#Query
degree of product of two polynomials equals to sum of degree
$\deg f*g = \deg f + \deg g$
`(f * g).degree = f.degree + g.degree`
#Retrieved theorem
001 Polynomial.degree_prod                                 |0
002 Polynomial.degree_list_prod                            |0
003 Polynomial.degree_mul                                  |2
004 Polynomial.degree_multiset_prod                        |0
005 Polynomial.natDegree_prod                              |0
006 Polynomial.natDegree_multiset_prod'                    |0
007 Polynomial.natDegree_prod'                             |0
008 Polynomial.natDegree_comp                              |0
009 Polynomial.degree_add_eq_of_leadingCoeff_add_ne_zero   |0
010 Polynomial.natDegree_mul_le                            |1
011 Polynomial.natDegree_prod_of_monic                     |0
012 Polynomial.natDegree_multiset_prod_of_monic            |0
013 MvPolynomial.totalDegree_mul                           |0
014 Polynomial.natDegree_multiset_prod                     |0
015 MvPolynomial.degrees_mul                               |1
016 Polynomial.degree_prod_le                              |0
017 Polynomial.degree_add_le                               |0
018 Polynomial.natDegree_add_le                            |0
019 Polynomial.degree_add_le_of_le                         |0
020 Polynomial.degree_sum_eq_of_disjoint                   |0
021 Polynomial.degree_add_eq_right_of_degree_lt            |0
022 Polynomial.natDegree_add_eq_right_of_natDegree_lt      |0
023 MvPolynomial.degrees_prod                              |0
024 MvPolynomial.totalDegree_add                           |0
025 Polynomial.degree_list_prod_le                         |0
026 Polynomial.degree_add_le_of_degree_le                  |0
027 Polynomial.degree_multiset_prod_le                     |0
028 MvPolynomial.totalDegree_finset_prod                   |0
029 Polynomial.natDegree_prod_le                           |0
030 Polynomial.natDegree_sum_eq_of_disjoint                |0
031 Polynomial.natDegree_add_eq_left_of_natDegree_lt       |0
032 Polynomial.degree_add_eq_left_of_degree_lt             |0
033 Polynomial.natDegree_list_prod_le                      |0
034 Polynomial.natDegree_mul'                              |1
035 Polynomial.degree_mul'                                 |2
036 MvPolynomial.totalDegree_add_eq_left_of_totalDegree_lt |0
037 Polynomial.natDegree_multiset_prod_le                  |0
038 Polynomial.natDegree_mul                               |1
039 Polynomial.Monic.natDegree_mul                         |1
040 Polynomial.degree_mul_le                               |1
041 Polynomial.degree_mul_le_of_le                         |1
042 MvPolynomial.totalDegree_finset_sum                    |0
043 MvPolynomial.totalDegree_list_prod                     |0
044 Polynomial.natDegree_C_mul_le                          |0
045 Polynomial.natDegree_comp_le                           |0
046 Polynomial.natDegree_multiset_prod_X_sub_C_eq_card     |0
047 Polynomial.Monic.degree_mul                            |1
048 MvPolynomial.degreeOf_mul_le                           |0
049 MvPolynomial.degrees_add                               |0
050 MvPolynomial.degreeOf_add_le                           |0



#Query
Every finite domain is a field.
Wedderburn's little theorem
#Retrieved theorem
001 Finite.isField_of_domain                                     |2
002 Algebra.isAlgebraic_of_finite                                |0
003 Algebra.isIntegral_of_finite                                 |0
004 Fintype.nonempty_field_iff                                   |0
005 FunLike.finite'                                              |0
006 FiniteDimensional.right                                      |0
007 FiniteField.isSquare_of_char_two                             |0
008 Fintype.isPrimePow_card_of_field                             |0
009 IsLocalization.AtPrime.isDedekindDomain                      |0
010 IsFractionRing.isDomain                                      |0
011 FiniteDimensional.left                                       |0
012 Algebra.IsIntegral.isField_iff_isField                       |0
013 IntermediateField.fg_adjoin_finset                           |0
014 FiniteField.cast_card_eq_zero                                |0
015 Polynomial.IsSplittingField.finiteDimensional                |0
016 Associates.finite_factors                                    |0
017 FiniteField.card'                                            |0
018 IntermediateField.fg_of_noetherian                           |0
019 isCyclic_of_subgroup_isDomain                                |1
020 IntermediateField.adjoin_finset_isCompactElement             |0
021 IntermediateField.adjoin_finite_isCompactElement             |0
022 isField_of_isIntegral_of_isField'                            |0
023 Subfield.closure_univ                                        |0
024 Polynomial.IsSplittingField.adjoin_rootSet                   |0
025 IntermediateField.adjoin_zero                                |0
026 FiniteDimensional.proper                                     |0
027 IntermediateField.exists_finset_of_mem_supr''                |0
028 IntermediateField.fg_bot                                     |0
029 FiniteDimensional.finiteDimensional_of_finrank               |0
030 FiniteDimensional.finite_of_finite                           |0
031 EuclideanDomain.lt_one                                       |0
032 isDedekindDomainInv_iff                                      |0
033 Field.exists_primitive_element_of_finite_bot                 |0
034 IntermediateField.finiteDimensional_iSup_of_finset'          |0
035 uniq_inv_of_isField                                          |0
036 IntermediateField.fg_def                                     |0
037 minpoly.isIntegrallyClosed_eq_field_fractions                |0
038 Field.toIsField                                              |0
039 Polynomial.irreducible_factor                                |0
