Submonoid 📖 | CompData | 1395 mathmath: unitary.spectrum.unitary_conjugate, Unitary.openPartialHomeomorph_source, ValuativeRel.ValueGroupWithZero.mk_one_one, Submonoid.inv_top, IsLocalization.AtPrime.coe_primeSpectrumOrderIso_symm_apply_asIdeal, Matrix.UnitaryGroup.toLin'_one, nonZeroDivisors_le_comap_nonZeroDivisors_of_injective, Submonoid.units_iInf, OreLocalization.OreSet.ore_eq, CStarRing.norm_mul_coe_unitary, Submonoid.mem_map_iff_mem, Submonoid.mem_toSubsemigroup, unitary.star_eq_inv', Submonoid.monotone_comap, Submonoid.mrange_inr', Submonoid.mem_map_equiv, Unitary.mul_left_inj, RatFunc.mk_def_of_ne, Submonoid.mem_closure_iff_of_fintype, Submonoid.map_inr, SymplecticGroup.transpose_mem_iff, Unitary.conjStarAlgAut_trans_conjStarAlgAut, RatFunc.mk_coe_def, MonoidHom.mker_snd, MonoidHom.coe_mrange, Submonoid.comap_iInf, Submonoid.le_op_iff, Submonoid.coe_ofClass, Subgroup.ofUnits_bot, Submonoid.subsemiringClosure_toAddSubmonoid, CStarRing.norm_unitary_smul, ValuativeRel.ValueGroupWithZero.mk_eq_one, AlgebraicGeometry.ProjectiveSpectrum.Proj.awayToSection_apply, Localization.one_rel, LaurentPolynomial.mk'_one_X_pow, FractionalIdeal.finprod_heightOneSpectrum_factorization_principal, Real.hasFDerivAt_fourierChar_neg_bilinear_left, IsOfFinOrder.natCard_powers_le_orderOf, Subgroup.equivSMul_apply_coe, Submonoid.equivOp_symm_apply_coe, Ideal.finite_setOf_absNorm_le₀, Submonoid.le_centralizer_centralizer, NumberField.mixedEmbedding.fundamentalCone.preimageOfMemIntegerSet_mixedEmbedding, isSMulRegular_iff_mem_nonZeroSMulDivisors, ClassGroup.mk0_integralRep, MonoidHom.eqOn_closureM, reindex_mem_doublyStochastic_iff, LaurentPolynomial.mk'_mul_T, LocalizedModule.r.isEquiv, ClassGroup.mk0_eq_mk0_iff, Valuation.val_mrange_zero, PrimeSpectrum.localization_specComap_range, Submonoid.op_iInf, Matrix.UnitaryGroup.toGL_mul, IsOfFinOrder.mem_powers_iff_mem_zpowers, spinGroup.star_eq_inv, Subgroup.toSubmonoid_le, Circle.exp_arg, Fintype.card_coeSort_mrange, Circle.injective_arg, mem_own_rightCoset, Submonoid.coe_invOrderIso_apply, Algebra.GrothendieckGroup.instFG, Ideal.IsHomogeneous.iff_exists, Submonoid.unop_eq_top, Submonoid.coe_pi, Localization.mk_le_mk, IsLocalization.smul_mem_finsetIntegerMultiple_span, Submonoid.powers_one, AlgebraicGeometry.ProjectiveSpectrum.Proj.toStalk_specStalkEquiv, Submonoid.val_mem_of_mem_units, Submonoid.smul_mem_pointwise_smul_iff₀, spinGroup.coe_star_mul_self, Unitary.conjStarAlgAut_symm, SymplecticGroup.coe_inv, unitary.map_id, Submonoid.IsLocalizationMap.map_units, Submonoid.mrange_inl', pinGroup.coe_star_mul_self, Unitary.mem_pathComponentOne_iff, Unitary.toUnits_injective, Submonoid.fg_iff_add_fg, Subring.addEquivOp_apply_coe, Real.fderiv_fourierChar_neg_bilinear_right_apply, Submonoid.unop_iInf, IsMulIndecomposable.mem_or_inv_mem_closure_baseOf, val_inv_unitarySubgroupUnitsEquiv_symm_apply_coe, OreLocalization.oreDiv_eq_iff, SymplecticGroup.coe_J, Submonoid.top_closure_mul_self_subset, Submonoid.closure_singleton_le_iff_mem, unitary.inv_mul_mem_iff, Submonoid.mem_pi, Submonoid.pi_top, Submonoid.prod_top, Matrix.reindex_mem_colStochastic_iff, OreLocalization.oreDiv_one_surjective_of_finite_right, Ring.ordFrac_eq_div, Submonoid.isClosed_topologicalClosure, Unitary.coe_zpow, Submonoid.mem_pos, Unitary.conjStarAlgEquiv_unitaryLinearIsometryEquiv, Matrix.specialUnitaryGroup_le_unitaryGroup, NumberField.mixedEmbedding.fundamentalCone.integerSetEquivNorm_apply_fst, unitsNonZeroDivisorsEquiv_apply, Submonoid.comap_sup_map_of_injective, Submonoid.mem_closure, finEquivPowers_apply, Submonoid.le_pi_iff, ofUnits_le_iff_le_units, selfAdjoint.expUnitaryPathToOne_apply, MonoidWithZeroHom.mrange_nontrivial, Unitary.continuousOn_argSelfAdjoint, Submonoid.coe_list_prod, Submonoid.coe_map, selfZPow_pow_sub, Submonoid.unop_iSup, Unitary.mapEquiv_symm_apply, IsLocalization.surj, Submonoid.mem_map, Con.le_iff, selfAdjoint.norm_sq_expUnitary_sub_one, Circle.ofConjDivSelf_coe, HomogeneousLocalization.eq_num_div_den, Submonoid.coe_inf, Submonoid.isScalarTower, spinGroup.star_eq_inv', IsLocalization.mk'_pow, Submonoid.instCanLiftSetCoeAndMemOfNatForallForallForallForallHMul, Submonoid.mem_top, pinGroup.val_inv_toUnits_apply, Circle.starRingEnd_addChar, Submonoid.closure_pow_le, IsLocalization.mk'_tmul, mem_doublyStochastic_iff_sum, LaurentPolynomial.mk'_one_X, Submonoid.unop_inf, Localization.localRingHom_mk', val_unitarySubgroupUnitsEquiv_symm_apply_coe, isRightRegular_iff_mem_nonZeroDivisorsRight, Submonoid.comap_surjective_of_injective, ValuationSubring.ofPrime_valuation_eq_one_iff_mem_primeCompl, Valuation.instNontrivialSubtypeUnitsMemSubmonoidValueMonoidOfIsNontrivial, Ideal.associatesNonZeroDivisorsEquivIsPrincipal_apply, Submonoid.subsemiringClosure_eq_closure, IsLocalization.lift_mem_adjoin_finsetIntegerMultiple, ValuativeRel.zero_srel_coe_posSubmonoid, Matrix.permMatrix_mem_rowStochastic, Submonoid.coe_invOrderIso_symm_apply, LocalizedModule.mk_add_mk, Monoid.fg_iff_submonoid_fg, Submonoid.map_comap_eq, Subgroup.mem_ofUnits_iff_exists_isUnit, Localization.liftOn_zero, IsLocalizedModule.iso_symm_apply, IsLocalization.isUnit_comp, SetLike.GradeZero.coe_submonoid, LocalizedModule.liftOn_mk, Submonoid.units_bot, pinGroup.star_mul_self, AddCircle.homeomorphCircle'_symm_apply, IsLocalizedModule.iso_apply_mk, Submonoid.mem_inv, OreLocalization.smul_cancel', IsFractionRing.mk'_eq_one_iff_eq, Subgroup.ofUnits_strictMono, Submonoid.le_comap_map, Rat.isFractionRingDen, MvPowerSeries.monomial_mem_nonzeroDivisors, LocalizedModule.divBy_apply, mem_unitarySubgroup_iff, MonoidWithZeroHom.coe_one, Ideal.absNorm_pos_iff_mem_nonZeroDivisors, unitary.coe_star_mul_self, IsLocalizedModule.mk'_eq_zero', IsLocalization.coe_primeSpectrumOrderIso_symm_apply_asIdeal, rotation_apply, toMatrix_rotation, IsFractionRing.mk'_