Subring 📖 | CompData | 419 mathmath: Subring.mem_center_iff, ValuationRing.mem_integer_iff, Subring.map_iSup, Subring.coe_unop, LocalSubring.isLocalRing, Subring.orderedSubtype_coe, Valuation.ltIdeal_le_leIdeal, Subring.coe_subtype, Subring.coe_comap, LocalSubring.range_toSubring, Subring.center.smulCommClass_right, NumberField.canonicalEmbedding.mem_span_latticeBasis, CommRingCat.closure_range_union_range_eq_top_of_isPushout, Subring.map_inf, Subring.pointwise_smul_def, PadicInt.mem_subring_iff, Valuation.mem_leSubmodule_iff, Subring.zero_mem, ValuationSubring.mem_toSubring, IsPurelyInseparable.inseparable', IsPurelyInseparable.elemExponent_min', Subfield.mem_toSubring, Subring.instCanLiftSetCoeAndMemOfNatForallForallForallForallHAddForallForallForallForallHMulForallForallNeg, Subring.toSubsemiring_injective, RingHom.rangeRestrict_surjective, RingHom.range_eq_top_of_surjective, Polynomial.Splits.mem_range_of_isRoot, Subring.instSMulCommClassSubtypeMemCenter, Subring.addEquivOp_apply_coe, Subring.centralizer_eq_top_iff_subset, Polynomial.monic_restriction, Subring.coe_op, Valued.integer.compactSpace_iff_completeSpace_and_isDiscreteValuationRing_and_finite_residueField, Valuation.mem_ltIdeal_iff, Subring.center.coe_div, Subring.closure_singleton_intCast, Valuation.leSubmodule_monotone, Subring.closure_singleton_one, IsPurelyInseparable.elemExponent_min, Algebra.coe_algebraMap_ofSubring, instIsDiscreteValuationRingSubtypeMemSubringIntegerWithZeroMultiplicativeIntValuation, IsPurelyInseparable.elemExponent_def', Subring.coe_toSubsemiring, IsPurelyInseparable.hasExponent_iff, Polynomial.eval₂_restriction, Algebra.toSubring_eq_top, Subring.center.coe_inv, Polynomial.Monic.eq_X_pow_char_pow_sub_C_pow_of_natSepDegree_eq_one, Subring.top_prod, Subring.toAddSubgroup_mono, Subring.centerCongr_apply_coe, Subring.op_le_op_iff, RingEquiv.ofLeftInverse_symm_apply, Subalgebra.toSubring_injective, ExpChar.expChar_center_iff, Subring.le_op_iff, instIsIntegrallyClosedInSubtypeMemSubringToSubringIntegralClosure, isLocalRing_top, RingCon.comapQuotientEquivRange_symm_mk, Algebra.mem_adjoin_iff, RingEquiv.ofLeftInverse_apply, Polynomial.restriction_one, Subring.unop_injective, Algebra.toSubring_bot, Subring.mem_pointwise_smul_iff_inv_smul_mem, IsNonarchimedeanLocalField.instIsDiscreteValuationRingSubtypeMemSubringIntegerValueGroupWithZeroValuation, IsPurelyInseparable.exists_pow_mem_range_tensorProduct, ValuationSubring.toSubring_top, Subring.mem_pointwise_smul_iff_inv_smul_mem₀, RingHom.ker_rangeRestrict, RingHom.mem_range, Subfield.mk_le_mk, Subring.ext_iff, Subring.toSubmonoid_mono, Subring.toSubsemiring_mono, Subring.center_eq_top, Valuation.Integers.mem_of_integral, Valuation.leIdeal_map_algebraMap_eq_leSubmodule_min, Subring.op_sup, Subring.mopRingEquivOp_apply_coe, Subring.comap_iInf, Subring.integralClosure_subring_le_iff, Subring.coe_mk', Subring.coe_matrix, IsDedekindDomain.HeightOneSpectrum.mem_integers_of_valuation_le_one, Subring.op_injective, Subring.closure_singleton_natCast, Valued.integer.exists_norm_lt_one, Polynomial.degree_restriction, Subring.closure_mono, LocalSubring.toSubring_mono, CharP.charP_center_iff, Subfield.mem_closure_iff, Subring.mem_closure, Subring.coe_toAddSubgroup, Subring.pointwise_smul_toAddSubgroup, Subring.coe_pointwise_smul, Subring.eq_top_iff', RingHom.mem_range_self, CharP.subring', Subring.card_top, NumberField.canonicalEmbedding.integerLattice.inter_ball_finite, Subring.instFaithfulSMulSubtypeMem, Subring.mem_inf, iInf_valuationSubring_superset, HahnSeries.mem_cardSuppLTSubring, Rat.HeightOneSpectrum.adicCompletion.padicEquiv_bijOn, Subring.mem_toSubsemiring, mem_perfectClosure_iff_pow_mem, Algebra.SubmersivePresentation.HasCoeffs.coeffs_subset_range, Subring.comap_map_eq, RingCon.comapQuotientEquivRange_mk, Valuation.integer.integers, minpoly.two_le_natDegree_iff, Subring.coe_mul, Valuation.Integers.isIntegrallyClosed_integers, Subring.closure_union, Ideal.image_subset_nonunits_valuationSubring, Subring.closure_eq, Subring.coe_inf, IsPurelyInseparable.HasExponent.has_exponent, Subring.unop_top, Subring.closure_le_centralizer_centralizer, Polynomial.restriction_zero, Subring.op_bot, is_noetherian_subring_closure, LocalSubring.eq_iInf_of_isIntegrallyClosedIn, Subring.mem_prod, Subring.instSubringClass, Subring.mem_inv_pointwise_smul_iff, isPurelyInseparable_iff_pow_mem, Subring.toSubsemiring_top, Subring.op_sSup, TopCat.Sheaf.objSupIsoProdEqLocus_hom_fst, Subring.centralizer_le, LocalSubring.instAtPrimeSubtypeMemSubringToSubringOfPrime, Subring.centerToMulOpposite_symm_apply_coe, Subring.pointwise_smul_subset_iff, Subring.mem_closure_iff, isNoetherianRing_range, Valuation.pow_Uniformizer_is_pow_generator, Subring.subtype_injective, Subring.mem_map, Subring.topEquiv_apply, CharP.subring, Subring.coe_set_mk, Subring.op_top, Subring.smulCommClass_right, Subring.mopRingEquivOp_symm_apply, ValuationSubring.pointwise_smul_toSubring, Subring.mem_top, Polynomial.coeff_ofSubring, PadicInt.Coe.ringHom_apply, mem_subalgebraOfSubring, Subring.coe_map, Subring.op_iInf, Subring.mem_smul_pointwise_iff_exists, Subring.prod_mono_left, Subring.top_prod_top, mem_perfectClosure_iff, IsPurelyInseparable.exponent_def', Subring.mem_map_equiv, RingHom.eqOn_set_closure, Subring.mem_bot, Valued.integer.norm_le_one, Algebra.top_toSubring, Subring.smul_bot, Subring.subset_closure, Subring.smul_closure, mem_range_of_deriv_eq_zero, Subring.map_bot, Subfield.subring_closure_le, Subring.op_sInf, Subring.prod_bot_sup_bot_prod, Subring.unop_eq_top, Subring.unop_sInf, Subring.instCovariantClassHSMulLe, IsNonarchimedeanLocalField.instFiniteResidueFieldSubtypeMemSubringIntegerValueGroupWithZeroValuation, Subfield.coe_set_mk, Valuation.ltSubmodule_monotone, FractionalIdeal.mem_dual, minpoly.mem_range_of_degree_eq_one, Subring.topEquiv_symm_apply_coe, Subring.mem_unop, RingCon.coe_comapQuotientEquivRange_mk, Subring.coe_eq_zero_iff, Subring.mem_op, Valuation.HasExtension.val_algebraMap, Subring.unop_le_unop_iff, Subring.instIsTopologicalRing, Subring.centerToMulOpposite_apply_coe, Subring.instIsDomainSubtypeMem, IsNonarchimedeanLocalField.instCompactSpaceSubtypeMemSubringIntegerValueGroupWithZeroValuation, IntermediateField.isPurelyInseparable_adjoin_iff_pow_mem, Valued.isClosed_integer, IsLocalRing.exists_factor_valuationRing, Subring.unop_inf, Subring.toSubsemiring_strictMono, Algebra.algebraMap_ofSubring, Subring.le_pointwise_smul_iff₀, IsSimpleRing.isField_center, Subring.op_eq_bot, Subring.op_le_iff, Subring.smul_mem_pointwise_smul_iff₀, Valued.integer.properSpace_iff_completeSpace_and_isDiscreteValuationRing_integer_and_finite_residueField, TopCat.Sheaf.objSupIsoProdEqLocus_inv_snd, Subfield.sInf_toSubring, Subring.le_topologicalClosure, Valuation.HasExtension.instIsLocalHomValuationInteger, Polynomial.filter_roots_map_range_eq_map_roots, LocalSubring.le_ofPrime, Subring.toAddSubgroup_strictMono, Valued.integer.coe_span_singleton_eq_closedBall, Subring.closure_preimage_le, Subring.map_comap_eq, IsDiscreteValuationRing.map_algebraMap_eq_valuationSubring, Subring.gc_map_comap, Polynomial.map_restriction, Subring.closure_empty, Valued.isClopen_integer, Subring.coe_top, IsInvariantSubring.coe_subtypeHom', Subring.unop_eq_bot, Subring.pointwise_central_scalar, Subring.pointwise_smul_le_pointwise_smul_iff, Subring.prod_top, Valuation.HasExtension.instIsTorsionFreeInteger, Subfield.mem_mk, Subring.sInf_toAddSubgroup, Subring.unop_bot, IsFractionRing.ringHom_fieldRange_eq_of_comp_eq, Valued.integer.isDiscreteValuationRing_of_compactSpace, Subring.isClosed_topologicalClosure, Valuation.ltSubmodule_le_leSubmodule, Valuation.leIdeal_mono, Subring.comap_top, Valued.integer.norm_unit, Valuation.leIdeal_zero, Subring.integralClosure_le_iff, Subring.pointwise_smul_le_iff₀, minpoly.ofSubring, Polynomial.mem_closure_X_union_C, LaurentSeries.powerSeriesEquivSubring_apply, Subring.mem_comap, Subring.coe_intCast, Subring.mk_le_mk, RingCon.range_mk', isIntegral_iff_isIntegral_closure_finite, Subring.mem_carrier, Subring.centerCongr_symm_apply_coe, isPurelyInseparable_iff, Subring.subtype_apply, Valued.integer.exists_nnnorm_lt_one, Subfield.coe_toSubring, RingHom.eqLocus_same, Subring.toSubmonoid_strictMono, Valued.integer.properSpace_iff_compactSpace_integer, Subring.ringEquivOpMop_symm_apply_coe, Polynomial.mem_map_range, Subring.coe_toSubmonoid, minpoly.degree_eq_one_iff, Subring.continuousSMul, Subring.coe_zero, IsPurelyInseparable.pow_mem, NumberField.mem_span_integralBasis, Polynomial.coeff_restriction, Subring.unop_sup, ValuationSubring.toSubring_injective, LaurentSeries.powerSeriesEquivSubring_coe_apply, Subring.op_iSup, isArtinianRing_range, RingHom.range_eq_map, TopCat.Sheaf.objSupIsoProdEqLocus_hom_snd, Valuation.integers_nontrivial, TopCat.Sheaf.objSupIsoProdEqLocus_inv_eq_iff, Subring.center.smulCommClass_left, Algebra.FormallyUnramified.range_eq_top_of_isPurelyInseparable, LinearMap.exists_mem_center_apply_eq_smul_of_forall_notLinearIndependent, Valuation.Integer.not_isUnit_iff_valuation_lt_one, Subring.coe_neg, minpoly.natSepDegree_eq_one_iff_pow_mem, Subring.smul_def, Valuation.exists_pow_Uniformizer, Submodule.mem_traceDual, Polynomial.toSubring_zero, Valued.isOpen_integer, Subring.pointwise_smul_le_pointwise_smul_iff₀, RingHom.range_eq_top, Subring.closure_singleton_zero, Valued.integer.finite_quotient_maximalIdeal_pow_of_finite_residueField, IsPurelyInseparable.exponent_min, Valuation.integer.coe_span_singleton_eq_setOf_le_v_coe, Valued.integer.mem_iff, Polynomial.lifts_iff_liftsRing, Subring.smul_sup, AlgebraicGeometry.Proj.valuativeCriterion_existence_aux, LocalSubring.instIsScalarTowerSubtypeMemSubringToSubringOfPrime, Polynomial.Monic.eq_X_pow_char_pow_sub_C_of_natSepDegree_eq_one_of_irreducible, Subring.mem_centralizer_iff, Subring.opEquiv_apply, Subring.instNontrivialSubtypeMem, instCharPLinearMapSubtypeMemSubringCenterId, Subring.mem_toAddSubgroup, Subring.instNoZeroDivisorsSubtypeMem, TopCat.Sheaf.objSupIsoProdEqLocus_inv_fst, Valued.integer.isUnit_iff_norm_eq_one, Subring.mem_closure_of_mem, Valued.integer.isPrincipalIdealRing_of_compactSpace, Subring.coe_prod, IsPurelyInseparable.hasExponent_iff', instIsLocalRingSubtypeMemSubringMapOfNontrivial, IsIntegral.mem_range_algebraMap_of_minpoly_splits, Valuation.mem_ltSubmodule_iff, LocalSubring.le_def, Polynomial.natDegree_restriction, Subring.sInf_toSubmonoid, Subring.mem_mk, minpoly.instIsScalarTowerSubtypeMemSubringSubalgebraIntegralClosure, minpoly.natDegree_eq_one_iff, Subring.opEquiv_symm_apply, Subring.mem_iInf, Subring.toAddSubgroup_eq_top, Valued.toNormedField.setOf_mem_integer_eq_closedBall, ValuationRing.instValuationRingInteger, Subring.closure_univ, Subring.subset_pointwise_smul_iff, Subring.op_inf, IsPurelyInseparable.exponent_min', Valuation.mem_leIdeal_iff, Subring.instSMulCommClassSubtypeMemCenter_1, bijective_rangeRestrict_comp_of_valuationRing, Subring.unop_iInf, Subring.coe_bot, Subring.unop_iSup, Polynomial.coeff_restriction', Valued.integer.totallyBounded_iff_finite_residueField, Valuation.Uniformizer.is_generator, instExpCharLinearMapSubtypeMemSubringCenterId, Subring.map_le_iff_le_comap, Subring.coe_center, Subring.mem_sInf, Polynomial.support_restriction, Subring.eq_iInf_of_isIntegrallyClosedIn, IsFractionRing.lift_fieldRange, Set.integer_eq, Subring.coe_ofClass, LinearMap.commute_transvections_iff_of_basis, Subring.op_eq_top, Subring.mem_toSubmonoid, Subalgebra.mem_toSubring, Subring.coe_equivMapOfInjective_apply, Valuation.ltIdeal_mono, Subring.range_subtype, Subring.coe_add, ValuationRing.coe_equivInteger_apply, Algebra.algebraMap_ofSubring_apply, RootPairing.toLinearMap_apply_apply_mem_range_algebraMap, AlgebraicGeometry.exists_lift_of_germInjective_aux, Valuation.leSubmodule_zero, Valuation.mem_integer_iff, Subring.map_iInf, Valued.integer.norm_coe_unit, Subring.closure_le, Subalgebra.coe_toSubring, Subring.unop_sSup, Subring.coe_pow, Subring.toAddSubgroup_injective, IsNonarchimedeanLocalField.instCompleteSpaceSubtypeMemSubringIntegerValueGroupWithZeroValuation, Subring.comap_inf, IsPurelyInseparable.exponent_def, Valuation.leSubmodule_comap_algebraMap_eq_leIdeal, Subring.smul_mem_pointwise_smul_iff, instIsPrincipalIdealRingSubtypeMemSubringIntegerWithZeroMultiplicativeIntValuation, Subring.coe_centralizer, IsPurelyInseparable.exists_pow_pow_mem_range_tensorProduct_of_expChar, Subring.ringEquivOpMop_apply, LinearMap.exists_mem_center_apply_eq_smul_of_forall_notLinearIndependent_of_basis, Subring.toSubsemiring_eq_top, Subring.pointwise_smul_toSubsemiring, Subring.map_sup, JacobsonNoether.exists_separable_and_not_isCentral, instIsIntegrallyClosedSubtypeMemSubring, FixedPoints.minpoly.eval₂, Polynomial.coeffs_ofSubring, Valuation.Uniformizer.valuation_gt_one, Subring.center_le_centralizer, Subring.mem_mk', Subalgebra.pointwise_smul_toSubring, Subring.toAddSubgroup_top, Differential.ContainConstants.mem_range_of_deriv_eq_zero, Subring.coe_sInf, IsPurelyInseparable.inseparable, Subring.mem_inv_pointwise_smul_iff₀, Valued.integer.exists_norm_coe_lt_one, Subring.coe_iInf, Subring.instIsScalarTowerSubtypeMem, Subring.smulCommClass_left, Subring.prod_mono_right, RingHom.closure_preimage_le, Subring.one_mem, RingHom.coe_range, Subring.coe_one, Subring.coe_natCast, RingHom.coe_rangeRestrict, Subring.addEquivOp_symm_apply_coe, Valuation.HasExtension.algebraMap_injective, ValuationRing.instIsFractionRingInteger, IsInvariantSubring.coe_subtypeHom, IsPurelyInseparable.elemExponent_def, LocalSubring.toSubring_injective, Subring.toSubmonoid_injective, RingHom.mem_eqLocus, IntermediateField.isPurelyInseparable_adjoin_simple_iff_pow_mem, Subring.closure_sUnion, Valuation.HasExtension.instIsScalarTowerInteger, Valuation.HasExtension.val_smul, Subring.closure_iUnion, Algebra.TensorProduct.closure_range_union_range_eq_top, Polynomial.toSubring_one
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