040 Field.exists_primitive_element_of_finite_top                 |0
041 FiniteDimensional.rank_lt_aleph0                             |0
042 AlgebraicClosureAux.isAlgebraic                              |0
043 IsField.nontrivial                                           |0
044 IsFractionRing.div_surjective                                |0
045 FractionalIdeal.inv_zero'                                    |0
046 MvPolynomial.exists_finset_rename                            |0
047 MvPolynomial.exists_fin_rename                               |0
048 FractionalIdeal.isNoetherian                                 |0
049 ax_grothendieck_of_locally_finite                            |0
050 IsIntegrallyClosed.eq_map_mul_C_of_dvd                       |0




#Query
the universal property of the localization of ring
Given ring homomorphism $f : A \to B$, if the image of $S \subset A$ is invertible in $B$, then $f$ factors through $S^{-1}A$
#Retrieved theorem
001 RingHom.RespectsIso.is_localization_away_iff                 |0
002 RingHom.PropertyIsLocal.ofLocalizationSpan                   |0
003 RingHom.PropertyIsLocal.is_local_affineLocally               |0
004 IsLocalization.card                                          |0
005 Localization.localRingHom_id                                 |0
006 Localization.localRingHom_mk'                                |0
007 Localization.localRingHom_unique                             |0
008 IsLocalization.isLocalization_of_base_ringEquiv              |0
009 IsLocalization.noZeroDivisors_of_le_nonZeroDivisors          |0
010 Ideal.isJacobson_localization                                |0
011 AlgebraicGeometry.StructureSheaf.localization_toBasicOpen    |0
012 Localization.neg_mk                                          |0
013 selfZpow_neg_mul                                             |0
014 IsLocalization.isInteger_one                                 |0
015 AlgebraicGeometry.StructureSheaf.localizationToStalk_stalkSpecializes_apply |0
016 IsLocalization.sec_spec                                      |0
017 RingHom.isIntegralElem_localization_at_leadingCoeff          |0
018 AlgebraicGeometry.StructureSheaf.localizationToStalk_stalkSpecializes_assoc |0
019 Localization.add_mk_self                                     |0
020 AlgebraicGeometry.StructureSheaf.localizationToStalk_stalkToFiberRingHom |0
021 Localization.algEquiv_symm_apply                             |0
022 AddLocalization.ind                                          |0
023 IsLocalizedModule.iso_apply_mk                               |0
024 AlgebraicGeometry.StructureSheaf.localizationToStalk_stalkToFiberRingHom_assoc |0
025 OreLocalization.add_left_neg                                 |0
026 Localization.algEquiv_mk'                                    |0
027 Localization.neg_def                                         |0
028 Localization.mapToFractionRing_apply                         |0
029 CategoryTheory.Localization.strictUniversalPropertyFixedTargetId_lift |0
030 CategoryTheory.Localization.strictUniversalPropertyFixedTargetQ_lift |0
031 Localization.toLocalizationMap_eq_monoidOf                   |0
032 IsLocalization.coeSubmodule_span                             |0
033 IsLocalization.localization_isScalarTower_of_submonoid_le    |0
034 AlgebraicGeometry.StructureSheaf.stalkIso_inv                |0
035 AlgebraicGeometry.StructureSheaf.stalkToFiberRingHom_localizationToStalk |0
036 UniversalEnvelopingAlgebra.lift_symm_apply                   |0
037 UniversalEnvelopingAlgebra.hom_ext                           |0
038 Localization.algEquiv_apply                                  |0
039 Subring.closure_univ                                         |0
040 AlgebraicGeometry.localRingHom_comp_stalkIso_apply           |0
041 Basis.localizationLocalization_span                          |0
042 Localization.mulEquivOfQuotient_monoidOf                     |0
043 AlgebraicGeometry.StructureSheaf.toStalk_comp_stalkToFiberRingHom |0
044 UniversalEnvelopingAlgebra.ι_comp_lift                       |0
045 Localization.mulEquivOfQuotient_symm_mk                      |0
046 Localization.mulEquivOfQuotient_symm_mk'                     |0
047 AddLocalization.addEquivOfQuotient_symm_addMonoidOf          |0
048 Localization.mkOrderEmbedding_apply                          |0
049 FreeLieAlgebra.universalEnvelopingEquivFreeAlgebra_symm_apply |0
050 UniversalEnvelopingAlgebra.lift_ι_apply                      |0
000 IsLocalization.lift_mk'  |2
000 IsLocalization.lift_comp |2
000 IsLocalization.lift_eq |2
000 IsLocalization.lift_mk'_spec |1
000 IsLocalization.lift_of_comp |1
000 IsLocalization.lift_eq_iff |1
000 IsLocalization.lift_unique |1
000 IsLocalization.lift_surjective_iff |1
000 IsLocalization.lift_injective_iff |1




#Query
derivative of sum = sum of derivatives
$f'(x) + g'(x) = (f + g)'(x)$
`deriv f x + deriv g x = deriv (f + g) x`
#Retrieved theorem
001 iteratedFDerivWithin_add_apply'                |0
002 deriv_add                                      |2
003 iteratedFDerivWithin_add_apply                 |0
004 iteratedFDeriv_add_apply'                      |0
005 derivWithin_add                                |1
006 iteratedFDeriv_add_apply                       |0
007 fderiv_tsum                                    |0
008 derivWithin_sum                                |1
009 iteratedFDeriv_tsum_apply                      |0
010 iteratedFDeriv_tsum                            |0
011 Derivation.map_sum                             |0
012 fderiv_tsum_apply                              |0
013 Polynomial.derivative_add                      |0
014 deriv_sum                                      |1
015 fderivWithin_sum                               |0
016 Complex.hasSum_deriv_of_summable_norm          |0
017 Polynomial.derivative_sum                      |0
018 HasFDerivWithinAt.sum                          |0
019 HasDerivWithinAt.sum                           |1
020 fderiv_add                                     |0