eq_div, Submonoid.pointwise_smul_le_pointwise_smul_iff₀, Submodule.mem_torsion_iff, Submodule.isTorsion'_powers_iff, Submonoid.mem_smul_pointwise_iff_exists, Submonoid.centralizer_eq_top_iff_subset, Unitary.coe_linearIsometryEquiv_apply, Unitary.inv_mul_mem_iff, Subgroup.toSubmonoid_mono, NumberField.mixedEmbedding.fundamentalCone.card_isPrincipal_dvd_norm_le, RatFunc.liftOn_ofFractionRing_mk, IsLocalization.smul_mk'_self, Submonoid.mem_closure_singleton_self, Localization.mk_algebraMap, Submonoid.instMeasurableSMul, Submonoid.topOrderMonoidIso_symm_apply_coe, Submonoid.comap_strictMono_of_surjective, AddSubmonoid.coe_toSubmonoid_apply, unitary.spectrum.unitary_conjugate', Pell.is_pell_solution_iff_mem_unitary, unitary.mem_iff_self_mul_star, Submonoid.subsemiringClosure_coe, Submonoid.map_sup_comap_of_surjective, Submonoid.pointwise_smul_le_pointwise_smul_iff, Submonoid.mem_sup, unitary.coe_div, finite_powers, mem_nonZeroDivisors_iff, IsLocalization.mk'_mul_cancel_left, OreLocalization.nonempty_oreSet_iff_of_noZeroDivisors, Unitary.linearIsometryEquiv_coe_apply, Matrix.IsHermitian.eigenvectorUnitary_col_eq, Submonoid.mem_oneLE, Equiv.Perm.mclosure_swap_castSucc_succ, MulEquiv.coe_submonoidMap_apply, IsLocalizedModule.exist_integer_multiples, Submonoid.top_prod, IsLocalization.AtPrime.coe_orderIsoOfPrime_symm_apply_coe, val_unitsCentralizerEquiv_apply_coe, LinearIsometryEquiv.trans_smul, Set.IsPWO.submonoid_closure, spinGroup.mem_iff, MvPowerSeries.monomial_mem_nonzeroDivisorsLeft, Submonoid.iSup_eq_closure, num_isRoot_scaleRoots_of_aeval_eq_zero, Submonoid.ofUnits_units_le, IsFractionRing.exists_reduced_fraction, NumberField.mixedEmbedding.fundamentalCone.card_isPrincipal_norm_eq_mul_torsion, Monoid.FG.fg_top, unitInterval.mem_unitIntervalSubmonoid, FractionalIdeal.isFractional_span_iff, Submonoid.mk_eq_bot, unitary.coe_zpow, LocalizedModule.divBy_mul_by, LocalizedModule.mk_mul_mk, isUnit_le_nonZeroDivisors, Submonoid.mrange_inl_sup_mrange_inr, Submonoid.prod_bot_sup_bot_prod, Subsemiring.addEquivOp_apply_coe, Complex.UnitDisc.coe_smul_circle, Subring.toSubmonoid_mono, LocalizedModule.mk_eq, Submonoid.le_topologicalClosure, Submonoid.mem_inv_pointwise_smul_iff, ValuativeRel.ValueGroupWithZero.mk_eq_div, unitary.map_comp, Submonoid.comap_iInf_map_of_injective, ValuativeRel.ValueGroupWithZero.exact, Submonoid.nontrivial_iff, OreLocalization.smul_oreDiv_one, ValuativeExtension.mapValueGroupWithZero_mk, NumberField.mixedEmbedding.fundamentalCone.integerSetQuotEquivAssociates_apply, Commute.exists_orderOf_eq_lcm, MonoidHom.mrange_eq_map, SymplecticGroup.mem_iff, Subring.mopRingEquivOp_apply_coe, Submonoid.log_mul, Submonoid.coe_bot, mem_nonZeroSMulDivisors_iff, IsLocalizedModule.mk'_add_mk', Monoid.Coprod.mrange_swap, Submonoid.mem_iSup, OreLocalization.oreDiv_mul_oreDiv, val_unitsCenterToCenterUnits_apply_coe, AlgebraicGeometry.ProjectiveSpectrum.Proj.awayToΓ_ΓToStalk, Submonoid.coe_center, Monoid.CoprodI.mrange_eq_iSup, spinGroup.coe_mul_star_self, LocalizedModule.mk_cancel, Subgroup.mem_mk, Submonoid.LocalizationMap.subsingleton_iff, pinGroup.val_toUnits_apply, Matrix.IsHermitian.eigenvectorUnitary_mulVec, unitary.mapEquiv_trans, QuadraticAlgebra.norm_eq_one_iff_mem_unitary, LocalizedModule.mem_ker_mkLinearMap_iff, Submonoid.iSup_map_mulSingle_le, IsIntegral.exists_multiple_integral_of_isLocalization, Submonoid.topEquiv_symm_apply_coe, IsLocalization.surj'', Ideal.absNorm_ne_zero_iff_mem_nonZeroDivisors, Monoid.Coprod.mrange_eq, Submonoid.lipschitzMul, LocalizedModule.equivTensorProduct_symm_apply_tmul_one, IsLocalization.map_eq_zero_iff, Subgroup.mem_ofUnits_iff, Unitary.spectrum_star_left_conjugate, IsLocalization.integerNormalization_map_to_map, Orientation.rotation_map_complex, Submonoid.op_eq_bot, Rat.associated_num_den, Subgroup.ofUnits_le_ofUnits_iff, spinGroup.val_inv_toUnits_apply, RatFunc.mk_eq_localization_mk, Submonoid.coe_pointwise_smul, Unitary.openPartialHomeomorph_symm_apply, Units.mul_inv_mem_unitary, Pell.Solution₁.coe_mk, instIsTopologicalGroupSubtypeMemSubmonoidUnitaryOfContinuousMulOfContinuousStar, IsLocalizedModule.exist_integer_multiples_of_finset, OreLocalization.mul_div_one, unitary.mem_iff, MonoidWithZeroHom.mem_valueMonoid, Submonoid.toAddSubmonoid_closure, Unitary.mem_iff_self_mul_star, Matrix.mem_unitaryGroup_iff, IsLocalizedModule.mk'_eq_mk'_iff, RatFunc.mk_eq_mk', OreLocalization.smul_cancel, BoundedContinuousFunction.mem_charPoly, NumberField.mixedEmbedding.fundamentalCone.integerSetEquiv_apply_fst, BoundedContinuousFunction.charAlgHom_apply, Submonoid.ext_iff, Circle.arg_eq_arg, unitary.coe_smul, Submonoid.sup_eq_closure_mul, Submonoid.eq_bot_iff_forall, Monoid.IsTorsion.torsionMulEquiv_apply, IsLocalizedModule.finsetIntegerMultiple_image, IsLocalization.mk'_sub, unitarySubgroupUnitsEquiv_apply_coe, Submonoid.centerToMulOpposite_apply_coe, LinearIsometryEquiv.smul_apply, OreLocalization.nonempty_oreSet_iff, IsLocalization.AtPrime.equivQuotientMapOfIsMaximal_symm_apply_mk, Localization.mk_intCast, fourier_one, Submonoid.coe_inv_val_mul_coe_val, Submonoid.comap_top, Unitary.map_coe, Submonoid.LocalizationMap.mk'_one, Module.FinitePresentation.exists_lift_of_isLocalizedModule, Matrix.mem_specialOrthogonalGroup_iff, orderOf_submonoid, Real.Angle.coe_toCircle, unitary.linearIsometryEquiv_coe_symm_apply, Submonoid.smul_mem_pointwise_smul_iff, unitary.toUnits_comp_map, Monoid.CoprodI.mclosure_iUnion_range_of, PresentedMonoid.closure_range_of, nonZeroDivisors_dvd_iff_dvd_coe, OreLocalization.div_eq_one, Submonoid.closure_eq_one_union, Ideal.disjoint_powers_iff_notMem, Matrix.reindex_mem_rowStochastic_iff, doublyStochastic_eq_rowStochastic_inf_colStochastic, MonoidHom.mker_inl, Submonoid.op_sSup, le_nonZeroDivisors_iff_isRegular, fourier_neg', MonoidHom.submonoidMap_apply_coe, Subgroup.top_toSubmonoid, Submonoid.op_sInf, MonoidHom.mrange_eq_top, Submonoid.LocalizationMap.top_injective_iff, MvPowerSeries.monomial_mem_nonzeroDivisorsRight, IsOfFinOrder.finite_powers, unitary.norm_sub_eq, Submonoid.map_le_iff_le_comap, Submonoid.centralizer_le, AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.mem_carrier_iff', Submonoid.mrange_inr, Algebra.GrothendieckGroup.lift_symm_apply, multiple_mem_adjoin_of_mem_localization_adjoin, Circle.smul_def, IsLocalization.mk'_spec', Submonoid.coe_val_mul_coe_inv_val, Localization.mk_one_eq_monoidOf_mk, IsDedekindDomain.HeightOneSpectrum.valuation_def, val_inv_unitsNonZeroDivisorsEquiv_symm_apply_coe, Subgroup.toSubmonoid_strictMono, Unitary.coe_symm_linearIsometryEquiv_apply, selfAdjoint.expUnitary_zero, fixedPoints_submonoid_sup, unitary.mem_iff_star_mul_self, IsOfFinOrder.mem_powers_iff_mem_range_orderOf, Submonoid.mem_square, mem_nonZeroDivisors_iff_right, Unitary.instLocPathConnectedSpace, Ideal.disjoint_map_primeCompl_iff_comap_le, Submonoid.mem_comap, OreLocalization.expand', Submonoid.LocalizationMap.mk'_eq_iff_eq, Submonoid.val_inv_unitsTypeEquivIsUnitSubmonoid_symm_apply, Submonoid.closure_univ, Unitary.coe_mul_star_self, Submonoid.smul_sup, Localization.mkHom_surjective, Subsemiring.coe_toSubmonoid, IsLocalizedModule.isRegular_of_smul_left_injective, IsLocalization.finsetIntegerMultiple_image, Submonoid.le_pointwise_smul_iff₀, LocalizedModule.mk_cancel_common_left, Submonoid.IsLocalizationMap.exists_of_eq, AlgebraicGeometry.StructureSheaf.isLocalizedModule_toPushforwardStalkAlgHom_aux, Ideal.disjoint_nonZeroDivisors_of_mem_minimalPrimes, Submonoid.LocalizationMap.nonZeroDivisors_le_comap, Con.mem_coe, OreLocalization.subsingleton_iff, Unitary.mul_star_self, Submonoid.inv_bot, AlgebraicGeometry.ProjectiveSpectrum.Proj.isLocalization_atPrime, IsLocalizedModule.exist_integer_multiples_of_finite, IsLocalization.isUnit_piRingHom_algebraMap_comp_piEvalRingHom, isLocalizedModule_iff, Submonoid.card_bot, Unitary.spectrum_star_right_conjugate, Monoid.CoprodI.lift_mrange_le, MonoidHom.mem_eqLocusM, Submonoid.mrange_subtype, Submonoid.FG.finset_sup, Submonoid.map_iSup, selfAdjoint.continuous_expUnitary, Subsemiring.toSubmonoid_mono, IsLocalization.algebraMap_mem_map_algebraMap_iff, Submonoid.pointwise_smul_le_iff₀, Unitary.conjStarAlgAut_ext_iff', Submonoid.coe_prod, Submonoid.map_inf, FractionalIdeal.canonicalEquiv_mk0, QuadraticAlgebra.star_mem_nonZeroDivisors_iff, selfAdjoint.unitarySelfAddISMul_coe, Submonoid.LocalizationMap.eq_iff_exists, Circle.ext_iff, ValuativeRel.ValueGroupWithZero.inv_mk, Submonoid.iSup_map_mulSingle, LaurentPolynomial.mk'_eq, Submonoid.LocalizationMap.mk'_eq_of_same, Submonoid.exists_minimal_closure_eq_top, Matrix.UnitaryGroup.one_apply, ValuativeRel.exists_valuation_posSubmonoid_div_valuation_posSubmonoid_eq, IsLocalizedModule.subsingleton_iff, QuadraticAlgebra.mem_unitary, coe_nonZeroDivisorsRight_eq, Submonoid.mem_sInf, unitary.mul_left_inj, Monoid.fg_range, FreeMonoid.closure_range_of, AddSubmonoid.toSubmonoid_closure, Submonoid.instSubmonoidClass, Submonoid.mem_bot, Polynomial.notMem_nonZeroDivisors_iff, Subgroup.mem_ofUnits_of_isUnit_of_unit_mem, OreLocalization.numeratorRingHom_apply, Submonoid.pi_le_iff, IsFractionRing.mk'_num_den', Subalgebra.linearEquivOp_apply_coe, IsLocalization.isPrime_iff_isPrime_disjoint, Subring.mem_closure_iff, LinearIsometryEquiv.conjStarAlgEquiv_ext_iff, Circle.coe_inv_eq_conj, MonoidHom.mem_mker, ValuativeRel.ValueGroupWithZero.mk_eq_mk, IsFractionRing.div_surjective, IsLocalization.mk'_algebraMap_eq_mk', Circle.coe_mul, Submonoid.mk_mul_mk_inv_eq_one, Unitary.map_comp, Unitary.mul_right_inj, Submonoid.map_sup, Submonoid.mk_inv_mul_mk_eq_one, Submonoid.unitsTypeEquivIsUnitSubmonoid_apply_coe, Unitary.mapEquiv_apply, HomogeneousLocalization.val_mk, unitInterval.coe_unitIntervalSubmonoid, mul_left_coe_nonZeroDivisors_eq_zero_iff, IsLocalization.localization_localization_surj, IsLocalizedModule.mem_ker_iff, unitary.toUnits_injective, powers_eq_zpowers, Real.Angle.arg_toCircle, Matrix.UnitaryGroup.one_val, OreLocalization.oreDiv_add_oreDiv, isLocalization_iff, Circle.coe_pow, ofMul_image_powers_eq_multiples_ofMul, Submonoid.inv_inf, dvd_cancel_right_coe_nonZeroDivisors, mul_right_coe_nonZeroDivisors_eq_zero_iff, Submonoid.map_comap_le, Subgroup.mk_le_mk, IsLocalization.algebraMap_isUnit_iff, Submonoid.mem_prod, Submonoid.unop_top, IsLocalization.mk'_self'', Ideal.span_singleton_nonZeroDivisors, Submonoid.instSMulCommClassSubtypeMemCenter_1, Submonoid.mem_closure_singleton, Submonoid.map_le_map_iff_of_injective, IsArtinianRing.isUnit_iff_mem_nonZeroDivisors, Submonoid.toSubsemigroup_injective, LinearMap.ker_toSpanSingleton_eq_bot_iff, Module.FinitePresentation.exists_free_localizedModule_powers, IsLocalization.ringEquivOfRingEquiv_eq_map, Submonoid.IsLocalizationMap.surj_pi_of_finite, IsLocalizedModule.fromLocalizedModule_mk, PNat.equivNonZeroDivisorsNat_symm_apply_coe, IsFractionRing.isUnit_map_of_injective, Matrix.convex_colStochastic, Submonoid.le_comap_mulSingle_pi, Subsemiring.addEquivOp_symm_apply_coe, Submonoid.op_iSup, Submonoid.units_top, Submonoid.coe_subtype, ValuativeRel.ValueGroupWithZero.mk_lt_mk, Submonoid.leftInv_leftInv_le, Submonoid.LocalizationMap.mk'_mul_cancel_left, pinGroup.star_eq_inv', OreLocalization.mul_cancel, MonoidHom.comap_bot', MulEquiv.ofLeftInverse'_apply, Pell.isPell_iff_mem_unitary, Submonoid.closure_mono, Nat.card_submonoidPowers, Circle.invOn_arg_exp, Submonoid.coe_op, Submonoid.LocalizationMap.mk'_cancel, mem_powers_iff_mem_range_orderOf, val_unitsNonZeroDivisorsEquiv_symm_apply_coe, NumberField.mixedEmbedding.fundamentalCone.integerSetToAssociates_apply, Submonoid.units_sInf, IsLocalization.smul_bijective, IsLocalization.sec_spec', Unitary.coe