafedSpace_hom_base_hom_apply_asIdeal, Polynomial.Splits.adjoin_rootSet_eq_range, IntermediateField.finrank_eq_one_iff, Setoid.top_def, AlgebraicGeometry.isIntegral_appTop_of_universallyClosed, IntermediateField.lift_top, AddSubmonoid.disjoint_def', Filter.isAtom_pure, AlgebraicGeometry.ΓSpec.adjunction_counit_app, IntermediateField.eq_bot_of_isAlgClosed_of_isAlgebraic, AlgebraicGeometry.tilde.isoTop_hom, CategoryTheory.Subfunctor.range_eq_top, borel_eq_top_of_discrete, IntermediateField.insepDegree_bot', MulAction.isCoatom_stabilizer_iff_preprimitive, Submonoid.FG.finset_sup, AlgebraicGeometry.Scheme.Opens.toSpecΓ_appTop, Filter.disjoint_atBot_atTop, CategoryTheory.Subfunctor.Subpresheaf.image_top, AddSubgroup.disjoint_def, AlgebraicGeometry.ProjectiveSpectrum.Proj.toStalk_stalkMap_toSpec_assoc, GradedAlgebra.exists_finset_adjoin_eq_top_and_homogeneous, AlgebraicGeometry.Γ_restrict_isLocalization, IntermediateField.restrictScalars_eq_top_iff, CategoryTheory.GrothendieckTopology.bot_covers, AffineSubspace.smul_bot, isDenseEmbedding_pure, TopologicalSpace.Opens.coe_eq_empty, CategoryTheory.top_apply, AlgebraicGeometry.ProjectiveSpectrum.Proj.toStalk_stalkMap_toSpec, NonUnitalAlgebra.toSubmodule_bot, Filter.disjoint_cofinite_right, Opens.pretopology_ofGrothendieck, AlgebraicGeometry.ΓSpecIso_obj_hom, IsLindelof.disjoint_nhdsSet_left, CategoryTheory.GrothendieckTopology.Cover.preOneHypercover_sieve₁, Filter.disjoint_map, AlgebraicGeometry.Spec.germ_stalkMapIso_hom_assoc, PrimeSpectrum.iInf_localization_eq_bot, AlgebraicGeometry.Scheme.Hom.preimage_basicOpen_top, IntermediateField.bot_eq_top_of_finrank_adjoin_eq_one, IsSimpleGroup.instIsSimpleOrderSubgroup, Setoid.mk_eq_top, MeasureTheory.OuterMeasure.comap_top, CategoryTheory.Sieve.overEquiv_top, ProbabilityTheory.iIndep_comap_mem_iff, IntermediateField.botContinuousSMul, Field.exists_primitive_element_of_finite_bot, CategoryTheory.GrothendieckTopology.close_eq_top_iff_mem, TopologicalSpace.Opens.functor_obj_map_obj, Subalgebra.unop_bot, AffineSubspace.parallel_iff_direction_eq_and_eq_bot_iff_eq_bot, CategoryTheory.Sieve.arrows_eq_top_iff, Filter.disjoint_cofinite_left, AlgebraicGeometry.exists_appTop_π_eq_of_isAffine_of_isLimit, CategoryTheory.Sieve.arrows_top, Frm.Iso.mk_inv, QuotientAddGroup.rightRel_eq_top, LieAlgebra.IsSemisimple.sSup_atoms_eq_top, AlgebraicGeometry.Scheme.OpenCover.iSup_opensRange, AffineSubspace.mem_top, Algebra.bijective_algebraMap_iff, Subalgebra.topEquiv_apply, CategoryTheory.topologyOfClosureOperator_sieves, Field.exists_primitive_element_of_finite_top, Subgroup.subgroupOf_eq_bot, IntermediateField.adjoin_simple_eq_top_iff_of_isAlgebraic, continuous_top, MulAction.IsPreprimitive.isCoatom_stabilizer_of_isPreprimitive, AlgebraicGeometry.Scheme.Γevaluation_naturality_assoc, IsCompact.disjoint_nhdsSet_left, essInf_const_top, AlgebraicGeometry.Scheme.IdealSheafData.coe_support_ofIdealTop, SSet.instHasDimensionLTToSSetBotSubcomplex, CategoryTheory.MorphismProperty.bot_mem_precoverage, LieAlgebra.IsSimple.isAtom_top, IntermediateField.rank_adjoin_eq_one_iff, iSupIndep_def'', AlgebraicGeometry.ΓSpec.toOpen_comp_locallyRingedSpaceAdjunction_homEquiv_app, AddGroupTopology.toTopologicalSpace_top, sSup_diff_singleton_bot, AlgebraicGeometry.Scheme.Opens.toSpecΓ_top, IsLindelof.disjoint_nhdsSet_right, TopologicalSpace.Closeds.coe_bot, IntermediateField.adjoin_one, IntermediateField.rank_top, CompleteDistribLattice.top_sdiff, CategoryTheory.sheafBotEquivalence_inverse_map_val, OrdinalApprox.lfpApprox_ord_eq_lfp, CategoryTheory.Sieve.generate_of_singleton_isSplitEpi, AlgebraicGeometry.AffineSpace.SpecIso_inv_appTop_coord, CategoryTheory.Sieve.functorPullback_top, AlgebraicGeometry.germ_stalkClosedPointIso_hom_assoc, TopologicalSpace.Opens.coe_finset_sup, NonUnitalAlgebra.top_toNonUnitalSubsemiring, TopologicalSpace.Opens.instIsSimpleOrderOfNonemptyOfIndiscreteTopology, Quotient.subsingleton_iff, Setoid.bot_def, CategoryTheory.Subfunctor.Subpresheaf.bot_obj, Subfield.bot_eq_of_zMod_algebra, ConvexCone.disjoint_hull_left_of_convex, PowerBasis.adjoin_gen_eq_top, Filter.not_disjoint_self_iff, isSimpleRing_iff, Filter.disjoint_of_disjoint_of_mem, StarSubalgebra.eq_top_iff, IntermediateField.adjoin_root_eq_top, AffineSubspace.smul_top, ConvexCone.toPointedCone_top, MeasurableSpace.measurableSet_bot_iff, Subfield.map_bot, iInf_ne_top_subtype, CategoryTheory.GrothendieckTopology.discrete_eq_top, TopCat.Presheaf.map_germ_eq_Γgerm_assoc, Submodule.comm_trans_rTensorOne, NonUnitalStarAlgebra.top_toNonUnitalSubalgebra, MeasureTheory.OuterMeasure.zero_caratheodory, ConvexCone.mem_top, CategoryTheory.sheafBotEquivalence_unitIso, IntermediateField.extendScalars_self, MeasureTheory.OuterMeasure.top_caratheodory, TopologicalSpace.Closeds.coe_finset_inf, AlgebraicGeometry.StructureSheaf.globalSectionsIso_hom, IntermediateField.finrank_top, CompleteSublattice.codisjoint_iff, iSupIndep_def', NonUnitalStarAlgebra.range_eq_top, separableClosure.eq_top_iff, ProbabilityTheory.iIndepSet.iIndep_comap_mem, Algebra.top_toSubring, isSemisimpleModule_iff, AlgebraicGeometry.ΓSpec.left_triangle, CategoryTheory.Subfunctor.eq_top_iff_isIso, AlgebraicGeometry.Scheme.Modules.toOpen_fromTildeΓ_app_assoc, CategoryTheory.sheafBotEquivalence_inverse_obj_val, AlgebraicGeometry.AffineSpace.comp_homOfVector, IntermediateField.top_toSubalgebra, Subfield.roots_X_pow_char_sub_X_bot, MeasureTheory.Measure.mutuallySingular_iff_disjoint, Finsupp.isCompl_range_lmapDomain_span, IntermediateField.relrank_top_left, Submodule.lTensorOne'_tmul, AlgebraicGeometry.Scheme.instIsOpenImmersionToSpecΓOfIsQuasiAffine, AlgebraicGeometry.Scheme.homOfLE_appTop, CompleteLatticeHomClass.toBoundedLatticeHomClass, CompleteSublattice.isCompl_iff, Submodule.rTensorOne'_tmul_one, IsGaloisGroup.fixedPoints_bot, isAtomic_of_complementedLattice, AlgebraicGeometry.Scheme.isoSpec_inv_naturality_assoc, AlgebraicGeometry.IsAffineOpen.fromSpec_app_self, nhds_top, Con.toSetoid_eq_bot, Subgroup.disjoint_def', MeasureTheory.condExp_bot, Submodule.rTensorOne'_tmul, separatedNhds_iff_disjoint, TopologicalSpace.Opens.mem_bot, AlgebraicGeometry.PresheafedSpace.restrictTopIso_hom, disjoint_nhdsSet_nhdsSet, exists_root_adjoin_eq_top_of_isCyclic, AlgebraicGeometry.AffineSpace.hom_ext_iff, iSupIndep.injOn, AlgebraicGeometry.ΓSpec.unop_locallyRingedSpaceAdjunction_counit_app', MeasurableSpace.generateFrom_singleton_univ, Quantale.mul_bot, Filter.nonneg_sub_iff, disjoint_nhds_atBot_iff, AlgebraicGeometry.Scheme.isoSpec_Spec_hom, AddMonoid.Coprod.codisjoint_mrange_inl_mrange_inr, sSup_atoms_eq_top, Real.disjoint_residual_ae, IntermediateField.finrank_eq_one_iff_eq_top, SSet.Subcomplex.iSup_ofSimplex_nonDegenerate_eq_top, TopologicalSpace.Opens.coe_finset_inf, AlgebraicGeometry.tilde.isIso_toOpen_top, RootPairing.isCompl_rootSpan_ker_rootForm, CompleteLinearOrder.himp_bot, complementedLattice_of_isAtomistic, Submodule.isCompl_orthogonal_of_hasOrthogonalProjection, iSupIndep_ne_bot, Subfield.extendScalars_top, AlgebraicGeometry.HasRingHomProperty.appTop, AffineSubspace.mk'_top, Algebra.zariskisMainProperty_iff_exists_saturation_eq_top, NumberField.is_primitive_element_of_infinitePlace_lt, clusterPt_iff_not_disjoint, StarSubalgebra.coe_bot, UniformSpace.toTopologicalSpace_bot, CompHausLike.LocallyConstant.locallyConstantIsoContinuousMap_hom, CategoryTheory.Presheaf.isLocallySurjective_iff_range_sheafify_eq_top, MonoidAlgebra.exists_finset_adjoin_eq_top, AlgebraicGeometry.Scheme.isNilpotent_iff_basicOpen_eq_bot, IntermediateField.finrank_adjoin_eq_one_iff, IntermediateField.bot_eq_top_of_rank_adjoin_eq_one, sSup_eq_bot', SeparationQuotient.inseparableSetoid_eq_top_iff, AlgebraicGeometry.Scheme.isPullback_toSpecΓ_toSpecΓ, AlgebraicGeometry.Scheme.ΓSpecIso_naturality, AlgebraicGeometry.Scheme.restrictFunctorΓ_hom_app, Partition.bot_notMem', AddSubgroup.codisjoint_addSubgroupOf_sup, IntermediateField.finrank_bot, Subgroup.disjoint_def, PowerBasis.adjoin_eq_top_of_gen_mem_adjoin, NonUnitalAlgebra.eq_top_iff, AlgebraicGeometry.SheafedSpace.Γ_map, IsIntegrallyClosed.integralClosure_eq_bot, Ideal.isSimpleOrder, IsIntegrallyClosed.integralClosure_eq_bot_iff, Polynomial.IsSplittingField.adjoin_rootSet', IntermediateField.sepDegree_bot, AlgebraicGeometry.germ_stalkClosedPointIso_hom, ProbabilityTheory.Kernel.iIndep_comap_mem_iff, AddQuantale.add_bot, AlgebraicGeometry.Scheme.toSpecΓ_isoSpec_inv, Submodule.rTensorOne_tmul, AlgebraicGeometry.IsFinite.instHasAffinePropertyAndIsAffineFiniteCarrierObjOppositeOpensCarrierCarrierCommRingCatPresheafOpOpensTopHomAppTop, IsSimpleAddGroup.instIsSimpleOrderAddSubgroup, AlgebraicGeometry.IsAffineOpen.instIsAffineToSchemeBasicOpen, IntermediateField.finInsepDegree_bot', OrdinalApprox.gfpApprox_ord_eq_gfp, Filter.HasBasis.disjoint_iff, CategoryTheory.Coverage.eq_top_pullback, IntermediateField.restrictScalars_bot_eq_self, DirectSum.IsInternal.isCompl, AlgebraicGeometry.Scheme.toIso_inv_ι_assoc, AlgebraicGeometry.SheafedSpace.restrictTopIso_inv, AlgebraicGeometry.Scheme.toSpecΓ_appTop, Order.Coframe.top_sdiff, IntermediateField.fg_bot, Filter.limsInf_top, AlgebraicGeometry.Scheme.IdealSheafData.support_eq_bot_iff, CategoryTheory.Presheaf.isLocallySurjective_iff_range_sheafify_eq_top', TopologicalSpace.Closeds.isAtom_iff, GroupTopology.toTopologicalSpace_bot, AlgHom.range_eq_top, Algebra.adjoin_singleton_one, CategoryTheory.Subfunctor.Subpresheaf.epi_iff_range_eq_top, NonUnitalAlgebra.mem_top, NonUnitalStarAlgebra.toNonUnitalSubalgebra_eq_top, SSet.Subcomplex.image_top, CompleteSublattice.coe_bot, AffineSubspace.affineSpan_eq_top_iff_vectorSpan_eq_top_of_nonempty, IntermediateField.adjoin_empty, AffineSubspace.topEquiv_symm_apply_coe, Submodule.coe_isComplEquivProj_apply, FrameHom.coe_toInfTopHom, Field.Emb.Cardinal.eq_bot_of_not_nonempty, AlgebraicGeometry.StructureSheaf.algebraMap_obj_top_bijective, TopologicalSpace.Opens.inclusion'_map_eq_top, IntermediateField.rank_bot, Subalgebra.centralizer_eq_top_iff_subset, CommMonoid.primaryComponent.disjoint, AlgebraicGeometry.IsAffineOpen.fromSpec_preimage_self, iSup_eq_dif, separableClosure_inf_perfectClosure, MeasureTheory.Measure.add_top, AlgebraicGeometry.LocallyRingedSpace.Γevaluation_naturality_apply, Algebra.top_toSubsemiring, t2Space_iff_disjoint_nhds, Subalgebra.rTensorBot_tmul_one, ConvexCone.disjoint_coe, CompletelyNormalSpace.completely_normal, AlgebraicGeometry.Spec_zeroLocus_eq_zeroLocus, MeasureTheory.Measure.mutuallySingular_iff_disjoint_ae, AlgebraicGeometry.Scheme.Opens.ι_appTop, Filter.disjoint_principal_principal, AddMonoid.Coprod.codisjoint_range_inl_range_inr, Subfield.bot_eq_of_charZero, RegularSpace.regular, IsIntegrallyClosedIn.integralClosure_eq_bot_iff, AffineMap.map_top_of_surjective, IsPGroup.disjoint_of_ne, AlgebraicGeometry.Scheme.map_basicOpen, CategoryTheory.Subfunctor.Subpresheaf.top_obj, sSupIndep.disjoint_sSup, Set.pairwiseDisjoint_nhds, NonUnitalAlgebra.adjoin_univ, AlgebraicGeometry.isRetrocompact_basicOpen, AlgebraicGeometry.Scheme.isoSpec_inv_naturality, Filter.disjoint_comap_iff_map, IntermediateField.bot_eq_top_iff_finrank_eq_one, FrameHom.map_sSup', IntermediateField.adjoin_eq_bot_iff, MeasureTheory.SimpleFunc.simpleFunc_bot', Submodule.rTensorOne_tmul_one, AlgebraicGeometry.Scheme.Hom.quasiFiniteLocus_eq_top_iff, AlgebraicGeometry.instQuasiSeparatedToSpecΓOfQuasiSeparatedSpaceCarrierCarrierCommRingCat, CategoryTheory.Sieve.pullback_eq_top_of_mem, CategoryTheory.GrothendieckTopology.trivial_eq_bot, AlgebraicGeometry.Scheme.comp_appTop_assoc, FirstOrder.Language.Substructure.closure_empty, AlgebraicGeometry.Proj.basicOpen_one, isCoatomistic_iff, Filter.limsup_bot, Filter.disjoint_pure_pure, IntermediateField.lift_sepDegree_bot', CompletelyDistribLattice.top_sdiff, LieIdeal.isCompl_killingCompl, Subalgebra.op_top, AlgebraicGeometry.Scheme.isoSpec_Spec, CategoryTheory.Subfunctor.Subpresheaf.range_id, AffineSubspace.coe_eq_bot_iff, iSup₂_eq_bot, essSup_measure_zero, Algebra.adjoin_singleton_natCast, IntermediateField.LinearDisjoint.inf_eq_bot, Field.exists_primitive_element_iff_finite_intermediateField, AlgebraicGeometry.IsClosedImmersion.hasAffineProperty, Field.Emb.Cardinal.filtration_apply, FirstOrder.Language.Substructure.map_bot, AlgebraicGeometry.Scheme.preimage_opensRange_toSpecΓ, IsSimpleRing.simple, indiscreteTopology_iff, AlgebraicGeometry.Scheme.Spec.algebraMap_residueFieldIso_inv, Subalgebra.bot_eq_top_iff_finrank_eq_one, Subalgebra.mulMap_bot_right_eq, AlgebraicGeometry.Scheme.eq_zeroLocus_of_isClosed_of_isAffine, AddMonoidAlgebra.exists_finset_adjoin_eq_top, AlgebraicGeometry.Scheme.Opens.ι_preimage_self, CategoryTheory.Subfunctor.instIsIsoFunctorTypeιTop, AddSubgroup.FG.finset_sup, CategoryTheory.Subfunctor.bot_obj, AlgebraicGeometry.Scheme.toSpecΓ_image_zeroLocus, AlgebraicGeometry.PresheafedSpace.ofRestrict_top_c, R1Space.specializes_or_disjoint_nhds, iSupIndep.sup_indep_univ, Finset.sup_eq_iSup, isSimpleModule_iff, LinearMap.eventually_isCompl_ker_pow_range_pow, AlgebraicGeometry.toSpecΓ_SpecMap_ΓSpecIso_inv_assoc, AlgebraicGeometry.SheafedSpace.Γ_obj_op, Algebra.isAlgebraic_iff, TopologicalSpace.Closeds.coe_top, Subalgebra.lTensorBot_symm_apply, AlgebraicGeometry.IsAffineOpen.fromSpec_app_self_apply, NonUnitalStarAlgebra.comap_top, LieModule.isCompl_genWeightSpace_zero_posFittingComp, FreeAlgebra.adjoin_range_ι, ContinuousMap.subalgebra_topologicalClosure_eq_top_of_separatesPoints, disjoint_nhds_nhds_iff_not_inseparable, Con.toSetoid_bot, MeasureTheory.sigmaFinite_bot_iff, SSet.Subcomplex.topIso_inv_ι_assoc, AlgebraicGeometry.Scheme.inv_appTop, IntermediateField.finrank_top', AlgebraicGeometry.isLocalization_away_of_isAffine, AlgebraicGeometry.IsAffineOpen.ι_basicOpen_preimage, iSup_neg, disjoint_nhds_nhdsSet, AffineSubspace.direction_top, Affine.Simplex.span_eq_top, IntermediateField.lift_bot, NumberField.mixedEmbedding.exists_primitive_element_lt_of_isReal, LieSubmodule.instIsAtomicOfIsArtinian, IsGalois.fixedField_top, WithBot.sInf_empty, TopologicalSpace.Opens.eq_bot_or_top, AlgebraicGeometry.morphismRestrict_appTop, IsLocalRing.closed_point_mem_iff, AlgebraicGeometry.Scheme.IdealSheafData.ofIdealTop_ideal, Algebra.IsAlgebraic.isNormalClosure_iff, Filter.not_one_le_div_iff, iInf_top, MeasureTheory.hittingAfter_eq_sInf, MeasureTheory.condLExp_bot_ae_eq, AlgebraicGeometry.Scheme.Hom.preimage_opensRange, StarSubalgebra.toSubalgebra_eq_top, AlgebraicGeometry.ΓSpecIso_hom_stalkClosedPointIso_inv, StarSubalgebra.mem_top, AlgebraicGeometry.AffineSpace.toSpecMvPolyIntEquiv_apply, IsPGroup.le_or_disjoint_of_coprime, CategoryTheory.GrothendieckTopology.bot_covering, Submodule.mulMap_one_left_eq, AffineSubspace.isEmpty_bot, AlgebraicGeometry.LocallyRingedSpace.Γ_map, IntermediateField.adjoin_simple_eq_bot_iff, IntermediateField.relfinrank_bot_left, AlgebraicGeometry.isCompactOpen_iff_eq_basicOpen_union, exists_subset_affineIndependent_affineSpan_eq_top, t1Space_iff_disjoint_pure_nhds, AlgebraicGeometry.Spec.algebraMap_stalkIso_inv, RingCon.matrix_bot, Algebra.adjoin_adjoin_coe_preimage, TopologicalSpace.Closeds.coe_finset_sup, AlgebraicGeometry.eq_top_of_sigmaSpec_subset_of_isCompact, Algebra.adjoin_top, TopologicalSpace.Opens.mk_empty, FrameHomClass.toInfTopHomClass, iSup_ne_bot_subtype, AlgebraicGeometry.PresheafedSpace.Γ_obj, AffineSubspace.perpBisector_eq_top, IntermediateField.adjoin_intCast, mem_codiscreteWithin, Algebra.adjoin_root_eq_top_of_isSplittingField, inf_sSup_eq_iSup_inf_sup_finset, AlgebraicGeometry.PresheafedSpace.toRestrictTop_c, Algebra.surjective_algebraMap_iff, IntermediateField.sepDegree_bot', CategoryTheory.Presheaf.isSheaf_bot, AlgebraicGeometry.Scheme.isoSpec_inv_toSpecΓ_assoc, IntermediateField.LinearDisjoint.bot_right, Submodule.isCompl_iff_disjoint, Filter.limsSup_bot, AffineSubspace.pointwise_vadd_bot, Field.primitive_element_iff_minpoly_degree_eq, Algebra.toSubmodule_eq_top, AlgHom.fieldRange_eq_top, Subgroup.map_eq_range_iff, iSup_emptyset, discreteTopology_bot, DualNumber.range_inlAlgHom_sup_adjoin_eps, AlgebraicGeometry.Scheme.toOpen_eq, isPurelyInseparable_iff_perfectClosure_eq_top, iSupIndep_def, RingCon.coe_bot, disjoint_pure_nhds, IntermediateField.rank_eq_one_iff, Module.End.invtSubmodule.isCompl_mk_iff, Subalgebra.fg_bot, Algebra.adjoin_univ, AlgHom.equalizer_same, IntermediateField.eq_bot_of_isPurelyInseparable_of_isSeparable, CategoryTheory.Subfunctor.image_top, AlgebraicGeometry.isCompl_opensRange_inl_inr, TopologicalSpace.Closeds.coe_eq_univ, Filter.disjoint_atTop_principal_Iic, AffineIndependent.affineSpan_eq_top_iff_card_eq_finrank_add_one, IntermediateField.adjoin_root_eq_top_of_isSplittingField, AlgebraicGeometry.ΓSpecIso_inv_ΓSpec_adjunction_homEquiv, algebraicClosure.eq_top_iff, AffineSubspace.not_sOppSide_bot, IsOpen.exists_between_affineIndependent_span_eq_top, Setoid.eq_top_iff, AffineSubspace.top_coe, disjoint_nhds_atTop, AlgebraicGeometry.toSpecΓ_SpecMap_ΓSpecIso_inv, Field.primitive_element_iff_minpoly_natDegree_eq, sInfHom.coe_top, AlgebraicGeometry.instIsIsoModulesSpecOfCarrierFromTildeΓUnitOpensCarrierCarrierCommRingCatRingCatSheaf, Subalgebra.fg_top, NonUnitalAlgebra.toSubmodule_eq_top, isDenseInducing_pure, AlgebraicGeometry.Scheme.Hom.inv_appTop, SSet.iSup_skeletonOfMono, affineSpan_coe_preimage_eq_top, AffineSubspace.pointwise_vadd_top, IntermediateField.isScalarTower_over_bot, ProbabilityTheory.condVar_bot', TwoSidedIdeal.top_ringCon, AffineSubspace.perpBisector_self, AlgebraicGeometry.Spec_zeroLocus, IntermediateField.bot_toSubfield, AlgebraicGeometry.Scheme.ΓSpecIso_inv_naturality, Metric.disjoint_cobounded_nhds, sSupIndep_pair, IntermediateField.topEquiv_symm_apply_coe, AffineSubspace.span_empty, NonUnitalStarAlgebra.map_bot, AffineSubspace.map_bot, affineSpan_eq_top_of_nonempty_interior, Subalgebra.lTensorBot_tmul, NonUnitalAlgebra.adjoin_empty, Filter.HasBasis.disjoint_iff_left, SSet.iSup_range_eq_top_of_isColimit, Subgroup.isComplement'_iff_card_mul_and_disjoint, IntermediateField.rank_top', Subalgebra.topEquiv_symm_apply_coe, AlgebraicGeometry.Scheme.SpecΓIdentity_inv_app, MeasureTheory.OuterMeasure.boundedBy_top, AlgebraicGeometry.Scheme.Pullback.diagonalCoverDiagonalRange_eq_top_of_injective, LinearMap.IsIdempotentElem.isCompl, iInf_eq_dif, CompleteSublattice.coe_top, CategoryTheory.Sieve.functorPushforward_bot, IntermediateField.topEquiv_apply, disjoint_principal_nhdsSet, ProbabilityTheory.condIndep_bot_right, regularSpace_generateFrom, Order.Frame.himp_bot, AffineSubspace.instIsSimpleOrderOfSubsingleton, mem_bot_iff_intCast, Filter.one_le_div_iff, AlgebraicGeometry.basicOpen_eq_of_affine', IntermediateField.adjoin_natCast, AlgebraicGeometry.AffineSpace.isoOfIsAffine_hom_appTop, TopologicalSpace.Opens.coe_bot, AffineSubspace.topEquiv_apply, AlgebraicGeometry.instIsDominantToSpecΓOfCompactSpaceCarrierCarrierCommRingCat, Submodule.exists_isCompl, AlgebraicGeometry.Scheme.preimage_eq_top_of_closedPoint_mem, iInf_extend_top, TopologicalSpace.Closeds.isClopen_singleton_bot, IsCyclotomicExtension.singleton_one, MeasureTheory.hittingBtwn_eq_sInf, SSet.finite_subcomplex_top_iff, AlgebraicGeometry.Scheme.isoSpec_Spec_inv, Algebra.adjoin_restrictScalars, alternatingGroup.isCoatom_stabilizer, NonUnitalAlgebra.coe_top, QuotientGroup.rightRel_eq_top, NonUnitalAlgebra.map_top, IntermediateField.isPurelyInseparable_bot, AddQuantale.bot_add, AlgebraicGeometry.Scheme.germ_stalkClosedPointTo_assoc, AlgebraicGeometry.Scheme.topIso_hom, Filter.disjoint_atTop_atBot, MeasurableSpace.generateFrom_singleton, AlgebraicGeometry.finite_appTop_of_universallyClosed, AffineSubspace.span_univ, IntermediateField.top_toSubfield, TopologicalSpace.IsOpenCover.iSup_eq_top, SeminormedGroup.disjoint_nhds_one, AffineSubspace.map_eq_bot_iff, NonUnitalAlgebra.to_subring_eq_top, Filter.disjoint_prod, regularSpace_iff, Filter.disjoint_atBot_principal_Ioi, RingOfIntegers.exponent_eq_one_iff, AlgebraicGeometry.specTargetImageFactorization_app_injective, Subalgebra.rTensorBot_symm_apply, Algebra.comap_top, AlgebraicGeometry.PresheafedSpace.Γ_map, NonUnitalAlgebra.toNonUnitalSubring_eq_top, ProbabilityTheory.indep_bot_right, AlgebraicGeometry.StructureSheaf.toOpenₗ_top_bijective, ProbabilityTheory.condVar_bot_ae_eq, AffineSubspace.direction_bot, TopologicalSpace.Closeds.isAtom_coe, CategoryTheory.Sieve.functorInclusion_top_isIso, instIsAtomicFilter, AddSubgroup.disjoint_iff_add_eq_zero, MaximalSpectrum.iInf_localization_eq_bot, Submodule.complementedLattice, CompleteBooleanAlgebra.top_le_sup_compl, AffineBasis.isUnit_toMatrix_iff, Algebra.IsCentral.center_eq_bot, IsGaloisGroup.fixingSubgroup_bot, Set.exists_seq_iSup_eq_top_iff_countable, AlgebraicGeometry.AffineSpace.isoOfIsAffine_hom, sInfHomClass.toInfTopHomClass, AffineSubspace.instNonemptySubtypeMemTop, Finset.sup_eq_sSup_image, isAtomistic_of_complementedLattice, ProbabilityTheory.indep_bot_left, Partition.disjoint, AlgebraicGeometry.affineAnd_apply, instIsSimpleOrderIdeal, IntermediateField.fg_top, instDiscreteMeasurableSpace, Submodule.lTensorOne_symm_apply, MeasureTheory.OuterMeasure.top_apply', AlgebraicGeometry.AffineSpace.reindex_appTop_coord, AlgebraicGeometry.Spec.algebraMap_stalkIso_inv_assoc, IntermediateField.finInsepDegree_bot, MvPolynomial.supported_empty, IsCyclotomicExtension.iff_adjoin_eq_top, AlgEquiv.fieldRange_eq_top, LinearOrder.bot_topologicalSpace_eq_generateFrom, Equiv.Perm.isCoatom_stabilizer, specializes_iff_not_disjoint, MeasureTheory.SimpleFunc.simpleFunc_bot, Field.Emb.Cardinal.iSup_adjoin_eq_top, ConvexCone.disjoint_hull_right_of_convex, AlgebraicGeometry.ΓSpec.toSpecΓ_of, affineSpan_eq_bot, ProbabilityTheory.condVar_bot, fixedPoints.lfp_eq_sSup_iterate, Filter.limsup_const_bot, sInf_diff_singleton_top, JacobsonNoether.exists_separable_and_not_isCentral', MeasureTheory.Measure.MutuallySingular.disjoint_ae, AlgebraicGeometry.exists_appTop_π_eq_of_isLimit, disjoint_nhds_pure, AlgebraicGeometry.IsAffine.affine, StarSubalgebra.coe_top, AlgebraicGeometry.AffineTargetMorphismProperty.IsLocal.to_basicOpen, AlgebraicGeometry.SheafedSpace.Γ_obj, Submodule.lTensorOne'_one_tmul, IntermediateField.finrank_bot', AlgebraicGeometry.Scheme.mem_basicOpen_top, ConvexCone.isGenerating_top, LieSubmodule.codisjoint_toSubmodule, Quantale.bot_mul, is_simple_module_of_finrank_eq_one, CategoryTheory.Pretopology.toGrothendieck_bot, Partition.bot_lt_of_mem, AlgebraicGeometry.Scheme.Opens.ι_image_basicOpen', AlgebraicGeometry.LocallyRingedSpace.Γevaluation_naturality_assoc, Algebra.eq_top_iff, Filter.liminf_bot, Setoid.mk_eq_bot, AffineSubspace.comap_top, fourierSubalgebra_closure_eq_top, Filter.limsInf_bot, AlgebraicGeometry.AffineSpace.homOverEquiv_apply, NonUnitalStarAlgebra.range_id, Subalgebra.rank_eq_one_iff, DiscreteTopology.eq_bot, Subalgebra.prod_top, CategoryTheory.Sieve.overEquiv_symm_top, NonUnitalAlgebra.comap_top, MeasureTheory.Measure.toOuterMeasure_top, NonUnitalStarAlgebra.map_top, AlgebraicGeometry.Scheme.comp_appTop, Finset.inf_eq_sInf_image, IntermediateField.botEquiv_def, AlgebraicGeometry.IsClosedImmersion.isAffine_surjective_of_isAffine, CategoryTheory.sieve₁'_toPreOneHypercover_eq_top, subalgebra_top_finrank_eq_submodule_top_finrank, IntermediateField.adjoin_eq_top_iff, Finset.inf_eq_iInf, Valuation.Integers.integralClosure, IntermediateField.finSepDegree_bot', IsCyclotomicExtension.Rat.adjoin_singleton_eq_top, AlgebraicGeometry.Scheme.fromSpecStalk_app, AlgebraicGeometry.AffineSpace.toSpecMvPolyIntEquiv_comp, Algebra.IsUnramifiedAt.exists_notMem_forall_ne_mem_and_adjoin_eq_top, AlgebraicGeometry.isField_of_universallyClosed, polynomialFunctions.topologicalClosure, sInf_empty, RingCon.range_mkₐ, AlgebraicGeometry.LocallyRingedSpace.