021 SmoothMap.coe_add                              |0
022 mfderiv_add                                    |0
023 HasDerivAtFilter.sum                           |0
024 HasDerivWithinAt.add                           |1
025 HasFDerivAt.sum                                |0
026 HasDerivAt.add                                 |1
027 HasFDerivAtFilter.sum                          |0
028 fderivWithin_add                               |0
029 fderiv_sum                                     |0
030 HasFDerivWithinAt.add                          |0
031 derivWithin_add_const                          |0
032 HasFDerivAt.add                                |0
033 fderivWithin_add_const                         |0
034 deriv_add_const                                |0
035 HasStrictDerivAt.add                           |1
036 HasStrictFDerivAt.add                          |0
037 fderiv_add_const                               |0
038 HasDerivAtFilter.add                           |1
039 HasFDerivAtFilter.add                          |0
040 HasDerivAt.sum                                 |1
041 SmoothMap.addGroup.proof_4                     |0
042 HasDerivAtFilter.const_add                     |0
043 HasDerivAtFilter.add_const                     |0
044 Complex.differentiableOn_tsum_of_summable_norm |0
045 HasStrictFDerivAt.sum                          |0
046 HasDerivWithinAt.clm_comp                      |0
047 HasMFDerivAt.add                               |0
048 Polynomial.derivative_prod                     |0
049 HasSum.add                                     |0
050 mfderiv_prod_eq_add                            |0



#Query
Integration by substitution on an interval
$\int_a^b g(f(x)) \cdot f(x) dx = \int_{f(a)}^{f(b)} g(y) dy$
#Retrieved theorem
001 intervalIntegral.integral_comp_sub_div                       |0
002 MeasureTheory.integral_sub_right_eq_self                     |0
003 intervalIntegral.integral_interval_sub_left                  |0
004 intervalIntegral.integral_comp_mul_sub                       |0
005 intervalIntegral.integral_comp_mul_deriv'''                  |2
006 intervalIntegral.integral_comp_sub_mul                       |0
007 intervalIntegral.integral_comp_smul_deriv'''                 |1
008 intervalIntegral.integral_comp_smul_deriv''                  |1
009 MeasureTheory.integral_fn_integral_sub                       |0
010 MeasureTheory.integrableOn_Iic_of_intervalIntegral_norm_bounded |0
011 MeasureTheory.integrableOn_Ioi_comp_rpow_iff                 |0
012 IntervalIntegrable.comp_sub_left                             |0
013 intervalIntegral.integral_eq_sub_of_hasDerivAt_of_tendsto    |0
014 MeasureTheory.integral_Icc_eq_integral_Ioo'                  |0
015 IntervalIntegrable.trans                                     |0
016 intervalIntegral.integral_eq_integral_of_support_subset      |0
017 intervalIntegral.inv_mul_integral_comp_sub_div               |0
018 intervalIntegral.integral_interval_sub_interval_comm         |0
019 intervalIntegral.integral_comp_mul_deriv''                   |2
020 intervalIntegral.integral_comp_sub_right                     |0
021 IntervalIntegrable.comp_mul_right                            |0
022 intervalIntegral.integral_comp_sub_left                      |0
023 intervalIntegral.integral_comp_div_add                       |0
024 MeasureTheory.lintegral_fn_integral_sub                      |0
025 intervalIntegral.integral_comp_mul_right                     |0
026 intervalIntegral.integral_comp_add_mul                       |0
027 MeasureTheory.integral_Icc_eq_integral_Ioc'                  |0
028 MeasureTheory.integral_Icc_eq_integral_Ioc                   |0
029 MeasureTheory.integral_Ico_eq_integral_Ioo'                  |0
030 MeasureTheory.set_integral_eq_of_subset_of_ae_diff_eq_zero_aux |0
031 MeasureTheory.set_integral_eq_of_subset_of_forall_diff_eq_zero |0
032 intervalIntegral.mul_integral_comp_sub_mul                   |0
033 MeasureTheory.integral_comp_smul_deriv_Ioi                   |0
034 intervalIntegral.integral_comp_add_div                       |0
035 intervalIntegral.inv_mul_integral_comp_add_div               |0
036 MeasureTheory.integral_union_eq_left_of_ae_aux               |0
037 intervalIntegral.integral_sub                                |0
038 MeasureTheory.integral_Icc_eq_integral_Ico'                  |0
039 MeasureTheory.integral_Icc_eq_integral_Ioo                   |0
040 MeasureTheory.integral_Icc_eq_integral_Ico                   |0
041 intervalIntegral.integral_comp_add_right                     |0
042 intervalIntegral.integral_comp_mul_add                       |0
043 integrableOn_Ico_iff_integrableOn_Ioo'                       |0
044 integrableOn_Ioc_iff_integrableOn_Ioo                        |0
045 integrableOn_Ico_iff_integrableOn_Ioo                        |0
046 integrableOn_Ioc_iff_integrableOn_Ioo'                       |0
047 MeasureTheory.integral_Ioc_eq_integral_Ioo'                  |0
048 MeasureTheory.integral_sub'                                  |0
049 MeasureTheory.integrableOn_Ioc_of_interval_integral_norm_bounded_left |0
050 MeasureTheory.integral_comp_mul_deriv_Ioi                    |0
000 intervalIntegral.integral_comp_smul_deriv'   |1
000 intervalIntegral.integral_comp_smul_deriv    |1
000 intervalIntegral.integral_comp_mul_deriv'    |2
000 intervalIntegral.integral_comp_mul_deriv     |2





#Query
Pi is greater than 3
$\pi > 3$
`Real.pi > 3`
#Retrieved theorem
001 Real.pi_gt_three                                             |2
002 Real.pi_gt_31415                                             |2
003 Real.pi_gt_314                                               |2
004 Real.pi_gt_3141592                                           |2
005 Real.pi_lt_3141593                                           |1
006 Real.pi_le_four                                              |1
007 Real.pi_lt_315                                               |1
008 Real.pi_lt_31416                                             |1