_smul, CStarAlgebra.norm_smul_two_inv_smul_add_four_unitary, MulAction.mem_stabilizerSubmonoid_iff, IsUnit.mem_submonoid_iff, IsDedekindDomain.HeightOneSpectrum.valuation_of_mk', Submonoid.inv_sup, IsLocalization.exists_isIntegral_smul_of_isIntegral_map, Submonoid.center_eq_top, Subgroup.toSubmonoid_zpowers, Subgroup.val_centerUnitsEquivUnitsCenter_symm_apply_coe, IsLocalization.mk'_add_eq_iff_add_mul_eq_mul, IsLocalizedModule.exists_isLocalizedModule_powers_of_finitePresentation, Monoid.IsTorsion.torsionMulEquiv_symm_apply_coe, Submonoid.mul_subset_closure, Submonoid.centerToMulOpposite_symm_apply_coe, Submonoid.eq_bot_of_subsingleton, LocalizedModule.mk_cancel_common_right, HomogeneousLocalization.den_mem, Submonoid.powers_le_zpowers, Submonoid.top_prod_top, Valuation.apply_posSubmonoid_pos, Matrix.IsHermitian.cfcAux_apply, Matrix.UnitaryGroup.det_isUnit, Algebra.GrothendieckGroup.inv_mk, Submonoid.LocalizationMap.map_eq_zero_iff, Submonoid.bot_or_exists_ne_one, notMem_nonZeroDivisors_iff, IsLocalization.mk'_mem_map_algebraMap_iff, unitsCenterToCenterUnits_injective, Monoid.exponent_submonoid_dvd, Submonoid.apply_coe_mem_map, Circle.star_addChar, Submonoid.op_le_iff, Submonoid.instSMulCommClassSubtypeMemCenter, Subgroup.topEquiv_symm_apply_coe, fixingSubmonoid_antitone, IsUnit.mem_unitary_of_star_mul_self, Unitary.linearIsometryEquiv_coe_symm_apply, IsLocalization.algEquivOfAlgEquiv_mk', MonoidHom.mrange_id, IsLocalizedModule.mk'_cancel', IsLocalization.algEquivOfAlgEquiv_eq_map, Algebra.adjoin_toSubmodule_le, TopCat.Presheaf.SubmonoidPresheaf.map, IsLocalization.map_integerMultiple, Submonoid.mem_inv_pointwise_smul_iff₀, NumberField.Ideal.tendsto_norm_le_div_atTop₀, IsLeftRegular.mem_nonZeroDivisorsLeft, Unitary.star_mul_self, Submonoid.closure_prod_one, Submonoid.units_surjective, IsLocalizedModule.mk'_one, Submonoid.coe_oneLE, Matrix.UnitaryGroup.inv_val, unitary.spectrum_subset_circle, noZeroDivisors_iff_forall_mem_nonZeroDivisorsRight, Submonoid.LocalizationMap.isUnit_comp, unitary.val_toUnits_apply, ValuativeRel.ValueGroupWithZero.mk_mul_mk, MonoidHom.mrangeRestrict_surjective, Submonoid.units_left_inverse, Unitary.mul_inv_mem_iff, Submonoid.subtype_injective, Unitary.instIsStarNormal, OreLocalization.ore_eq, MonoidHom.coe_mker, Valued.integer.mulArchimedean_mrange_of_isCompact_integer, Subgroup.coe_set_mk, Algebra.algebraMapSubmonoid_le_comap, Submonoid.mem_iSup_prop, Subgroup.mem_ofUnits_iff_toUnits_mem, Subsemiring.toSubmonoid_strictMono, IsLocalization.localization_localization_map_units, Submonoid.closure_eq, Submonoid.faithfulSMul, Submodule.annihilator_top_inter_nonZeroDivisors, Subgroup.ofUnits_inf, Submonoid.coe_top, Module.Finite.exists_smul_of_comp_eq_of_isLocalizedModule, Algebra.GrothendieckGroup.mk_div_mk, Unitary.conjStarAlgAut_apply, Monoid.Coprod.mrange_lift, IsFractionRing.inv_def, Submonoid.map_iSup_comap_of_surjective, FractionalIdeal.absNorm_eq, Polynomial.Monic.mem_nonZeroDivisors, IsArtinianRing.isUnit_iff_nonZeroDivisor_of_isIntegral, Submonoid.centerCongr_symm_apply_coe, IsLocalization.mk'_self', ofUnits_units_gc, LinearIsometryEquiv.toLinearEquiv_smul, unitary.mapEquiv_symm, ValuativeRel.zero_vlt_coe_posSubmonoid, Unitary.mem_iff_star_mul_self, Polynomial.mem_nonZeroDivisors_iff, Unitary.toAlgEquiv_conjStarAlgAut, le_nonZeroDivisorsLeft_iff_isLeftRegular, Submonoid.comap_iSup_map_of_injective, pinGroup.units_mem_iff, selfAdjoint.realPart_unitarySelfAddISMul, MonoidHom.submonoidComap_apply_coe, Subgroup.exponent_toSubmonoid, Submonoid.prod_eq_bot_iff, den_dvd_of_is_root, ValuativeRel.one_apply_posSubmonoid, Submonoid.op_sup, Circle.leftInverse_exp_arg, Subgroup.le_closure_toSubmonoid, IsRightRegular.mem_nonZeroDivisorsRight, IsFractionRing.associated_num_den_inv, Submonoid.coe_comap, IsLocalization.mk'_eq_iff_eq, Matrix.transpose_mem_rowStochastic_iff_mem_colStochastic, Monoid.closure_finset_fg, Submonoid.smul_def, OreLocalization.cardinalMk_le_max, CommMonoid.primaryComponent.disjoint, Submonoid.coe_inv, mk_mem_nonZeroDivisors_associates, noZeroDivisors_iff_forall_mem_nonZeroDivisors, Matrix.mem_specialUnitaryGroup_iff, LocalizedModule.subsingleton_iff, Submonoid.one_mem, Submonoid.bot_prod_bot, Submonoid.map_inf_comap_of_surjective, Unitary.openPartialHomeomorph_target, Submonoid.LocalizationMap.exists_of_sec_mk', IsLocalization.mem_invSubmonoid_iff_exists_mk', Unitary.inv_mem_iff, Ideal.card_norm_le_eq_card_norm_le_add_one, Submonoid.units_inf, OreLocalization.one_div_mul, Submonoid.inv_iSup, spinGroup.mul_star_self, Localization.mk_one_eq_algebraMap, unitary.coe_mul_star_self, Submonoid.LocalizationMap.map_nonZeroDivisors_le, Submonoid.mem_pointwise_smul_iff_inv_smul_mem₀, Submonoid.exponent_top, LocalizedModule.smul_def, IsCyclic.exists_monoid_generator, Submonoid.coe_one, Submonoid.card_le_one_iff_eq_bot, Submonoid.coe_unitSphere, Submonoid.fromLeftInv_eq_iff, Circle.norm_coe, IsLocalization.nonZeroDivisors_le_comap, val_unitsCentralizerEquiv_symm_apply_coe, IsLocalizedModule.mk'_cancel_right, Matrix.UnitaryGroup.mul_val, fourier_apply, Subgroup.ofUnits_mono, Submonoid.mem_mk, Matrix.IsHermitian.eigenvectorUnitary_apply, Matrix.mem_rowStochastic_iff_sum, Ideal.associatesNonZeroDivisorsEquivIsPrincipal_mul, IsLocalization.mk'_sec, Submonoid.leftInv_eq_inv, Unitary.coe_isStarNormal, Subgroup.ofUnits_inf_units, Subgroup.val_centerUnitsEquivUnitsCenter_apply_coe, IsLocalization.mk'_eq_iff_eq_mul, mem_nonZeroDivisors_iff', OreLocalization.smul_one_oreDiv_one_smul, multiple_mem_span_of_mem_localization_span, fixedPoints_submonoid_iSup, Localization.mk_natCast, Submonoid.pow_right_injective_iff_pow_injective, Monoid.IsTorsion.torsion_eq_top, Submonoid.le_prod_iff, Matrix.UnitaryGroup.mul_apply, OreLocalization.zero_def, mem_doublyStochastic_iff_mem_rowStochastic_and_mem