Γevaluation_eq_zero_iff_notMem_basicOpen, disjoint_lift'_closure_nhds, stronglyMeasurable_bot_iff, AlgebraicGeometry.Scheme.Spec.algebraMap_residueFieldIso_inv_assoc, IntermediateField.eq_bot_of_isSepClosed_of_isSeparable, WellFoundedLT.finite_ne_bot_of_iSupIndep, MeasurableSpace.map_top, iSup_eq_if, FirstOrder.Language.Substructure.elementarySkolem₁Reduct.instSmall, Algebra.map_top, CategoryTheory.Subfunctor.top_obj, TopologicalSpace.Opens.coe_disjoint, Subalgebra.eq_bot_of_rank_le_one, AlgebraicGeometry.Scheme.Hom.smoothLocus_eq_top, CategoryTheory.Sieve.functorPushforward_top, IsSimpleModule.toIsSimpleOrder, NonUnitalStarSubalgebra.center_eq_top, AlgebraicGeometry.IsAffineOpen.fromSpec_top, IntermediateField.bot_eq_top_of_finrank_adjoin_le_one, AddSubmonoid.disjoint_def, CategoryTheory.Presieve.isSeparatedFor_top, Submodule.lTensorOne_tmul, AddCon.toSetoid_eq_bot, IntermediateField.relrank_bot_right, AlgebraicGeometry.Scheme.IdealSheafData.support_top, Subalgebra.comm_trans_lTensorBot, InfiniteGalois.fixedField_bot, Subgroup.codisjoint_subgroupOf_sup, CompletelyDistribLattice.himp_bot, NonUnitalAlgebra.map_bot, AlgebraicGeometry.Scheme.map_basicOpen_map, ωScottContinuous.top, StarAlgHom.range_eq_map_top, IsSepClosed.separableClosure_eq_bot_iff, Subfield.closure_empty, AlgebraicGeometry.Scheme.Hom.comp_appTop, Directed.disjoint_iSup_left, NonUnitalStarAlgebra.eq_top_iff, Subalgebra.bot_eq_top_of_finrank_eq_one, AlgebraicGeometry.IsIntegralHom.hasAffineProperty, RingCon.toCon_eq_top, AlgebraicGeometry.Scheme.germ_stalkClosedPointTo_Spec, MeasureTheory.OuterMeasure.map_top_of_surjective, SemidirectProduct.mulEquivSubgroup_apply, Filter.disjoint_cocompact_right, AddSubgroup.disjoint_def', complementedLattice_iff_isAtomistic, AlgebraicGeometry.instHasAffinePropertyIsomorphismsSchemeAndIsAffineIsIsoCommRingCatAppTop, eq_bot_of_singletons_open, MvPolynomial.supported_univ, conductor_eq_top_iff_adjoin_eq_top, fixedPoints.gfp_eq_sInf_iterate, Subalgebra.isAlgebraic_bot_iff, Filter.disjoint_pure_atBot, TwoSidedIdeal.bot_ringCon, r1Space_iff_specializes_or_disjoint_nhds, AdjoinRoot.adjoinRoot_eq_top, AlgebraicGeometry.Scheme.IsLocallyDirected.V_self, borel_eq_top_of_countable, Subalgebra.isSimpleOrder_of_finrank_prime, CategoryTheory.GrothendieckTopology.top_mem', TopCat.Presheaf.Γgerm_res_apply, IntermediateField.extendScalars_top, perfectClosure.eq_bot_of_isSeparable, ConvexCone.coe_top, MeasurableSpace.measurableSet_top, CategoryTheory.Presheaf.equalizerSieve_eq_top_iff, ProbabilityTheory.Kernel.indep_bot_left, IntermediateField.fixedField_bot, NonUnitalAlgebra.range_eq_top, SemidirectProduct.mulEquivSubgroup_symm_apply, AlgebraicGeometry.LocallyRingedSpace.Γevaluation_ne_zero_iff_mem_basicOpen, Equiv.Perm.isCoatom_stabilizer_of_ncard_lt_ncard_compl, AlgebraicGeometry.basicOpen_eq_of_affine, MeasureTheory.sigmaFinite_trim_bot_iff, AlgebraicGeometry.LocallyRingedSpace.Γ_map_op, OrdinalApprox.lfp_mem_range_lfpApprox, LinearMap.isCompl_span_singleton_orthogonal, FrameHom.toFun_eq_coe, Filter.compl_diagonal_mem_prod, AlgebraicGeometry.Scheme.ΓSpecIso_naturality_assoc, AlgebraicGeometry.IsAffineOpen.fromSpec_toSpecΓ_assoc, Complex.adjoin_I, SSet.Subcomplex.range_eq_top_iff, Con.toSetoid_top, Partition.coe_removeBot, AlgebraicGeometry.PresheafedSpace.toRestrictTop_base, AlgebraicGeometry.LocallyRingedSpace.isUnit_res_toΓSpecMapBasicOpen, Finset.exists_inf_eq_iInf, AlgebraicGeometry.ΓSpec_adjunction_homEquiv_eq, ContinuousMap.elemental_id_eq_top, isCompactElement_iff_exists_le_sSup_of_le_sSup, CompleteSublattice.disjoint_iff, Opens.toPretopology_grothendieckTopology, MeasureTheory.OuterMeasure.map_top, Algebra.FiniteType.out, polynomialFunctions_closure_eq_top', Finset.inf_univ_eq_iInf, AffineSubspace.affineSpan_eq_top_iff_vectorSpan_eq_top_of_nontrivial, Projectivization.Subspace.span_univ, AlgebraicGeometry.Scheme.toIso_inv_ι, FirstOrder.Language.Substructure.instIsEmptySubtypeMemBotOfConstants, IntermediateField.insepDegree_bot, Submodule.isCompl_span_singleton_of_isCoatom_of_notMem, IsAlgClosed.algebraicClosure_eq_bot_iff, LinearMap.isCompl_of_proj, IntermediateField.rank_bot', OpenSubgroup.toOpens_top, iSupIndep_iff_supIndep, iInf_eq_if, isAtomistic_iff, TopologicalSpace.Opens.map_top, CategoryTheory.Functor.imageSieve_map, AlgebraicGeometry.LocallyRingedSpace.toΓSpec_preimage_zeroLocus_eq, AlgebraicGeometry.LocallyRingedSpace.SpecΓIdentity_hom_app, CategoryTheory.Sieve.pullback_top, IntermediateField.insepDegree_top, ProjectiveSpectrum.basicOpen_one, Filter.blimsup_false, AlgebraicGeometry.Scheme.Γevaluation_naturality, MeasureTheory.condExp_bot', IsGalois.tfae, Subfield.card_bot, iSupIndep_iff_supIndep_univ, RingCon.coe_top, OpenAddSubgroup.toOpens_top, AlgebraicGeometry.Scheme.IdealSheafData.ker_glueDataObjι_appTop, CategoryTheory.Sieve.uliftFunctorInclusion_top_isIso, AlgebraicGeometry.Spec.germ_stalkMapIso_hom, IntermediateField.isSimpleOrder_of_finrank_prime, Finset.sup_univ_eq_iSup, IntermediateField.LinearDisjoint.eq_bot_of_self, NonUnitalStarSubalgebra.prod_top, ProbabilityTheory.Kernel.indep_bot_right, separableClosure.separableClosure_eq_bot, AlgebraicGeometry.Spec.fromSpecStalk_eq, NonUnitalAlgebra.top_toSubring, ContinuousMapZero.elemental_eq_top, AlgebraicGeometry.AffineSpace.isoOfIsAffine_inv_over, IntermediateField.fg_top_iff, CategoryTheory.GrothendieckTopology.top_covers, CategoryTheory.Presieve.factorsThru_top, Submonoid.disjoint_def', NonUnitalAlgebra.mem_bot, AlgebraicGeometry.LocallyRingedSpace.Γevaluation_naturality, AlgebraicGeometry.Proj.iSup_basicOpen_eq_top, AlgebraicGeometry.IsAffineOpen.fromSpec_preimage_basicOpen', pairwise_disjoint_nhds, induced_top, AlgebraicGeometry.Scheme.Hom.opensRange_of_isIso, AlgebraicGeometry.LocallyRingedSpace.toΓSpecMapBasicOpen_eq, Submodule.lTensorOne_one_tmul, IntermediateField.relfinrank_top_left, iSup_of_empty, AlgebraicGeometry.Scheme.Opens.instIsIsoCommRingCatAppLEιTopToScheme, disjoint_nhds_atTop_iff, sInf_eq_top, iInf_neg, AlgebraicGeometry.IsAffineOpen.ΓSpecIso_hom_fromSpec_app, CategoryTheory.sheafBotEquivalence_counitIso, MeasureTheory.Measure.MutuallySingular.disjoint, GroupTopology.toTopologicalSpace_top, iInf_lt_top, Con.toSetoid_eq_top, IsCompact.disjoint_nhdsSet_right, MeasureTheory.OuterMeasure.dirac_caratheodory, AddSubgroup.addSubgroupOf_eq_bot, IntermediateField.finSepDegree_top, SSet.N.iSup_subcomplex_eq_top, LieAlgebra.InvariantForm.atomistic, IsOpen.affineSpan_eq_top, CategoryTheory.Sieve.id_mem_iff_eq_top, AffineSubspace.not_wOppSide_bot, AlgebraicGeometry.instQuasiCompactToSpecΓOfCompactSpaceCarrierCarrierCommRingCat, Filter.HasBasis.disjoint_cobounded_iff, IsAdjoinRoot.primitive_element_root, Monoid.Coprod.codisjoint_range_inl_range_inr, AlgebraicGeometry.AffineSpace.comp_homOfVector_assoc, Filter.disjoint_atBot_principal_Ici, AlgebraicGeometry.Scheme.Opens.ι_image_top, Metric.disjoint_nhdsSet_cobounded, Subfield.extendScalars_self, Field.Emb.Cardinal.adjoin_basis_eq_top, NonUnitalAlgebra.top_toSubmodule, iSup_eq_bot, Algebra.TensorProduct.adjoin_tmul_eq_top, AlgebraicGeometry.Scheme.Γ_obj_op, ContinuousMap.starSubalgebra_topologicalClosure_eq_top_of_separatesPoints, disjoint_nhdsSet_nhds, IntermediateField.relrank_bot_left, SSet.Subcomplex.topIso_hom, CompleteBooleanAlgebra.inf_compl_le_bot, Subalgebra.LinearDisjoint.eq_bot_of_commute_of_self, CategoryTheory.Sieve.pullback_ofObjects_eq_top, AlgebraicGeometry.HasRingHomProperty.iff_of_source_openCover, AddSubmonoid.FG.finset_sup, LieAlgebra.IsSemisimple.sSupIndep_isAtom, Subalgebra.finrank_bot, AlgebraicGeometry.Scheme.Opens.topIso_hom, MeasureTheory.OuterMeasure.top_apply, Submonoid.disjoint_def, Subgroup.disjoint_iff_mul_eq_one, CategoryTheory.presheafIsGeneratedBy_of_isFinite, ZMod.fieldRange_castHom_eq_bot, Subalgebra.restrictScalars_top, QuotientGroup.leftRel_eq_top, PrimeSpectrum.basicOpen_one, Submodule.closedComplemented_iff_isClosed_exists_isClosed_isCompl, AlgebraicGeometry.Scheme.Hom.comp_appTop_assoc, AlgebraicGeometry.IsAffineOpen.fromSpec_app_of_le, AffineSubspace.eq_bot_or_nonempty, RootPairing.isCompl_corootSpan_ker_corootForm, MeasureTheory.Measure.top_add, AlgebraicGeometry.AffineSpace.map_appTop_coord, Filter.disjoint_pure_atTop, AlgebraicGeometry.Scheme.toSpecΓ_preimage_basicOpen, PrimeSpectrum.iSup_basicOpen_eq_top_iff', Filter.bliminf_false, MeasureTheory.hitting_eq_sInf, AlgebraicGeometry.isIso_fromTildeΓ_iff, IntermediateField.mem_bot, AlgebraicGeometry.PresheafedSpace.restrictTopIso_inv, Algebra.IsCentral.out, polynomialFunctions_closure_eq_top, AlgebraicGeometry.Scheme.openCoverBasicOpenTop_f, IsDedekindDomain.integer_univ, SeminormedAddGroup.disjoint_nhds, LinearIndependent.isCompl_span_image, AlgebraicGeometry.LocallyRingedSpace.toΓSpecSheafedSpace_hom_c_app, IsCompact.disjoint_nhdsSet_nhds, IsAdjoinRoot.adjoin_root_eq_top, NonUnitalAlgebra.toNonUnitalSubring_top, iInf_emptyset, AlgebraicGeometry.isField_of_isIntegral_of_subsingleton, AlgebraicGeometry.ProjectiveSpectrum.Proj.toOpen_toSpec_val_c_app, Finset.sup_id_eq_sSup, CategoryTheory.PreOneHypercover.sieve₀_trivial, CategoryTheory.Subfunctor.preimage_eq_top_iff, IntermediateField.fixingSubgroup_bot, CategoryTheory.Subfunctor.Subpresheaf.eq_top_iff_isIso, CliffordAlgebra.evenOdd_isCompl, Finset.exists_sup_eq_iSup, CategoryTheory.Presieve.isSheafFor_top, AlgebraicGeometry.IsAffineOpen.fromSpec_toSpecΓ, Subalgebra.bot_eq_top_iff_rank_eq_one, IntermediateField.coe_top, AlgebraicGeometry.AffineSpace.homOfVector_appTop_coord, isGalois_bot, AlgebraicGeometry.Scheme.IsQuasiAffine.isBasis_basicOpen, Subalgebra.rTensorBot_tmul, AlgebraicGeometry.Scheme.isoSpec_image_zeroLocus, DirectSum.isInternal_submodule_iff_isCompl, iSup_extend_bot, Algebra.range_id, sSupHomClass.toSupBotHomClass, Polynomial.SplittingField.adjoin_rootSet, AlgebraicGeometry.isCompact_and_isOpen_iff_finite_and_eq_biUnion_basicOpen, isSplittingField_iff_intermediateField, iSup_bot, AlgebraicGeometry.PresheafedSpace.restrict_top_presheaf, isGalois_iff_isGalois_bot, DirectedOn.disjoint_sSup_right, TrivSqZeroExt.range_inlAlgHom_sup_adjoin_range_inr, Polynomial.adjoin_X, eq_sSup_atoms, LieAlgebra.IsSimple.isAtom_iff_eq_top, AffineSubspace.bot_parallel_iff_eq_bot, FirstOrder.Language.Substructure.fg_bot, Specializes.not_disjoint, TopologicalSpace.Opens.coe_top, LinearMap.IsPerfectCompl.isCompl_left, CategoryTheory.Subfunctor.epi_iff_range_eq_top, IntermediateField.finrank_adjoin_simple_eq_one_iff, ωScottContinuous.bot, CategoryTheory.PreOneHypercover.sieve₁_trivial, StarSubalgebra.top_toSubalgebra, Monoid.Coprod.codisjoint_mrange_inl_mrange_inr, CategoryTheory.Subfunctor.Subpresheaf.range_eq_top, AlgebraicGeometry.LocallyRingedSpace.toΓSpec_preimage_basicOpen_eq, instIndiscreteTopology, Subalgebra.LinearDisjoint.eq_bot_of_self, NonUnitalAlgebra.toNonUnitalSubsemiring_eq_top, TopologicalSpace.Opens.coe_eq_univ, Module.End.isSemisimple_iff, Field.primitive_element_iff_algHom_eq_of_eval, AlgebraicGeometry.IsAffineOpen.isoSpec_hom_appTop, separableClosure.eq_bot_iff, Subalgebra.LinearDisjoint.bot_left, Subalgebra.rank_bot, AlgebraicGeometry.instIsIsoModulesSpecOfCarrierFromTildeΓFreeOpensCarrierCarrierCommRingCat, CategoryTheory.Subfunctor.range_id, LinearMap.BilinForm.isCompl_orthogonal_of_restrict_nondegenerate, Filter.isCompl_principal, Subalgebra.LinearDisjoint.bot_right, IntermediateField.rank_adjoin_simple_eq_one_iff, separableClosure.adjoin_eq_of_isAlgebraic_of_isSeparable, ContinuousMap.idealOfSet_isMaximal_iff, AddCon.toSetoid_eq_top, disjoint_nhds_atBot, MonoidHom.noncommCoprod_injective, notMem_iff_exists_ne_and_isConjRoot, AlgebraicGeometry.instIsAffineHomιBasicOpen, LieIdeal.comap_incl_eq_bot, Algebra.adjoin_empty, Subalgebra.fg_bot_toSubmodule, Algebra.botEquiv_symm_apply, MeasurableSpace.generateFrom_singleton_empty, NonUnitalStarAlgebra.toNonUnitalSubalgebra_bot, AffineSubspace.notMem_bot, AddAction.isCoatom_stabilizer_iff_preprimitive, AddAction.IsPreprimitive.isCoatom_stabilizer_of_isPreprimitive, MeasureTheory.condExp_bot_ae_eq, CategoryTheory.Sieve.generate_of_contains_isSplitEpi, Subalgebra.rank_top, AlgebraicGeometry.HasRingHomProperty.iff_of_isAffine, Subalgebra.bot_eq_top_of_rank_eq_one, RingCon.toCon_bot, sInfHom.top_apply, CategoryTheory.Subfunctor.Subpresheaf.range_toRange, AlgebraicGeometry.SheafedSpace.restrictTopIso_hom, IsGalois.mem_bot_iff_fixed, Subalgebra.finrank_eq_one_iff, AlgebraicGeometry.ΓSpec.toSpecΓ_unop, MeasureTheory.OuterMeasure.toMeasure_top, IntermediateField.map_bot, subalgebra_top_rank_eq_submodule_top_rank, AlgebraicGeometry.Scheme.IdealSheafData.vanishingIdeal_bot, AlgebraicGeometry.SheafedSpace.Γ_map_op, RingCon.toCon_top, Subgroup.IsComplement'.disjoint, Ultrafilter.isAtom, disjoint_nhds_nhds, isCyclic_tfae, separableClosure.eq_bot_of_isPurelyInseparable, Subalgebra.eq_bot_of_isPurelyInseparable_of_isSeparable, Subalgebra.finite_bot, IntermediateField.finSepDegree_bot, affineSpan_eq_top_iff_nonempty_of_subsingleton, AlgebraicGeometry.SpecMap_ΓSpecIso_inv_toSpecΓ_assoc, Filter.disjoint_principal_right, CategoryTheory.GrothendieckTopology.top_covering, alternatingGroup.isCoatom_stabilizer_of_ncard_lt_ncard_compl, Filter.limsSup_top, AlgebraicGeometry.LocallyRingedSpace.Γ_obj, AddGroupTopology.toTopologicalSpace_bot, AlgebraicGeometry.Scheme.Spec_fromSpecStalk, LinearMap.BilinForm.isCompl_span_singleton_orthogonal, IntermediateField.sepDegree_top, IsGaloisGroup.fixedPoints_eq_bot, AlgebraicGeometry.exists_etale_isCompl_of_quasiFiniteAt, LieModule.disjoint_genWeightSpace, LieModule.isCompl_genWeightSpaceOf_zero_posFittingCompOf, Subgroup.isCoatom_comap, SSet.skeleton_zero, Ultrafilter.disjoint_iff_not_le, AlgebraicGeometry.Scheme.toSpecΓ_base
toCompleteSemilatticeInf 📖	CompOp	591 math math: sInf_univ, biInf_gt_eq_iInf, biSup_inf_le_inf_biSup, Set.pairwiseDisjoint_pair_insert, fixingSubgroup_antitone, iInf_inf, Set.disjoint_ordT5Nhd, LowerAdjoint.closure_iUnion₂_closure, iInf_sum, Set.preimage_kernImage, Language.reverse_iInf, iUnion_Iic_eq_Iio_iSup, Set.map_finite_iInf, le_iInf₂_add, CategoryTheory.MorphismProperty.isomorphisms_le_of_containsIdentities, CategoryTheory.MorphismProperty.isStableUnderLimitsOfShape_iff_limitsOfShape_le, CompletelyDistribLattice.MinimalAxioms.iSup_iInf_eq, CategoryTheory.ObjectProperty.preservesLimitsOfShape_eq_iSup, CategoryTheory.Sieve.generate_mono, SSet.horn₃₁.desc.multicofork_π_two, CategoryTheory.MorphismProperty.hasOfPostcompProperty_iff_le_diagonal, Nucleus.himp_apply, sInf_pair, LowerAdjoint.mem_iff, Set.iUnionLift_unary, ClosureOperator.closure_iSup₂_closure, iInf_iInf_eq_right, CompleteLatticeHom.dual_symm_apply_toFun, lowerPolar_anti, sInf_union, biSup_inf_le_biSup_inf, sInf_lt_iff, LowerAdjoint.closure_union_closure, iInf_psigma', UpperSet.sdiff_eq_left, iInf_of_empty, CategoryTheory.Sieve.generate_functorPullback_le, scottContinuous_inf_right, iInf_unpair, iInf_sup_iInf, inf_eq_iInf, iInf_false, Set.Iio_disjoint_Ioi_of_not_lt, bot_lt_iSup, iInf_eq_top, compl_sInf, Interval.coe_sInf, CategoryTheory.Sieve.functorPushforward_ofObjects_le, iInf_prod', LowerAdjoint.le_iff_subset, biInf_lt_eq_iInf, Interval.coe_iInf₂, CategoryTheory.Sieve.pullback_monotone, LowerSet.disjoint_coe, le_iInf_const, Language.instMulRightMono, Frm.Iso.mk_hom, CategoryTheory.MorphismProperty.pushouts_le, CategoryTheory.Limits.CompleteLattice.colimit_eq_iSup, Set.Iic_iInf, CategoryTheory.Localization.LeftBousfield.galoisConnection, SSet.horn₃₂.desc.multicofork_π_one, CategoryTheory.ObjectProperty.isomorphisms_le_isoModSerre, sup_biInf_le_biInf_sup, iInf_iSup_ge_nat_add, biInf_sigma, sSup_disjoint_iff, Set.Iio_disjoint_Ioi_iff, CategoryTheory.MorphismProperty.colimitsOfShape_le_coproducts, Set.BijOn.iInf_comp, CategoryTheory.MorphismProperty.IsInvertedBy.iff_map_le_isomorphisms, inf_biInf, isLUB_iSup, CategoryTheory.MorphismProperty.monotone_map, sInf_sup_le_iInf_sup, iInf_ite, Finset.inf_id_eq_sInf, Set.Iic_disjoint_Ioi, Nucleus.map_himp_le, Set.disjoint_pi_univ_Ioc_update_left_right, AddCon.coe_iInf, iInf_eq_iInf_finset', OrderHom.iInf_apply, CategoryTheory.Presieve.map_functorPullback, iInf_option, CategoryTheory.MorphismProperty.colimitsOfShape_le, LowerSet.Iic_iInf, Quantale.rightMulResiduation_le_iff_mul_le, inf_iInf_nat_succ, UpperSet.Ici_iSup, CompleteLatticeHom.toFun_eq_coe, iSup_iInf_le_iInf_iSup, CompleteLatticeHom.map_sSup', GaloisCoinsertion.u_iInf_l, CategoryTheory.MorphismProperty.coproducts_monotone, biSup_mono, PUnit.sInf_eq, CategoryTheory.Sieve.pullback_pushforward_le, biSup_symmDiff_biSup_le, iInf₂_eq_top, ScottContinuous.inf₂, HomotopicalAlgebra.trivialCofibrations_sub_cofibrations, compl_sSup', Set.Ioc_disjoint_Ioc, Finset.iInf_insert, OrderHom.sSup_apply, UpperSet.codisjoint_coe, CategoryTheory.MorphismProperty.colimitsOfShape_discrete_le_llp_rlp, CategoryTheory.Limits.LimitPresentation.self_π, iSup_inf_le_inf_iSup, gc_upperPolar_lowerPolar, Concept.isCompl_extent_intent, le_iSup₂, Set.disjoint_right_ordSeparatingSet, CategoryTheory.MorphismProperty.isStableUnderRetracts_iff_retracts_le, OrderIso.map_sInf_eq_sInf_symm_preimage, LowerSet.sdiff_eq_left, gc_upperBounds_lowerBounds, Order.Coframe.MinimalAxioms.iInf_sup_eq, CategoryTheory.MorphismProperty.multiplicativeClosure_monotone, SSet.horn₃₁.desc.multicofork_pt, CategoryTheory.Functor.relativelyRepresentable.isomorphisms_le, biInf_le_eq_iInf, Set.pairwise_disjoint_vadd_iff, OrderIso.map_iSup, iInf₂_le, CategoryTheory.MorphismProperty.retracts_le, Language.instMulLeftMono, iInf_range, Set.BijOn.iInf_congr, Set.disjoint_iUnion_left, HomotopicalAlgebra.trivialCofibrations_sub_weakEquivalences, Set.PairwiseDisjoint.exists_mem_filter, CategoryTheory.Localization.LeftBousfield.le_W_iff, compl_sSup, HomotopicalAlgebra.trivialFibrations_sub_fibrations, biInf_le_eq_inf, fixingAddSubmonoid_fixedPoints_gc, Language.sub_iSup, Language.le_iff, UpperSet.Ici_sSup, le_iInf_add, sInf_le_sInf_of_subset_insert_top, MulAction.disjoint_image_image_iff, sInf_image, iInf_option_elim, Order.Coframe.MinimalAxioms.sup_iInf_eq, CategoryTheory.MorphismProperty.isStableUnderCobaseChange_iff_pushouts_le, Set.disjoint_smul_set, biInf_lt_iff, LowerSet.coe_iicsInfHom, iInf_subtype, Filter.disjoint_iff, CategoryTheory.MorphismProperty.universally_le, compl_eq_sSup_disjoint, CategoryTheory.Limits.CompleteLattice.colimitCocone_cocone_pt, Filter.monotone_mem, Set.disjoint_sUnion_left, CategoryTheory.Presieve.ofArrows_le_iff, LowerSet.sdiff_lt_left, iSup_eq_top, le_iInf_mul, isGLB_biInf, iInf_const_mono, le_iInf_iff, MulticoequalizerDiagram.multispanIndex_right, CompletelyDistribLattice.MinimalAxioms.iInf_iSup_eq', CategoryTheory.Sieve.le_generate, CategoryTheory.Limits.CompleteLattice.colimitCocone_cocone_ι_app, MulticoequalizerDiagram.multispanIndex_fst, QuotientAddGroup.strictMono_comap_prod_image, SSet.horn₃₁.desc.multicofork_π_two_assoc, iInf_sigma', CategoryTheory.MorphismProperty.le_pushouts, iInf_iSup_eq, disjoint_iSup_iff, OrderIso.map_sSup_eq_sSup_symm_preimage, Set.Ioi_disjoint_Iio_iff, iSup_inf_le_sSup_inf, SSet.horn₃₁.desc.multicofork_π_zero_assoc, LowerSet.iicsInfHom_apply, CategoryTheory.MorphismProperty.strictMap_multiplicativeClosure_le, iInf_Prop_eq, Set.PairwiseDisjoint.exists_mem_filter_basis, le_iSup, Set.kernImage_mono, Frm.Iso.mk_inv, CategoryTheory.Limits.ColimitPresentation.self_diag, lt_iInf_iff, OrderIso.map_iInf, CategoryTheory.Presieve.ofArrows_comp_le, Order.Coframe.MinimalAxioms.sup_iInf₂_eq, iInf_subtype'', CategoryTheory.Limits.CompleteLattice.hasLimits_of_completeLattice, CategoryTheory.Sieve.le_pullback_bind, CategoryTheory.MorphismProperty.isoClosure_le_iff, iInf_subtype', isNowhereDense_iff_disjoint, GaloisInsertion.l_biInf_u, AddQuantale.leftAddResiduation_le_iff_add_le, CategoryTheory.MorphismProperty.coproducts_le_iff, le_sInf_inter, iInf_ne_top_subtype, CategoryTheory.Limits.CompleteLattice.limitCone_isLimit_lift, CategoryTheory.ObjectProperty.galoisConnection_isColocal, unary_relation_sInf_iff, CategoryTheory.MorphismProperty.map_inverseImage_le, CategoryTheory.MorphismProperty.le_isLocal_isLocal, MulticoequalizerDiagram.multispanIndex_left, Set.pairwiseDisjoint_prod_left, le_iInf_comp, CategoryTheory.MorphismProperty.gc_llp_rlp, sInf_sUnion, Finset.iInf_biUnion, Set.disjoint_iUnion_right, HomotopicalAlgebra.trivialFibrations_sub_weakEquivalences, Set.image_preimage, sInf_image2, GaloisInsertion.l_iInf_u, iInf_le_iInf₂, iSup₂_add_le, LowerAdjoint.closure_iSup₂_closure, iInf_sigma, Filter.antitone_seq_of_seq, Set.Iic_iInf₂, fixingSubmonoid_antitone, SimpleGraph.Subgraph.verts_monotone, CategoryTheory.Limits.LimitPresentation.self_diag, iInf_eq_iInf_finset, CategoryTheory.Sieve.le_pushforward_pullback, iInf_sup_iInf_le, Con.coe_iInf, iSup_disjoint_iff, fixedPoints_addSubgroup_antitone, CategoryTheory.Sieve.functorPushforward_monotone, compl_iSup, le_iSup_inf_iSup, compl_iInf, CategoryTheory.MorphismProperty.map_le_iff, biInf_prod, Interval.coe_iInf, biInf_finsetSigma, LowerAdjoint.closure_union_closure_subset, Set.disjoint_iUnion₂_right, infsInfHom_apply, CategoryTheory.ObjectProperty.le_isLocal_iff, sSup_lowerBounds_eq_sInf, LowerSet.Iic_iInf₂, CategoryTheory.MorphismProperty.le_coproducts, Filter.HasBasis.disjoint_iff, UpperSet.coe_icisSupHom, le_iInf₂_iff, hnot_eq_sInf_codisjoint, SimpleGraph.pairwise_disjoint_supp_connectedComponent, Language.instOrderedSub, sInf_sup_eq, biInf_finsetSigma', Finset.iInf_finset_image, Set.Ioi_disjoint_Iio_of_le, sInf_eq_iInf, Concept.codisjoint_extent_intent, CategoryTheory.MorphismProperty.le_pullbacks, CategoryTheory.ObjectProperty.epimorphisms_le_epiModSerre, CategoryTheory.MorphismProperty.antitone_rlp, LowerAdjoint.subset_closure, UpperSet.lt_sdiff_left, Set.Ici_iSup₂, AddCon.coe_sInf, iInf_bool_eq, Finset.intervalGapsWithin_pairwiseDisjoint_Ioc, iSup_const_mono, CategoryTheory.MorphismProperty.le_llp_rlp, sSup_le_sSup_of_subset_insert_bot, OrderHom.coe_iInf, sdiff_eq_sInf, Filter.disjoint_principal_principal, iInf_image, iSup₂_disjoint_iff, iInf_split, CategoryTheory.MorphismProperty.le_llp_iff_le_rlp, CategoryTheory.MorphismProperty.coproducts_le, CategoryTheory.Subfunctor.Subpresheaf.iInf_obj, CategoryTheory.Sieve.functorPullback_monotone, LowerAdjoint.closure_union_closure_right, AddSubgroup.quotientiInfEmbedding_apply, le_sdiff_iff, Set.Iic_disjoint_Ioc, inf_iInf, iSup_inf_le_inf_sSup, ClosureOperator.closure_iSup_closure, sup_iInf_le_iInf_sup, CategoryTheory.ObjectProperty.le_isColocal_iff, biInf_ge_eq_inf, Set.Ioc_disjoint_Ioi, iInf_psigma, QuotientGroup.strictMono_comap_prod_image, Filter.countable_biInf_eq_iInf_seq', CategoryTheory.Subfunctor.sInf_obj, GaloisInsertion.l_sInf_u_image, Set.Iic_disjoint_Ici, Set.Ici_sSup, gc_upperClosure_coe, RingCon.coe_iInf, CategoryTheory.MorphismProperty.le_retracts, Finset.iInf_union, GaloisCoinsertion.u_sInf_l_image, UpperSet.icisSupHom_apply, CategoryTheory.ObjectProperty.monomorphisms_le_monoModSerre, Interval.disjoint_coe, scottContinuous_inf_left, Set.Ioi_disjoint_Iio_of_not_lt, CategoryTheory.Sieve.ofObjects_mono, Nucleus.sInf_apply, CategoryTheory.MorphismProperty.coproducts_le_llp_rlp, CategoryTheory.Limits.CompleteLattice.hasColimits_of_completeLattice, iInf₂_comm, CategoryTheory.Sieve.functorPushforward_pullback_le, WithBot.sInf_empty, Set.Ico_disjoint_Ico, Finset.iInf_insert_update, iInf_top, LowerSet.Iic_sInf, iInf_lt_iff, Subgroup.quotientiInfEmbedding_apply, iInf_union, disjoint_sSup_iff, Set.Ioi_disjoint_Iio_same, RingCon.coe_sInf, iInf_dite, IsLowerSet.disjoint_upperClosure_left, Finset.iInf_singleton, SSet.horn₃₂.desc.multicofork_π_zero_assoc, MulticoequalizerDiagram.multispanIndex_snd, FreeGroup.startsWith.disjoint_iff_ne, Set.disjoint_vadd_set, Set.disjoint_sUnion_right, CategoryTheory.MorphismProperty.limitsOfShape_le, Filter.ker_mono, CompleteLat.Iso.mk_hom, CategoryTheory.MorphismProperty.retracts_le_iff, iInf₂_sup_eq, iSup_inf_le_iSup_inf, LowerAdjoint.closure_iSup_closure, sInfHom.coe_top, OrderIsoClass.tosInfHomClass, IsLowerSet.disjoint_upperClosure_right, CategoryTheory.Sieve.le_functorPushforward_pullback, iInf_and, Concept.strictAnti_intent, CategoryTheory.MorphismProperty