009 Real.pi_pos                                                  |1
010 NNReal.pi_pos                                                |0
011 Real.pi_div_two_pos                                          |0
012 Real.pi_gt_sqrtTwoAddSeries                                  |0
013 Real.pi_div_two_le_two                                       |0
014 Real.pi_ne_zero                                              |0
015 Real.pi_lt_sqrtTwoAddSeries                                  |0
016 Real.Angle.pi_ne_zero                                        |0
017 Real.two_le_pi                                               |1
018 Real.Wallis.W_pos                                            |0
019 Real.Angle.coe_pi_add_coe_pi                                 |0
020 Real.cos_pi_div_six                                          |0
021 Real.sin_pi_div_three                                        |0
022 Real.Angle.neg_coe_pi                                        |0
023 Real.cos_pi_div_three                                        |0
024 Real.Angle.cos_coe_pi                                        |0
025 Real.one_le_pi_div_two                                       |0
026 Real.Angle.pi_div_two_ne_zero                                |0
027 Real.cos_pi                                                  |0
028 NNReal.pi_ne_zero                                            |0
029 Real.sin_pi                                                  |0
030 NNReal.coe_real_pi                                           |0
031 Real.sq_sin_pi_div_three                                     |0
032 Real.arccos_lt_pi_div_two                                    |0
033 Real.arcsin_of_one_le                                        |0
034 Real.Angle.two_nsmul_coe_pi                                  |0
035 Real.Angle.two_zsmul_coe_pi                                  |0
036 Real.sin_nonneg_of_nonneg_of_le_pi                           |0
037 Real.sq_cos_pi_div_six                                       |0
038 EuclideanGeometry.angle_lt_pi_of_not_collinear               |0
039 Real.neg_pi_div_two_lt_arctan                                |0
040 EuclideanGeometry.left_ne_of_oangle_eq_pi                    |0
041 Real.sin_pos_of_pos_of_lt_pi                                 |0
042 Complex.cos_two_pi                                           |0
043 Real.arcsin_lt_pi_div_two                                    |0
044 Real.Angle.two_nsmul_neg_pi_div_two                          |0
045 Real.Angle.two_zsmul_neg_pi_div_two                          |0
046 Real.pi_upper_bound_start                                    |0
047 Real.arcsin_eq_pi_div_two                                    |0
048 Real.mul_le_sin                                              |0
049 Real.Angle.sin_coe_pi                                        |0
050 Complex.two_pi_I_ne_zero                                     |0




#Query
Order of an element divides the order of the group
$\text{Ord}(x) \mid |G|, \forall x \in G$
If $G$ is a group, $g \in G$, `orderOf g ∣ G.card`
#Retrieved theorem
001 orderOf_dvd_nat_card                        |2
002 addOrderOf_dvd_card_univ                    |2
003 orderOf_dvd_of_mem_zpowers                  |0
004 addOrderOf_dvd_nat_card                     |2
005 addOrderOf_dvd_of_mem_zmultiples            |0
006 addOrderOf_eq_card_zmultiples               |0
007 ZMod.orderOf_dvd_card_sub_one               |0
008 orderOf_pos_iff                             |0
009 add_orderOf_fst_dvd_add_orderOf             |0
010 orderOf_eq_card_of_forall_mem_zpowers       |0
011 addOrderOf_dvd_iff_nsmul_eq_zero            |0
012 orderOf_dvd_iff_zpow_eq_one                 |0
013 orderOf_dvd_of_pow_eq_one                   |0
014 SubgroupClass.coe_div                       |0
015 add_orderOf_snd_dvd_add_orderOf             |0
016 order_eq_card_zpowers'                      |0
017 addOrderOf_eq_card_of_forall_mem_zmultiples |0
018 orderOf_dvd_card_univ                       |2
019 addOrderOf_smul_dvd                         |0
020 Subgroup.card_subgroup_dvd_card             |0
021 orderOf_eq_card_zpowers                     |0
022 orderOf_dvd_iff_pow_eq_one                  |0
023 min_div_div_left'                           |0
024 IsOfFinOrder.of_mem_zpowers                 |0
025 div_eq_div_mul_div                          |0
026 addOrderOf_dvd_iff_zsmul_eq_zero            |0
027 div_eq_mul_inv                              |0
028 orderOf_eq_of_pow_and_pow_div_prime         |0
029 div_eq_self                                 |0
030 dvd_sub_self_left                           |0
031 div_right_inj                               |0
032 le_div_iff_mul_le                           |0
033 addOrderOf_addSubgroup                      |0
034 ZMod.addOrderOf_coe'                        |0
035 ZMod.orderOf_units_dvd_card_sub_one         |0
036 div_lt_div_right'                           |0
037 IsOfFinOrder.inv                            |0
038 div_eq_iff                                  |0
039 orderOf_pow                                 |0
040 div_le_div_right'                           |0
041 div_le_div_right₀                           |0
042 orderOf_subgroup                            |0
043 card_sylow_dvd_index                        |1
044 IsOfFinAddOrder.mono                        |0
045 div_mul_div_cancel'                         |0
046 dvd_sub_right                               |0
047 LatticeOrderedGroup.pos_div_neg             |0
048 Monoid.order_dvd_exponent                   |0
049 div_self'                                   |0
050 Set.div_mem_center₀                         |0
000 AddSubgroup.addOrderOf_dvd_natCard  |1
000 Subgroup.orderOf_dvd_natCard   |1
000 card_nsmul_eq_zero'   |1
000 pow_card_eq_one'     |1
000 card_nsmul_eq_zero   |1
000 pow_card_eq_one      |1



#Query
The Schur-Zassenhaus Theorem
Schur-Zassenhaus
Let $G$ be a finite subgroup, $H$ be a normal subgroup of $G$, then there exist a subgroup $K$ such that $K$ is the complement of $H$ in $G$.