_colStochastic, Submonoid.pointwise_smul_subset_iff, IsArtinianRing.isUnit_iff_nonZeroDivisor_of_isIntegral', Subsemiring.mopRingEquivOp_apply_coe, IsLocalization.toInvSubmonoid_mul, Unitary.conjStarAlgAut_symm_apply, Submonoid.pow_apply, ClassGroup.mk_mk0, selfAdjoint.star_coe_unitarySelfAddISMul, IsLocalization.scaleRoots_commonDenom_mem_lifts, Orientation.kahler_rotation_left', Submonoid.unop_eq_bot, OreLocalization.mul_cancel', ValuativeRel.left_cancel_posSubmonoid, Submonoid.closure_le_centralizer_centralizer, isLeftRegular_iff_mem_nonZeroDivisorsLeft, MulEquiv.submonoidMap_symm_apply, Unitary.toRingEquiv_conjStarAlgAut, OreLocalization.numeratorHom_apply, SymplecticGroup.mem_iff', Unitary.val_toUnits_apply, Submonoid.mem_pointwise_smul_iff_inv_smul_mem, ClassGroup.mk0_eq_mk0_iff_exists_fraction_ring, IsDedekindDomain.HeightOneSpectrum.intValuation_zero_lt, Submonoid.nontrivial_iff_exists_ne_one, Matrix.UnitaryGroup.ext_iff, Subgroup.bot_toSubmonoid, Localization.mk_self, Ideal.absNorm_pos_of_nonZeroDivisors, Submonoid.unop_le_unop_iff, pinGroup.toUnits_injective, MonoidHom.submonoidComap_surjective_of_surjective, Monoid.fg_iff, pinGroup.mul_star_self, Submonoid.mem_carrier, Localization.r_iff_exists, doublyStochastic_eq_convexHull_permMatrix, unitary.coe_inv, IsLocalizedModule.exists_integer_multiple, NumberField.mixedEmbedding.fundamentalCone.mixedEmbedding_preimageOfMemIntegerSet, Matrix.IsHermitian.spectral_theorem, Ideal.mem_primeCompl_iff, MonoidHom.mker_inr, Submonoid.LocalizationMap.mk'_eq_iff_eq', Submonoid.op_le_op_iff, MonoidHom.mclosure_preimage_le, Submonoid.LocalizationMap.sec_spec', unitary.coe_map, Submonoid.unop_sup, Unitary.coe_div, MonoidHom.noncommPiCoprod_mrange, Submonoid.comap_inf, Submonoid.equivOp_apply_coe, IsLocalizedModule.smul_inj, pinGroup.star_mem_iff, Submonoid.pointwise_isCentralScalar, IsLocalization.toInvSubmonoid_surjective, Submonoid.FG.iSup, OreLocalization.mul_inv, rotationOf_coe, IsLocalizedModule.iso_symm_apply_aux, Submonoid.comap_le_comap_iff_of_surjective, isUnit_iff_mem_nonZeroDivisors_of_finite, IsLocalizedModule.mk'_surjective, Submonoid.closure_mul_le, LocalizedModule.mk_mul_mk', Submonoid.center.smulCommClass_left, MonoidHom.restrict_mker, Unitary.val_inv_toUnits_apply, IsLocalization.mem_localizationLocalizationSubmodule, Matrix.IsHermitian.eigenvectorUnitary_coe, HomogeneousLocalization.NumDenSameDeg.den_mem, Unitary.norm_map, Submonoid.inv_val_mem_of_mem_units, orderOf_eq_card_powers, Subsemiring.mem_toSubmonoid, IsLocalizedModule.mk'_smul_mk', dense_submonoidClosure_iff_subgroupClosure, closure_submonoidClosure_eq_closure_subgroupClosure, powers_le_nonZeroDivisors_of_noZeroDivisors, ValuativeRel.exists_valuation_div_valuation_eq, neg_one_mem_torsion, Unitary.toUnits_comp_map, IsLocalizedModule.exists_of_eq, IsFractionRing.num_mul_den_eq_num_mul_den_iff_eq, Module.FinitePresentation.exists_lift_equiv_of_isLocalizedModule, Submonoid.mk_le_mk, FractionalIdeal.den_mul_self_eq_num, Unitary.coe_map_star, Monoid.Coprod.mker_swap, MonoidHom.mker_fst, Submonoid.smulCommClass_left, IsLocalization.localization_localization_exists_of_eq, unitary.norm_map, Algebra.adjoin_eq_span, Submonoid.isOfFinOrder_coe, Real.hasDerivAt_fourierChar, Matrix.star_eq_inv, Submonoid.comap_inf_map_of_injective, IsLocalization.mk'_cancel, Submonoid.coe_sup, LocalizedModule.mul_by_divBy, Orientation.kahler_rotation_left, Submonoid.prod_eq_top_iff, Submonoid.fromLeftInv_mul, IsLocalization.exist_integer_multiples_of_finset, IsLocalization.smul_mem_iff, IsLocalization.orderIsoOfPrime_symm_apply_coe, mul_cancel_right_coe_nonZeroDivisors, IsLocalization.sec_spec, Submonoid.inv_le_inv, Submonoid.center.smulCommClass_right, Submonoid.op_inf, Localization.mk_prod, HomogeneousLocalization.Away.val_mk, Submonoid.eq_top_iff', Submonoid.toIsOrderedMonoid, Monoid.CoprodI.iSup_mrange_of, IsLocalization.map_units_map_submonoid, IsLocalization.exist_integer_multiples_of_finite, Unitary.instSMulCommClassSubtypeMemSubmonoidUnitary, IsLocalization.disjoint_comap_iff, Submonoid.mem_units_iff, Submonoid.unop_sInf, Localization.mk_one, Matrix.mem_orthogonalGroup_iff, NumberField.exists_ideal_in_class_of_norm_le, Submonoid.LocalizationMap.eq', Submonoid.mem_centralizer_iff, IsLocalization.toInvSubmonoid_eq_mk', Submonoid.mem_divPairs, scaleRoots_aeval_eq_zero_of_aeval_mk'_eq_zero, Submonoid.mem_center_iff, Subring.toSubmonoid_strictMono, Circle.coe_eq_one, unitary.coe_map_star, HomogeneousLocalization.val_awayMap_mk, Submonoid.pi_bot, Submonoid.one_def, extremePoints_doublyStochastic, coe_nonZeroDivisorsLeft_eq, Subring.ringEquivOpMop_symm_apply_coe, Submonoid.instCovariantClassHSMulLe, Submonoid.opEquiv_apply, mem_own_leftCoset, Real.fourierChar_apply, IsLocalization.tmul_mk', mem_nonZeroDivisorsLeft_iff, FractionalIdeal.spanSingleton_mul_coeIdeal_eq_coeIdeal, Submonoid.map_surjective_of_surjective, Commute.expUnitary_add, nonZeroDivisors.associated_coe, Submonoid.isLocalizationMap_iff, Subring.coe_toSubmonoid, Submonoid.LocalizationMap.mk'_mul_cancel_right, Submonoid.LocalizationMap.mk'_spec', MonoidWithZeroHom.valueGroup_def, Unitary.isPathConnected_ball, IsArtinianRing.isUnit_iff_mem_nonZeroDivisors_of_mulOpposite, Unitary.map_injective, PNat.equivNonZeroDivisorsNat_apply_coe, Unitary.mapEquiv_symm, HomogeneousLocalization.awayMapAux_mk, Matrix.of_mem_specialOrthogonalGroup_fin_two_iff, fixingSubmonoid_union, fixingSubmonoid_fixedPoints_gc, ValuativeRel.right_cancel_posSubmonoid, IsLocalization.mk'_one, Matrix.mem_colStochastic, instContinuousStarSubtypeMemSubmonoidUnitary, Subsemiring.ringEquivOpMop_symm_apply_coe, Submonoid.unop_sSup, Submonoid.mul_def, Subgroup.mem_ofUnits, Localization.coe_algEquiv, Monoid.Coprod.mrange_mk, Matrix.mem_colStochastic_iff_sum, NumberField.mixedEmbedding.fundamentalCone.idealSetEquiv_symm_apply, Submonoid.gc_map_comap, Localization.mk_pow, Subsemiring.coe_closure_eq, Valued.integer.locallyFiniteOrder_units_mrange_of_isCompact_integer, Submonoid.map_injective_of_injective, Unitary.conjStarAlgAut_mul_apply, Submonoid.continuousSMul, IsLocalization.ker_map, Submonoid.powLogEquiv_apply, Submonoid.op_eq_top, AlgebraicGeometry.StructureSheaf.Localizations.comapFun_mk, Submonoid.monotone_map, Submonoid.subset_pointwise_smul_iff, cfc_unitary_iff, IsLocalizedModule.smul_injective, le_isUnit_iff_zero_notMem, IsLocalization.mk'_mul_mk'_eq_one, Ideal.span_singleton_mem_isPrincipalSubmonoid, AddSubmonoid.coe_toSubmonoid_symm_apply, unitary.toMonoidHom_mapEquiv, Submonoid.closure_empty, isRegular_iff_mem_nonZeroDivisors, ValuativeRel.ValueGroupWithZero.embed_mk, Unitary.map_id, Submonoid.unop_bot, IsUnit.mem_nonZeroDivisors, MonoidWithZeroHom.valueMonoid_eq_valueGroup', Polynomial.X_mem_nonzeroDivisors, Submonoid.coe_pathComponentOne, IsStarProjection.two_mul_sub_one_mem_unitary, IsLocalization.submonoid_map_le_is_unit, Unitary.mapEquiv_trans, biUnion_associatedPrimes_eq_compl_nonZeroDivisors, MulAction.orbit_submonoid_subset, pinGroup.mem_iff, LinearIsometryEquiv.smul_trans, IsLocalization.mk'_eq_zero_iff, MonoidWithZeroHom.val_mrange_zero, Submonoid.continuousMul, Circle.coe_inv, Localization.add_mk, Unitary.inner_map_map, Matrix.specialUnitaryGroup.coe_star, OreLocalization.oreDiv_smul_oreDiv, OrthonormalBasis.toMatrix_orthonormalBasis_mem_unitary, Submonoid.mem_closure_finset, Unitary.norm_sub_eq, Submonoid.inv_le, unitary_iff_isStarNormal_and_spectrum_subset_unitary, IsFractionRing.num_mul_den_eq_num_iff_eq, pinGroup.star_eq_inv, Submonoid.FG.sup, mem_doublyStochastic, Submonoid.instSubsingletonUnits, MulEquiv.ofLeftInverse'_symm_apply, unitary.coe_star, mem_nonZeroDivisors_of_ne_zero, Matrix.mem_unitaryGroup_iff', IsLocalization.map_units, LocalizedModule.mk'_smul_mk, Subfield.mem_toSubmonoid, Unitary.coe_map, Matrix.convex_rowStochastic, Submonoid.mrange_snd, ClassGroup.exists_mk0_eq_mk0, Localization.mk_eq_mk_iff', Submonoid.mem_closure_range_iff, Submonoid.subsingleton_iff, Submonoid.coe_topologicalClosure, Submonoid.mem_powers_iff, unitary.mul_inv_mem_iff, Valuation.Integers.wellFounded_gt_on_v_iff_discrete_mrange, Submonoid.FG.finite_irreducible_mem_submonoidClosure, Real.differentiable_fourierChar, nonZeroDivisorsEquivUnits_apply, Submonoid.LocalizationMap.mk'_mul, Ideal.nonempty_inter_nonZeroDivisors_of_faithfulSMul, ClassGroup.mk0_surjective, Localization.mkHom_apply, Submonoid.smul_closure, IsLocalization.injective_iff_isRegular, mem_nonZeroDivisors_iff_ne_zero, fixingSubmonoid_iUnion, lipschitzGroup.coe_mem_iff_mem, Submonoid.coe_equivMapOfInjective_apply, Subsemiring.mem_mk, OreLocalization.oreDiv_mul_oreDiv_comm, Submonoid.LocalizationMap.mulEquivOfMulEquiv_mk', Ideal.disjoint_primeCompl_of_liesOver, Submonoid.instMeasurableConstSMul, Submonoid.mem_iInf, Submonoid.mrange_inl, Circle.normSq_coe, ValuativeRel.posSubmonoid_def, Subgroup.toSubmonoid_injective, MonoidHom.mrange_eq_top_of_surjective, Submonoid.IsLocalizationMap.surj, IsIdempotentElem.coe_powers, Matrix.permMatrix_mem_colStochastic, Submonoid.mulSingle_mem_pi, Submonoid.units_iInf₂, MeasureTheory.Measure.instSMulInvariantMeasureSubtypeMemSubmonoidOfIsMulLeftInvariant, Subgroup.val_mem_ofUnits_iff_mem, MonoidHom.coe_mgraph, MvPolynomial.isLocalization_C_mk', Submonoid.op_injective, Matrix.mem_specialOrthogonalGroup_fin_two_iff, Submonoid.unop_injective, Submonoid.mrange_fst, Submonoid.coe_toAddSubmonoid_symm_apply, Submonoid.mem_op, Submonoid.coe_toAddSubmonoid_apply, OreLocalization.numerator_isUnit, Ideal.mem_isPrincipalSubmonoid_iff, fourier_zero', ZMod.toCircle_intCast, IsLocalization.exists_integer_multiple, unitary.mapEquiv_refl, Submonoid.mem_closure_range_iff_of_fintype, ZMod.toCircle_natCast, nonZeroDivisorsEquivUnits_symm_apply_coe, Ideal.primeCompl_le_nonZeroDivisors, Submonoid.toIsOrderedCancelMonoid, Real.deriv_fourierChar, Monoid.powers_fg, ClassGroup.exists_min, mem_powers_of_prime_card, Subsemiring.coe_set_mk, CategoryTheory.ActionCategory.stabilizerIsoEnd_apply, IsSMulRegular.mem_nonZeroSMulDivisors, LocalizedModule.mk_smul_mk, MonoidHom.coe_mrangeRestrict, Submonoid.coe_square, convex_doublyStochastic, NumberField.mixedEmbedding.fundamentalCone.preimage_of_IdealSetMap, AlgebraicGeometry.ProjectiveSpectrum.Proj.toSpec_base_apply_eq_comap, Module.End.mem_submonoidCenter_iff, LinearIsometryEquiv.symm_units_smul, Submonoid.map_iInf, Submonoid.pi_eq_bot_iff, OreLocalization.one_def, AlgebraicGeometry.ProjectiveSpectrum.Proj.awayToSection_germ, IsLocalizedModule.fromLocalizedModule'_mk, Unitary.mem_iff, IsLocalization.AtPrime.isUnit_mk'_iff, Subgroup.ofUnits_injective, Real.differentiable_fourierChar_neg_bilinear_right, Unitary.instIsScalarTowerSubtypeMemSubmonoidUnitary, Subalgebra.linearEquivOp_symm_apply_coe, QuadraticAlgebra.coe_mem_nonZeroDivisors_iff, Localization.mk_mul, Submonoid.map_bot, Circle.coeHom_apply, Localization.mk_lt_mk, rootsOfUnityCircleEquiv_apply, Subgroup.ofUnits_iSup₂, FractionalIdeal.map_canonicalEquiv_mk0, Submonoid.closure_union, Subring.sInf_toSubmonoid, OreLocalization.oreDiv_one_surjective_of_finite_left, IsOfFinOrder.powers_eq_image_range_orderOf, Submonoid.mem_closure_inv, LinearIsometryEquiv.symm_smul_apply, Submonoid.mem_unop, Monoid.Coprod.mclosure_range_inl_union_inr, Circle.coe_inj, IsLocalization.mk'_mul_cancel_right, IsLocalization.exists_integer_multiple', IsRegular.mem_nonZeroDivisors, IsLocalizedModule.mk'_mul_mk', IsSelfAdjoint.self_add_I_smul_cfcSqrt_sub_sq_mem_unitary, Submonoid.LocalizationMap.mk'_eq_zero_iff, StarAlgebra.adjoin_eq_span, Orientation.kahler_rotation_right, Submonoid.leftInv_le_isUnit, Submonoid.toAddSubmonoid'_closure, mul_cancel_left_coe_nonZeroDivisors, Unitary.coe_neg, SymplecticGroup.coe_inv', Matrix.IsHermitian.star_mul_self_mul_eq_diagonal, Submonoid.inv_iInf, ValuativeExtension.mapPosSubmonoid_apply_coe, fourier_coe_apply', Unitary.star_eq_inv, Subsemiring.closure_submonoid_closure, IsLocalizedModule.map_units, OreLocalization.smul_div_one, CStarRing.norm_coe_unitary_mul, Submonoid.mem_powers, Matrix.IsHermitian.eigenvectorUnitary_transpose_apply, RatFunc.liftOn_def, Submonoid.map_equiv_top, instContinuousInvSubtypeMemSubmonoidUnitaryOfContinuousStar, Submonoid.mem_closure_finset', Localization.le_comap_primeCompl_iff, Matrix.UnitaryGroup.toLin'_mul, MonoidWithZeroHom.one_mem_valueMonoid, Submonoid.prod_le_iff, Real.hasFDerivAt_fourierChar_neg_bilinear_right, Subgroup.mem_toSubmonoid, AlgebraicGeometry.StructureSheaf.comap_basicOpen, zero_notMem_nonZeroDivisorsRight, IsLocalization.AtPrime.isUnit_to_map_iff, Monoid.closure_finite_fg, Submonoid.coe_iInf, ValuativeRel.ValueGroupWithZero.mk_le_mk, ZMod.toCircle_apply, IsLocalization.mul_toInvSubmonoid, IsLocalization.algebraMap_mk', exists_bijective_map_powers, Submonoid.LocalizationMap.eq_mk'_iff_mul_eq, Subgroup.coe_toSubmonoid, Circle.arg_exp, IsLocalization.mk'_eq_iff_eq', IsLocalization.span_invSubmonoid, MonoidHom.mker_one, Unitary.conjStarAlgAut_symm_unitaryLinearIsometryEquiv, Submonoid.LocalizationMap.map_units, Subgroup.ofUnits_sSup, pinGroup.coe_star, irreducible_mem_submonoidClosure_subset, Matrix.mem_orthogonalGroup_iff', powers_eq_top_of_prime_card, Submonoid.smulCommClass_right, ValuativeRel.ValueGroupWithZero.lift_zero, CommMonoid.primaryComponent.exists_orderOf_eq_prime_pow, Submonoid.closure_singleton_one, Real.differentiable_fourierChar_neg_bilinear_left, OreLocalization.one_div_smul, Submonoid.fromLeftInv_one, IsLocalizedModule.eq_iff_exists, transpose_mem_doublyStochastic_iff, FractionalIdeal.equivNum_apply, MvPowerSeries.X_mem_nonzeroDivisors, IsLocalization.invertible_mk'_one_invOf, ClassGroup.equiv_mk0, unitary.isPathConnected_ball, ValuativeRel.ValueGroupWithZero.mk_self, Subgroup.ofUnits_right_inverse, Submonoid.subsemiringClosure_mem, IsLocalizedModule.smul_mem_finsetIntegerMultiple_span, Matrix.UnitaryGroup.toGL_one, MonoidWithZeroHom.mem_valueMonoid_iff, dvd_cancel_left_coe_nonZeroDivisors, Submonoid.instNontrivial, Circle.coe_exp, NumberField.mixedEmbedding.fundamentalCone.mem_idealSet, NumberField.Ideal.tendsto_norm_le_and_mk_eq_div_atTop, Submonoid.top_closure_mul_self_eq, Submonoid.closure_iUnion, IsLocalization.integerNormalization_spec, Submonoid.divPairs_comap, OrthonormalBasis.toMatrix_orthonormalBasis_mem_orthogonal, Submonoid.mem_closure_of_mem, Submonoid.fromCommLeftInv_apply, IsLocalization.eq_mk'_iff_mul_eq, NumberField.mixedEmbedding.fundamentalCone.integerSetToAssociates_surjective, fourier_add', Subsemiring.sInf_toSubmonoid, NumberField.mixedEmbedding.fundamentalCone.idealSetEquiv_apply, BoundedContinuousFunction.charMonoidHom_apply, Submonoid.map_inl, FractionalIdeal.den_mul_self_eq_num', IsFractionRing.num_den_reduced, QuadraticAlgebra.algebraMap_mem_nonZeroDivisors_iff, pinGroup.coe_mul_star_self, Subgroup.equivSMul_symm_apply_coe, Localization.r_iff_oreEqv_r, Matrix.mem_rowStochastic, Submonoid.coe_multiset_prod, Zsqrtd.norm_eq_one_iff_mem_unitary, IsFractionRing.associated_den_num_inv, ValuativeRel.ValueGroupWithZero.lift_valuation, MonoidHom.subgroupComap_apply_coe, Submonoid.coe_centralizer, CoxeterSystem.submonoid_closure_range_simple, Submonoid.coe_set_mk, IsUnit.mem_unitary_of_mul_star_self, Ideal.exists_disjoint_powers_of_span_eq_top, IsLocalization.mk'_add, Submonoid.mem_inf, Submonoid.closure_closure_coe_preimage, Subring.mem_toSubmonoid, Equiv.Perm.mclosure_isSwap, ClassGroup.mk0_eq_mk0_inv_iff, noZeroDivisors_iff_forall_mem_nonZeroDivisorsLeft, Submonoid.disjoint_def', Ideal.associatesNonZeroDivisorsEquivIsPrincipal_map_one, IsLocalizedModule.map_integerMultiple, Algebra.GrothendieckGroup.lift_apply, IsFractionRing.self_iff_nonZeroDivisors_le_isUnit, Ideal.associatesNonZeroDivisorsEquivIsPrincipal_coe, Submonoid.topEquiv_toMonoidHom, MonoidHom.restrict_mrange, Submonoid.LocalizationMap.mk'_self', Submonoid.closure_inv, Circle.toUnits_apply, Matrix.transpose_mem_colStochastic_iff_mem_rowStochastic, Submonoid.fromLeftInv_eq_inv, comap_nonZeroDivisors_le_of_injective, IsLocalization.AtPrime.equivQuotMaximalIdeal_symm_apply_mk, IsLocalization.mem_map_algebraMap_iff, Submonoid.FG.biSup_finset, PrimeSpectrum.localization_comap_range, IsLocalization.orderIsoOfPrime_apply_coe, Unitary.coe_star_mul_self, ValuativeRel.ValueGroupWithZero.lift_one, unitary.map_injective, Unitary.instStarModuleSubtypeMemSubmonoidUnitary, Submonoid.LocalizationMap.surj, permMatrix_mem_doublyStochastic, Submonoid.val_unitsTypeEquivIsUnitSubmonoid_symm_apply, IsLocalization.exist_integer_multiples, Unitary.toMonoidHom_mapEquiv, IsUnit.mem_unitary_iff_mul_star_self, Unitary.star_mem_iff, Submonoid.sup_eq_range, Algebra.GrothendieckGroup.of_injective, Submonoid.LocalizationMap.mk'_sec, Submonoid.topOrderMonoidIso_apply, Submonoid.closure_irreducible, Submonoid.IsUnit.Submonoid.coe_inv, mem_powers_iff_mem_zpowers, Unitary.openPartialHomeomorph_apply, Real.probChar_apply, powersEquivPowers_apply, fixedPoints_antitone, Matrix.IsHermitian.conjStarAlgAut_star_eigenvectorUnitary, Unitary.argSelfAdjoint_coe, Submonoid.disjoint_def, Subgroup.ofUnits_iSup, Submonoid.pi_empty, Subsemiring.toSubmonoid_injective, Submonoid.subset_closure, Submonoid.surjOn_iff_le_map, Submonoid.FG.bot, Circle.coe_injective, unitary.mul_star_self, Submonoid.centerCongr_apply_coe, spinGroup.coe_star, CommMonoid.torsion.isTorsion, Submonoid.coe_sInf, Circle.coe_zpow, MonoidHom.mrangeRestrict_mker, unitary.star_mem_iff, IsDedekindDomain.HeightOneSpectrum.adicCompletion.mul_nonZeroDivisor_mem_adicCompletionIntegers, Submonoid.map_iInf_comap_of_surjective, QuadraticAlgebra.norm_mem_nonZeroDivisors_iff, OreLocalization.nontrivial_iff, Submonoid.map_strictMono_of_injective, FixedPoints.mem_submonoid, FractionalIdeal.finprod_heightOneSpectrum_factorization_principal_fraction, Con.mrange_mk', Submonoid.coe_unop, zero_notMem_nonZeroDivisorsLeft, IsFractionRing.num_mul_den_eq_num_iff_eq', Submonoid.LocalizationMap.mk'_eq_iff_eq_mul, Submonoid.coe_pos, CStarAlgebra.exists_sum_four_unitary, Localization.awayLift_mk, Submonoid.powLogEquiv_symm_apply, Subalgebra.mopAlgEquivOp_apply_coe, Unitary.coe_star, MonoidHom.subgroupMap_apply_coe, IsLocalization.eq, zero_notMem_nonZeroDivisors, TopCat.Presheaf.submonoidPresheafOfStalk_obj, Submonoid.subsemiringClosure_toNonUnitalSubsemiring, IsLocalizedModule.mk'_cancel, Matrix.UnitaryGroup.star_mul_self, NormedSpace.exp_mem_unitary_of_mem_skewAdjoint, Subalgebra.algEquivOpMop_symm_apply_coe, IsLocalization.map_algebraMap_ne_top_iff_disjoint, LocalizedModule.liftOn₂_mk, Submonoid.op_top, Submonoid.FG.exists_minimal_closure_eq, Unitary.conjStarAlgAut_star_apply, ClassGroup.integralRep_mem_nonZeroDivisors, Circle.coe_div, notMem_nonZeroDivisorsLeft_iff, Submonoid.LocalizationMap.exists_of_eq, Submonoid.center_le_centralizer, Circle.coe_one, notMem_nonZeroDivisors_iff_right, Monoid.Coprod.mrange_inl_sup_mrange_inr, IsMulIndecomposable.apply_ne_one_iff_mem_closure, mul_mem_nonZeroDivisors, Submonoid.bot_or_nontrivial, unitary.coe_neg, le_nonZeroDivisorsRight_iff_isRightRegular, RingHom.toKerIsLocalization_isLocalizedModule, LinearIsometryEquiv.toContinuousLinearEquiv_smul, FractionalIdeal.den_mem_inv, unitary.star_eq_inv, unitary.mem_pathComponentOne_iff, Subgroup.topEquiv_apply, Submonoid.LocalizationMap.sec_spec, isNilpotent_iff_zero_mem_powers, Real.fderiv_fourierChar_neg_bilinear_left_apply, Submonoid.mk_eq_top, Submonoid.op_bot, Submonoid.opEquiv_symm_apply, Submonoid.coe_mul_self_eq, RatFunc.laurentAux_ofFractionRing_mk, Unitary.mapEquiv_refl, Submonoid.LocalizationMap.mk'_spec, FractionalIdeal.coe_mk0, CStarRing.norm_coe_unitary, Algebra.mem_algebraMapSubmonoid_of_mem, Monoid.Coprod.codisjoint_mrange_inl_mrange_inr, mem_fixingSubmonoid_iff, Unitary.star_eq_inv', Submonoid.units_mono, finEquivPowers_symm_apply, Subsemiring.mem_closure_iff, IsLocalization.map_id_mk', FractionRing.mk_eq_div, Submonoid.sup_eq_closure, Submonoid.closure_le, IsLocalization.surj₂, Matrix.IsHermitian.star_eigenvectorUnitary_mulVec, Submonoid.mem_closure_pair, LocalizedModule.mkLinearMap_apply, Localization.sub_mk, ofAdd_image_multiples_eq_powers_ofAdd, IsLocalizedModule.mk'_cancel_left, Circle.argEquiv_apply_coe, ValuativeRel.ValueGroupWithZero.mk_eq_valuation, AddSubmonoid.fg_iff_mul_fg, IsLocalization.exists_of_eq, CStarAlgebra.span_unitary, IsLocalization.coe_primeSpectrumOrderIso_apply_coe_asIdeal, SymplecticGroup.J_mem, MonoidHom.eqLocusM_same, Submonoid.mem_closure_iff_exists_finset_subset, Submonoid.coe_mul, Submonoid.exists_mem_sup, IsFractionRing.isUnit_den_zero, IsUnit.mem_unitary_iff_star_mul_self, Submonoid.powers_le, Submonoid.topEquiv_apply, CategoryTheory.ActionCategory.stabilizerIsoEnd_symm_apply, Submonoid.pow_coe, IsLocalization.eq_iff_exists, OreLocalization.oreDiv_one_smul, Subgroup.ofUnits_sup_units, notMem_nonZeroDivisorsRight_iff, Unitary.conjStarAlgAut_ext_iff, IsOfFinOrder.powers_eq_zpowers, Unitary.spectrum_subset_circle, IsLocalizedModule.mk'_eq_iff, spinGroup.toUnits_injective, Submonoid.eq_bot_iff_card, Submonoid.mem_closure_image_one_lt_iff, Submonoid.coe_finset_prod, Subfield.coe_toSubmonoid, Module.FinitePresentation.exists_basis_localizedModule_powers, AddSubmonoid.toSubmonoid'_closure, infinite_powers, Unitary.isUnit_coe, Complex.UnitDisc.coe_circle_smul, ZMod.stdAddChar_apply, Submonoid.coe_powers, Valuation.Integers.isPrincipalIdealRing_iff_not_denselyOrdered_mrange, unitary.mul_right_inj, IsFractionRing.lift_mk', Submonoid.LocalizationMap.sec_zero_fst, IsLocalization.smul_toInvSubmonoid, IsLocalization.subsingleton_iff, Commute.expUnitary, Subgroup.pointwise_smul_toSubmonoid, exists_mem_doublyStochastic_eq_smul_iff, Submonoid.smul_bot, IsLocalization.mk'_spec, Circle.argPartialEquiv_apply, MeasureTheory.charFun_eq_integral_probChar, Monoid.fg_def, LocalizedModule.oreEqv_eq_r, unitary.star_mul_self, Submonoid.subtype_apply, Units.inv_mul_mem_unitary, Unitary.coe_inv, MonoidHom.mem_mrange, Subring.addEquivOp_symm_apply_coe, Submonoid.comap_injective_of_surjective, Submonoid.FG.biSup, selfAdjoint.expUnitary_coe, MonoidHom.mem_mgraph, IsFractionRing.isUnit_den_iff, Subring.toSubmonoid_injective, Complex.rotation, isClosed_unitary, Matrix.UnitaryGroup.inv_apply, Submonoid.closure_eq_image_prod, IsLocalizedModule.eq_zero_iff, spinGroup.star_mul_self, IsLocalizedModule.surj', spinGroup.val_toUnits_apply, spinGroup.star_mem_iff, IsLocalizedModule.mk'_mul_mk'_of_map_mul, MonoidHom.submonoidMap_surjective, IsLocalizedModule.surj, Submonoid.closure_one_prod, unitary.inner_map_map, BoundedContinuousFunction.char_apply, IsLocalizedModule.iso_mk_one, Submonoid.mul_fromLeftInv, selfAdjoint.joined_one_expUnitary, Submonoid.LocalizationMap.surj₂, NumberField.mixedEmbedding.fundamentalCone.intNorm_idealSetEquiv_apply, Localization.coe_algEquiv_symm, IsLocalization.mk'_mul, IsDedekindDomain.HeightOneSpectrum.adicAbv_of_mk', IsLocalization.ringEquivOfRingEquiv_mk', unitary.continuousOn_argSelfAdjoint, Localization.mk_eq_mk_iff, CommMonoid.coe_primaryComponent, IsLocalizedModule.mk'_sub_mk', IsLocalization.map_nonZeroDivisors_le, mem_nonZeroDivisorsRight_iff, IsLocalizedModule.injective_iff_isRegular, OreLocalization.inv_def, unitary.linearIsometryEquiv_coe_apply, Submonoid.LocalizationMap.eq
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