.IsInvertedBy.iff_le_inverseImage_isomorphisms, iSup_iInf_le, iInf_unique, biInf_sup_le_biInf_sup, CategoryTheory.MorphismProperty.isomorphisms_le_pushouts, iSup₂_le_iff, fixingSubmonoid_fixedPoints_gc, CompleteLatticeHom.coe_tosInfHom, iInf_eq_dif, SSet.horn₃₁.desc.multicofork_π_three_assoc, CategoryTheory.Sieve.sInf_apply, CategoryTheory.MorphismProperty.le_leftBousfieldW_isLocal, himp_iInf_eq, GaloisInsertion.l_biInf_of_ul_eq_self, sInf_insert, SSet.horn₃₁.desc.multicofork_π_zero, CategoryTheory.MorphismProperty.colimitsOfShape_le_of_final, iInf_extend_top, Set.Iio_disjoint_Ioi_same, CategoryTheory.Sieve.galoisConnection, CategoryTheory.Subfunctor.Subpresheaf.sInf_obj, iInf_iInf_eq_left, fixedPoints_subgroup_antitone, CategoryTheory.LocalizerMorphism.map, CategoryTheory.Presieve.bind_ofArrows_le_bindOfArrows, Set.iUnionLift_binary, himp_le_iff, sup_sInf_le_iInf_sup, CategoryTheory.Sieve.functor_galoisConnection, Nucleus.iInf_apply, iSup_himp_eq, iInf_iUnion, iInf_and', fixingAddSubmonoid_antitone, biInf_sigma', iInf_image2, Filter.countable_biInf_eq_iInf_seq, iInf_univ, sSup_eq_top, biInf_sup_biInf, iInf_sup_eq, disjoint_memPartition, Set.monotone_preimage, iInf_comm, CategoryTheory.ObjectProperty.preservesColimitsOfShape_eq_iSup, iSup_le_iSup_of_subset, SSet.horn₃₂.desc.multicofork_π_three, fixedPoints_antitone_addSubmonoid, AddSubgroup.quotientiInfAddSubgroupOfEmbedding_apply, CategoryTheory.MorphismProperty.retracts_le_llp_rlp, Set.Iic_sInf, IsUpperSet.disjoint_lowerClosure_left, Set.disjoint_vadd_set_left, ScottContinuous.of_map_sSup, fixingAddSubgroup_fixedPoints_gc, Set.Ioc_disjoint_Ioc_of_le, CategoryTheory.Sieve.pushforward_le_bind_of_mem, CategoryTheory.MorphismProperty.antitone_llp, sInf_diff_singleton_top, Set.pairwiseDisjoint_vadd_iff, Subgroup.quotientiInfSubgroupOfEmbedding_apply, iInf_or, Composition.disjoint_range, iInf_singleton, biInf_inf, le_iInf_iSup, OrderIso.map_iInf₂, OrderIso.map_sSup, Set.Ici_disjoint_Iic, biInf_const, lt_sSup_iff, fixingSubgroup_fixedPoints_gc, Finset.inf_eq_sInf_image, CategoryTheory.Subfunctor.iInf_obj, fixingAddSubgroup_antitone, biInf_prod', iInf_sup_le_iInf_sup, biInf_mono, Finset.inf_eq_iInf, Con.coe_sInf, MulticoequalizerDiagram.multicofork_pt, sInf_empty, Set.pairwiseDisjoint_iff, CategoryTheory.MorphismProperty.le_isColocal_isColocal, scottContinuous_iff_map_sSup, CategoryTheory.Sieve.functorPullback_pushforward_le, AddAction.disjoint_image_image_iff, sup_sInf_eq, CategoryTheory.Sieve.functorPushforward_le_iff_le_functorPullback, CompleteLatticeHom.dual_apply_toFun, SSet.horn₃₂.desc.multicofork_pt, Filter.IsAntitoneBasis.antitone, iInf_prod, Set.disjoint_vadd_set_right, iInf_eq_bot, Set.Iio_disjoint_Ici, fixedPoints.gfp_eq_sInf_iterate, iInf_nat_gt_zero_eq, sInf_upperBounds_eq_csSup, iInf₂_eq_bot, Set.pairwise_disjoint_smul_iff, sInf_upperBounds_eq_sSup, lt_biSup_iff, iSup_le_iff, CategoryTheory.Limits.CompleteLattice.limit_eq_iInf, isGLB_iInf, Set.Ico_disjoint_Ico_same, Filter.HasAntitoneBasis.antitone, OrderIso.map_sInf, biInf_le_biSup, iSup_add_le, OrderHom.map_sInf_subset_fixedPoints_le, Disjoint.edgeSet, Finset.exists_inf_eq_iInf, CategoryTheory.MorphismProperty.monotone_isoClosure, OrderIso.map_iSup₂, Finset.inf_univ_eq_iInf, OrderHom.iSup_apply, Equiv.biInf_comp, iSup_const_le, iInf_eq_if, CategoryTheory.MorphismProperty.le_def, sInf_Prop_eq, Filter.exists_antitone_seq, CategoryTheory.MorphismProperty.pushouts_le_llp_rlp, CategoryTheory.Limits.CompleteLattice.limitCone_cone_π_app, Pairwise.exists_mem_filter_basis_of_disjoint, Set.pairwiseDisjoint_smul_iff, Concept.strictMono_extent, Set.Ioo_disjoint_Ioo, iSup_symmDiff_iSup_le, sup_iInf₂_eq, AddQuantale.instAddLeftMono, iInf_ge_eq_iInf_nat_add, isLUB_biSup, OrderHom.sInf_apply, iInf_insert, CategoryTheory.MorphismProperty.isStableUnderColimitsOfShape_iff_colimitsOfShape_le, le_iInf₂_mul, upperPolar_anti, CategoryTheory.MorphismProperty.le_isoClosure, Quantale.instMulLeftMono, Set.disjoint_smul_set_left, sSup_inter_le, le_biSup, CategoryTheory.ObjectProperty.galoisConnection_isLocal, sInf_eq_top, compl_sInf', iInf_neg, iInf_lt_top, OrderIso.toCompleteLatticeHom_toFun, Filter.NeBot.not_disjoint, CategoryTheory.MorphismProperty.pullbacks_monotone, Set.disjoint_smul_set_right, CategoryTheory.MorphismProperty.universally_mono, CategoryTheory.MorphismProperty.presheaf_monomorphisms_le_monomorphisms, CategoryTheory.MorphismProperty.pushouts_le_iff, Quantale.instMulRightMono, iSup₂_le_iSup, CategoryTheory.MorphismProperty.isStableUnderBaseChange_iff_pullbacks_le, iSup_iInf_ge_nat_add, iUnion_Iic_eq_Iic_iSup, CategoryTheory.Sieve.pushforward_monotone, iSup₂_mul_le, Pairwise.exists_mem_filter_of_disjoint, iInf_sUnion, Set.disjoint_left_ordSeparatingSet, fixedPoints_antitone, AddQuantale.rightAddResiduation_le_iff_add_le, CategoryTheory.Limits.CompleteLattice.colimitCocone_isColimit_desc, Finset.iInf_option_toFinset, Set.Iio_disjoint_Ioi_of_le, CategoryTheory.Sieve.generate_le_iff, Finset.pairwiseDisjoint_pair_insert, gc_lowerClosure_coe, sup_iInf_eq, iInf_exists, iSup_comp_le, SSet.horn₃₂.desc.multicofork_π_zero, SSet.horn₃₂.desc.multicofork_π_three_assoc, iInf_le_iSup, sInf_le_sSup, iInf_emptyset, Set.sUnion_powerset_gc, SSet.horn₃₂.desc.multicofork_π_one_assoc, CategoryTheory.MorphismProperty.pullbacks_le, IsUpperSet.disjoint_lowerClosure_right, iInf_le, Quantale.leftMulResiduation_le_iff_mul_le, Disjoint.exists_mem_filter_basis, binary_relation_sInf_iff, iSup₂_eq_top, iSup_mul_le, iInf_inf_eq, CategoryTheory.ConcreteCategory.injective_le_monomorphisms, SimpleGraph.deleteEdges_eq_self, Set.Ici_iSup, iInf_pos, Set.disjoint_iUnion₂_left, CategoryTheory.MorphismProperty.pushouts_monotone, Order.Coframe.MinimalAxioms.sup_sInf_eq, CategoryTheory.MorphismProperty.le_colimitsOfShape_punit, OrderHom.coe_iSup, GaloisInsertion.l_iInf_of_ul_eq_self, CategoryTheory.MorphismProperty.retracts_monotone, sdiff_iSup_eq, CategoryTheory.ConcreteCategory.surjective_le_epimorphisms, iInf_le_iInf_of_subset, iInf_const, SSet.horn₃₁.desc.multicofork_π_three, iInf_pair, CategoryTheory.MorphismProperty.le_multiplicativeClosure, sInf_eq_bot, CompleteLat.Iso.mk_inv, disjoint_iSup₂_iff, iInf_split_single, lowerClosure_mono, Set.map_finite_biInf, Set.Ioc_disjoint_Ioi_same, iSup_lt_iff, Order.Coframe.MinimalAxioms.sInf_sup_eq, iSup_iInf_eq, sInfHom.top_apply, CategoryTheory.Sieve.ofArrows_le_ofObjects, Concept.disjoint_extent_intent, iInf_true, CompleteLatticeHomClass.tosInfHomClass, biInf_ge_eq_iInf, Filter.monotone_principal, LowerAdjoint.closure_union_closure_left, biInf_le, Setoid.sInf_def, CategoryTheory.MorphismProperty.multiplicativeClosure_le_iff, upperClosure_anti, LowerAdjoint.closure_iUnion_closure, sInf_sup_sInf, AddQuantale.instAddRightMono, UpperSet.Ici_iSup₂, CategoryTheory.Limits.CompleteLattice.limitCone_cone_pt, CategoryTheory.Limits.ColimitPresentation.self_ι, himp_eq_sSup, lt_iSup_iff
toCompleteSemilatticeSup 📖	CompOp	503 math math: Finset.iSup_biUnion, iSup_insert, sInf_image2_eq_sInf_sSup, Subsemiring.mem_sSup_of_directedOn, Order.Frame.MinimalAxioms.iSup_inf_eq, AddCon.sSup_eq_addConGen, biSup_inf_le_inf_biSup, Subring.coe_sSup_of_directedOn, iSup_pair, Subring.map_iSup, Language.iSup_add, biSup_le_eq_sup, unary_relation_sSup_iff, GaloisCoinsertion.u_biSup_l, CompletelyDistribLattice.MinimalAxioms.iSup_iInf_eq, NonUnitalSubsemiring.mem_iSup_of_directed, Language.iSup_sub, AddSubsemigroup.closure_iUnion, AddSubsemigroup.op_iSup, iSup₂_le, sSup_univ, ClosureOperator.closure_iSup₂_closure, CompleteLatticeHom.coe_tosSupHom, iSup_psigma, iSup_bool_eq, biSup_lt_eq_iSup, fixedPoints_subgroup_iSup, iSup_split, Antitone.le_map_iInf₂, Subsemigroup.unop_iSup, biSup_inf_le_biSup_inf, Subgroup.normalCore_eq_iSup, AddSubgroup.iSup_eq_closure, Subgroup.op_iSup, sSup_eq_bot, Subgroup.mem_biSup_of_directedOn, Subring.mem_sSup_of_directedOn, Submonoid.unop_iSup, iSup_false, Subgroup.mem_iSup_of_mem, bot_lt_iSup, compl_sInf, sSup_Prop_eq, Submonoid.coe_iSup_of_directed, RingCon.sSup_def, sSup_inf_sSup, iSup_unpair, CategoryTheory.Limits.CompleteLattice.colimit_eq_iSup, sup_iSup, Subsemigroup.mem_sSup_of_mem, iSup₂_mono, Encodable.iSup_decode₂, TwoSidedIdeal.iSup_ringCon, iSup_image, iInf_iSup_ge_nat_add, Language.mem_iSup, sSup_disjoint_iff, IsLUB.iSup_eq, PUnit.sSup_eq, fixedPoints_addSubmonoid_iSup, iInf_iSup_of_antitone, iSup₂_inf_eq, isLUB_iSup, sSup_empty, NonUnitalSubring.map_iSup, Language.reverse_iSup, NonUnitalSubring.mem_sSup_of_directedOn, AddSubsemigroup.coe_iSup_of_directed, Antitone.sSup, sSup_insert, Submonoid.iSup_eq_closure, Subgroup.map_iSup, Submonoid.coe_sSup_of_directedOn, iSup_and', UpperSet.Ici_iSup, sSup_union, CategoryTheory.Subfunctor.Subpresheaf.iSup_obj, sSupHom.coe_bot, Subring.coe_iSup_of_directed, Subsemigroup.mem_iSup_prop, iSup_iInf_le_iInf_iSup, AddSubsemigroup.unop_sSup, CompleteLatticeHom.map_sSup', biSup_mono, biSup_le_eq_iSup, iSup_subtype, Submonoid.mem_iSup, AddSubsemigroup.unop_iSup, sSup_image2_eq_sInf_sInf, biSup_symmDiff_biSup_le, compl_sSup', biSup_le_eq_of_monotone, biSup_inf_biSup, OrderHom.sSup_apply, AddSubgroup.coe_iSup_of_directed, iSup_exists, CategoryTheory.Sieve.sSup_apply, Submonoid.iSup_map_mulSingle_le, iSup_inf_le_inf_iSup, AddSubsemigroup.map_iSup, Subsemiring.mem_iSup_of_directed, Subsemiring.coe_iSup_of_directed, le_iSup₂, Set.map_finite_iSup, CategoryTheory.ObjectProperty.instIsClosedUnderIsomorphismsISup, iSup_iInf_of_monotone, Order.Frame.MinimalAxioms.sSup_inf_eq, OrderIso.map_iSup, iSup_subtype', sSup_image2, NonUnitalSubsemiring.coe_iSup_of_directed, biSup_sigma', Finset.iSup_singleton, sSupHom.bot_apply, biSup_finsetSigma', GaloisCoinsertion.u_iSup_of_lu_eq_self, Subgroup.mem_iSup_prop, AddSubgroup.mem_iSup_of_directed, Set.Finite.iSup_biInf_of_antitone, CategoryTheory.Subfunctor.isGeneratedBy_iff, iSup_mono', GaloisCoinsertion.u_biSup_of_lu_eq_self, CategoryTheory.Subfunctor.Subpresheaf.isGeneratedBy_iff, AddSubsemigroup.mem_iSup, Antitone.map_iSup₂_le, Submonoid.mem_sSup_of_directedOn, compl_sSup, Language.sub_iSup, AddSubsemigroup.mem_biSup_of_directedOn, iSup_subtype'', Submonoid.op_sSup, CategoryTheory.Subfunctor.sSup_obj, UpperSet.Ici_sSup, sup_eq_iSup, Subsemigroup.mem_iSup_of_directed, Subsemigroup.mem_sSup_of_directed_on, sup_biSup, Subsemiring.op_iSup, inf_iSup₂_eq, IsQuantale.sSup_mul_distrib, Subgroup.mem_sSup_of_directedOn, iSup_range, compl_eq_sSup_disjoint, Set.Finite.iSup_biInf_of_monotone, CategoryTheory.Limits.CompleteLattice.colimitCocone_cocone_pt, Language.add_iSup, AddSubgroup.mem_sSup_of_directedOn, Equiv.biSup_comp, FirstOrder.Language.DirectLimit.partialEquivLimit_comp_inclusion, AddSubgroup.unop_iSup, NonUnitalSubring.mem_iSup_of_directed, iSup_eq_top, Set.iSup_iInf_of_monotone, Monotone.le_map_iSup₂, iSup_sdiff_eq, CompletelyDistribLattice.MinimalAxioms.iInf_iSup_eq', biSup_ge_eq_sup, CategoryTheory.Limits.CompleteLattice.colimitCocone_cocone_ι_app, Setoid.sSup_def, Monotone.le_map_iSup, Submonoid.map_iSup, iInf_iSup_eq, disjoint_iSup_iff, RingCon.sSup_eq_ringConGen, iSup_comm, CategoryTheory.Pairwise.cocone_pt, OrderIso.map_sSup_eq_sSup_symm_preimage, iSup_inf_le_sSup_inf, iSup_split_single, AddSubmonoid.iSup_map_single_le, AddSubsemigroup.mem_iSup_of_mem, Subring.op_sSup, le_iSup, iSup_eq_iSup_finset, AlgebraicGeometry.Scheme.IsLocallyDirected.homOfLE_tAux, inf_sSup_eq, Set.Finite.iInf_biSup_of_antitone, OrderHom.le_map_sSup_subset_fixedPoints, NonUnitalSubring.closure_sUnion, AddQuantale.add_iSup_distrib, binary_relation_sSup_iff, biSup_gt_eq_iSup, CategoryTheory.ObjectProperty.isoClosure_iSup, sSup_diff_singleton_bot, Subsemigroup.mem_biSup_of_directedOn, GaloisConnection.l_sSup, AddSubmonoid.unop_iSup, Submonoid.op_iSup, Antitone.map_sSup_le, iSup_pos, Antitone.le_map_iInf, Set.pairwiseDisjoint_prod_left, iSup_dite, AddSubmonoid.mem_iSup, iSup_eq_iSup_finset', iSup_inf_eq, GaloisConnection.l_iSup₂, CategoryTheory.Subfunctor.iSup_min, Subsemigroup.comap_iSup_map_of_injective, iSup₂_add_le, LowerAdjoint.closure_iSup₂_closure, NonUnitalSubsemiring.map_iSup, Subgroup.iSup_comap_le, GaloisInsertion.l_sSup_u_image, AddSubsemigroup.op_sSup, CategoryTheory.MorphismProperty.iSup_iff, iSup_sup_eq, iSup_disjoint_iff, biSup_iInter_of_pairwise_disjoint, compl_iSup, le_iSup_inf_iSup, compl_iInf, sSup_eq_bot', supsSupHom_apply, ωScottContinuous.iSup, CategoryTheory.Pairwise.cocone_ι_app, iSup_ite, Submonoid.mem_iSup_prop, biSup_prod, sSup_lowerBounds_eq_sInf, Subsemiring.closure_iUnion, IsQuantale.mul_sSup_distrib, OrderIsoClass.tosSupHomClass, Subgroup.mem_sSup_of_mem, iSup_option_elim, sSup_image2_eq_sSup_sSup, UpperSet.coe_icisSupHom, biSup_ge_eq_of_antitone, Submonoid.map_iSup_comap_of_surjective, inf_iSup_eq, Con.sSup_eq_conGen, AddSubmonoid.coe_iSup_of_directed, AddSubmonoid.coe_sSup_of_directedOn, Subsemiring.closure_sUnion, Submonoid.comap_iSup_map_of_injective, AddSubsemigroup.comap_iSup_map_of_injective, AddSubmonoid.unop_sSup, CategoryTheory.ObjectProperty.instEssentiallySmallISupOfSmall, Antitone.map_iSup_le, iSup_prod', sSup_iUnion, Set.Ici_iSup₂, CategoryTheory.Subfunctor.IsGeneratedBy.iSup_eq, iSup_univ, iSup_eq_dif, AddSubsemigroup.iSup_eq_closure, Subgroup.coe_iSup_of_directed, iSup_const_mono, iSup_image2, sSup_le_sSup_of_subset_insert_bot, iSup₂_disjoint_iff, Set.map_finite_biSup, ωScottContinuous.sSup, AddSubgroup.op_sSup, GaloisConnection.l_iSup, iSup₂_mono', FrameHom.map_sSup', iSup_inf_le_inf_sSup, ClosureOperator.closure_iSup_closure, Order.Frame.MinimalAxioms.inf_iSup₂_eq, fixedPoints_submonoid_iSup, iSup_iUnion, iSup₂_eq_bot, Order.Frame.MinimalAxioms.inf_iSup_eq, TwoSidedIdeal.sSup_ringCon, Set.Ici_sSup, sup_iSup_nat_succ, Subsemigroup.op_iSup, Submonoid.mem_iSup_of_directed, AddCon.sSup_def, sSup_eq_iSup, FrameHomClass.map_sSup, Finset.sup_eq_iSup, UpperSet.icisSupHom_apply, Order.Frame.MinimalAxioms.inf_sSup_eq, AddSubmonoid.op_sSup, Language.iSup_mul, iSup_singleton, iSup_neg, add_sSup_distrib, Finset.iSup_insert, Subring.mem_iSup_of_directed, GaloisInsertion.l_biSup_u, AddSubmonoid.op_iSup, NonUnitalSubring.coe_sSup_of_directedOn, sInf_image2_eq_sSup_sSup, Subsemigroup.closure_iUnion, Subsemigroup.mem_iSup_of_mem, AddSubsemigroup.map_iSup_comap_of_surjective, Subsemiring.map_iSup, iSup_ne_bot_subtype, NonUnitalSubsemiring.closure_sUnion, disjoint_sSup_iff, iInf_iSup_of_monotone, Quantale.iSup_mul_distrib, NonUnitalSubsemiring.closure_iUnion, AddSubmonoid.mem_iSup_of_directed, iSup_emptyset, Subsemiring.op_sSup, NonUnitalSubring.closure_iUnion, CategoryTheory.Subfunctor.Subpresheaf.iSup_min, biSup_const, iSup₂_comm, Monotone.iSup_comp_eq, AddSubgroup.closure_iUnion, iSup_inf_le_iSup_inf, Set.Finite.iInf_biSup_of_monotone, LowerAdjoint.closure_iSup_closure, sInf_image2_eq_sSup_sInf, le_iSup_of_le, iSup_true, sSup_mul_distrib, AddSubsemigroup.mem_sSup_of_mem, Subgroup.unop_iSup, iSup_iInf_le, Subgroup.mem_iSup_of_directed, Antitone.le_map_sInf, iSup₂_le_iff, Submonoid.unop_sSup, iSup_Prop_eq, Subring.op_iSup, iSup_sup, Subsemigroup.mem_iSup, AddSubgroup.mem_iSup_of_mem, iSup_sigma', Monotone.le_map_sSup, Antitone.iSup, Finset.iSup_insert_update, iSup_iSup_eq_right, iSup_himp_eq, iSup_unique, sSup_eq_top, GaloisCoinsertion.u_iSup_l, Subsemiring.coe_sSup_of_directedOn, Setoid.sSup_eq_eqvGen, Set.exists_seq_iSup_eq_top_iff_countable, iSup_sigma, Finset.sup_eq_sSup_image, sSup_image, iSup_le_iSup_of_subset, AddSubgroup.mem_sSup_of_mem, sSup_sUnion, AddSubmonoid.map_iSup, biSup_prod', AddSubsemigroup.mem_iSup_prop, CompleteLatticeHomClass.map_sSup, CategoryTheory.Subfunctor.image_iSup, sSup_image2_eq_sSup_sInf, fixedPoints.lfp_eq_sSup_iterate, iSup_le, mul_sSup_distrib, biSup_sigma, AddSubgroup.mem_iSup_prop, Finset.iSup_finset_image, CategoryTheory.MorphismProperty.sSup_iff, le_iInf_iSup, OrderIso.map_sSup, sSup_pair, Set.BijOn.iSup_congr, AddSubgroup.op_iSup, CategoryTheory.Subfunctor.Subpresheaf.sSup_obj, lt_sSup_iff, AddSubsemigroup.mem_sSup_of_directed_on, AddSubsemigroup.coe_sSup_of_directed_on, Subgroup.unop_sSup, Language.mul_iSup, Submonoid.mem_biSup_of_directedOn, fixedPoints_addSubgroup_iSup, iSup_eq_if, Subsemigroup.map_iSup, Subsemigroup.map_iSup_comap_of_surjective, scottContinuous_iff_map_sSup, Subsemigroup.coe_iSup_of_directed, Con.sSup_def, AddSubmonoid.closure_iUnion, iSup_psigma', AddSubmonoid.mem_sSup_of_mem, AlgebraicGeometry.Scheme.IsLocallyDirected.exists_of_pullback_V_V, FirstOrder.Language.DirectLimit.Equiv_isup_of_apply, iSup_iInf_of_antitone, Submonoid.mem_sSup_of_mem, Set.BijOn.iSup_comp, FrameHomClass.tosSupHomClass, sInf_upperBounds_eq_csSup, sInf_upperBounds_eq_sSup, lt_biSup_iff, iSup_le_iff, AddSubsemigroup.mem_iSup_of_directed, Set.iSup_iInf_of_antitone, Subring.unop_iSup, iSup_nat_gt_zero_eq, biInf_le_biSup, iSup_add_le, CategoryTheory.Subfunctor.iSup_obj, sSup_image2_eq_sInf_sSup, IsAddQuantale.add_sSup_distrib, NonUnitalSubring.coe_iSup_of_directed, Submonoid.mem_iSup_of_mem, Set.iInf_iSup_of_monotone, sSup_eq_of_forall_le_of_forall_lt_exists_gt, OrderIso.map_iSup₂, iSup_option, OrderHom.iSup_apply, AddSubgroup.iSup_comap_le, iSup_const_le, iSup_ge_eq_iSup_nat_add, Submonoid.closure_iUnion, iSup_const, Subsemiring.unop_sSup, GaloisCoinsertion.u_sSup_l_image, le_iSup₂_of_le, iSup_symmDiff_iSup_le, Quantale.mul_iSup_distrib, AddSubgroup.unop_sSup, iSup_inf_of_monotone, Subgroup.closure_iUnion, Finset.sup_univ_eq_iSup, isLUB_biSup, Subsemigroup.iSup_eq_closure, NonUnitalSubsemiring.coe_sSup_of_directedOn, sSup_inter_le, le_biSup, iSup_of_empty, compl_sInf', Finset.iSup_option_toFinset, biSup_sup, AddSubmonoid.mem_iSup_of_mem, AddSubmonoid.map_iSup_comap_of_surjective, iSup₂_le_iSup, iSup_eq_bot, FirstOrder.Language.DirectLimit.Equiv_isup_symm_inclusion_apply, MeasureTheory.Submartingale.exists_ae_trim_tendsto_of_bdd, CategoryTheory.Subfunctor.Subpresheaf.image_iSup, iSup_iInf_ge_nat_add, IsAddQuantale.sSup_add_distrib, Subring.unop_sSup, Subgroup.op_sSup, iSup₂_mul_le, NonUnitalSubsemiring.mem_sSup_of_directedOn, AddSubmonoid.comap_iSup_map_of_injective, CategoryTheory.Limits.CompleteLattice.colimitCocone_isColimit_desc, AlgebraicGeometry.Scheme.IsLocallyDirected.homOfLE_tAux_assoc, GaloisInsertion.l_iSup_u, MulticoequalizerDiagram.iSup_eq, iSup_comp_le, Subsemigroup.coe_sSup_of_directed_on, AddSubmonoid.mem_biSup_of_directedOn, sSup_inf_eq, Monotone.iSup, iInf_le_iSup, sInf_le_sSup, Subsemigroup.unop_sSup, AddSubgroup.map_iSup, Language.kstar_eq_iSup_pow, Subsemigroup.op_sSup, Finset.sup_id_eq_sSup, iSup₂_eq_top, iSup_mul_le, Finset.exists_sup_eq_iSup, Set.Ici_iSup, iSup_union, biSup_inter_of_pairwise_disjoint, iSup_extend_bot, OrderHom.coe_iSup, iSup_bot, Subsemiring.unop_iSup, CategoryTheory.ObjectProperty.instSmallISupOfSmall, Subgroup.iSup_eq_closure, iSup_inf_of_antitone, sdiff_iSup_eq, Monotone.sSup, AddQuantale.iSup_add_distrib, Monotone.iSup_nat_add, iSup_sum, CategoryTheory.Subfunctor.Subpresheaf.IsGeneratedBy.iSup_eq, iSup_sUnion, disjoint_iSup₂_iff, biSup_ge_eq_iSup, iSup_or, AddSubmonoid.mem_sSup_of_directedOn, iSup_lt_iff, sSup_add_distrib, iSup_iInf_eq, AddSubgroup.mem_biSup_of_directedOn, iSup_mono, iSup_and, iSup_eq_of_forall_le_of_forall_lt_exists_gt, AddSubmonoid.iSup_eq_closure, Finset.iSup_union, iSup_iSup_eq_left, ScottContinuous.map_sSup, biSup_finsetSigma, iSup_inf_iSup, AddSubmonoid.mem_iSup_prop, Subring.closure_sUnion, iSup_prod, Set.iInf_iSup_of_antitone, CategoryTheory.MorphismProperty.toSet_iSup, Subring.closure_iUnion, UpperSet.Ici_iSup₂, CategoryTheory.ObjectProperty.prop_iSup_iff, himp_eq_sSup, lt_iSup_iff
toInfSet 📖	CompOp	7 math math: le_sInf, CompleteAtomicBooleanAlgebra.iInf_iSup_eq, CompletelyDistribLattice.iInf_iSup_eq, CompletelyDistribLattice.MinimalAxioms.iInf_iSup_eq, CompleteDistribLattice.MinimalAxioms.iInf_sup_le_sup_sInf, sInf_le, Order.Coframe.MinimalAxioms.iInf_sup_le_sup_sInf
toLattice 📖	CompOp	1423 math math: AlgebraicGeometry.Scheme.toSpecΓ_apply, AlgebraicGeometry.Scheme.Opens.ι_image_basicOpen, AlgebraicGeometry.Γ_map_morphismRestrict, iSup_insert, AlgebraicGeometry.Scheme.Hom.smoothLocus_eq_top_iff, IsCyclotomicExtension.adjoin_primitive_root_eq_top, Algebra.range_ofId, NonUnitalSubalgebra.center_eq_top, sInf_univ, CategoryTheory.sheafBotEquivalence_functor, Subalgebra.LinearDisjoint.inf_eq_bot, AlgebraicGeometry.Scheme.ΓSpecIso_inv_naturality_assoc, Submodule.rTensorOne_symm_apply, Order.Frame.MinimalAxioms.iSup_inf_eq, AlgebraicGeometry.Scheme.Hom.toPartialMap_hom, biSup_inf_le_inf_biSup, AlgebraicGeometry.Scheme.map_PrimeSpectrum_basicOpen_of_affine, coinduced_bot, AlgebraicGeometry.Scheme.bot_mem_grothendieckTopology, IntermediateField.finInsepDegree_top, iSup_pair, SSet.Subcomplex.preimage_eq_top_iff, CategoryTheory.ObjectProperty.InheritedFromTarget.instMin, iInf_inf, measurable_from_top, AlgebraicGeometry.IsAffineOpen.isoSpec_inv_appTop, AlgebraicGeometry.Scheme.germ_stalkClosedPointTo, CompleteLatticeHom.coe_toBoundedLatticeHom, IntermediateField.adjoin_eq_top_iff_of_isAlgebraic, iSupIndep.injOn_iInf, Set.smul_set_symmDiff, IntermediateField.LinearDisjoint.iff_inf_eq_bot, iInf_sum, Subgroup.exists_mem_sup, AlgebraicGeometry.ΓSpec.locallyRingedSpaceAdjunction_counit_app, MeasurableSpace.comap_const, AlgebraicGeometry.Scheme.IsQuasiAffine.toIsImmersion, AlgebraicGeometry.StructureSheaf.globalSectionsIso_inv, biSup_le_eq_sup, TopologicalSpace.Opens.mem_top, AlgebraicGeometry.iSup_affineOpens_eq_top, AlgebraicGeometry.isAffineOpen_top, AddQuantale.add_sup_distrib, AlgebraicGeometry.AffineSpace.isoOfIsAffine_inv, MeasureTheory.Measure.sub_top, AddSubgroup.map_eq_map_iff, Submodule.comm_trans_lTensorOne, IntermediateField.coe_algebraMap_over_bot, RingCon.toCon_eq_bot, AddSubgroup.exists_mem_sup, NonUnitalStarAlgebra.mem_bot, CategoryTheory.MorphismProperty.universally_inf, Nucleus.himp_apply, IsDedekindDomain.HeightOneSpectrum.iInf_localization_eq_bot, AddSubmonoid.sup_eq_closure, Algebra.adjoin_singleton_intCast, Finset.inf_id_set_eq_sInter, sInf_pair, CategoryTheory.ObjectProperty.instIsClosedUnderIsomorphismsMin, Subalgebra.pi_top, AlgebraicGeometry.Scheme.Opens.toSpecΓ_SpecMap_presheaf_map_top, AlgebraicGeometry.LocallyRingedSpace.notMem_prime_iff_unit_in_stalk, sSup_univ, Monoid.Coprod.range_inl_sup_range_inr, AlgebraicGeometry.Scheme.restrictFunctorΓ_inv_app, Nucleus.map_himp_apply, AlgebraicGeometry.instIsDomainCarrierObjOppositeOpensCarrierCarrierCommRingCatPresheafOpOpensTopOfIsIntegral, MeasurableSpace.comap_bot, MeasureTheory.OuterMeasure.mkMetric_top, GradedAlgebra.exists_finset_adjoin_eq_top_and_homogeneous_ne_zero, AffineBasis.tot, Algebra.mem_top, AlgebraicGeometry.AffineSpace.toSpecMvPolyIntEquiv_symm_apply, Filter.liminf_const_top, iSup_bool_eq, AlgebraicGeometry.Scheme.fromSpecStalk_toSpecΓ_assoc, MulticoequalizerDiagram.eq_inf, AlgebraicGeometry.AffineSpace.SpecIso_hom_appTop, OrderHom.le_map_sup_fixedPoints, iSup_split, IsOpen.exists_subset_affineIndependent_span_eq_top, SSet.iSup_subcomplexOfSimplex_prod_eq_top, IntermediateField.coe_bot, sInf_union, Submodule.mulMap_one_right_eq, OrdinalApprox.gfp_mem_range_gfpApprox, CompleteBooleanAlgebra.himp_eq, biSup_inf_le_biSup_inf, Subgroup.map_sup, AlgebraicGeometry.Scheme.zeroLocus_eq_univ_iff_subset_nilradical, AlgebraicGeometry.Scheme.Opens.topIso_inv, CategoryTheory.Sieve.mem_iff_pullback_eq_top, IntermediateField.lift_insepDegree_bot', CategoryTheory.Subfunctor.Subpresheaf.max_obj, AddCon.sup_def, NonUnitalAlgebra.coe_bot, Subsemiring.map_sup, Submonoid.comap_sup_map_of_injective, iInf_of_empty, scottContinuous_inf_right, sSup_eq_bot, inseparable_top, TensorAlgebra.adjoin_range_ι, NonUnitalStarAlgebra.coe_bot, NonUnitalStarAlgebra.mem_top, AlgHom.equalizer_eq_top, iSup_false, algebraicClosure.algebraicClosure_eq_bot, iInf_sup_iInf, inf_eq_iInf, AffineSubspace.comap_bot, Subgroup.map_eq_map_iff, iInf_false, AlgebraicGeometry.Scheme.Hom.id_appTop, CategoryTheory.Presheaf.imageSieve_app, AddSubgroup.mem_sup_left, bot_lt_iSup, iInf_eq_top, IsGaloisGroup.fixingSubgroup_top, CategoryTheory.Presieve.isSheaf_bot, Con.sup_eq_conGen, AddSubmonoid.op_sup, MeasurableSpace.generateFrom_empty, IsGaloisGroup.fixedPoints_top, RingCon.comap_bot, MeasureTheory.condLExp_bot', sSup_inf_sSup, CategoryTheory.MorphismProperty.ContainsIdentities.inf, AlgebraicGeometry.Scheme.ideal_ker_le_ker_ΓSpecIso_inv_comp, Algebra.toSubsemiring_eq_top, IntermediateField.botEquiv_symm, Subalgebra.op_bot, AddSubgroup.comap_map_eq, MeasureTheory.OuterMeasure.trim_top, IsLocalRing.closedPoint_mem_iff, AlgebraicGeometry.Scheme.isoSpec_inv_toSpecΓ, QuotientAddGroup.leftRel_eq_top, Subsemiring.op_sup, Algebra.adjoin_singleton_algebraMap, RingCon.matrix_top, SSet.Subcomplex.range_eq_top, AlgHom.fieldRange_eq_map, IntermediateField.relfinrank_bot_right, Field.exists_primitive_element, isGalois_iff_isGalois_top, AlgebraicGeometry.isIso_ΓSpec_adjunction_unit_app_basicOpen, AlgebraicGeometry.isNoetherian_iff_of_finite_affine_openCover, AlgebraicGeometry.Scheme.