#Retrieved theorem
001 Subgroup.exists_left_complement'_of_coprime_of_fintype       |2
002 Subgroup.exists_right_complement'_of_coprime_of_fintype      |2
003 Subgroup.exists_right_complement'_of_coprime                 |2
004 Subgroup.exists_left_complement'_of_coprime                  |2
005 Subgroup.card_commutator_le_of_finite_commutatorSet          |0
006 Subgroup.SchurZassenhausInduction.step7                      |1
007 WithSeminorms.banach_steinhaus                               |0
008 MeasureTheory.Measure.sub_mem_nhds_zero_of_addHaar_pos       |0
009 Subgroup.closure_mul_image_eq_top                            |0
010 MeasureTheory.Measure.div_mem_nhds_one_of_haar_pos           |0
011 KummerDedekind.normalizedFactors_ideal_map_eq_normalizedFactors_min_poly_mk_map |0
012 IsCyclotomicExtension.zeta_pow                               |0
013 KummerDedekind.multiplicity_factors_map_eq_multiplicity      |0
014 Subgroup.closure_mul_image_eq                                |0
015 Sylow.exists_subgroup_card_pow_prime                         |0
016 IsCyclotomicExtension.zeta_spec                              |0
017 Sylow.exists_subgroup_card_pow_succ                          |0
018 Subgroup.closure_mul_image_eq_top'                           |0
019 Sylow.card_quotient_normalizer_modEq_card_quotient           |0
020 banach_steinhaus                                             |0
021 HahnSeries.SummableFamily.lsum_apply                         |0
022 Sylow.exists_subgroup_card_pow_prime_le                      |0
023 banach_steinhaus_iSup_nnnorm                                 |0
024 ZMod.units_pow_card_sub_one_eq_one                           |0
025 IsCyclotomicExtension.zeta_isRoot                            |0
026 IsCyclotomicExtension.aeval_zeta                             |0
027 MeasurableSet.image_of_continuousOn_injOn                    |0
028 IsPrimitiveRoot.unique                                       |0
029 MeasureTheory.exists_pair_mem_lattice_not_disjoint_vadd      |0
030 padicValNat_choose                                           |0
031 locally_integrable_zetaKernel₂                               |0
032 isRoot_of_unity_iff                                          |0
033 Sylow.card_normalizer_modEq_card                             |0
034 AddCommGroup.equiv_directSum_zmod_of_fintype                 |0
035 HahnSeries.single_zero_one                                   |0
036 HahnSeries.zero_coeff                                        |0
037 Sylow.coe_subgroup_smul                                      |0
038 padicValNat_choose'                                          |0
039 locally_integrable_zetaKernel₁                               |0
040 IsPrimitiveRoot.nnnorm_eq_one                                |0
041 ContinuousMap.continuousMap_mem_subalgebra_closure_of_separatesPoints |0
042 MeasureTheory.Measure.haar.haarContent_apply                 |0
043 zeta_eq_tsum_one_div_nat_cpow                                |0
044 zeta_nat_eq_tsum_of_gt_one                                   |0
045 Sylow.pow_dvd_card_of_pow_dvd_card                           |0
046 ContinuousMap.exists_mem_subalgebra_near_continuous_of_separatesPoints |0
047 MeasureTheory.AnalyticSet.measurableSet_of_compl             |0
048 Set.integer_eq                                               |0
049 ContinuousMap.exists_mem_subalgebra_near_continuousMap_of_separatesPoints |0
050 Sylow.ext                                                    |0



#Query
the number of Sylow $p$ subgroup equals $1 \mod p$
Sylow’s third theorem
`Fintype.card (Sylow G p)` equals $1 \mod p$
#Retrieved theorem
001 card_sylow_modEq_one                                     |2
002 Sylow.card_eq_multiplicity                               |0
003 Sylow.ext                                                |0
004 card_sylow_eq_index_normalizer                           |1
005 card_sylow_eq_card_quotient_normalizer                   |1
006 IsPGroup.exists_le_sylow                                 |0
007 Sylow.subsingleton_of_normal                             |0
008 Sylow.card_quotient_normalizer_modEq_card_quotient       |0
009 Sylow.dvd_card_of_dvd_card                               |0
010 not_dvd_index_sylow'                                     |0
011 IsPGroup.sylow_mem_fixedPoints_iff                       |0
012 Sylow.normal_of_all_max_subgroups_normal                 |0
013 Sylow.coe_smul                                           |0
014 Sylow.card_normalizer_modEq_card                         |0
015 card_sylow_dvd_index                                     |1
016 Sylow.coe_ofCard                                         |0
017 Sylow.coe_subgroup_smul                                  |0
018 Sylow.exists_subgroup_card_pow_succ                      |0
019 Sylow.pow_dvd_card_of_pow_dvd_card                       |0
020 Sylow.card_coprime_index                                 |0
021 Sylow.prime_dvd_card_quotient_normalizer                 |0
022 Sylow.exists_subgroup_card_pow_prime                     |0
023 Sylow.smul_eq_of_normal                                  |0
024 Sylow.conj_eq_normalizer_conj_of_mem_centralizer         |0
025 Sylow.ext_iff                                            |0
026 Sylow.prime_pow_dvd_card_normalizer                      |0
027 Sylow.smul_def                                           |0
028 MonoidHom.transferSylow_eq_pow_aux                       |0
029 Sylow.normalizer_normalizer                              |0
030 Sylow.coe_subtype                                        |0
031 Sylow.smul_subtype                                       |0
032 Sylow.smul_le                                            |0
033 Sylow.conj_eq_normalizer_conj_of_mem                     |0
034 Subgroup.index_top                                       |0
035 Sylow.orbit_eq_top                                       |0
036 not_dvd_index_sylow                                      |0
037 Sylow.characteristic_of_normal                           |0
038 Subgroup.sylow_mem_fixedPoints_iff                       |0
039 Sylow.subtype_injective                                  |0
040 Sylow.ne_bot_of_dvd_card                                 |0
041 Sylow.exists_subgroup_card_pow_prime_le                  |0
042 Sylow.exists_comap_subtype_eq                            |0
043 Sylow.exists_comap_eq_of_ker_isPGroup                    |0
044 Sylow.exists_comap_eq_of_injective                       |0
045 Sylow.smul_eq_iff_mem_normalizer                         |0
046 IsPGroup.inf_normalizer_sylow                            |0
047 Sylow.normal_of_normalizer_normal                        |0
048 Sylow.pointwise_smul_def                                 |0
049 Submodule.coe_toSubalgebra                               |0
050 Sylow.coe_comapOfKerIsPGroup                             |0
000 not_dvd_card_sylow |1




#Query
The trace of the product of two matrices is independent of order of multiplication
tr($AB$) = tr($BA$)
`(A * B).trace = (B * A).trace`
#Retrieved theorem
001 Matrix.trace_mul_comm                                   |2
002 Matrix.trace_col_mul_row                                |1
003 Algebra.traceMatrix_of_matrix_mulVec                    |0
004 Matrix.trace_mul_cycle'                                 |1
005 Matrix.trace_transpose_mul                              |1
006 Matrix.trace_kronecker                                  |0
007 Matrix.trace_add                                        |0
008 Matrix.trace_mul_cycle                                  |1
009 Matrix.sum_hadamard_eq                                  |0
010 Matrix.trace_kroneckerTMul                              |0
011 Matrix.trace_smul                                       |0
012 Matrix.trace_sub                                        |0
013 LinearMap.trace_tensorProduct'                          |0
014 Matrix.fromColumns_mul_fromRows_eq_one_comm             |0
015 Matrix.traceLinearMap_apply                             |0
016 Matrix.trace_multiset_sum                               |0
017 Algebra.traceMatrix_of_matrix_vecMul                    |0
018 Matrix.mul_assoc                                        |0
019 Matrix.mul_eq_one_comm                                  |0
020 Matrix.mul_right_inj_of_invertible                      |0
021 Matrix.trace_sum                                        |0
022 Matrix.mul_add                                          |0
023 Matrix.traceAddMonoidHom_apply                          |0
024 Matrix.det_mul_comm                                     |0
025 Matrix.mul_fin_three                                    |0
026 Matrix.smul_mulVec_assoc                                |0
027 Matrix.mulVec_add                                       |0
028 Matrix.rank_mul_eq_left_of_isUnit_det                   |0
029 Matrix.mulVec_smul_assoc                                |0
030 Matrix.rank_mul_eq_right_of_isUnit_det                  |0
031 Matrix.trace_transpose                                  |0
032 Matrix.mul_fromColumns                                  |0
033 Matrix.circulant_mul_comm                               |0
034 Algebra.trace_prod                                      |0
035 Matrix.hadamard_comm                                    |0
036 Matrix.mul_sum                                          |0
037 Matrix.add_mulVec                                       |0
038 Matrix.trace_zero                                       |0
039 LinearMap.trace_eq_matrix_trace_of_finset               |0
040 Matrix.equiv_compl_fromColumns_mul_fromRows_eq_one_comm |0
041 Algebra.traceForm_apply                                 |0
042 Matrix.GeneralLinearGroup.coe_mul                       |0
043 Matrix.smul_mul                                         |0
044 Matrix.trace_fin_one                                    |0
045 Matrix.adjugate_mul_distrib                             |0
046 Matrix.self_mul_conjTranspose_mul_eq_zero               |0
047 Matrix.mul_self_mul_conjTranspose_eq_zero               |0
048 Matrix.two_mul_expl                                     |0
049 Matrix.isSymm_mul_transpose_self                        |0
050 Matrix.TransvectionStruct.inv_mul                       |0




#Query
All eigenvalues of a self-adjoint matrix are real
If $A$ is symmetric, eigenvalues of $A$ are real.
#Retrieved theorem
001 LinearMap.IsSymmetric.conj_eigenvalue_eq_self                |2
002 LinearMap.IsSymmetric.eigenvalues_def                        |1
003 IsSelfAdjoint.mem_spectrum_eq_re                             |2
004 LinearMap.IsSymmetric.orthogonalFamily_eigenspaces           |1
005 Matrix.eigenvalues_conjTranspose_mul_self_nonneg             |0
006 IsSelfAdjoint.val_re_map_spectrum                            |1
007 Matrix.IsHermitian.isSelfAdjoint                             |0
008 selfAdjoint.val_re_map_spectrum                              |1
009 LinearMap.IsSymmetric.hasEigenvector_eigenvectorBasis        |0
010 Matrix.IsHermitian.coe_re_diag                               |0
011 LinearMap.IsSymmetric.eigenvectorBasis_def                   |0
012 LinearMap.IsSymmetric.invariant_orthogonalComplement_eigenspace |1
013 Matrix.IsHermitian.eigenvalues_eq                            |0
014 Matrix.eigenvalues_self_mul_conjTranspose_nonneg             |0
015 Matrix.isHermitian_iff_isSymmetric                           |0
016 Matrix.PosSemidef.eigenvalues_nonneg                         |0
017 Matrix.IsHermitian.coe_re_apply_self                         |0
018 LinearMap.IsSymmetric.hasEigenvalue_eigenvalues              |0
019 Matrix.isHermitian_diagonal_of_self_adjoint                  |0
020 IsSelfAdjoint.imaginaryPart                                  |0
021 IsSelfAdjoint.coe_realPart                                   |0
022 span_selfAdjoint                                             |0
023 Module.End.hasEigenvalue_of_hasGeneralizedEigenvalue         |0
024 LinearMap.IsSymmetric.direct_sum_isInternal                  |0
025 Matrix.PosDef.eigenvalues_pos                                |0
026 Matrix.IsHermitian.rank_eq_card_non_zero_eigs                |0
027 LinearMap.IsSymmetric.orthogonalComplement_iSup_eigenspaces_eq_bot |0
028 Complex.selfAdjointEquiv_apply                               |0
029 realPart_comp_subtype_selfAdjoint                            |0
030 ContinuousLinearMap.IsPositive.isSelfAdjoint                 |0
031 Matrix.IsHermitian.eigenvectorMatrix_apply                   |0
032 IsSelfAdjoint.spectralRadius_eq_nnnorm                       |0
033 mem_selfAdjointMatricesSubmodule'                            |0
034 Module.End.eigenvectors_linearIndependent                    |0
035 LinearMap.IsSymmetric.diagonalization_symm_apply             |0
036 mem_selfAdjointMatricesSubmodule                             |0
037 IsSelfAdjoint.all                                            |0
038 IsSelfAdjoint.smul_mem_skewAdjoint                           |0
039 continuous_selfAdjointPart                                   |0
040 Commute.expUnitary                                           |0
041 isSelfAdjoint_smul_of_mem_skewAdjoint                        |0
042 LinearMap.IsSymmetric.isSelfAdjoint                          |0
043 IsSelfAdjoint.skewAdjointPart_apply                          |0
044 ContinuousLinearMap.IsPositive.adjoint_conj                  |0
045 selfAdjointPart_comp_subtype_selfAdjoint                     |0
046 BilinForm.mem_selfAdjointSubmodule                           |0
047 ContinuousLinearMap.IsPositive.orthogonalProjection_comp     |0
048 Complex.selfAdjointEquiv_symm_apply                          |0
049 selfAdjoint.val_div                                          |0
050 ContinuousLinearMap.isPositive_one                           |0
000 LinearMap.IsSymmetric.orthogonalFamily_eigenspaces' |1



#Query
expansion of determinant of 2*2 matrix
$\begin{vmatrix} a & b \\ c & d \end{vmatrix} = a*c-b*d $
If $A$ is a $2*2$ matrix, $\det A = a_{11}*a_{22} - a_{12}*a_{21}$.