ι_toIso_inv, CategoryTheory.Subfunctor.range_toRange, UnitAddTorus.mFourierSubalgebra_closure_eq_top, InfiniteGalois.mem_bot_iff_fixed, Frm.Iso.mk_hom, Algebra.toSubring_eq_top, AddCon.toSetoid_top, sup_iSup, ProbabilityTheory.condIndep_bot_left, NumberField.adjoin_eq_top_of_infinitePlace_lt, Polynomial.IsSplittingField.splits_iff, sup_biInf_le_biInf_sup, AlgebraicGeometry.Scheme.IdealSheafData.equivOfIsAffine_apply, Partition.bot_notMem, SSet.Subcomplex.topIso_inv_app_coe, bot_lt_affineSpan, polynomialFunctions.starClosure_topologicalClosure, AlgebraicGeometry.Scheme.IdealSheafData.equivOfIsAffine_symm_apply, CategoryTheory.GrothendieckTopology.trivial_covering, AddSubgroup.mem_sup_of_normal_left, AffineEquiv.span_eq_top_iff, AlgebraicGeometry.Scheme.SpecΓIdentity_hom_app, AffineSubspace.direction_eq_top_iff_of_nonempty, StarSubalgebra.bot_toSubalgebra, Subfield.splits_bot, induced_const, Algebra.toSubring_bot, affineSpan_singleton_union_vadd_eq_top_of_span_eq_top, integralClosure_eq_top_iff, interior_convexHull_nonempty_iff_affineSpan_eq_top, OrderHom.gfp_const_inf_le, CompleteSublattice.top_mem, AlgebraicGeometry.Scheme.isoSpec_inv_preimage_zeroLocus, iSup₂_inf_eq, Subalgebra.LinearDisjoint.inf_eq_bot_of_commute, inf_biInf, TopologicalSpace.eq_top_iff_forall_inseparable, sSup_empty, AlgebraicGeometry.ProjectiveSpectrum.Proj.toOpen_toSpec_val_c_app_assoc, AlgebraicGeometry.PresheafedSpace.Γ_obj_op, IntermediateField.adjoin_zero, Submonoid.map_sup_comap_of_surjective, sInf_sup_le_iInf_sup, iInf_ite, Nucleus.restrict_toFun, IntermediateField.adjoin_univ, Finset.inf_id_eq_sInf, AlgebraicGeometry.Scheme.Modules.toOpen_fromTildeΓ_app, TopCat.GlueData.MkCore.V_id, AffineSubspace.linear_topEquiv, Subalgebra.algebra_isAlgebraic_bot_right, Submonoid.mem_sup, UniformSpace.toTopologicalSpace_top, Nucleus.map_himp_le, CategoryTheory.Subfunctor.Subpresheaf.preimage_eq_top_iff, Subalgebra.comm_trans_rTensorBot, AffineSubspace.not_wSameSide_bot, AlgebraicGeometry.Scheme.bot_mem_precoverage, CategoryTheory.Sieve.top_apply, Subalgebra.algebra_isAlgebraic_bot_left_iff, Algebra.EssFiniteType.adjoin_mem_finset, AlgebraicGeometry.Scheme.Hom.preimage_top, sSup_insert, Polynomial.IsSplittingField.adjoin_rootSet, AddCon.toSetoid_bot, AlgebraicGeometry.Scheme.isoSpec_hom, AlgebraicGeometry.Scheme.ι_toIso_inv_assoc, CategoryTheory.MorphismProperty.IsMultiplicative.inf, TopologicalSpace.Closeds.coe_eq_empty, iInf_option, AlgebraicGeometry.Proj.basicOpenToSpec_app_top, NonUnitalSubalgebra.prod_top, AlgebraicGeometry.Scheme.Hom.image_top_eq_opensRange, NonUnitalSubsemiring.closure_union, inf_iInf_nat_succ, Submonoid.mrange_inl_sup_mrange_inr, MeasurableSpace.map_const, sSup_union, AddSubsemigroup.map_sup, Submonoid.prod_bot_sup_bot_prod, NonUnitalStarAlgebra.coe_top, AlgebraicGeometry.Scheme.toSpecΓ_naturality_assoc, sSupHom.coe_bot, Sublocale.top_mem, AlgebraicGeometry.SurjectiveOnStalks.iff_of_isAffine, AlgebraicGeometry.SpecMap_ΓSpecIso_hom, Subalgebra.center_eq_top, CompleteBooleanAlgebra.sdiff_eq, NumberField.mixedEmbedding.exists_primitive_element_lt_of_isComplex, IsIntegrallyClosedIn.integralClosure_eq_bot, SSet.iSup_skeleton, integralClosure_idem, Subgroup.normal_subgroupOf_sup_of_le_normalizer, PrimeSpectrum.iSup_basicOpen_eq_top_iff, IsNormalClosure.adjoin_rootSet, instIsAbelianGaloisSubtypeMemIntermediateFieldBot, AffineBasis.tot', Finset.sup_set_eq_biUnion, Subring.op_sup, CategoryTheory.Subfunctor.Subpresheaf.min_obj, CategoryTheory.Sieve.functorPullback_bot, TopologicalSpace.Opens.isOpenEmbedding_obj_top, AlgebraicGeometry.LocallyRingedSpace.Γ_obj_op, Algebra.mem_bot, AlgebraicGeometry.AffineSpace.isoOfIsAffine_inv_appTop_coord, Subsemigroup.mul_mem_sup, AddMonoid.Coprod.mrange_lift, AlgebraicGeometry.AffineSpace.isoOfIsAffine_inv_over_assoc, biSup_symmDiff_biSup_le, CategoryTheory.Sieve.generate_bot, SSet.Subcomplex.eq_top_iff_of_hasDimensionLT, AlgebraicGeometry.Scheme.isoSpec_hom_naturality, RingCon.sup_def, CategoryTheory.Subfunctor.max_obj, iInf₂_eq_top, induced_topology_pure, ScottContinuous.inf₂, AlgebraicGeometry.Scheme.toSpecΓ_isoSpec_inv_assoc, FirstOrder.Language.Substructure.small_bot, Finset.iInf_insert, CompletelyDistribLattice.sdiff_le_iff, biSup_inf_biSup, WellFoundedGT.finite_ne_bot_of_iSupIndep, essInf_measure_zero, Polynomial.SplittingFieldAux.adjoin_rootSet, RingCon.subsingleton_quotient, iSup_inf_le_inf_iSup, CategoryTheory.Sieve.generate_top, StarSubalgebra.mem_bot, SSet.Subcomplex.topIso_inv_ι, AlgebraicGeometry.Scheme.toSpecΓ_naturality, AlgebraicGeometry.Scheme.Hom.quasiFiniteLocus_eq_top, Monoid.Coprod.mrange_eq, IntermediateField.fixingSubgroup_top, AffineSubspace.coe_eq_univ_iff, CategoryTheory.Sieve.equalizer_self, IntermediateField.bot_toSubalgebra, TopologicalSpace.Opens.mk_univ, MonoidHom.noncommCoprod_range, Algebra.toSubmodule_bot, AlgebraicGeometry.LocallyRingedSpace.Γ_Spec_left_triangle, Algebra.map_bot, Order.Coframe.MinimalAxioms.iInf_sup_eq, Subfield.mem_bot_iff_pow_eq_self, AlgebraicGeometry.RingedSpace.mem_top_basicOpen, Subfield.isTotallyReal_bot, PMF.toOuterMeasure_caratheodory, RingCon.inf_iff_and, Order.Frame.MinimalAxioms.sSup_inf_eq, CategoryTheory.MorphismProperty.DescendsAlong.inf, IntermediateField.relfinrank_top_right, CategoryTheory.MorphismProperty.CodescendsAlong.inf, IndiscreteTopology.eq_top, Algebra.adjoin_singleton_zero, AlgebraicGeometry.Scheme.Hom.toPartialMap_domain, CategoryTheory.Subfunctor.max_min, Algebra.coe_bot, CompleteLinearOrder.top_sdiff, Setoid.injective_iff_ker_bot, Submodule.toConvexCone_top, CategoryTheory.MorphismProperty.IsStableUnderBaseChange.inf, MeasureTheory.condLExp_bot, Con.coe_inf, FrameHomClass.toBoundedLatticeHomClass, sSupHom.bot_apply, AlgebraicGeometry.IsAffineOpen.isoSpec_inv, AlgebraicGeometry.Scheme.Γ_obj, CategoryTheory.Presieve.isSheafFor_top_sieve, AlgebraicGeometry.isLocallyNoetherian_iff_of_affine_openCover, SSet.hasDimensionLT_subcomplex_top_iff, IntermediateField.restrictScalars_top, SSet.Subcomplex.instSubsingletonHomToSSetBot, EuclideanGeometry.Sphere.orthRadius_center, TopologicalSpace.isOpen_top_iff, AlgebraicGeometry.IsLocallyArtinian.isArtinianRing_of_isAffine, AlgebraicGeometry.StructureSheaf.toPushforwardStalkAlgHom_apply, AlgebraicGeometry.Scheme.preimage_basicOpen_top, Submonoid.mem_sup_left, MvPolynomial.adjoin_range_X, AlgebraicGeometry.Scheme.ΓSpecIso_inv, NonUnitalAlgebra.range_id, IntermediateField.mem_top, AlgebraicGeometry.Scheme.isoSpec_hom_naturality_assoc, AlgebraicGeometry.Scheme.isoSpec_inv_image_zeroLocus, Field.primitive_element_iff_algHom_eq_of_eval', CategoryTheory.GrothendieckTopology.top_mem, IsDedekindDomain.integer_empty, AlgebraicGeometry.ΓSpec.locallyRingedSpaceAdjunction_counit_app', AffineSubspace.not_sSameSide_bot, adjoin_eq_top_of_conductor_eq_top, biInf_le_eq_inf, Subalgebra.eq_bot_of_finrank_one, AlgebraicGeometry.Scheme.Opens.ι_image_basicOpen_topIso_inv, IntermediateField.relrank_top_right, AddSubgroup.add_mem_sup, AlgebraicGeometry.SpecMap_ΓSpecIso_inv_toSpecΓ, AffineBasis.affineSpan_eq_top_of_toMatrix_left_inv, AlgebraicGeometry.Scheme.toSpecΓ_preimage_zeroLocus, continuous_bot, AddSubmonoid.closure_union, Subring.comap_map_eq, Algebra.coe_top, IntermediateField.LinearDisjoint.bot_left, TopologicalSpace.Opens.inclusion'_top_functor, FirstOrder.Language.Substructure.cg_bot, IntermediateField.isTotallyReal_bot, MeasureTheory.Measure.mkMetric_top, sup_eq_iSup, SSet.Subcomplex.eq_top_iff_contains_nonDegenerate, AlgebraicGeometry.Scheme.fromSpecStalk_appTop, Subalgebra.unop_top, AlgebraicGeometry.PresheafedSpace.Γ_map_op, Subalgebra.lTensorBot_one_tmul, Projectivization.Subspace.span_empty, AlgebraicGeometry.Scheme.id_appTop, sSup_compact_eq_top, Subalgebra.mulMap_bot_left_eq, iInf_option_elim, Order.Coframe.MinimalAxioms.sup_iInf_eq, AlgebraicGeometry.Scheme.Opens.toSpecΓ_appTop_assoc, sup_biSup, fixedPoints_submonoid_sup, AffineSubspace.bot_coe, inf_iSup₂_eq, Convex.interior_nonempty_iff_affineSpan_eq_top, Subring.closure_union, IntermediateField.instFiniteSubtypeMemBot, RingCon.coe_inf, Polynomial.adjoin_rootSet_eq_range, AffineSubspace.parallel_bot_iff_eq_bot, Subsemigroup.map_sup, CategoryTheory.Presheaf.equalizerSieve_self_eq_top, essSup_const_bot, iInf_sup_of_monotone, AlgebraicGeometry.Scheme.fromSpecStalk_toSpecΓ, AlgebraicGeometry.Scheme.ker_toSpecΓ, TopCat.Presheaf.map_germ_eq_Γgerm, Algebra.top_toSubmodule, SSet.Subcomplex.preimage_range, AddSubsemigroup.comap_sup_map_of_injective, AlgebraicGeometry.Scheme.ker_of_isAffine, Subalgebra.fg_of_submodule_fg, Subsemiring.unop_sup, AddSubsemigroup.mem_sup_left, AlgebraicGeometry.IsAffineOpen.algebraMap_Spec_obj, AlgebraicGeometry.isBasis_basicOpen, CompleteSublattice.bot_mem, AlgebraicGeometry.Scheme.Γevaluation_naturality_apply, CliffordAlgebra.adjoin_range_ι, AlgebraicGeometry.Flat.flat_and_surjective_iff_faithfullyFlat_of_isAffine, AlgebraicGeometry.LocallyRingedSpace.coe_toΓSpecSheafedSpace_hom_base_hom_apply_asIdeal, AddSubgroup.mem_sup', Polynomial.Splits.adjoin_rootSet_eq_range, IntermediateField.finrank_eq_one_iff, Setoid.top_def, AlgebraicGeometry.isIntegral_appTop_of_universallyClosed, IntermediateField.lift_top, AlgebraicGeometry.ΓSpec.adjunction_counit_app, IntermediateField.eq_bot_of_isAlgClosed_of_isAlgebraic, AlgebraicGeometry.tilde.isoTop_hom, CategoryTheory.Subfunctor.range_eq_top, borel_eq_top_of_discrete, IntermediateField.insepDegree_bot', AddSubsemigroup.unop_sup, Subgroup.comap_sup_eq, biSup_ge_eq_sup, AlgebraicGeometry.Scheme.Opens.toSpecΓ_appTop, CategoryTheory.Subfunctor.Subpresheaf.image_top, AlgebraicGeometry.ProjectiveSpectrum.Proj.toStalk_stalkMap_toSpec_assoc, GradedAlgebra.exists_finset_adjoin_eq_top_and_homogeneous, AlgebraicGeometry.Γ_restrict_isLocalization, IntermediateField.restrictScalars_eq_top_iff, CategoryTheory.GrothendieckTopology.bot_covers, AffineSubspace.smul_bot, isDenseEmbedding_pure, TopologicalSpace.Opens.coe_eq_empty, CategoryTheory.top_apply, CategoryTheory.Pairwise.cocone_pt, AlgebraicGeometry.ProjectiveSpectrum.Proj.toStalk_stalkMap_toSpec, AddSubgroup.map_le_map_iff, iSup_inf_le_sSup_inf, NonUnitalAlgebra.toSubmodule_bot, AddCon.inf_iff_and, iSup_split_single, AlgebraicGeometry.ΓSpecIso_obj_hom, CategoryTheory.GrothendieckTopology.Cover.preOneHypercover_sieve₁, Nucleus.comp_eq_right_iff_le, AlgebraicGeometry.Spec.germ_stalkMapIso_hom_assoc, PrimeSpectrum.iInf_localization_eq_bot, AlgebraicGeometry.Scheme.Hom.preimage_basicOpen_top, IntermediateField.bot_eq_top_of_finrank_adjoin_eq_one, Setoid.mk_eq_top, MeasureTheory.OuterMeasure.comap_top, CategoryTheory.Sieve.overEquiv_top, ProbabilityTheory.iIndep_comap_mem_iff, IntermediateField.botContinuousSMul, Field.exists_primitive_element_of_finite_bot, CategoryTheory.GrothendieckTopology.close_eq_top_iff_mem, TopologicalSpace.Opens.functor_obj_map_obj, Subalgebra.unop_bot, AffineSubspace.parallel_iff_direction_eq_and_eq_bot_iff_eq_bot, CategoryTheory.Sieve.arrows_eq_top_iff, inf_sSup_eq, AlgebraicGeometry.exists_appTop_π_eq_of_isAffine_of_isLimit, Setoid.sup_eq_eqvGen, NonUnitalSubring.map_sup, CategoryTheory.Sieve.arrows_top, Frm.Iso.mk_inv, QuotientAddGroup.rightRel_eq_top, AlgebraicGeometry.Scheme.OpenCover.iSup_opensRange, CategoryTheory.Sieve.functorPullback_inter, AddMonoid.Coprod.range_eq, AffineSubspace.mem_top, Algebra.bijective_algebraMap_iff, Subalgebra.topEquiv_apply, CategoryTheory.topologyOfClosureOperator_sieves, AddSubmonoid.sup_eq_range, Field.exists_primitive_element_of_finite_top, IntermediateField.adjoin_simple_eq_top_iff_of_isAlgebraic, continuous_top, AlgebraicGeometry.Scheme.Γevaluation_naturality_assoc, essInf_const_top, AddSubgroup.op_sup, AlgebraicGeometry.Scheme.IdealSheafData.coe_support_ofIdealTop, Order.Coframe.MinimalAxioms.sup_iInf₂_eq, SSet.instHasDimensionLTToSSetBotSubcomplex, CategoryTheory.MorphismProperty.bot_mem_precoverage, IntermediateField.rank_adjoin_eq_one_iff, CategoryTheory.ObjectProperty.instEssentiallySmallMax, Submonoid.map_sup, AlgebraicGeometry.ΓSpec.toOpen_comp_locallyRingedSpaceAdjunction_homEquiv_app, AddGroupTopology.toTopologicalSpace_top, sSup_diff_singleton_bot, AlgebraicGeometry.Scheme.Opens.toSpecΓ_top, TopologicalSpace.Closeds.coe_bot, IntermediateField.adjoin_one, IntermediateField.rank_top, CompleteDistribLattice.top_sdiff, CategoryTheory.sheafBotEquivalence_inverse_map_val, CategoryTheory.ObjectProperty.instSmallMin, OrdinalApprox.lfpApprox_ord_eq_lfp, CategoryTheory.Sieve.generate_of_singleton_isSplitEpi, AlgebraicGeometry.AffineSpace.SpecIso_inv_appTop_coord, CategoryTheory.Sieve.functorPullback_top, AlgebraicGeometry.germ_stalkClosedPointIso_hom_assoc, NonUnitalAlgebra.top_toNonUnitalSubsemiring, Quotient.subsingleton_iff, Setoid.bot_def, CategoryTheory.Subfunctor.Subpresheaf.bot_obj, Subfield.bot_eq_of_zMod_algebra, PowerBasis.adjoin_gen_eq_top, FrameHom.coe_toLatticeHom, StarSubalgebra.eq_top_iff, le_sInf_inter, IntermediateField.adjoin_root_eq_top, AffineSubspace.smul_top, ConvexCone.toPointedCone_top, Subgroup.mem_sup_of_normal_right, MeasurableSpace.measurableSet_bot_iff, Subfield.map_bot, iInf_ne_top_subtype, CategoryTheory.GrothendieckTopology.discrete_eq_top, TopCat.Presheaf.map_germ_eq_Γgerm_assoc, Submodule.comm_trans_rTensorOne, NonUnitalStarAlgebra.top_toNonUnitalSubalgebra, AddSubgroup.comap_sup_eq_of_le_range, MeasureTheory.OuterMeasure.zero_caratheodory, ConvexCone.mem_top, AddSubsemigroup.op_sup, CategoryTheory.sheafBotEquivalence_unitIso, IntermediateField.extendScalars_self, MeasureTheory.OuterMeasure.top_caratheodory, AlgebraicGeometry.StructureSheaf.globalSectionsIso_hom, IntermediateField.finrank_top, CategoryTheory.ObjectProperty.instIsStableUnderShiftByMin, CategoryTheory.ObjectProperty.instIsClosedUnderIsomorphismsMax, iSup_dite, NonUnitalStarAlgebra.range_eq_top, separableClosure.eq_top_iff, ProbabilityTheory.iIndepSet.iIndep_comap_mem, Algebra.top_toSubring, Subgroup.mem_sup_left, AlgebraicGeometry.ΓSpec.left_triangle, CategoryTheory.Subfunctor.eq_top_iff_isIso, iSup_inf_eq, AlgebraicGeometry.Scheme.Modules.toOpen_fromTildeΓ_app_assoc, CategoryTheory.sheafBotEquivalence_inverse_obj_val, AlgebraicGeometry.AffineSpace.comp_homOfVector, IntermediateField.top_toSubalgebra, Subfield.roots_X_pow_char_sub_X_bot, CategoryTheory.Subfunctor.iSup_min, Subsemigroup.closure_union, CompleteBooleanAlgebra.le_sup_inf, Subsemigroup.unop_sup, IntermediateField.relrank_top_left, Submodule.lTensorOne'_tmul, Subsemigroup.comap_sup_map_of_injective, AlgebraicGeometry.Scheme.instIsOpenImmersionToSpecΓOfIsQuasiAffine, CategoryTheory.ObjectProperty.prop_sup_iff, AlgebraicGeometry.Scheme.homOfLE_appTop, CompleteLatticeHomClass.toBoundedLatticeHomClass, Order.Frame.le_himp_iff, Submodule.rTensorOne'_tmul_one, IsGaloisGroup.fixedPoints_bot, AlgebraicGeometry.Scheme.isoSpec_inv_naturality_assoc, AlgebraicGeometry.IsAffineOpen.fromSpec_app_self, nhds_top, Con.toSetoid_eq_bot, CategoryTheory.MorphismProperty.instIsStableUnderRetractsMin, MeasureTheory.condExp_bot, Submodule.rTensorOne'_tmul, TopologicalSpace.Opens.mem_bot, AlgebraicGeometry.PresheafedSpace.restrictTopIso_hom, exists_root_adjoin_eq_top_of_isCyclic, AlgebraicGeometry.AffineSpace.hom_ext_iff, iSupIndep.injOn, AlgebraicGeometry.ΓSpec.unop_locallyRingedSpaceAdjunction_counit_app', MeasurableSpace.generateFrom_singleton_univ, Quantale.mul_bot, AlgebraicGeometry.Scheme.isoSpec_Spec_hom, sSup_atoms_eq_top, IntermediateField.finrank_eq_one_iff_eq_top, SSet.Subcomplex.iSup_ofSimplex_nonDegenerate_eq_top, AlgebraicGeometry.tilde.isIso_toOpen_top, CompleteLinearOrder.himp_bot, AddSubmonoid.exists_mem_sup, fixedPoints_addSubmonoid_sup, iSupIndep_ne_bot, Subfield.extendScalars_top, AlgebraicGeometry.HasRingHomProperty.appTop, AddSubgroup.unop_sup, AffineSubspace.mk'_top, iInf_sup_iInf_le, Algebra.zariskisMainProperty_iff_exists_saturation_eq_top, iSup_sup_eq, NumberField.is_primitive_element_of_infinitePlace_lt, StarSubalgebra.coe_bot, Subring.prod_bot_sup_bot_prod, UniformSpace.toTopologicalSpace_bot, CompHausLike.LocallyConstant.locallyConstantIsoContinuousMap_hom, CategoryTheory.Presheaf.isLocallySurjective_iff_range_sheafify_eq_top, MonoidAlgebra.exists_finset_adjoin_eq_top, AlgebraicGeometry.Scheme.isNilpotent_iff_basicOpen_eq_bot, IntermediateField.finrank_adjoin_eq_one_iff, le_iSup_inf_iSup, IntermediateField.bot_eq_top_of_rank_adjoin_eq_one, CategoryTheory.Subfunctor.min_obj, sSup_eq_bot', SeparationQuotient.inseparableSetoid_eq_top_iff, AlgebraicGeometry.Scheme.isPullback_toSpecΓ_toSpecΓ, CompleteLinearOrder.le_himp_iff, AlgebraicGeometry.Scheme.ΓSpecIso_naturality, supsSupHom_apply, AlgebraicGeometry.Scheme.restrictFunctorΓ_hom_app, Subsemigroup.mem_sup_left, Partition.bot_notMem', AddSubgroup.codisjoint_addSubgroupOf_sup, Nucleus.range_subset_range, IntermediateField.finrank_bot, Subgroup.subgroupOf_sup, CategoryTheory.Pairwise.cocone_ι_app, PowerBasis.adjoin_eq_top_of_gen_mem_adjoin, iSup_ite, NonUnitalAlgebra.eq_top_iff, AlgebraicGeometry.SheafedSpace.Γ_map, IsIntegrallyClosed.integralClosure_eq_bot, IsIntegrallyClosed.integralClosure_eq_bot_iff, infsInfHom_apply, Subgroup.comap_sup_comap_le, Polynomial.IsSplittingField.adjoin_rootSet', IntermediateField.sepDegree_bot, AlgebraicGeometry.germ_stalkClosedPointIso_hom, ProbabilityTheory.Kernel.iIndep_comap_mem_iff, AddQuantale.add_bot, AlgebraicGeometry.Scheme.toSpecΓ_isoSpec_inv, Submodule.rTensorOne_tmul, AlgebraicGeometry.IsFinite.instHasAffinePropertyAndIsAffineFiniteCarrierObjOppositeOpensCarrierCarrierCommRingCatPresheafOpOpensTopHomAppTop, Finset.inf_set_eq_iInter, AlgebraicGeometry.IsAffineOpen.instIsAffineToSchemeBasicOpen, IntermediateField.finInsepDegree_bot', iSup_option_elim, OrdinalApprox.gfpApprox_ord_eq_gfp, CategoryTheory.Coverage.eq_top_pullback, Monoid.Coprod.mrange_lift, Subsemigroup.op_sup, inf_iSup_eq, CategoryTheory.MorphismProperty.RespectsLeft.inf, IntermediateField.restrictScalars_bot_eq_self, sInf_sup_eq, AlgebraicGeometry.Scheme.toIso_inv_ι_assoc, AlgebraicGeometry.SheafedSpace.restrictTopIso_inv, AlgebraicGeometry.Scheme.toSpecΓ_appTop, Order.Coframe.top_sdiff, IntermediateField.fg_bot, Filter.limsInf_top, AlgebraicGeometry.Scheme.IdealSheafData.support_eq_bot_iff, CategoryTheory.Presheaf.isLocallySurjective_iff_range_sheafify_eq_top', AddSubmonoid.add_mem_sup, GroupTopology.toTopologicalSpace_bot, AlgHom.range_eq_top, CategoryTheory.MorphismProperty.bijective_eq_sup, Algebra.adjoin_singleton_one, CategoryTheory.Subfunctor.Subpresheaf.epi_iff_range_eq_top, NonUnitalAlgebra.mem_top, NonUnitalStarAlgebra.toNonUnitalSubalgebra_eq_top, SSet.Subcomplex.image_top, CompleteSublattice.coe_bot, Submonoid.op_sup, AffineSubspace.affineSpan_eq_top_iff_vectorSpan_eq_top_of_nonempty, IntermediateField.adjoin_empty, AffineSubspace.topEquiv_symm_apply_coe, Subgroup.map_le_map_iff, FrameHom.coe_toInfTopHom, Field.Emb.Cardinal.eq_bot_of_not_nonempty, AlgebraicGeometry.StructureSheaf.algebraMap_obj_top_bijective, TopologicalSpace.Opens.inclusion'_map_eq_top, IntermediateField.rank_bot, Subalgebra.centralizer_eq_top_iff_subset, AlgebraicGeometry.IsAffineOpen.fromSpec_preimage_self, iSup_eq_dif, separableClosure_inf_perfectClosure, iInf_bool_eq, MeasureTheory.Measure.add_top, AlgebraicGeometry.LocallyRingedSpace.Γevaluation_naturality_apply, Algebra.top_toSubsemiring, Subalgebra.rTensorBot_tmul_one, Quantale.sup_mul_distrib, Subgroup.comap_map_eq, AlgebraicGeometry.Spec_zeroLocus_eq_zeroLocus, sdiff_eq_sInf, AlgebraicGeometry.Scheme.Opens.ι_appTop, Subgroup.mem_sup, iInf_split, Subfield.bot_eq_of_charZero, AddSubmonoid.mrange_inl_sup_mrange_inr, Monoid.Coprod.range_lift, IsIntegrallyClosedIn.integralClosure_eq_bot_iff, CompletelyDistribLattice.le_himp_iff, AffineMap.map_top_of_surjective, AlgebraicGeometry.Scheme.map_basicOpen, CategoryTheory.Subfunctor.Subpresheaf.top_obj, NonUnitalAlgebra.adjoin_univ, AlgebraicGeometry.isRetrocompact_basicOpen, AlgebraicGeometry.Scheme.isoSpec_inv_naturality, Con.sup_def, IntermediateField.bot_eq_top_iff_finrank_eq_one, FrameHom.map_sSup', inf_iInf, IntermediateField.adjoin_eq_bot_iff, iSup_inf_le_inf_sSup, MeasureTheory.SimpleFunc.simpleFunc_bot', Submodule.rTensorOne_tmul_one, sup_iInf_le_iInf_sup, AlgebraicGeometry.Scheme.Hom.quasiFiniteLocus_eq_top_iff, AlgebraicGeometry.instQuasiSeparatedToSpecΓOfQuasiSeparatedSpaceCarrierCarrierCommRingCat, CategoryTheory.Sieve.pullback_eq_top_of_mem, Order.Frame.MinimalAxioms.inf_iSup₂_eq, CategoryTheory.GrothendieckTopology.trivial_eq_bot, biInf_ge_eq_inf, AlgebraicGeometry.Scheme.comp_appTop_assoc, FirstOrder.Language.Substructure.closure_empty, AlgebraicGeometry.Proj.basicOpen_one, Subgroup.map_le_map_iff', Filter.limsup_bot, IntermediateField.lift_sepDegree_bot', CompletelyDistribLattice.top_sdiff, Subalgebra.op_top, AlgebraicGeometry.Scheme.isoSpec_Spec, CategoryTheory.Subfunctor.Subpresheaf.range_id, AffineSubspace.coe_eq_bot_iff, iSup₂_eq_bot, essSup_measure_zero, Order.Frame.MinimalAxioms.inf_iSup_eq, Algebra.adjoin_singleton_natCast, sup_iSup_nat_succ, IntermediateField.LinearDisjoint.inf_eq_bot, Subgroup.sup_subgroupOf_eq, Field.exists_primitive_element_iff_finite_intermediateField, Filter.EventuallyEq.symmDiff, AlgebraicGeometry.IsClosedImmersion.hasAffineProperty, Field.Emb.Cardinal.filtration_apply, FirstOrder.Language.Substructure.map_bot, AlgebraicGeometry.Scheme.preimage_opensRange_toSpecΓ, CategoryTheory.Sieve.pullback_inter, indiscreteTopology_iff, AlgebraicGeometry.Scheme.Spec.algebraMap_residueFieldIso_inv, Subalgebra.bot_eq_top_iff_finrank_eq_one, Subalgebra.mulMap_bot_right_eq, AlgebraicGeometry.Scheme.eq_zeroLocus_of_isClosed_of_isAffine, Finset.iInf_union, CategoryTheory.MorphismProperty.instHasOfPrecompPropertyMin, AddMonoidAlgebra.exists_finset_adjoin_eq_top, AlgebraicGeometry.Scheme.Opens.ι_preimage_self, CategoryTheory.Subfunctor.instIsIsoFunctorTypeιTop, CategoryTheory.Subfunctor.bot_obj, AlgebraicGeometry.Scheme.toSpecΓ_image_zeroLocus, AlgebraicGeometry.PresheafedSpace.ofRestrict_top_c, Finset.sup_eq_iSup, AlgebraicGeometry.toSpecΓ_SpecMap_ΓSpecIso_inv_assoc, Order.Frame.MinimalAxioms.inf_sSup_eq, AlgebraicGeometry.SheafedSpace.Γ_obj_op, scottContinuous_inf_left, Algebra.isAlgebraic_iff, TopologicalSpace.Closeds.coe_top, AddSubgroup.mem_sup_right, AddSubgroup.sup_eq_closure, Subalgebra.lTensorBot_symm_apply, AlgebraicGeometry.IsAffineOpen.fromSpec_app_self_apply, NonUnitalStarAlgebra.comap_top, Submonoid.unop_sup, FreeAlgebra.adjoin_range_ι, CategoryTheory.MorphismProperty.inf, ContinuousMap.subalgebra_topologicalClosure_eq_top_of_separatesPoints, Nucleus.sInf_apply, Con.toSetoid_bot, MeasureTheory.sigmaFinite_bot_iff, SSet.Subcomplex.topIso_inv_ι_assoc, CategoryTheory.MorphismProperty.RespectsRight.inf, AlgebraicGeometry.Scheme.inv_appTop, IntermediateField.finrank_top', Order.Coframe.sdiff_le_iff, AlgebraicGeometry.isLocalization_away_of_isAffine, AlgebraicGeometry.IsAffineOpen.ι_basicOpen_preimage, iSup_neg, CategoryTheory.ObjectProperty.InheritedFromSource.instMin, AffineSubspace.direction_top, Affine.Simplex.span_eq_top, ωScottContinuous.sup, IntermediateField.lift_bot, NumberField.mixedEmbedding.exists_primitive_element_lt_of_isReal, Finset.iSup_insert, IsGalois.fixedField_top, WithBot.sInf_empty, AddSubsemigroup.mem_sup_right, TopologicalSpace.Opens.eq_bot_or_top, AlgebraicGeometry.morphismRestrict_appTop, IsLocalRing.closed_point_mem_iff, CompleteLinearOrder.compare_eq_compareOfLessAndEq, AlgebraicGeometry.Scheme.IdealSheafData.ofIdealTop_ideal, Finset.iInf_insert_update, Algebra.IsAlgebraic.isNormalClosure_iff, iInf_top, AddCon.sup_eq_addConGen, MeasureTheory.hittingAfter_eq_sInf, MeasureTheory.condLExp_bot_ae_eq, NonUnitalSubring.closure_union, AlgebraicGeometry.Scheme.Hom.preimage_opensRange, StarSubalgebra.toSubalgebra_eq_top, Submonoid.mul_mem_sup, AlgebraicGeometry.ΓSpecIso_hom_stalkClosedPointIso_inv, StarSubalgebra.mem_top, AlgebraicGeometry.AffineSpace.toSpecMvPolyIntEquiv_apply, CategoryTheory.GrothendieckTopology.bot_covering, Submodule.mulMap_one_left_eq, AffineSubspace.isEmpty_bot, AlgebraicGeometry.LocallyRingedSpace.Γ_map, Setoid.inf_def, IntermediateField.adjoin_simple_eq_bot_iff, IntermediateField.relfinrank_bot_left, AlgebraicGeometry.isCompactOpen_iff_eq_basicOpen_union, exists_subset_affineIndependent_affineSpan_eq_top, AlgebraicGeometry.Spec.algebraMap_stalkIso_inv, RingCon.matrix_bot, CompleteLinearOrder.le_total, Algebra.adjoin_adjoin_coe_preimage, iInf_union, AlgebraicGeometry.eq_top_of_sigmaSpec_subset_of_isCompact, Algebra.adjoin_top, Subgroup.mul_mem_sup, TopologicalSpace.Opens.mk_empty, FrameHomClass.toInfTopHomClass, iSup_ne_bot_subtype, AlgebraicGeometry.PresheafedSpace.Γ_obj, AffineSubspace.perpBisector_eq_top, IntermediateField.adjoin_intCast, Algebra.adjoin_root_eq_top_of_isSplittingField, AlgebraicGeometry.PresheafedSpace.toRestrictTop_c, Algebra.surjective_algebraMap_iff, IntermediateField.sepDegree_bot', iInf_dite, CategoryTheory.Presheaf.isSheaf_bot, AlgebraicGeometry.Scheme.isoSpec_inv_toSpecΓ_assoc, IntermediateField.LinearDisjoint.bot_right, Filter.limsSup_bot, ωScottContinuous.inf, AffineSubspace.pointwise_vadd_bot, Field.primitive_element_iff_minpoly_degree_eq, Algebra.toSubmodule_eq_top, AlgHom.fieldRange_eq_top, iSup_emptyset, AddSubgroup.mem_sup_of_normal_right, discreteTopology_bot, DualNumber.range_inlAlgHom_sup_adjoin_eps, AlgebraicGeometry.Scheme.toOpen_eq, isPurelyInseparable_iff_perfectClosure_eq_top, RingCon.coe_bot, IntermediateField.rank_eq_one_iff, Subalgebra.fg_bot, Algebra.adjoin_univ, AlgHom.equalizer_same, Subgroup.comap_sup_eq_of_le_range, AddMonoid.Coprod.range_lift, IntermediateField.eq_bot_of_isPurelyInseparable_of_isSeparable, CategoryTheory.Subfunctor.Subpresheaf.iSup_min, CategoryTheory.Subfunctor.image_top, TopologicalSpace.Closeds.coe_eq_univ, iInf₂_sup_eq, AffineIndependent.affineSpan_eq_top_iff_card_eq_finrank_add_one, IntermediateField.adjoin_root_eq_top_of_isSplittingField, CompleteDistribLattice.sdiff_le_iff, AlgebraicGeometry.ΓSpecIso_inv_ΓSpec_adjunction_homEquiv, algebraicClosure.eq_top_iff, AffineSubspace.not_sOppSide_bot, CategoryTheory.Sieve.functorPushforward_union, Quantale.mul_sup_distrib, CompleteDistribLattice.MinimalAxioms.iInf_sup_le_sup_sInf, IsOpen.exists_between_affineIndependent_span_eq_top, Setoid.eq_top_iff, iSup_inf_le_iSup_inf, AffineSubspace.top_coe, Setoid.sup_def, AlgebraicGeometry.toSpecΓ_SpecMap_ΓSpecIso_inv, Field.primitive_element_iff_minpoly_natDegree_eq, sInfHom.coe_top, AlgebraicGeometry.instIsIsoModulesSpecOfCarrierFromTildeΓUnitOpensCarrierCarrierCommRingCatRingCatSheaf, Subalgebra.fg_top, NonUnitalAlgebra.toSubmodule_eq_top, isDenseInducing_pure, AlgebraicGeometry.Scheme.Hom.inv_appTop, SSet.iSup_skeletonOfMono, affineSpan_coe_preimage_eq_top, AffineSubspace.pointwise_vadd_top, IntermediateField.isScalarTower_over_bot, ProbabilityTheory.condVar_bot', AddSubgroup.mem_sup, TwoSidedIdeal.top_ringCon, Nucleus.mem_range, AffineSubspace.perpBisector_self, AlgebraicGeometry.Spec_zeroLocus, IntermediateField.bot_toSubfield, AlgebraicGeometry.Scheme.ΓSpecIso_inv_naturality, IntermediateField.topEquiv_symm_apply_coe, AffineSubspace.span_empty, biInf_sup_le_biInf_sup, NonUnitalStarAlgebra.map_bot, AffineSubspace.map_bot, affineSpan_eq_top_of_nonempty_interior, Subalgebra.lTensorBot_tmul, NonUnitalAlgebra.adjoin_empty, SSet.iSup_range_eq_top_of_isColimit, Subring.unop_sup, IntermediateField.rank_top', Subalgebra.topEquiv_symm_apply_coe, AlgebraicGeometry.Scheme.SpecΓIdentity_inv_app, MeasureTheory.OuterMeasure.boundedBy_top, AlgebraicGeometry.Scheme.Pullback.diagonalCoverDiagonalRange_eq_top_of_injective, CategoryTheory.MorphismProperty.IsStableUnderComposition.inf, iInf_eq_dif, Language.mem_inf, AddMonoid.Coprod.mrange_inl_sup_mrange_inr, CompleteSublattice.coe_top, CategoryTheory.ObjectProperty.isoClosure_sup, CategoryTheory.Sieve.functorPushforward_bot, iSup_sup, IntermediateField.topEquiv_apply, ProbabilityTheory.condIndep_bot_right, Order.Frame.himp_bot, mem_bot_iff_intCast, AlgebraicGeometry.basicOpen_eq_of_affine', IntermediateField.adjoin_natCast, AlgebraicGeometry.AffineSpace.isoOfIsAffine_hom_appTop, sInf_insert, TopologicalSpace.Opens.coe_bot, AffineSubspace.topEquiv_apply, AlgebraicGeometry.instIsDominantToSpecΓOfCompactSpaceCarrierCarrierCommRingCat, AlgebraicGeometry.Scheme.preimage_eq_top_of_closedPoint_mem, iInf_extend_top, TopologicalSpace.Closeds.isClopen_singleton_bot, IsCyclotomicExtension.singleton_one, MeasureTheory.hittingBtwn_eq_sInf, SSet.finite_subcomplex_top_iff, AlgebraicGeometry.Scheme.isoSpec_Spec_inv, Algebra.adjoin_restrictScalars, TwoSidedIdeal.sup_ringCon, fixedPoints_subgroup_sup, NonUnitalAlgebra.coe_top, QuotientGroup.rightRel_eq_top, NonUnitalAlgebra.map_top, IntermediateField.isPurelyInseparable_bot, AddQuantale.bot_add, AlgebraicGeometry.Scheme.germ_stalkClosedPointTo_assoc, AlgebraicGeometry.Scheme.topIso_hom, MeasurableSpace.generateFrom_singleton, sup_sInf_le_iInf_sup, AlgebraicGeometry.finite_appTop_of_universallyClosed, AffineSubspace.span_univ, IntermediateField.top_toSubfield, Finset.iSup_insert_update, TopologicalSpace.IsOpenCover.iSup_eq_top, CategoryTheory.MorphismProperty.instHasOfPostcompPropertyMin, CategoryTheory.ObjectProperty.instIsStableUnderShiftMin, Nucleus.iInf_apply, CategoryTheory.Sieve.pushforward_union, AffineSubspace.map_eq_bot_iff, Submonoid.mem_sup_right, NonUnitalAlgebra.to_subring_eq_top, RingOfIntegers.exponent_eq_one_iff, Subsemigroup.mem_sup_right, AddSubsemigroup.map_sup_comap_of_surjective, AlgebraicGeometry.specTargetImageFactorization_app_injective, Subalgebra.rTensorBot_symm_apply, Algebra.comap_top, AlgebraicGeometry.PresheafedSpace.Γ_map, NonUnitalAlgebra.toNonUnitalSubring_eq_top, ProbabilityTheory.indep_bot_right, AlgebraicGeometry.StructureSheaf.toOpenₗ_top_bijective, ProbabilityTheory.condVar_bot_ae_eq, AffineSubspace.direction_bot, biInf_sup_biInf, iInf_sup_eq, CategoryTheory.Sieve.functorInclusion_top_isIso, MaximalSpectrum.iInf_localization_eq_bot, CompleteBooleanAlgebra.top_le_sup_compl, AffineBasis.isUnit_toMatrix_iff, Algebra.IsCentral.center_eq_bot, IsGaloisGroup.fixingSubgroup_bot, Set.exists_seq_iSup_eq_top_iff_countable, AlgebraicGeometry.AffineSpace.isoOfIsAffine_hom, sInfHomClass.toInfTopHomClass, AffineSubspace.instNonemptySubtypeMemTop, Finset.sup_eq_sSup_image, ProbabilityTheory.indep_bot_left, Subsemiring.closure_union, AlgebraicGeometry.affineAnd_apply, IntermediateField.fg_top, instDiscreteMeasurableSpace, AddSubmonoid.mem_sup, Submodule.lTensorOne_symm_apply, MeasureTheory.OuterMeasure.top_apply', AlgebraicGeometry.AffineSpace.reindex_appTop_coord, AlgebraicGeometry.Spec.algebraMap_stalkIso_inv_assoc, IntermediateField.finInsepDegree_bot, MvPolynomial.supported_empty, IsCyclotomicExtension.iff_adjoin_eq_top, AddSubmonoid.map_sup_comap_of_surjective, AlgEquiv.fieldRange_eq_top, LinearOrder.bot_topologicalSpace_eq_generateFrom, MeasureTheory.SimpleFunc.simpleFunc_bot, Field.Emb.Cardinal.iSup_adjoin_eq_top, AlgebraicGeometry.ΓSpec.toSpecΓ_of, affineSpan_eq_bot, TopCat.Presheaf.IsSheaf.isSheafPairwiseIntersections, ProbabilityTheory.condVar_bot, fixedPoints.lfp_eq_sSup_iterate, Filter.limsup_const_bot, sInf_diff_singleton_top, JacobsonNoether.exists_separable_and_not_isCentral', AlgebraicGeometry.exists_appTop_π_eq_of_isLimit, TopCat.Presheaf.isGluing_iff_pairwise, AlgebraicGeometry.IsAffine.affine, StarSubalgebra.coe_top, AlgebraicGeometry.AffineTargetMorphismProperty.IsLocal.to_basicOpen, AlgebraicGeometry.SheafedSpace.Γ_obj, Submodule.lTensorOne'_one_tmul, Subsemiring.prod_bot_sup_bot_prod, IntermediateField.finrank_bot', iInf_or, AlgebraicGeometry.Scheme.mem_basicOpen_top, ConvexCone.isGenerating_top, Quantale.bot_mul, CategoryTheory.Pretopology.toGrothendieck_bot, Partition.bot_lt_of_mem, AlgebraicGeometry.Scheme.Opens.ι_image_basicOpen', AlgebraicGeometry.LocallyRingedSpace.Γevaluation_naturality_assoc, biInf_inf, Algebra.eq_top_iff, Filter.liminf_bot, Setoid.mk_eq_bot, AffineSubspace.comap_top, fourierSubalgebra_closure_eq_top, Filter.limsInf_bot, AlgebraicGeometry.AffineSpace.homOverEquiv_apply, NonUnitalStarAlgebra.range_id, Subalgebra.rank_eq_one_iff, DiscreteTopology.eq_bot, sSup_pair, Subalgebra.prod_top, CategoryTheory.Sieve.overEquiv_symm_top, NonUnitalAlgebra.comap_top, MeasureTheory.Measure.toOuterMeasure_top, NonUnitalStarAlgebra.map_top, AlgebraicGeometry.Scheme.comp_appTop, Finset.inf_eq_sInf_image, IntermediateField.botEquiv_def, iInf_sup_le_iInf_sup, AlgebraicGeometry.IsClosedImmersion.isAffine_surjective_of_isAffine, CategoryTheory.sieve₁'_toPreOneHypercover_eq_top, subalgebra_top_finrank_eq_submodule_top_finrank, IntermediateField.adjoin_eq_top_iff, Finset.inf_eq_iInf, AddSubsemigroup.closure_union, Valuation.Integers.integralClosure, Con.inf_iff_and, IntermediateField.finSepDegree_bot', IsCyclotomicExtension.Rat.adjoin_singleton_eq_top, AlgebraicGeometry.Scheme.fromSpecStalk_app, Subgroup.op_sup, CategoryTheory.ObjectProperty.instContainsZeroMinOfIsClosedUnderIsomorphisms, AlgebraicGeometry.AffineSpace.toSpecMvPolyIntEquiv_comp, Algebra.IsUnramifiedAt.exists_notMem_forall_ne_mem_and_adjoin_eq_top, AlgebraicGeometry.isField_of_universallyClosed, polynomialFunctions.topologicalClosure, sInf_empty, RingCon.range_mkₐ, AlgebraicGeometry.LocallyRingedSpace.Γevaluation_eq_zero_iff_notMem_basicOpen, stronglyMeasurable_bot_iff, AlgebraicGeometry.Scheme.Spec.algebraMap_residueFieldIso_inv_assoc, IntermediateField.eq_bot_of_isSepClosed_of_isSeparable, WellFoundedLT.finite_ne_bot_of_iSupIndep, MeasurableSpace.map_top, iSup_eq_if, FirstOrder.Language.Substructure.elementarySkolem₁Reduct.instSmall, Algebra.map_top, CategoryTheory.Subfunctor.top_obj, Subalgebra.eq_bot_of_rank_le_one, AlgebraicGeometry.Scheme.Hom.smoothLocus_eq_top, CategoryTheory.Sieve.functorPushforward_top, AddMonoid.Coprod.mrange_eq, NonUnitalStarSubalgebra.center_eq_top, AlgebraicGeometry.IsAffineOpen.fromSpec_top, IntermediateField.bot_eq_top_of_finrank_adjoin_le_one, CategoryTheory.Presieve.isSeparatedFor_top, Submodule.lTensorOne_tmul, AddQuantale.sup_add_distrib, AddCon.toSetoid_eq_bot, IntermediateField.relrank_bot_right, Submonoid.closure_union, AlgebraicGeometry.Scheme.IdealSheafData.support_top, Subalgebra.comm_trans_lTensorBot, InfiniteGalois.fixedField_bot, Subgroup.codisjoint_subgroupOf_sup, CompletelyDistribLattice.himp_bot, NonUnitalAlgebra.map_bot, sup_sInf_eq, AlgebraicGeometry.Scheme.map_basicOpen_map, ωScottContinuous.top, sInf_le, AddSubgroup.comap_sup_eq, StarAlgHom.range_eq_map_top, IsSepClosed.separableClosure_eq_bot_iff, Subfield.closure_empty, AlgebraicGeometry.Scheme.Hom.comp_appTop, NonUnitalStarAlgebra.eq_top_iff, Subalgebra.bot_eq_top_of_finrank_eq_one, AlgebraicGeometry.IsIntegralHom.hasAffineProperty, RingCon.toCon_eq_top, AlgebraicGeometry.Scheme.germ_stalkClosedPointTo_Spec, MeasureTheory.OuterMeasure.map_top_of_surjective, AlgebraicGeometry.instHasAffinePropertyIsomorphismsSchemeAndIsAffineIsIsoCommRingCatAppTop, eq_bot_of_singletons_open, NonUnitalSubsemiring.map_sup, MvPolynomial.supported_univ, conductor_eq_top_iff_adjoin_eq_top, fixedPoints.gfp_eq_sInf_iterate, Subalgebra.isAlgebraic_bot_iff, Subgroup.mem_sup_of_normal_left, TwoSidedIdeal.bot_ringCon, CategoryTheory.MorphismProperty.instHasTwoOutOfThreePropertyMin, CategoryTheory.Sieve.union_apply, AdjoinRoot.adjoinRoot_eq_top, AlgebraicGeometry.Scheme.IsLocallyDirected.V_self, borel_eq_top_of_countable, CategoryTheory.GrothendieckTopology.top_mem', TopCat.Presheaf.Γgerm_res_apply, IntermediateField.extendScalars_top, perfectClosure.eq_bot_of_isSeparable, AddSubsemigroup.add_mem_sup, ConvexCone.coe_top, MeasurableSpace.measurableSet_top, CategoryTheory.Presheaf.equalizerSieve_eq_top_iff, ProbabilityTheory.Kernel.indep_bot_left, IntermediateField.fixedField_bot, Order.Frame.MinimalAxioms.inf_sSup_le_iSup_inf, NonUnitalAlgebra.range_eq_top, Set.vadd_set_symmDiff, AlgebraicGeometry.LocallyRingedSpace.Γevaluation_ne_zero_iff_mem_basicOpen, AlgebraicGeometry.basicOpen_eq_of_affine, AlgebraicGeometry.LocallyRingedSpace.Γ_map_op, OrdinalApprox.lfp_mem_range_lfpApprox, FrameHom.toFun_eq_coe, AlgebraicGeometry.Scheme.ΓSpecIso_naturality_assoc, AlgebraicGeometry.IsAffineOpen.fromSpec_toSpecΓ_assoc, Complex.adjoin_I, CategoryTheory.Sieve.inter_apply, SSet.Subcomplex.range_eq_top_iff, CategoryTheory.MorphismProperty.IsStableUnderCobaseChange.inf, Con.toSetoid_top, Partition.coe_removeBot, AlgebraicGeometry.PresheafedSpace.toRestrictTop_base, AlgebraicGeometry.LocallyRingedSpace.isUnit_res_toΓSpecMapBasicOpen, Finset.exists_inf_eq_iInf, Monoid.Coprod.range_eq, AlgebraicGeometry.ΓSpec_adjunction_homEquiv_eq, ContinuousMap.elemental_id_eq_top, Subgroup.closure_union, MeasureTheory.OuterMeasure.map_top, Algebra.FiniteType.out, iSup_option, polynomialFunctions_closure_eq_top', Finset.inf_univ_eq_iInf, AffineSubspace.affineSpan_eq_top_iff_vectorSpan_eq_top_of_nontrivial, Projectivization.Subspace.span_univ, AlgebraicGeometry.Scheme.toIso_inv_ι, FirstOrder.Language.Substructure.instIsEmptySubtypeMemBotOfConstants, IntermediateField.insepDegree_bot, IsAlgClosed.algebraicClosure_eq_bot_iff, IntermediateField.rank_bot', OpenSubgroup.toOpens_top, iInf_eq_if, TopologicalSpace.Opens.map_top, CategoryTheory.Functor.imageSieve_map, AddSubmonoid.map_sup, AlgebraicGeometry.LocallyRingedSpace.toΓSpec_preimage_zeroLocus_eq, AlgebraicGeometry.LocallyRingedSpace.SpecΓIdentity_hom_app, CategoryTheory.Sieve.pullback_top, IntermediateField.insepDegree_top, ProjectiveSpectrum.basicOpen_one, Filter.blimsup_false, AlgebraicGeometry.Scheme.Γevaluation_naturality, MeasureTheory.condExp_bot', iSup_symmDiff_iSup_le, sup_iInf₂_eq, IsGalois.tfae, Subfield.card_bot, RingCon.coe_top, OpenAddSubgroup.toOpens_top, AlgebraicGeometry.Scheme.IdealSheafData.ker_glueDataObjι_appTop, CategoryTheory.Sieve.uliftFunctorInclusion_top_isIso, AlgebraicGeometry.Spec.germ_stalkMapIso_hom, iSup_inf_of_monotone, CategoryTheory.ObjectProperty.instSmallMax, Finset.sup_univ_eq_iSup, IntermediateField.LinearDisjoint.eq_bot_of_self, NonUnitalStarSubalgebra.prod_top, CategoryTheory.ObjectProperty.prop_inf_iff, ProbabilityTheory.Kernel.indep_bot_right, separableClosure.separableClosure_eq_bot, iInf_insert, AlgebraicGeometry.Spec.fromSpecStalk_eq, NonUnitalAlgebra.top_toSubring, CategoryTheory.ObjectProperty.instSmallMin_1, AddMonoid.Coprod.range_inl_sup_range_inr, ContinuousMapZero.elemental_eq_top, AlgebraicGeometry.AffineSpace.isoOfIsAffine_inv_over, IntermediateField.fg_top_iff, CategoryTheory.MorphismProperty.toSet_max, fixedPoints_addSubgroup_sup, CategoryTheory.GrothendieckTopology.top_covers, CategoryTheory.Presieve.factorsThru_top, NonUnitalAlgebra.mem_bot, AlgebraicGeometry.LocallyRingedSpace.Γevaluation_naturality, AlgebraicGeometry.Proj.iSup_basicOpen_eq_top, AlgebraicGeometry.IsAffineOpen.fromSpec_preimage_basicOpen', le_sSup, induced_top, AlgebraicGeometry.Scheme.Hom.opensRange_of_isIso, AlgebraicGeometry.LocallyRingedSpace.toΓSpecMapBasicOpen_eq, Submodule.lTensorOne_one_tmul, sSup_inter_le, IntermediateField.relfinrank_top_left, CompleteLinearOrder.sdiff_le_iff, iSup_of_empty, AlgebraicGeometry.Scheme.Opens.instIsIsoCommRingCatAppLEιTopToScheme, sInf_eq_top, iInf_neg, AlgebraicGeometry.IsAffineOpen.ΓSpecIso_hom_fromSpec_app, CategoryTheory.sheafBotEquivalence_counitIso, GroupTopology.toTopologicalSpace_top, biSup_sup, iInf_lt_top, Con.toSetoid_eq_top, MeasureTheory.OuterMeasure.dirac_caratheodory, MulticoequalizerDiagram.min_eq, IntermediateField.finSepDegree_top, SSet.N.iSup_subcomplex_eq_top, IsOpen.affineSpan_eq_top, CategoryTheory.Sieve.id_mem_iff_eq_top, AffineSubspace.not_wOppSide_bot, AlgebraicGeometry.instQuasiCompactToSpecΓOfCompactSpaceCarrierCarrierCommRingCat, IsAdjoinRoot.primitive_element_root, AlgebraicGeometry.AffineSpace.comp_homOfVector_assoc, Subgroup.mem_sup_right, AlgebraicGeometry.Scheme.Opens.ι_image_top, AddSubgroup.comap_sup_comap_le, Subfield.extendScalars_self, Field.Emb.Cardinal.adjoin_basis_eq_top, NonUnitalAlgebra.top_toSubmodule, Subgroup.mem_sup', iSup_eq_bot, Algebra.TensorProduct.adjoin_tmul_eq_top, iInf_sup_of_antitone, AlgebraicGeometry.Scheme.Γ_obj_op, ContinuousMap.starSubalgebra_topologicalClosure_eq_top_of_separatesPoints, Submonoid.sup_eq_range, IntermediateField.relrank_bot_left, SSet.Subcomplex.topIso_hom, CompleteBooleanAlgebra.inf_compl_le_bot, Subalgebra.LinearDisjoint.eq_bot_of_commute_of_self, CategoryTheory.Sieve.pullback_ofObjects_eq_top, AlgebraicGeometry.HasRingHomProperty.iff_of_source_openCover, Subalgebra.finrank_bot, CategoryTheory.Subfunctor.Subpresheaf.max_min, AlgebraicGeometry.Scheme.Opens.topIso_hom, MeasureTheory.OuterMeasure.top_apply, CategoryTheory.presheafIsGeneratedBy_of_isFinite, ZMod.fieldRange_castHom_eq_bot, Subalgebra.restrictScalars_top, QuotientGroup.leftRel_eq_top, Subsemigroup.map_sup_comap_of_surjective, PrimeSpectrum.basicOpen_one, AlgebraicGeometry.Scheme.Hom.comp_appTop_assoc, AlgebraicGeometry.IsAffineOpen.fromSpec_app_of_le, AffineSubspace.eq_bot_or_nonempty, MeasureTheory.Measure.top_add, AlgebraicGeometry.AffineSpace.map_appTop_coord, AlgebraicGeometry.Scheme.toSpecΓ_preimage_basicOpen, PrimeSpectrum.iSup_basicOpen_eq_top_iff', Filter.bliminf_false, MeasureTheory.hitting_eq_sInf, AlgebraicGeometry.isIso_fromTildeΓ_iff, OrderHom.map_inf_fixedPoints_le, sup_iInf_eq, IntermediateField.mem_bot, AlgebraicGeometry.PresheafedSpace.restrictTopIso_inv, Algebra.IsCentral.out, polynomialFunctions_closure_eq_top, AlgebraicGeometry.Scheme.openCoverBasicOpenTop_f, sSup_inf_eq, IsDedekindDomain.integer_univ, AddSubmonoid.prod_bot_sup_bot_prod, AlgebraicGeometry.LocallyRingedSpace.toΓSpecSheafedSpace_hom_c_app, IsAdjoinRoot.adjoin_root_eq_top, NonUnitalAlgebra.toNonUnitalSubring_top, iInf_emptyset, AlgebraicGeometry.isField_of_isIntegral_of_subsingleton, AlgebraicGeometry.ProjectiveSpectrum.Proj.toOpen_toSpec_val_c_app, Finset.sup_id_eq_sSup, Subgroup.unop_sup, CategoryTheory.PreOneHypercover.sieve₀_trivial, CategoryTheory.Subfunctor.preimage_eq_top_iff, AddSubgroup.closure_union, IntermediateField.fixingSubgroup_bot, CategoryTheory.Subfunctor.Subpresheaf.eq_top_iff_isIso, iInf_inf_eq, Finset.exists_sup_eq_iSup, CategoryTheory.Presieve.isSheafFor_top, AlgebraicGeometry.IsAffineOpen.fromSpec_toSpecΓ, Subalgebra.bot_eq_top_iff_rank_eq_one, IntermediateField.coe_top, AlgebraicGeometry.AffineSpace.homOfVector_appTop_coord, isGalois_bot, Monoid.Coprod.mrange_inl_sup_mrange_inr, iSup_union, AlgebraicGeometry.Scheme.IsQuasiAffine.isBasis_basicOpen, Subalgebra.rTensorBot_tmul, AddSubmonoid.unop_sup, AlgebraicGeometry.Scheme.isoSpec_image_zeroLocus, biSup_inter_of_pairwise_disjoint, TwoSidedIdeal.inf_ringCon, iSup_extend_bot, Order.Coframe.MinimalAxioms.sup_sInf_eq, Algebra.range_id, sSupHomClass.toSupBotHomClass, Polynomial.SplittingField.adjoin_rootSet, AlgebraicGeometry.isCompact_and_isOpen_iff_finite_and_eq_biUnion_basicOpen, isSplittingField_iff_intermediateField, RingCon.sup_eq_ringConGen, iSup_bot, AlgebraicGeometry.PresheafedSpace.restrict_top_presheaf, isGalois_iff_isGalois_bot, Subring.map_sup, TrivSqZeroExt.range_inlAlgHom_sup_adjoin_range_inr, Polynomial.adjoin_X, AddSubgroup.addSubgroupOf_sup, AffineSubspace.bot_parallel_iff_eq_bot, AddSubmonoid.mem_sup_left, iSup_inf_of_antitone, FirstOrder.Language.Substructure.fg_bot, CompleteDistribLattice.MinimalAxioms.inf_sSup_le_iSup_inf, AddSubgroup.map_sup, Finset.sup_id_set_eq_sUnion, TopologicalSpace.Opens.coe_top, CategoryTheory.Subfunctor.epi_iff_range_eq_top, IntermediateField.finrank_adjoin_simple_eq_one_iff, iSup_sum, ωScottContinuous.bot, CategoryTheory.PreOneHypercover.sieve₁_trivial, StarSubalgebra.top_toSubalgebra, Order.Coframe.MinimalAxioms.iInf_sup_le_sup_sInf, iInf_pair, CategoryTheory.Subfunctor.Subpresheaf.range_eq_top, AlgebraicGeometry.LocallyRingedSpace.toΓSpec_preimage_basicOpen_eq, instIndiscreteTopology, Subalgebra.LinearDisjoint.eq_bot_of_self, NonUnitalAlgebra.toNonUnitalSubsemiring_eq_top, Submonoid.sup_eq_closure, TopologicalSpace.Opens.coe_eq_univ, Set.smul_set_symmDiff₀, Field.primitive_element_iff_algHom_eq_of_eval, AlgebraicGeometry.IsAffineOpen.isoSpec_hom_appTop, separableClosure.eq_bot_iff, Subalgebra.LinearDisjoint.bot_left, Subalgebra.rank_bot, AlgebraicGeometry.instIsIsoModulesSpecOfCarrierFromTildeΓFreeOpensCarrierCarrierCommRingCat, CategoryTheory.Subfunctor.range_id, Subalgebra.LinearDisjoint.bot_right, Submonoid.exists_mem_sup, IntermediateField.rank_adjoin_simple_eq_one_iff, iInf_split_single, separableClosure.adjoin_eq_of_isAlgebraic_of_isSeparable, AddCon.toSetoid_eq_top, notMem_iff_exists_ne_and_isConjRoot, AlgebraicGeometry.instIsAffineHomιBasicOpen, Algebra.adjoin_empty, Subalgebra.fg_bot_toSubmodule, Algebra.botEquiv_symm_apply, MeasurableSpace.generateFrom_singleton_empty, NonUnitalStarAlgebra.toNonUnitalSubalgebra_bot, AffineSubspace.notMem_bot, iSup_or, Order.Coframe.MinimalAxioms.sInf_sup_eq, MeasureTheory.condExp_bot_ae_eq, AddSubgroup.map_le_map_iff', AddSubmonoid.comap_sup_map_of_injective, CategoryTheory.Sieve.generate_of_contains_isSplitEpi, Subalgebra.rank_top, AlgebraicGeometry.HasRingHomProperty.iff_of_isAffine, Subalgebra.bot_eq_top_of_rank_eq_one, RingCon.toCon_bot, sInfHom.top_apply, CategoryTheory.Subfunctor.Subpresheaf.range_toRange, AlgebraicGeometry.SheafedSpace.restrictTopIso_hom, IsGalois.mem_bot_iff_fixed, Subalgebra.finrank_eq_one_iff, AlgebraicGeometry.ΓSpec.toSpecΓ_unop, MeasureTheory.OuterMeasure.toMeasure_top, AddSubmonoid.mem_sup_right, IntermediateField.map_bot, subalgebra_top_rank_eq_submodule_top_rank, AddSubgroup.normal_addSubgroupOf_sup_of_le_normalizer, AlgebraicGeometry.Scheme.IdealSheafData.vanishingIdeal_bot, AlgebraicGeometry.SheafedSpace.Γ_map_op, RingCon.toCon_top, Finset.iSup_union, iSup_inf_iSup, isCyclic_tfae, separableClosure.eq_bot_of_isPurelyInseparable, Subalgebra.eq_bot_of_isPurelyInseparable_of_isSeparable, Subalgebra.finite_bot, IntermediateField.finSepDegree_bot, affineSpan_eq_top_iff_nonempty_of_subsingleton, AddCon.coe_inf, CategoryTheory.Sieve.functorPullback_union, AlgebraicGeometry.SpecMap_ΓSpecIso_inv_toSpecΓ_assoc, CategoryTheory.GrothendieckTopology.top_covering, Filter.limsSup_top, AlgebraicGeometry.LocallyRingedSpace.Γ_obj, AddGroupTopology.toTopologicalSpace_bot, AlgebraicGeometry.Scheme.Spec_fromSpecStalk, IntermediateField.sepDegree_top, IsGaloisGroup.fixedPoints_eq_bot, Subgroup.sup_eq_closure, sInf_sup_sInf, SSet.skeleton_zero, himp_eq_sSup, AlgebraicGeometry.Scheme.toSpecΓ_base
toPartialOrder' 📖	CompOp	—
toSupSet 📖	CompOp	7 math math: CompleteAtomicBooleanAlgebra.iInf_iSup_eq, CompletelyDistribLattice.iInf_iSup_eq, CompletelyDistribLattice.MinimalAxioms.iInf_iSup_eq, Order.Frame.MinimalAxioms.inf_sSup_le_iSup_inf, le_sSup, sSup_le, CompleteDistribLattice.MinimalAxioms.inf_sSup_le_iSup_inf