#Retrieved theorem
001 Matrix.det_succ_row                                          |1
002 Matrix.det_succ_column                                       |1
003 Matrix.det_succ_row_zero                                     |1
004 Matrix.det_succ_column_zero                                  |1
005 Matrix.det_add_col_mul_row                                   |0
006 IsUnit.det_zpow                                              |0
007 Matrix.det_updateColumn_smul'                                |0
008 Matrix.det_smul_adjugate_adjugate                            |0
009 Matrix.det_isEmpty                                           |0
010 Matrix.det_fromBlocks_zero₂₁                                 |0
011 Matrix.det_eq_of_eq_mul_det_one                              |0
012 Matrix.det_smul_of_tower                                     |0
013 Matrix.det_smul                                              |0
014 Matrix.det_mul_row                                           |0
015 Matrix.det_add_mul                                           |0
016 Matrix.det_mul_column                                        |0
017 Gamma0_det                                                   |0
018 Matrix.det_mul_comm                                          |0
019 UpperHalfPlane.det_coe'                                      |0
020 Matrix.det_eq_of_eq_det_one_mul                              |0
021 Matrix.eval_det                                              |0
022 Matrix.det_adjugate                                          |0
023 Matrix.det_eq_one_of_card_eq_zero                            |0
024 Matrix.det_one                                               |0
025 Matrix.det_mul_left_comm                                     |0
026 Matrix.det_eq_sum_mul_adjugate_row                           |0
027 Matrix.det_fromBlocks_one₂₂                                  |0
028 Matrix.det_eq_sum_mul_adjugate_col                           |0
029 Matrix.isUnit_det_transpose                                  |0
030 Matrix.det_pow                                               |0
031 Matrix.det_neg                                               |0
032 Matrix.det_diagonal                                          |0
033 Matrix.det_neg_eq_smul                                       |0
034 Matrix.det_fromBlocks_zero₁₂                                 |0
035 Matrix.det_eq_of_forall_row_eq_smul_add_const                |0
036 Matrix.det_fin_zero                                          |1
037 Matrix.det_fromBlocks_one₁₁                                  |0
038 Matrix.det_permute                                           |0
039 Polynomial.coeff_det_X_add_C_card                            |0
040 Matrix.det_transpose                                         |0
041 Matrix.det_fin_three                                         |1
042 OrthonormalBasis.det_to_matrix_orthonormalBasis_of_same_orientation |0
043 Matrix.isUnit_det_zpow_iff                                   |0
044 Matrix.det_one_sub_mul_comm                                  |0
045 Polynomial.leadingCoeff_det_X_one_add_C                      |0
046 Matrix.det_fin_one_of                                        |1
047 Matrix.coe_detMonoidHom                                      |0
048 OrthonormalBasis.det_to_matrix_orthonormalBasis_real         |0
049 OrthonormalBasis.det_to_matrix_orthonormalBasis_of_opposite_orientation |0
050 Matrix.det_permutation                                       |0
000 Matrix.det_fin_two |2
000 Matrix.det_fin_two_of |2
000 Matrix.det_fin_one |1



#Query
For integers $a,b,c$, if $a$ divides $b$ and $c$, then $a$ divides the greatest common divisor of $b$ and $c$
For $a, b, c \in \mathbb{Z}$, $a \mid b$ and $a \mid c$ implies $a \mid \text{gcd}(b, c)$
`(a b c : ℤ) : a ∣ b → a ∣ c → a ∣ gcd b c`
#Retrieved theorem
001 gcd_eq_of_dvd_sub_left                 |0
002 EuclideanDomain.dvd_gcd                |2
003 gcd_eq_of_dvd_sub_right                |0
004 Int.gcd_dvd_iff                        |0
005 Int.gcd_greatest                       |1
006 Nat.gcd_mul_of_coprime_of_dvd          |0
007 Int.dvd_of_dvd_mul_right_of_gcd_one    |0
008 Tactic.NormNum.int_gcd_helper'         |0
009 ZNum.gcd_to_nat                        |0
010 Int.gcd_eq_natAbs                      |0
011 Int.dvd_of_dvd_mul_left_of_gcd_one     |0
012 gcd_dvd_gcd                            |1
013 Int.dvd_gcd                            |2
014 Int.gcd_dvd_gcd_of_dvd_right           |0
015 Int.gcd_dvd_gcd_of_dvd_left            |0
016 Nat.gcd_eq_iff                         |1
017 PNat.Coprime.factor_eq_gcd_right       |0
018 Nat.gcd_greatest                       |1
019 PNat.Coprime.factor_eq_gcd_left        |0
020 EuclideanDomain.gcd_eq_left            |0
021 PNat.gcd_rel_left                      |0
022 EuclideanDomain.gcd_dvd_left           |0
023 Int.gcd_div                            |1
024 EuclideanDomain.gcd_dvd_right          |0
025 EuclideanDomain.gcd_dvd                |0
026 Int.gcd_dvd_right                      |0
027 Int.gcd_dvd_left                       |0
028 Int.natAbs_euclideanDomain_gcd         |0
029 Int.coe_gcd                            |0
030 Nat.gcd_dvd_gcd_of_dvd_right           |0
031 Nat.gcd_dvd_right                      |0
032 Tactic.NormNum.int_gcd_helper          |0
033 PNat.dvd_gcd                           |0
034 PNat.Coprime.factor_eq_gcd_right_right |0
035 PNat.Coprime.factor_eq_gcd_left_right  |0
036 Nat.gcd_dvd_gcd_of_dvd_left            |0
037 Int.gcd_eq_left                        |0