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
le_sInf 📖	mathematical	Preorder.toLE PartialOrder.toPreorder SemilatticeSup.toPartialOrder Lattice.toSemilatticeSup toLattice	InfSet.sInf toInfSet	—	—
le_sSup 📖	mathematical	Set Set.instMembership	Preorder.toLE PartialOrder.toPreorder SemilatticeSup.toPartialOrder Lattice.toSemilatticeSup toLattice SupSet.sSup toSupSet	—	—
sInf_le 📖	mathematical	Set Set.instMembership	Preorder.toLE PartialOrder.toPreorder SemilatticeSup.toPartialOrder Lattice.toSemilatticeSup toLattice InfSet.sInf toInfSet	—	—
sSup_le 📖	mathematical	Preorder.toLE PartialOrder.toPreorder SemilatticeSup.toPartialOrder Lattice.toSemilatticeSup toLattice	SupSet.sSup toSupSet	—	—

CompleteLinearOrder

Definitions

Name	Category	Theorems
toBiheytingAlgebra 📖	CompOp	6 math math: iSup_eq_top, sSup_eq_top, iInf_eq_bot, iInf₂_eq_bot, iSup₂_eq_top, sInf_eq_bot
toCompl 📖	CompOp	1 math math: himp_bot
toCompleteLattice 📖	CompOp	18 math math: sInf_lt_iff, top_sdiff, biInf_lt_iff, iSup_eq_top, himp_bot, le_himp_iff, compare_eq_compareOfLessAndEq, iInf_lt_iff, le_total, sSup_eq_top, lt_sSup_iff, iInf_eq_bot, iInf₂_eq_bot, lt_biSup_iff, sdiff_le_iff, iSup₂_eq_top, sInf_eq_bot, lt_iSup_iff
toDecidableEq 📖	CompOp	1 math math: compare_eq_compareOfLessAndEq
toDecidableLE 📖	CompOp	—
toDecidableLT 📖	CompOp	1 math math: compare_eq_compareOfLessAndEq
toHImp 📖	CompOp	2 math math: himp_bot, le_himp_iff
toHNot 📖	CompOp	1 math math: top_sdiff
toLinearOrder 📖	CompOp	—
toOrd 📖	CompOp	1 math math: compare_eq_compareOfLessAndEq
toSDiff 📖	CompOp	2 math math: top_sdiff, sdiff_le_iff

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
compare_eq_compareOfLessAndEq 📖	mathematical	—	toOrd Preorder.toLT PartialOrder.toPreorder SemilatticeSup.toPartialOrder Lattice.toSemilatticeSup CompleteLattice.toLattice toCompleteLattice toDecidableLT toDecidableEq	—	—
himp_bot 📖	mathematical	—	HImp.himp toHImp Bot.bot OrderBot.toBot Preorder.toLE PartialOrder.toPreorder SemilatticeSup.toPartialOrder Lattice.toSemilatticeSup CompleteLattice.toLattice toCompleteLattice BoundedOrder.toOrderBot CompleteLattice.toBoundedOrder Compl.compl toCompl	—	—
le_himp_iff 📖	mathematical	—	Preorder.toLE PartialOrder.toPreorder SemilatticeSup.toPartialOrder Lattice.toSemilatticeSup CompleteLattice.toLattice toCompleteLattice HImp.himp toHImp SemilatticeInf.toMin Lattice.inf Lattice.inf_le_left Lattice.inf_le_right Lattice.le_inf	—	—
le_total 📖	mathematical	—	Preorder.toLE PartialOrder.toPreorder SemilatticeSup.toPartialOrder Lattice.toSemilatticeSup CompleteLattice.toLattice toCompleteLattice	—	—
sdiff_le_iff 📖	mathematical	—	Preorder.toLE PartialOrder.toPreorder SemilatticeSup.toPartialOrder Lattice.toSemilatticeSup CompleteLattice.toLattice toCompleteLattice toSDiff SemilatticeSup.toMax	—	—
top_sdiff 📖	mathematical	—	toSDiff Top.top OrderTop.toTop Preorder.toLE PartialOrder.toPreorder SemilatticeSup.toPartialOrder Lattice.toSemilatticeSup CompleteLattice.toLattice toCompleteLattice BoundedOrder.toOrderTop CompleteLattice.toBoundedOrder HNot.hnot toHNot	—	—

CompleteSemilatticeInf

Definitions

Name	Category	Theorems
toInfSet 📖	CompOp	323 math math: sInf_univ, sInf_image2_eq_sInf_sSup, biInf_gt_eq_iInf, iInf_inf, Antitone.iInf, iInf_sum, IsGLB.sInf_eq, Language.reverse_iInf, Set.map_finite_iInf, le_iInf₂_add, Antitone.sInf, CompletelyDistribLattice.MinimalAxioms.iSup_iInf_eq, CategoryTheory.ObjectProperty.preservesLimitsOfShape_eq_iSup, Nucleus.himp_apply, sInf_pair, le_sInf_iff, iInf_iInf_eq_right, CompleteLatticeHom.dual_symm_apply_toFun, Antitone.le_map_iInf₂, sInf_union, sInf_lt_iff, iInf_psigma', iInf_of_empty, iInf_unpair, iInf_sup_iInf, inf_eq_iInf, iInf_false, iInf_eq_top, compl_sInf, Interval.coe_sInf, Antitone.iInf_nat_add, iInf_prod', biInf_lt_eq_iInf, Interval.coe_iInf₂, le_iInf_const, Set.Iic_iInf, sup_biInf_le_biInf_sup, iInf_iSup_ge_nat_add, biInf_sigma, Set.BijOn.iInf_comp, iInf_iSup_of_antitone, inf_biInf, sInf_sup_le_iInf_sup, iInf_ite, Finset.inf_id_eq_sInf, Filter.HasBasis.biInf_mem, AddCon.coe_iInf, iInf_eq_iInf_finset', OrderHom.iInf_apply, Monotone.sInf, iInf_option, LowerSet.Iic_iInf, inf_iInf_nat_succ, biInf_le_eq_of_antitone, CompleteLatticeHom.toFun_eq_coe, iSup_iInf_le_iInf_iSup, CompleteLatticeHom.map_sSup', GaloisCoinsertion.u_iInf_l, PUnit.sInf_eq, sSup_image2_eq_sInf_sInf, iInf₂_eq_top, compl_sSup', Finset.iInf_insert, OrderIso.map_sInf_eq_sInf_symm_preimage, Order.Coframe.MinimalAxioms.iInf_sup_eq, iSup_iInf_of_monotone, biInf_le_eq_iInf, iInf₂_le, iInf_range, Set.BijOn.iInf_congr, sInf_image2_eq_sInf_sInf, Set.Finite.iSup_biInf_of_antitone, Antitone.map_iSup₂_le, sInf_le_sInf, compl_sSup, biInf_le_eq_inf, Language.sub_iSup, le_iInf, le_iInf_add, sInf_le_sInf_of_subset_insert_top, sInf_image, iInf_option_elim, Order.Coframe.MinimalAxioms.sup_iInf_eq, biInf_lt_iff, LowerSet.coe_iicsInfHom, iInf_subtype, iInf_sup_of_monotone, Set.Finite.iSup_biInf_of_monotone, biInf_ge_eq_of_monotone, GaloisConnection.u_iInf, le_iInf_mul, Set.iSup_iInf_of_monotone, isGLB_biInf, iInf_const_mono, le_iInf_iff, CompletelyDistribLattice.MinimalAxioms.iInf_iSup_eq', iInf_sigma', iInf_iSup_eq, LowerSet.iicsInfHom_apply, iInf₂_le_of_le, iInf_Prop_eq, Set.Finite.iInf_biSup_of_antitone, iInf_le_iff, lt_iInf_iff, Monotone.iInf_comp_eq, OrderIso.map_iInf, Order.Coframe.MinimalAxioms.sup_iInf₂_eq, iInf_subtype'', iInf_subtype', GaloisInsertion.l_biInf_u, le_sInf, iInf_mono, le_sInf_inter, iInf_ne_top_subtype, CategoryTheory.Limits.CompleteLattice.limitCone_isLimit_lift, unary_relation_sInf_iff, Antitone.map_sSup_le, Antitone.le_map_iInf, le_iInf_comp, sInf_sUnion, Finset.iInf_biUnion, sInf_image2, GaloisInsertion.l_iInf_u, iInf_le_iInf₂, iInf_sigma, Set.Iic_iInf₂, iInf_eq_iInf_finset, iInf_sup_iInf_le, Con.coe_iInf, Monotone.map_iInf_le, biSup_iInter_of_pairwise_disjoint, compl_iSup, compl_iInf, biInf_prod, Interval.coe_iInf, biInf_finsetSigma, infsInfHom_apply, iInf_eq_of_forall_ge_of_forall_gt_exists_lt, isGLB_iff_sInf_eq, sSup_lowerBounds_eq_sInf, LowerSet.Iic_iInf₂, le_iInf₂_iff, hnot_eq_sInf_codisjoint, sInf_sup_eq, biInf_finsetSigma', Finset.iInf_finset_image, sInf_eq_iInf, gc_Ici_sInf, Antitone.map_iSup_le, AddCon.coe_sInf, Order.PFilter.sInf_gc, iInf_bool_eq, OrderHom.coe_iInf, sdiff_eq_sInf, iInf_image, iInf_split, IsGLB.iInf_eq, CategoryTheory.Subfunctor.Subpresheaf.iInf_obj, sInf_le_sInf_of_isCoinitialFor, le_iInf₂, AddSubgroup.quotientiInfEmbedding_apply, inf_iInf, sup_iInf_le_iInf_sup, biInf_ge_eq_inf, iInf_psigma, Filter.countable_biInf_eq_iInf_seq', CategoryTheory.Subfunctor.sInf_obj, GaloisInsertion.l_sInf_u_image, isGLB_sInf, RingCon.coe_iInf, Finset.iInf_union, GaloisCoinsertion.u_sInf_l_image, ClosureOperator.sInf_isClosed, Nucleus.sInf_apply, iInf₂_comm, WithBot.sInf_empty, iInf₂_mono', Finset.iInf_insert_update, iInf_top, LowerSet.Iic_sInf, iInf_lt_iff, sInf_image2_eq_sSup_sSup, Subgroup.quotientiInfEmbedding_apply, iInf_union, sInf_le, iInf₂_mono, iInf_iSup_of_monotone, RingCon.coe_sInf, iInf_dite, Finset.iInf_singleton, CompleteLat.Iso.mk_hom, iInf₂_sup_eq, sInf_le, Set.Finite.iInf_biSup_of_monotone, sInfHom.coe_top, OrderIsoClass.tosInfHomClass, sInf_image2_eq_sSup_sInf, iInf_and, iSup_iInf_le, iInf_unique, Antitone.le_map_sInf, biInf_sup_le_biInf_sup, CompleteLatticeHom.coe_tosInfHom, iInf_eq_dif, CategoryTheory.Sieve.sInf_apply, himp_iInf_eq, GaloisInsertion.l_biInf_of_ul_eq_self, sInf_insert, iInf_extend_top, CategoryTheory.Subfunctor.Subpresheaf.sInf_obj, iInf_iInf_eq_left, sup_sInf_le_iInf_sup, Nucleus.iInf_apply, iSup_himp_eq, iInf_iUnion, iInf_and', biInf_sigma', iInf_image2, Filter.countable_biInf_eq_iInf_seq, iInf_univ, biInf_sup_biInf, iInf_sup_eq, iInf_comm, CategoryTheory.ObjectProperty.preservesColimitsOfShape_eq_iSup, AddSubgroup.quotientiInfAddSubgroupOfEmbedding_apply, Set.Iic_sInf, sSup_image2_eq_sSup_sInf, sInf_diff_singleton_top, le_sInf, Subgroup.quotientiInfSubgroupOfEmbedding_apply, iInf_or, Monotone.map_sInf_le, iInf_singleton, biInf_inf, le_iInf_iSup, OrderIso.map_iInf₂, biInf_const, Finset.inf_eq_sInf_image, CategoryTheory.Subfunctor.iInf_obj, biInf_prod', iInf_sup_le_iInf_sup, biInf_mono, Finset.inf_eq_iInf, Con.coe_sInf, sInf_empty, sInf_eq_of_forall_ge_of_forall_gt_exists_lt, sup_sInf_eq, CompleteLatticeHom.dual_apply_toFun, iInf_prod, iSup_iInf_of_antitone, iInf_eq_bot, fixedPoints.gfp_eq_sInf_iterate, iInf_nat_gt_zero_eq, sInf_upperBounds_eq_csSup, iInf₂_eq_bot, sInf_upperBounds_eq_sSup, CategoryTheory.Limits.CompleteLattice.limit_eq_iInf, isGLB_iInf, Set.iSup_iInf_of_antitone, OrderIso.map_sInf, biInf_le_biSup, sSup_image2_eq_sInf_sSup, OrderHom.map_sInf_subset_fixedPoints_le, Finset.exists_inf_eq_iInf, Set.iInf_iSup_of_monotone, Finset.inf_univ_eq_iInf, Equiv.biInf_comp, iInf_eq_if, sInf_Prop_eq, GaloisConnection.u_iInf₂, CategoryTheory.Limits.CompleteLattice.limitCone_cone_π_app, iInf_le_of_le, sup_iInf₂_eq, iInf_ge_eq_iInf_nat_add, OrderHom.sInf_apply, iInf_insert, le_iInf₂_mul, sInf_le_of_le, sInf_eq_top, compl_sInf', iInf_neg, iInf_lt_top, OrderIso.toCompleteLatticeHom_toFun, Monotone.map_iInf₂_le, iInf_sup_of_antitone, iSup_iInf_ge_nat_add, iInf_sUnion, Finset.iInf_option_toFinset, sup_iInf_eq, iInf_exists, iInf_le_iSup, sInf_le_sSup, iInf_emptyset, iInf_le, binary_relation_sInf_iff, iInf_inf_eq, iInf_pos, sInf_singleton, GaloisConnection.u_sInf, Order.Coframe.MinimalAxioms.sup_sInf_eq, GaloisInsertion.l_iInf_of_ul_eq_self, sdiff_iSup_eq, iInf_le_iInf_of_subset, iInf_const, Monotone.iInf, iInf_pair, sInf_eq_bot, CompleteLat.Iso.mk_inv, iInf_split_single, Set.map_finite_biInf, iInf_mono', Order.Coframe.MinimalAxioms.sInf_sup_eq, iSup_iInf_eq, sInfHom.top_apply, iInf_true, CompleteLatticeHomClass.tosInfHomClass, biInf_ge_eq_iInf, biInf_le, Setoid.sInf_def, sInf_le_iff, Set.iInf_iSup_of_antitone, sInf_sup_sInf, CategoryTheory.Limits.CompleteLattice.limitCone_cone_pt
toPartialOrder 📖	CompOp	425 math math: biSup_inf_le_inf_biSup, Set.pairwiseDisjoint_pair_insert, fixingSubgroup_antitone, Set.disjoint_ordT5Nhd, LowerAdjoint.closure_iUnion₂_closure, Set.preimage_kernImage, iUnion_Iic_eq_Iio_iSup, le_iInf₂_add, CategoryTheory.MorphismProperty.isomorphisms_le_of_containsIdentities, CategoryTheory.MorphismProperty.isStableUnderLimitsOfShape_iff_limitsOfShape_le, CategoryTheory.Sieve.generate_mono, SSet.horn₃₁.desc.multicofork_π_two, CategoryTheory.MorphismProperty.hasOfPostcompProperty_iff_le_diagonal, Nucleus.himp_apply, le_sInf_iff, LowerAdjoint.mem_iff, ClosureOperator.ofCompletePred_apply, Set.iUnionLift_unary, ClosureOperator.closure_iSup₂_closure, lowerPolar_anti, biSup_inf_le_biSup_inf, sInf_lt_iff, LowerAdjoint.closure_union_closure, UpperSet.sdiff_eq_left, CategoryTheory.Sieve.generate_functorPullback_le, scottContinuous_inf_right, Set.Iio_disjoint_Ioi_of_not_lt, bot_lt_iSup, Interval.coe_sInf, CategoryTheory.Sieve.functorPushforward_ofObjects_le, LowerAdjoint.le_iff_subset, Interval.coe_iInf₂, CategoryTheory.Sieve.pullback_monotone, LowerSet.disjoint_coe, le_iInf_const, Language.instMulRightMono, Frm.Iso.mk_hom, CategoryTheory.MorphismProperty.pushouts_le, CategoryTheory.Limits.CompleteLattice.colimit_eq_iSup, Set.Iic_iInf, CategoryTheory.Localization.LeftBousfield.galoisConnection, SSet.horn₃₂.desc.multicofork_π_one, CategoryTheory.ObjectProperty.isomorphisms_le_isoModSerre, sup_biInf_le_biInf_sup, sSup_disjoint_iff, Set.Iio_disjoint_Ioi_iff, CategoryTheory.MorphismProperty.colimitsOfShape_le_coproducts, CategoryTheory.MorphismProperty.IsInvertedBy.iff_map_le_isomorphisms, isLUB_iSup, CategoryTheory.MorphismProperty.monotone_map, sInf_sup_le_iInf_sup, Set.Iic_disjoint_Ioi, Nucleus.map_himp_le, Set.disjoint_pi_univ_Ioc_update_left_right, OrderHom.iInf_apply, CategoryTheory.Presieve.map_functorPullback, CategoryTheory.MorphismProperty.colimitsOfShape_le, LowerSet.Iic_iInf, Quantale.rightMulResiduation_le_iff_mul_le, UpperSet.Ici_iSup, iSup_iInf_le_iInf_iSup, CategoryTheory.MorphismProperty.coproducts_monotone, biSup_mono, CategoryTheory.Sieve.pullback_pushforward_le, biSup_symmDiff_biSup_le, ScottContinuous.inf₂, HomotopicalAlgebra.trivialCofibrations_sub_cofibrations, Set.Ioc_disjoint_Ioc, OrderHom.sSup_apply, UpperSet.codisjoint_coe, CategoryTheory.MorphismProperty.colimitsOfShape_discrete_le_llp_rlp, CategoryTheory.Limits.LimitPresentation.self_π, iSup_inf_le_inf_iSup, gc_upperPolar_lowerPolar, Concept.isCompl_extent_intent, le_iSup₂, Set.disjoint_right_ordSeparatingSet, CategoryTheory.MorphismProperty.isStableUnderRetracts_iff_retracts_le, OrderIso.map_sInf_eq_sInf_symm_preimage, LowerSet.sdiff_eq_left, gc_upperBounds_lowerBounds, CategoryTheory.MorphismProperty.multiplicativeClosure_monotone, SSet.horn₃₁.desc.multicofork_pt, CategoryTheory.Functor.relativelyRepresentable.isomorphisms_le, Set.pairwise_disjoint_vadd_iff, OrderIso.map_iSup, iInf₂_le, CategoryTheory.MorphismProperty.retracts_le, Language.instMulLeftMono, Set.disjoint_iUnion_left, HomotopicalAlgebra.trivialCofibrations_sub_weakEquivalences, Set.PairwiseDisjoint.exists_mem_filter, CategoryTheory.Localization.LeftBousfield.le_W_iff, sInf_le_sInf, HomotopicalAlgebra.trivialFibrations_sub_fibrations, fixingAddSubmonoid_fixedPoints_gc, Language.le_iff, UpperSet.Ici_sSup, le_iInf_add, sInf_le_sInf_of_subset_insert_top, MulAction.disjoint_image_image_iff, CategoryTheory.MorphismProperty.isStableUnderCobaseChange_iff_pushouts_le, Set.disjoint_smul_set, biInf_lt_iff, LowerSet.coe_iicsInfHom, Filter.disjoint_iff, CategoryTheory.MorphismProperty.universally_le, compl_eq_sSup_disjoint, CategoryTheory.Limits.CompleteLattice.colimitCocone_cocone_pt, Filter.monotone_mem, Set.disjoint_sUnion_left, CategoryTheory.Presieve.ofArrows_le_iff, LowerSet.sdiff_lt_left, iSup_eq_top, le_iInf_mul, isGLB_biInf, iInf_const_mono, le_iInf_iff, CompleteLattice.MulticoequalizerDiagram.multispanIndex_right, CategoryTheory.Sieve.le_generate, CategoryTheory.Limits.CompleteLattice.colimitCocone_cocone_ι_app, CompleteLattice.MulticoequalizerDiagram.multispanIndex_fst, QuotientAddGroup.strictMono_comap_prod_image, SSet.horn₃₁.desc.multicofork_π_two_assoc, CategoryTheory.MorphismProperty.le_pushouts, disjoint_iSup_iff, OrderIso.map_sSup_eq_sSup_symm_preimage, Set.Ioi_disjoint_Iio_iff, iSup_inf_le_sSup_inf, SSet.horn₃₁.desc.multicofork_π_zero_assoc, LowerSet.iicsInfHom_apply, CategoryTheory.MorphismProperty.strictMap_multiplicativeClosure_le, Set.PairwiseDisjoint.exists_mem_filter_basis, le_iSup, Set.kernImage_mono, Frm.Iso.mk_inv, CategoryTheory.Limits.ColimitPresentation.self_diag, iInf_le_iff, lt_iInf_iff, OrderIso.map_iInf, CategoryTheory.Presieve.ofArrows_comp_le, CategoryTheory.Limits.CompleteLattice.hasLimits_of_completeLattice, CategoryTheory.Sieve.le_pullback_bind, CategoryTheory.MorphismProperty.isoClosure_le_iff, isNowhereDense_iff_disjoint, AddQuantale.leftAddResiduation_le_iff_add_le, CategoryTheory.MorphismProperty.coproducts_le_iff, le_sInf_inter, CategoryTheory.Limits.CompleteLattice.limitCone_isLimit_lift, CategoryTheory.ObjectProperty.galoisConnection_isColocal, CategoryTheory.MorphismProperty.map_inverseImage_le, CategoryTheory.MorphismProperty.le_isLocal_isLocal, CompleteLattice.MulticoequalizerDiagram.multispanIndex_left, Set.pairwiseDisjoint_prod_left, le_iInf_comp, CategoryTheory.MorphismProperty.gc_llp_rlp, Set.disjoint_iUnion_right, HomotopicalAlgebra.trivialFibrations_sub_weakEquivalences, Set.image_preimage, iInf_le_iInf₂, iSup₂_add_le, LowerAdjoint.closure_iSup₂_closure, Filter.antitone_seq_of_seq, Set.Iic_iInf₂, fixingSubmonoid_antitone, SimpleGraph.Subgraph.verts_monotone, CategoryTheory.Limits.LimitPresentation.self_diag, CategoryTheory.Sieve.le_pushforward_pullback, iInf_sup_iInf_le, iSup_disjoint_iff, fixedPoints_addSubgroup_antitone, CategoryTheory.Sieve.functorPushforward_monotone, ClosureOperator.ofCompletePred_isClosed, le_iSup_inf_iSup, CategoryTheory.MorphismProperty.map_le_iff, Interval.coe_iInf, LowerAdjoint.closure_union_closure_subset, Set.disjoint_iUnion₂_right, CategoryTheory.ObjectProperty.le_isLocal_iff, isGLB_iff_sInf_eq, sSup_lowerBounds_eq_sInf, LowerSet.Iic_iInf₂, CategoryTheory.MorphismProperty.le_coproducts, Filter.HasBasis.disjoint_iff, UpperSet.coe_icisSupHom, le_iInf₂_iff, hnot_eq_sInf_codisjoint, SimpleGraph.pairwise_disjoint_supp_connectedComponent, Language.instOrderedSub, Set.Ioi_disjoint_Iio_of_le, Concept.codisjoint_extent_intent, CategoryTheory.MorphismProperty.le_pullbacks, CategoryTheory.ObjectProperty.epimorphisms_le_epiModSerre, CategoryTheory.MorphismProperty.antitone_rlp, gc_Ici_sInf, LowerAdjoint.subset_closure, UpperSet.lt_sdiff_left, Set.Ici_iSup₂, Order.PFilter.sInf_gc, Finset.intervalGapsWithin_pairwiseDisjoint_Ioc, iSup_const_mono, CategoryTheory.MorphismProperty.le_llp_rlp, sSup_le_sSup_of_subset_insert_bot, OrderHom.coe_iInf, sdiff_eq_sInf, Filter.disjoint_principal_principal, iSup₂_disjoint_iff, CategoryTheory.MorphismProperty.le_llp_iff_le_rlp, CategoryTheory.MorphismProperty.coproducts_le, CategoryTheory.Sieve.functorPullback_monotone, LowerAdjoint.closure_union_closure_right, le_sdiff_iff, Set.Iic_disjoint_Ioc, iSup_inf_le_inf_sSup, ClosureOperator.closure_iSup_closure, sup_iInf_le_iInf_sup, CategoryTheory.ObjectProperty.le_isColocal_iff, Set.Ioc_disjoint_Ioi, QuotientGroup.strictMono_comap_prod_image, Set.Iic_disjoint_Ici, Set.Ici_sSup, isGLB_sInf, gc_upperClosure_coe, CategoryTheory.MorphismProperty.le_retracts, UpperSet.icisSupHom_apply, CategoryTheory.ObjectProperty.monomorphisms_le_monoModSerre, Interval.disjoint_coe, scottContinuous_inf_left, Set.Ioi_disjoint_Iio_of_not_lt, CategoryTheory.Sieve.ofObjects_mono, CategoryTheory.MorphismProperty.coproducts_le_llp_rlp, CategoryTheory.Limits.CompleteLattice.hasColimits_of_completeLattice, CategoryTheory.Sieve.functorPushforward_pullback_le, WithBot.sInf_empty, Set.Ico_disjoint_Ico, LowerSet.Iic_sInf, iInf_lt_iff, sInf_le, disjoint_sSup_iff, Set.Ioi_disjoint_Iio_same, IsLowerSet.disjoint_upperClosure_left, SSet.horn₃₂.desc.multicofork_π_zero_assoc, CompleteLattice.MulticoequalizerDiagram.multispanIndex_snd, FreeGroup.startsWith.disjoint_iff_ne, Set.disjoint_vadd_set, Set.disjoint_sUnion_right, CategoryTheory.MorphismProperty.limitsOfShape_le, Filter.ker_mono, CompleteLat.Iso.mk_hom, CategoryTheory.MorphismProperty.retracts_le_iff, iSup_inf_le_iSup_inf, sInf_le, LowerAdjoint.closure_iSup_closure, IsLowerSet.disjoint_upperClosure_right, CategoryTheory.Sieve.le_functorPushforward_pullback, Concept.strictAnti_intent, CategoryTheory.MorphismProperty.IsInvertedBy.iff_le_inverseImage_isomorphisms, iSup_iInf_le, biInf_sup_le_biInf_sup, CategoryTheory.MorphismProperty.isomorphisms_le_pushouts, iSup₂_le_iff, fixingSubmonoid_fixedPoints_gc, SSet.horn₃₁.desc.multicofork_π_three_assoc, CategoryTheory.MorphismProperty.le_leftBousfieldW_isLocal, SSet.horn₃₁.desc.multicofork_π_zero, CategoryTheory.MorphismProperty.colimitsOfShape_le_of_final, Set.Iio_disjoint_Ioi_same, CategoryTheory.Sieve.galoisConnection, fixedPoints_subgroup_antitone, CategoryTheory.LocalizerMorphism.map, CategoryTheory.Presieve.bind_ofArrows_le_bindOfArrows, Set.iUnionLift_binary, himp_le_iff, sup_sInf_le_iInf_sup, CategoryTheory.Sieve.functor_galoisConnection, fixingAddSubmonoid_antitone, sSup_eq_top, disjoint_memPartition, Set.monotone_preimage, iSup_le_iSup_of_subset, SSet.horn₃₂.desc.multicofork_π_three, fixedPoints_antitone_addSubmonoid, CategoryTheory.MorphismProperty.retracts_le_llp_rlp, Set.Iic_sInf, IsUpperSet.disjoint_lowerClosure_left, Set.disjoint_vadd_set_left, ScottContinuous.of_map_sSup, fixingAddSubgroup_fixedPoints_gc, Set.Ioc_disjoint_Ioc_of_le, CategoryTheory.Sieve.pushforward_le_bind_of_mem, CategoryTheory.MorphismProperty.antitone_llp, Set.pairwiseDisjoint_vadd_iff, Composition.disjoint_range, le_iInf_iSup, OrderIso.map_iInf₂, OrderIso.map_sSup, Set.Ici_disjoint_Iic, lt_sSup_iff, fixingSubgroup_fixedPoints_gc, fixingAddSubgroup_antitone, iInf_sup_le_iInf_sup, biInf_mono, CompleteLattice.MulticoequalizerDiagram.multicofork_pt, iUnion_Ici_eq_Ioi_iInf, Set.pairwiseDisjoint_iff, CategoryTheory.MorphismProperty.le_isColocal_isColocal, scottContinuous_iff_map_sSup, CategoryTheory.Sieve.functorPullback_pushforward_le, AddAction.disjoint_image_image_iff, CategoryTheory.Sieve.functorPushforward_le_iff_le_functorPullback, SSet.horn₃₂.desc.multicofork_pt, Filter.IsAntitoneBasis.antitone, Set.disjoint_vadd_set_right, iInf_eq_bot, Set.Iio_disjoint_Ici, sInf_upperBounds_eq_csSup, iInf₂_eq_bot, Set.pairwise_disjoint_smul_iff, sInf_upperBounds_eq_sSup, lt_biSup_iff, iSup_le_iff, CategoryTheory.Limits.CompleteLattice.limit_eq_iInf, isGLB_iInf, Set.Ico_disjoint_Ico_same, Filter.HasAntitoneBasis.antitone, OrderIso.map_sInf, biInf_le_biSup, iSup_add_le, Disjoint.edgeSet, CategoryTheory.MorphismProperty.monotone_isoClosure, OrderIso.map_iSup₂, OrderHom.iSup_apply, iSup_const_le, CategoryTheory.MorphismProperty.le_def, Filter.exists_antitone_seq, CategoryTheory.MorphismProperty.pushouts_le_llp_rlp, CategoryTheory.Limits.CompleteLattice.limitCone_cone_π_app, Pairwise.exists_mem_filter_basis_of_disjoint, Set.pairwiseDisjoint_smul_iff, Concept.strictMono_extent, Set.Ioo_disjoint_Ioo, iSup_symmDiff_iSup_le, AddQuantale.instAddLeftMono, isLUB_biSup, OrderHom.sInf_apply, CategoryTheory.MorphismProperty.isStableUnderColimitsOfShape_iff_colimitsOfShape_le, le_iInf₂_mul, upperPolar_anti, CategoryTheory.MorphismProperty.le_isoClosure, Quantale.instMulLeftMono, Set.disjoint_smul_set_left, sSup_inter_le, le_biSup, CategoryTheory.ObjectProperty.galoisConnection_isLocal, iInf_lt_top, OrderIso.toCompleteLatticeHom_toFun, Filter.NeBot.not_disjoint, CategoryTheory.MorphismProperty.pullbacks_monotone, Set.disjoint_smul_set_right, CategoryTheory.MorphismProperty.universally_mono, CategoryTheory.MorphismProperty.presheaf_monomorphisms_le_monomorphisms, CategoryTheory.MorphismProperty.pushouts_le_iff, Quantale.instMulRightMono, iSup₂_le_iSup, CategoryTheory.MorphismProperty.isStableUnderBaseChange_iff_pullbacks_le, iUnion_Iic_eq_Iic_iSup, CategoryTheory.Sieve.pushforward_monotone, iSup₂_mul_le, Pairwise.exists_mem_filter_of_disjoint, Set.disjoint_left_ordSeparatingSet, fixedPoints_antitone, AddQuantale.rightAddResiduation_le_iff_add_le, CategoryTheory.Limits.CompleteLattice.colimitCocone_isColimit_desc, iUnion_Ici_eq_Ici_iInf, Set.Iio_disjoint_Ioi_of_le, CategoryTheory.Sieve.generate_le_iff, Finset.pairwiseDisjoint_pair_insert, gc_lowerClosure_coe, iSup_comp_le, SSet.horn₃₂.desc.multicofork_π_zero, SSet.horn₃₂.desc.multicofork_π_three_assoc, iInf_le_iSup, sInf_le_sSup, Set.sUnion_powerset_gc, SSet.horn₃₂.desc.multicofork_π_one_assoc, CategoryTheory.MorphismProperty.pullbacks_le, IsUpperSet.disjoint_lowerClosure_right, iInf_le, Quantale.leftMulResiduation_le_iff_mul_le, Disjoint.exists_mem_filter_basis, iSup₂_eq_top, iSup_mul_le, CategoryTheory.ConcreteCategory.injective_le_monomorphisms, SimpleGraph.deleteEdges_eq_self, Set.Ici_iSup, Set.disjoint_iUnion₂_left, CategoryTheory.MorphismProperty.pushouts_monotone, CategoryTheory.MorphismProperty.le_colimitsOfShape_punit, OrderHom.coe_iSup, CategoryTheory.MorphismProperty.retracts_monotone, CategoryTheory.ConcreteCategory.surjective_le_epimorphisms, iInf_le_iInf_of_subset, SSet.horn₃₁.desc.multicofork_π_three, CategoryTheory.MorphismProperty.le_multiplicativeClosure, sInf_eq_bot, CompleteLat.Iso.mk_inv, disjoint_iSup₂_iff, lowerClosure_mono, Set.Ioc_disjoint_Ioi_same, iSup_lt_iff, CategoryTheory.Sieve.ofArrows_le_ofObjects, Concept.disjoint_extent_intent, Filter.monotone_principal, LowerAdjoint.closure_union_closure_left, biInf_le, CategoryTheory.MorphismProperty.multiplicativeClosure_le_iff, sInf_le_iff, upperClosure_anti, LowerAdjoint.closure_iUnion_closure, AddQuantale.instAddRightMono, UpperSet.Ici_iSup₂, CategoryTheory.Limits.CompleteLattice.limitCone_cone_pt, CategoryTheory.Limits.ColimitPresentation.self_ι, himp_eq_sSup, lt_iSup_iff

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
le_sInf 📖	mathematical	Preorder.toLE PartialOrder.toPreorder toPartialOrder	InfSet.sInf toInfSet	—	—
sInf_le 📖	mathematical	Set Set.instMembership	Preorder.toLE PartialOrder.toPreorder toPartialOrder InfSet.sInf toInfSet	—	—

CompleteSemilatticeSup

Definitions

Name	Category	Theorems
toPartialOrder 📖	CompOp	11 math math: isLUB_iff_sSup_eq, le_sSup, gc_sSup_Iic, le_sSup_iff, isLUB_sSup, le_sSup, sSup_le_iff, le_iSup_iff, isOrderRightAdjoint_sSup, MeasureTheory.Submartingale.exists_ae_trim_tendsto_of_bdd, sSup_le_sSup
toSupSet 📖	CompOp	519 math math: Finset.iSup_biUnion, iSup_insert, sInf_image2_eq_sInf_sSup, Subsemiring.mem_sSup_of_directedOn, Order.Frame.MinimalAxioms.iSup_inf_eq, AddCon.sSup_eq_addConGen, biSup_inf_le_inf_biSup, Subring.coe_sSup_of_directedOn, iSup_pair, Subring.map_iSup, Language.iSup_add, biSup_le_eq_sup, unary_relation_sSup_iff, GaloisCoinsertion.u_biSup_l, CompletelyDistribLattice.MinimalAxioms.iSup_iInf_eq, NonUnitalSubsemiring.mem_iSup_of_directed, Language.iSup_sub, AddSubsemigroup.closure_iUnion, AddSubsemigroup.op_iSup, iSup₂_le, sSup_univ, ClosureOperator.closure_iSup₂_closure, CompleteLatticeHom.coe_tosSupHom, iSup_psigma, isLUB_iff_sSup_eq, iSup_bool_eq, biSup_lt_eq_iSup, fixedPoints_subgroup_iSup, iSup_split, Antitone.le_map_iInf₂, Subsemigroup.unop_iSup, biSup_inf_le_biSup_inf, Subgroup.normalCore_eq_iSup, AddSubgroup.iSup_eq_closure, Subgroup.op_iSup, sSup_eq_bot, Subgroup.mem_biSup_of_directedOn, Subring.mem_sSup_of_directedOn, Submonoid.unop_iSup, iSup_false, Subgroup.mem_iSup_of_mem, bot_lt_iSup, compl_sInf, sSup_Prop_eq, Submonoid.coe_iSup_of_directed, RingCon.sSup_def, sSup_inf_sSup, iSup_unpair, CategoryTheory.Limits.CompleteLattice.colimit_eq_iSup, sup_iSup, Subsemigroup.mem_sSup_of_mem, iSup₂_mono, Encodable.iSup_decode₂, TwoSidedIdeal.iSup_ringCon, iSup_image, iInf_iSup_ge_nat_add, Language.mem_iSup, sSup_disjoint_iff, IsLUB.iSup_eq, PUnit.sSup_eq, fixedPoints_addSubmonoid_iSup, iInf_iSup_of_antitone, iSup₂_inf_eq, isLUB_iSup, sSup_empty, NonUnitalSubring.map_iSup, Language.reverse_iSup, NonUnitalSubring.mem_sSup_of_directedOn, AddSubsemigroup.coe_iSup_of_directed, Antitone.sSup, sSup_insert, Submonoid.iSup_eq_closure, Subgroup.map_iSup, Submonoid.coe_sSup_of_directedOn, iSup_and', UpperSet.Ici_iSup, sSup_union, CategoryTheory.Subfunctor.Subpresheaf.iSup_obj, sSupHom.coe_bot, Subring.coe_iSup_of_directed, Subsemigroup.mem_iSup_prop, iSup_iInf_le_iInf_iSup, AddSubsemigroup.unop_sSup, CompleteLatticeHom.map_sSup', biSup_mono, biSup_le_eq_iSup, iSup_subtype, Submonoid.mem_iSup, AddSubsemigroup.unop_iSup, sSup_image2_eq_sInf_sInf, biSup_symmDiff_biSup_le, le_sSup, compl_sSup', biSup_le_eq_of_monotone, biSup_inf_biSup, OrderHom.sSup_apply, AddSubgroup.coe_iSup_of_directed, iSup_exists, CategoryTheory.Sieve.sSup_apply, Submonoid.iSup_map_mulSingle_le, iSup_inf_le_inf_iSup, AddSubsemigroup.map_iSup, Subsemiring.mem_iSup_of_directed, Subsemiring.coe_iSup_of_directed, le_iSup₂, Set.map_finite_iSup, CategoryTheory.ObjectProperty.instIsClosedUnderIsomorphismsISup, iSup_iInf_of_monotone, Order.Frame.MinimalAxioms.sSup_inf_eq, OrderIso.map_iSup, iSup_subtype', sSup_image2, NonUnitalSubsemiring.coe_iSup_of_directed, biSup_sigma', Finset.iSup_singleton, sSupHom.bot_apply, biSup_finsetSigma', GaloisCoinsertion.u_iSup_of_lu_eq_self, Subgroup.mem_iSup_prop, AddSubgroup.mem_iSup_of_directed, Set.Finite.iSup_biInf_of_antitone, CategoryTheory.Subfunctor.isGeneratedBy_iff, iSup_mono', GaloisCoinsertion.u_biSup_of_lu_eq_self, CategoryTheory.Subfunctor.Subpresheaf.isGeneratedBy_iff, AddSubsemigroup.mem_iSup, Antitone.map_iSup₂_le, Submonoid.mem_sSup_of_directedOn, sSup_singleton, compl_sSup, Language.sub_iSup, AddSubsemigroup.mem_biSup_of_directedOn, iSup_subtype'', Submonoid.op_sSup, CategoryTheory.Subfunctor.sSup_obj, UpperSet.Ici_sSup, sup_eq_iSup, Subsemigroup.mem_iSup_of_directed, Subsemigroup.mem_sSup_of_directed_on, sup_biSup, Subsemiring.op_iSup, inf_iSup₂_eq, IsQuantale.sSup_mul_distrib, Subgroup.mem_sSup_of_directedOn, iSup_range, compl_eq_sSup_disjoint, Set.Finite.iSup_biInf_of_monotone, CategoryTheory.Limits.CompleteLattice.colimitCocone_cocone_pt, Language.add_iSup, AddSubgroup.mem_sSup_of_directedOn, Equiv.biSup_comp, FirstOrder.Language.DirectLimit.partialEquivLimit_comp_inclusion, AddSubgroup.unop_iSup, NonUnitalSubring.mem_iSup_of_directed, iSup_eq_top, Set.iSup_iInf_of_monotone, Monotone.le_map_iSup₂, iSup_sdiff_eq, CompletelyDistribLattice.MinimalAxioms.iInf_iSup_eq', biSup_ge_eq_sup, CategoryTheory.Limits.CompleteLattice.colimitCocone_cocone_ι_app, Setoid.sSup_def, Monotone.le_map_iSup, Submonoid.map_iSup, iInf_iSup_eq, disjoint_iSup_iff, RingCon.sSup_eq_ringConGen, iSup_comm, CategoryTheory.Pairwise.cocone_pt, OrderIso.map_sSup_eq_sSup_symm_preimage, iSup_inf_le_sSup_inf, iSup_split_single, AddSubmonoid.iSup_map_single_le, AddSubsemigroup.mem_iSup_of_mem, Subring.op_sSup, le_iSup, iSup_eq_iSup_finset, AlgebraicGeometry.Scheme.IsLocallyDirected.homOfLE_tAux, inf_sSup_eq, Set.Finite.iInf_biSup_of_antitone, OrderHom.le_map_sSup_subset_fixedPoints, NonUnitalSubring.closure_sUnion, AddQuantale.add_iSup_distrib, binary_relation_sSup_iff, biSup_gt_eq_iSup, CategoryTheory.ObjectProperty.isoClosure_iSup, sSup_diff_singleton_bot, Subsemigroup.mem_biSup_of_directedOn, GaloisConnection.l_sSup, IsLUB.sSup_eq, AddSubmonoid.unop_iSup, Submonoid.op_iSup, Antitone.map_sSup_le, iSup_pos, Antitone.le_map_iInf, Set.pairwiseDisjoint_prod_left, iSup_dite, AddSubmonoid.mem_iSup, iSup_eq_iSup_finset', iSup_inf_eq, GaloisConnection.l_iSup₂, CategoryTheory.Subfunctor.iSup_min, Subsemigroup.comap_iSup_map_of_injective, iSup₂_add_le, LowerAdjoint.closure_iSup₂_closure, NonUnitalSubsemiring.map_iSup, Subgroup.iSup_comap_le, GaloisInsertion.l_sSup_u_image, AddSubsemigroup.op_sSup, CategoryTheory.MorphismProperty.iSup_iff, le_sSup_of_le, iSup_sup_eq, iSup_disjoint_iff, biSup_iInter_of_pairwise_disjoint, compl_iSup, le_iSup_inf_iSup, compl_iInf, sSup_eq_bot', supsSupHom_apply, CompleteLattice.ωScottContinuous.iSup, CategoryTheory.Pairwise.cocone_ι_app, iSup_ite, Submonoid.mem_iSup_prop, biSup_prod, sSup_lowerBounds_eq_sInf, Subsemiring.closure_iUnion, IsQuantale.mul_sSup_distrib, OrderIsoClass.tosSupHomClass, Subgroup.mem_sSup_of_mem, iSup_option_elim, sSup_image2_eq_sSup_sSup, UpperSet.coe_icisSupHom, biSup_ge_eq_of_antitone, Submonoid.map_iSup_comap_of_surjective, inf_iSup_eq, Con.sSup_eq_conGen, AddSubmonoid.coe_iSup_of_directed, AddSubmonoid.coe_sSup_of_directedOn, Subsemiring.closure_sUnion, Submonoid.comap_iSup_map_of_injective, AddSubsemigroup.comap_iSup_map_of_injective, AddSubmonoid.unop_sSup, CategoryTheory.ObjectProperty.instEssentiallySmallISupOfSmall, Antitone.map_iSup_le, iSup_prod', sSup_iUnion, Set.Ici_iSup₂, CategoryTheory.Subfunctor.IsGeneratedBy.iSup_eq, sSup_le, iSup_univ, iSup_eq_dif, AddSubsemigroup.iSup_eq_closure, Subgroup.coe_iSup_of_directed, iSup_const_mono, iSup_image2, sSup_le_sSup_of_subset_insert_bot, iSup₂_disjoint_iff, Set.map_finite_biSup, CompleteLattice.ωScottContinuous.sSup, AddSubgroup.op_sSup, GaloisConnection.l_iSup, iSup₂_mono', FrameHom.map_sSup', iSup_inf_le_inf_sSup, ClosureOperator.closure_iSup_closure, Order.Frame.MinimalAxioms.inf_iSup₂_eq, fixedPoints_submonoid_iSup, iSup_iUnion, iSup₂_eq_bot, Order.Frame.MinimalAxioms.inf_iSup_eq, TwoSidedIdeal.sSup_ringCon, Set.Ici_sSup, sup_iSup_nat_succ, Subsemigroup.op_iSup, Submonoid.mem_iSup_of_directed, AddCon.sSup_def, sSup_eq_iSup, FrameHomClass.map_sSup, Finset.sup_eq_iSup, UpperSet.icisSupHom_apply, Order.Frame.MinimalAxioms.inf_sSup_eq, AddSubmonoid.op_sSup, Language.iSup_mul, iSup_singleton, iSup_neg, add_sSup_distrib, Finset.iSup_insert, Subring.mem_iSup_of_directed, GaloisInsertion.l_biSup_u, AddSubmonoid.op_iSup, NonUnitalSubring.coe_sSup_of_directedOn, sInf_image2_eq_sSup_sSup, Subsemigroup.closure_iUnion, Subsemigroup.mem_iSup_of_mem, AddSubsemigroup.map_iSup_comap_of_surjective, Subsemiring.map_iSup, iSup_ne_bot_subtype, NonUnitalSubsemiring.closure_sUnion, disjoint_sSup_iff, iInf_iSup_of_monotone, Quantale.iSup_mul_distrib, NonUnitalSubsemiring.closure_iUnion, AddSubmonoid.mem_iSup_of_directed, iSup_emptyset, Subsemiring.op_sSup, NonUnitalSubring.closure_iUnion, CategoryTheory.Subfunctor.Subpresheaf.iSup_min, biSup_const, iSup₂_comm, Monotone.iSup_comp_eq, AddSubgroup.closure_iUnion, iSup_inf_le_iSup_inf, Set.Finite.iInf_biSup_of_monotone, LowerAdjoint.closure_iSup_closure, sInf_image2_eq_sSup_sInf, le_iSup_of_le, gc_sSup_Iic, iSup_true, sSup_mul_distrib, AddSubsemigroup.mem_sSup_of_mem, Subgroup.unop_iSup, iSup_iInf_le, Subgroup.mem_iSup_of_directed, Antitone.le_map_sInf, le_sSup_iff, iSup₂_le_iff, Submonoid.unop_sSup, iSup_Prop_eq, Subring.op_iSup, iSup_sup, Subsemigroup.mem_iSup, AddSubgroup.mem_iSup_of_mem, iSup_sigma', Monotone.le_map_sSup, Antitone.iSup, Finset.iSup_insert_update, iSup_iSup_eq_right, iSup_himp_eq, iSup_unique, sSup_eq_top, GaloisCoinsertion.u_iSup_l, Subsemiring.coe_sSup_of_directedOn, Setoid.sSup_eq_eqvGen, Set.exists_seq_iSup_eq_top_iff_countable, iSup_sigma, Finset.sup_eq_sSup_image, sSup_image, iSup_le_iSup_of_subset, AddSubgroup.mem_sSup_of_mem, sSup_sUnion, AddSubmonoid.map_iSup, biSup_prod', AddSubsemigroup.mem_iSup_prop, CompleteLatticeHomClass.map_sSup, CategoryTheory.Subfunctor.image_iSup, sSup_image2_eq_sSup_sInf, fixedPoints.lfp_eq_sSup_iterate, iSup_le, mul_sSup_distrib, biSup_sigma, AddSubgroup.mem_iSup_prop, Finset.iSup_finset_image, CategoryTheory.MorphismProperty.sSup_iff, le_iInf_iSup, OrderIso.map_sSup, sSup_pair, Set.BijOn.iSup_congr, AddSubgroup.op_iSup, CategoryTheory.Subfunctor.Subpresheaf.sSup_obj, lt_sSup_iff, AddSubsemigroup.mem_sSup_of_directed_on, AddSubsemigroup.coe_sSup_of_directed_on, isLUB_sSup, Subgroup.unop_sSup, Language.mul_iSup, Submonoid.mem_biSup_of_directedOn, fixedPoints_addSubgroup_iSup, iSup_eq_if, Subsemigroup.map_iSup, Subsemigroup.map_iSup_comap_of_surjective, scottContinuous_iff_map_sSup, Subsemigroup.coe_iSup_of_directed, Con.sSup_def, AddSubmonoid.closure_iUnion, iSup_psigma', AddSubmonoid.mem_sSup_of_mem, AlgebraicGeometry.Scheme.IsLocallyDirected.exists_of_pullback_V_V, FirstOrder.Language.DirectLimit.Equiv_isup_of_apply, iSup_iInf_of_antitone, Submonoid.mem_sSup_of_mem, le_sSup, Set.BijOn.iSup_comp, FrameHomClass.tosSupHomClass, sInf_upperBounds_eq_csSup, sInf_upperBounds_eq_sSup, lt_biSup_iff, iSup_le_iff, AddSubsemigroup.mem_iSup_of_directed, Set.iSup_iInf_of_antitone, Subring.unop_iSup, iSup_nat_gt_zero_eq, biInf_le_biSup, iSup_add_le, CategoryTheory.Subfunctor.iSup_obj, sSup_image2_eq_sInf_sSup, IsAddQuantale.add_sSup_distrib, NonUnitalSubring.coe_iSup_of_directed, Submonoid.mem_iSup_of_mem, Set.iInf_iSup_of_monotone, sSup_eq_of_forall_le_of_forall_lt_exists_gt, OrderIso.map_iSup₂, iSup_option, OrderHom.iSup_apply, AddSubgroup.iSup_comap_le, iSup_const_le, iSup_ge_eq_iSup_nat_add, Submonoid.closure_iUnion, iSup_const, sSup_le_iff, Subsemiring.unop_sSup, GaloisCoinsertion.u_sSup_l_image, le_iSup₂_of_le, iSup_symmDiff_iSup_le, Quantale.mul_iSup_distrib, AddSubgroup.unop_sSup, iSup_inf_of_monotone, Subgroup.closure_iUnion, Finset.sup_univ_eq_iSup, isLUB_biSup, Subsemigroup.iSup_eq_closure, NonUnitalSubsemiring.coe_sSup_of_directedOn, le_iSup_iff, sSup_le, sSup_inter_le, isOrderRightAdjoint_sSup, le_biSup, iSup_of_empty, compl_sInf', Finset.iSup_option_toFinset, biSup_sup, AddSubmonoid.mem_iSup_of_mem, AddSubmonoid.map_iSup_comap_of_surjective, iSup₂_le_iSup, iSup_eq_bot, FirstOrder.Language.DirectLimit.Equiv_isup_symm_inclusion_apply, MeasureTheory.Submartingale.exists_ae_trim_tendsto_of_bdd, CategoryTheory.Subfunctor.Subpresheaf.image_iSup, iSup_iInf_ge_nat_add, IsAddQuantale.sSup_add_distrib, Subring.unop_sSup, Subgroup.op_sSup, iSup₂_mul_le, NonUnitalSubsemiring.mem_sSup_of_directedOn, AddSubmonoid.comap_iSup_map_of_injective, CategoryTheory.Limits.CompleteLattice.colimitCocone_isColimit_desc, AlgebraicGeometry.Scheme.IsLocallyDirected.homOfLE_tAux_assoc, GaloisInsertion.l_iSup_u, CompleteLattice.MulticoequalizerDiagram.iSup_eq, iSup_comp_le, Subsemigroup.coe_sSup_of_directed_on, AddSubmonoid.mem_biSup_of_directedOn, sSup_le_sSup, sSup_inf_eq, Monotone.iSup, iInf_le_iSup, sInf_le_sSup, Subsemigroup.unop_sSup, AddSubgroup.map_iSup, Language.kstar_eq_iSup_pow, Subsemigroup.op_sSup, Finset.sup_id_eq_sSup, iSup₂_eq_top, iSup_mul_le, Finset.exists_sup_eq_iSup, Set.Ici_iSup, iSup_union, biSup_inter_of_pairwise_disjoint, iSup_extend_bot, OrderHom.coe_iSup, iSup_bot, Subsemiring.unop_iSup, CategoryTheory.ObjectProperty.instSmallISupOfSmall, Subgroup.iSup_eq_closure, iSup_inf_of_antitone, sdiff_iSup_eq, Monotone.sSup, AddQuantale.iSup_add_distrib, Monotone.iSup_nat_add, iSup_sum, CategoryTheory.Subfunctor.Subpresheaf.IsGeneratedBy.iSup_eq, iSup_sUnion, disjoint_iSup₂_iff, biSup_ge_eq_iSup, iSup_or, AddSubmonoid.mem_sSup_of_directedOn, iSup_lt_iff, sSup_add_distrib, sSup_le_sSup_of_isCofinalFor, iSup_iInf_eq, AddSubgroup.mem_biSup_of_directedOn, iSup_mono, iSup_and, iSup_eq_of_forall_le_of_forall_lt_exists_gt, AddSubmonoid.iSup_eq_closure, Finset.iSup_union, iSup_iSup_eq_left, ScottContinuous.map_sSup, biSup_finsetSigma, iSup_inf_iSup, AddSubmonoid.mem_iSup_prop, Subring.closure_sUnion, iSup_prod, Set.iInf_iSup_of_antitone, CategoryTheory.MorphismProperty.toSet_iSup, Subring.closure_iUnion, UpperSet.Ici_iSup₂, CategoryTheory.ObjectProperty.prop_iSup_iff, himp_eq_sSup, lt_iSup_iff

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
le_sSup 📖	mathematical	Set Set.instMembership	Preorder.toLE PartialOrder.toPreorder toPartialOrder SupSet.sSup toSupSet	—	—
sSup_le 📖	mathematical	Preorder.toLE PartialOrder.toPreorder toPartialOrder	SupSet.sSup toSupSet	—	—

IsGLB

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
sInf_eq 📖	mathematical	IsGLB Preorder.toLE PartialOrder.toPreorder CompleteSemilatticeInf.toPartialOrder	InfSet.sInf CompleteSemilatticeInf.toInfSet	—	isGLB_iff_sInf_eq

IsLUB

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
sSup_eq 📖	mathematical	IsLUB Preorder.toLE PartialOrder.toPreorder CompleteSemilatticeSup.toPartialOrder	SupSet.sSup CompleteSemilatticeSup.toSupSet	—	isLUB_iff_sSup_eq

OrderDual

Definitions

Name	Category	Theorems
infSet 📖	CompOp	10 math math: sSupHom.symm_dual_id, ofDual_iInf, toDual_sSup, sSupHom.dual_symm_apply_toFun, ofDual_sInf, sSupHom.dual_apply_toFun, sSupHom.dual_comp, sSupHom.symm_dual_comp, toDual_iSup, sSupHom.dual_id
instCompleteLattice 📖	CompOp	7 math math: CompleteLatticeHom.dual_symm_apply_toFun, CompleteLatticeHom.dual_id, CompleteLatticeHom.symm_dual_id, CompleteLat.dual_map, CompleteLatticeHom.dual_apply_toFun, CompleteLatticeHom.dual_comp, CompleteLatticeHom.symm_dual_comp
instCompleteLinearOrder 📖	CompOp	—
supSet 📖	CompOp	10 math math: sInfHom.dual_id, toDual_sInf, sInfHom.dual_apply_toFun, ofDual_iSup, sInfHom.dual_comp, sInfHom.dual_symm_apply_toFun, sInfHom.symm_dual_comp, ofDual_sSup, toDual_iInf, sInfHom.symm_dual_id

(root)

Definitions

Name	Category	Theorems
CompleteSemilatticeInf 📖	CompData	—
CompleteSemilatticeSup 📖	CompData	—
completeLatticeOfCompleteSemilatticeInf 📖	CompOp	—
completeLatticeOfCompleteSemilatticeSup 📖	CompOp	—
completeLatticeOfInf 📖	CompOp	—
completeLatticeOfSup 📖	CompOp	—
instCompleteSemilatticeInfOrderDualOfCompleteSemilatticeSup 📖	CompOp	—
instCompleteSemilatticeSupOrderDualOfCompleteSemilatticeInf 📖	CompOp	—

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
biInf_lt_iff 📖	mathematical	—	Preorder.toLT PartialOrder.toPreorder CompleteSemilatticeInf.toPartialOrder CompleteLattice.toCompleteSemilatticeInf CompleteLinearOrder.toCompleteLattice iInf CompleteSemilatticeInf.toInfSet Set Set.instMembership	—	—
iInf_le_iff 📖	mathematical	—	Preorder.toLE PartialOrder.toPreorder CompleteSemilatticeInf.toPartialOrder iInf CompleteSemilatticeInf.toInfSet	—	sInf_le_iff Set.mem_range
iInf_lt_iff 📖	mathematical	—	Preorder.toLT PartialOrder.toPreorder CompleteSemilatticeInf.toPartialOrder CompleteLattice.toCompleteSemilatticeInf CompleteLinearOrder.toCompleteLattice iInf CompleteSemilatticeInf.toInfSet	—	sInf_lt_iff Set.exists_range_iff
isGLB_iff_sInf_eq 📖	mathematical	—	IsGLB Preorder.toLE PartialOrder.toPreorder CompleteSemilatticeInf.toPartialOrder InfSet.sInf CompleteSemilatticeInf.toInfSet	—	IsGLB.unique isGLB_sInf
isGLB_sInf 📖	mathematical	—	IsGLB Preorder.toLE PartialOrder.toPreorder CompleteSemilatticeInf.toPartialOrder InfSet.sInf CompleteSemilatticeInf.toInfSet	—	sInf_le le_sInf
isLUB_iff_sSup_eq 📖	mathematical	—	IsLUB Preorder.toLE PartialOrder.toPreorder CompleteSemilatticeSup.toPartialOrder SupSet.sSup CompleteSemilatticeSup.toSupSet	—	IsLUB.unique isLUB_sSup
isLUB_sSup 📖	mathematical	—	IsLUB Preorder.toLE PartialOrder.toPreorder CompleteSemilatticeSup.toPartialOrder SupSet.sSup CompleteSemilatticeSup.toSupSet	—	le_sSup sSup_le
le_iSup_iff 📖	mathematical	—	Preorder.toLE PartialOrder.toPreorder CompleteSemilatticeSup.toPartialOrder iSup CompleteSemilatticeSup.toSupSet	—	—
le_sInf 📖	mathematical	Preorder.toLE PartialOrder.toPreorder CompleteSemilatticeInf.toPartialOrder	InfSet.sInf CompleteSemilatticeInf.toInfSet	—	CompleteSemilatticeInf.le_sInf
le_sInf_iff 📖	mathematical	—	Preorder.toLE PartialOrder.toPreorder CompleteSemilatticeInf.toPartialOrder InfSet.sInf CompleteSemilatticeInf.toInfSet	—	le_isGLB_iff isGLB_sInf
le_sSup 📖	mathematical	Set Set.instMembership	Preorder.toLE PartialOrder.toPreorder CompleteSemilatticeSup.toPartialOrder SupSet.sSup CompleteSemilatticeSup.toSupSet	—	CompleteSemilatticeSup.le_sSup
le_sSup_iff 📖	mathematical	—	Preorder.toLE PartialOrder.toPreorder CompleteSemilatticeSup.toPartialOrder SupSet.sSup CompleteSemilatticeSup.toSupSet	—	le_trans sSup_le le_sSup
le_sSup_of_le 📖	mathematical	Set Set.instMembership Preorder.toLE PartialOrder.toPreorder CompleteSemilatticeSup.toPartialOrder	SupSet.sSup CompleteSemilatticeSup.toSupSet	—	le_trans le_sSup
lt_biSup_iff 📖	mathematical	—	Preorder.toLT PartialOrder.toPreorder CompleteSemilatticeInf.toPartialOrder CompleteLattice.toCompleteSemilatticeInf CompleteLinearOrder.toCompleteLattice iSup CompleteSemilatticeSup.toSupSet CompleteLattice.toCompleteSemilatticeSup Set Set.instMembership	—	—
lt_iSup_iff 📖	mathematical	—	Preorder.toLT PartialOrder.toPreorder CompleteSemilatticeInf.toPartialOrder CompleteLattice.toCompleteSemilatticeInf CompleteLinearOrder.toCompleteLattice iSup CompleteSemilatticeSup.toSupSet CompleteLattice.toCompleteSemilatticeSup	—	lt_sSup_iff Set.exists_range_iff
lt_sSup_iff 📖	mathematical	—	Preorder.toLT PartialOrder.toPreorder CompleteSemilatticeInf.toPartialOrder CompleteLattice.toCompleteSemilatticeInf CompleteLinearOrder.toCompleteLattice SupSet.sSup CompleteSemilatticeSup.toSupSet CompleteLattice.toCompleteSemilatticeSup Set Set.instMembership	—	lt_isLUB_iff isLUB_sSup
ofDual_iInf 📖	mathematical	—	DFunLike.coe Equiv OrderDual EquivLike.toFunLike Equiv.instEquivLike OrderDual.ofDual iInf OrderDual.infSet iSup	—	—
ofDual_iSup 📖	mathematical	—	DFunLike.coe Equiv OrderDual EquivLike.toFunLike Equiv.instEquivLike OrderDual.ofDual iSup OrderDual.supSet iInf	—	—
ofDual_sInf 📖	mathematical	—	DFunLike.coe Equiv OrderDual EquivLike.toFunLike Equiv.instEquivLike OrderDual.ofDual InfSet.sInf OrderDual.infSet SupSet.sSup Set.preimage OrderDual.toDual	—	—
ofDual_sSup 📖	mathematical	—	DFunLike.coe Equiv OrderDual EquivLike.toFunLike Equiv.instEquivLike OrderDual.ofDual SupSet.sSup OrderDual.supSet InfSet.sInf Set.preimage OrderDual.toDual	—	—
sInf_eq_bot 📖	mathematical	—	InfSet.sInf CompleteSemilatticeInf.toInfSet CompleteLattice.toCompleteSemilatticeInf CompleteLinearOrder.toCompleteLattice Bot.bot OrderBot.toBot Preorder.toLE PartialOrder.toPreorder SemilatticeSup.toPartialOrder Lattice.toSemilatticeSup GeneralizedHeytingAlgebra.toLattice HeytingAlgebra.toGeneralizedHeytingAlgebra BiheytingAlgebra.toHeytingAlgebra CompleteLinearOrder.toBiheytingAlgebra HeytingAlgebra.toOrderBot Set Set.instMembership Preorder.toLT CompleteSemilatticeInf.toPartialOrder	—	sSup_eq_top
sInf_le 📖	mathematical	Set Set.instMembership	Preorder.toLE PartialOrder.toPreorder CompleteSemilatticeInf.toPartialOrder InfSet.sInf CompleteSemilatticeInf.toInfSet	—	CompleteSemilatticeInf.sInf_le
sInf_le_iff 📖	mathematical	—	Preorder.toLE PartialOrder.toPreorder CompleteSemilatticeInf.toPartialOrder InfSet.sInf CompleteSemilatticeInf.toInfSet	—	ge_trans le_sInf sInf_le
sInf_le_of_le 📖	mathematical	Set Set.instMembership Preorder.toLE PartialOrder.toPreorder CompleteSemilatticeInf.toPartialOrder	InfSet.sInf CompleteSemilatticeInf.toInfSet	—	ge_trans sInf_le
sInf_le_sInf 📖	mathematical	Set Set.instHasSubset	Preorder.toLE PartialOrder.toPreorder CompleteSemilatticeInf.toPartialOrder InfSet.sInf CompleteSemilatticeInf.toInfSet	—	IsGLB.mono isGLB_sInf
sInf_lt_iff 📖	mathematical	—	Preorder.toLT PartialOrder.toPreorder CompleteSemilatticeInf.toPartialOrder CompleteLattice.toCompleteSemilatticeInf CompleteLinearOrder.toCompleteLattice InfSet.sInf CompleteSemilatticeInf.toInfSet Set Set.instMembership	—	isGLB_lt_iff isGLB_sInf
sSup_eq_top 📖	mathematical	—	SupSet.sSup CompleteSemilatticeSup.toSupSet CompleteLattice.toCompleteSemilatticeSup CompleteLinearOrder.toCompleteLattice Top.top OrderTop.toTop Preorder.toLE PartialOrder.toPreorder SemilatticeSup.toPartialOrder Lattice.toSemilatticeSup GeneralizedCoheytingAlgebra.toLattice CoheytingAlgebra.toGeneralizedCoheytingAlgebra BiheytingAlgebra.toCoheytingAlgebra CompleteLinearOrder.toBiheytingAlgebra CoheytingAlgebra.toOrderTop Set Set.instMembership Preorder.toLT CompleteSemilatticeInf.toPartialOrder CompleteLattice.toCompleteSemilatticeInf	—	lt_sSup_iff LT.lt.trans_eq top_unique le_of_not_gt LT.lt.false LT.lt.trans_le le_sSup
sSup_le 📖	mathematical	Preorder.toLE PartialOrder.toPreorder CompleteSemilatticeSup.toPartialOrder	SupSet.sSup CompleteSemilatticeSup.toSupSet	—	CompleteSemilatticeSup.sSup_le
sSup_le_iff 📖	mathematical	—	Preorder.toLE PartialOrder.toPreorder CompleteSemilatticeSup.toPartialOrder SupSet.sSup CompleteSemilatticeSup.toSupSet	—	isLUB_le_iff isLUB_sSup
sSup_le_sSup 📖	mathematical	Set Set.instHasSubset	Preorder.toLE PartialOrder.toPreorder CompleteSemilatticeSup.toPartialOrder SupSet.sSup CompleteSemilatticeSup.toSupSet	—	IsLUB.mono isLUB_sSup
toDual_iInf 📖	mathematical	—	DFunLike.coe Equiv OrderDual EquivLike.toFunLike Equiv.instEquivLike OrderDual.toDual iInf iSup OrderDual.supSet	—	—
toDual_iSup 📖	mathematical	—	DFunLike.coe Equiv OrderDual EquivLike.toFunLike Equiv.instEquivLike OrderDual.toDual iSup iInf OrderDual.infSet	—	—
toDual_sInf 📖	mathematical	—	DFunLike.coe Equiv OrderDual EquivLike.toFunLike Equiv.instEquivLike OrderDual.toDual InfSet.sInf SupSet.sSup OrderDual.supSet Set.preimage OrderDual.ofDual	—	—
toDual_sSup 📖	mathematical	—	DFunLike.coe Equiv OrderDual EquivLike.toFunLike Equiv.instEquivLike OrderDual.toDual SupSet.sSup InfSet.sInf OrderDual.infSet Set.preimage OrderDual.ofDual	—	—

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