038 Nat.dvd_gcd                            |1
039 Finset.dvd_gcd_iff                     |1
040 Nat.dvd_gcd_iff                        |1
041 Nat.gcd_div                            |0
042 Nat.gcd_dvd                            |0
043 Int.gcd_eq_right                       |0
044 dvd_mul_gcd_of_dvd_mul                 |0
045 dvd_gcd_mul_of_dvd_mul                 |0
046 Int.exists_gcd_one                     |0
047 PNat.gcd_eq_left_iff_dvd               |0
048 Int.gcd_assoc                          |0
049 PNat.gcd_dvd_left                      |0
050 Multiset.gcd_mono                      |0



#Query
fundamental identity of ramification and inertia
$\Sum e_i f_i =$ extension degree, where $e_i$ is ramification index and $f_i$ is inertia degree
#Retrieved theorem
001 Ideal.sum_ramification_inertia                               |2
002 Ideal.Factors.finrank_pow_ramificationIdx                    |1
003 Ideal.inertiaDeg_algebraMap                                  |0
004 Ideal.inertiaDeg_of_subsingleton                             |0
005 Ideal.ramificationIdx_eq_zero                                |0
006 Ideal.ramificationIdx_ne_zero                                |0
007 Ideal.Factors.ramificationIdx_ne_zero                        |1
008 Ideal.ramificationIdx_eq_find                                |0
009 Ideal.le_pow_ramificationIdx                                 |0
010 Ideal.Factors.inertiaDeg_ne_zero                             |1
011 Ideal.IsDedekindDomain.ramificationIdx_ne_zero               |1
012 Ideal.ramificationIdx_spec                                   |0
013 Ideal.IsDedekindDomain.ramificationIdx_eq_normalizedFactors_count |1
014 Ideal.IsDedekindDomain.ramificationIdx_eq_factors_count      |1
015 QuadraticForm.equivalent_one_zero_neg_one_weighted_sum_squared |0
016 Ideal.ramificationIdx_bot                                    |0
017 FundamentalGroupoid.id_eq_path_refl                          |0
018 Ideal.ramificationIdx_lt                                     |0
019 Ideal.Quotient.algebraMap_quotient_of_ramificationIdx_neZero |0
020 Ideal.rank_prime_pow_ramificationIdx                         |1
021 Ideal.finrank_prime_pow_ramificationIdx                      |1
022 Ideal.ramificationIdx_of_not_le                              |0
023 ModuleCat.MonoidalCategory.tensor_id                         |0
024 FundamentalGroupoidFunctor.prodIso_inv                       |0
025 ModularGroup.eq_smul_self_of_mem_fdo_mem_fdo                 |0
026 MeasureTheory.IsFundamentalDomain.measure_zero_of_invariant  |0
027 MeasureTheory.IsAddFundamentalDomain.hasFiniteIntegral_on_iff |0
028 Pell.IsFundamental.zpow_ne_neg_zpow                          |0
029 MeasureTheory.fundamentalInterior_union_fundamentalFrontier  |0
030 MeasureTheory.mem_fundamentalInterior                        |0
031 FundamentalGroupoidFunctor.piIso_inv                         |0
032 FundamentalGroupoidFunctor.proj_map                          |0
033 Ideal.quotientToQuotientRangePowQuotSucc_injective           |0
034 MeasureTheory.disjoint_fundamentalInterior_fundamentalFrontier |0
035 MeasureTheory.IsFundamentalDomain.map_restrict_quotient      |0
036 FundamentalGroupoidFunctor.prodIso_hom                       |0
037 MeasureTheory.IsFundamentalDomain.smul_of_comm               |0
038 MeasureTheory.sdiff_fundamentalInterior                      |0
039 MeasureTheory.fundamentalInterior_subset                     |0
040 MeasureTheory.IsFundamentalDomain.image_of_equiv             |0
041 MeasureTheory.fundamentalInterior_smul                       |0
042 Algebra.TensorProduct.one_def                                |0
043 Pell.IsFundamental.zpow_eq_one_iff                           |0
044 MeasureTheory.fundamentalFrontier_union_fundamentalInterior  |0
045 CategoryTheory.LocalizerMorphism.id_functor                  |0
046 MeasureTheory.disjoint_addFundamentalInterior_addFundamentalFrontier |0
047 MeasureTheory.addFundamentalFrontier_union_addFundamentalInterior |0
048 CategoryTheory.MonoidalOfChosenFiniteProducts.tensor_id      |0
049 MeasureTheory.addFundamentalInterior_union_addFundamentalFrontier |0
050 Bimod.whisker_assoc_bimod                                    |0
000 Ideal.Factors.piQuotientEquiv_map |1
000 Ideal.Factors.piQuotientEquiv_mk |1



#Query
For any Galois extension, the fixed field of fixing subgroup of any intermediate field is itself.
Let $L/K$ be a Galois extension, $F$ be an intermidiate field, then $L^{\{\sigma \in \text{Gal}(L/K) | \sigma x = x , \forall x \in F \} } = F$
#Retrieved theorem
001 IsGalois.fixedField_fixingSubgroup                      |2
002 IsGalois.of_fixedField_eq_bot                           |1
003 IntermediateField.fixingSubgroup_fixedField             |1
004 IntermediateField.top_toSubalgebra                      |0
005 top_fixedByFinite                                       |0
006 IsGalois.card_fixingSubgroup_eq_finrank                 |1
007 fixingSubgroup_fixedPoints_gc                           |0
008 IntermediateField.equivOfEq_rfl                         |0
009 IntermediateField.le_iff_le                             |1
010 IntermediateField.fixingSubgroup.antimono               |0
011 IntermediateField.fixingSubgroup.bot                    |0
012 IsGalois.tfae                                           |1
013 IsGalois.of_separable_splitting_field_aux               |0
014 IntermediateField.finrank_fixedField_eq_card            |1
015 IntermediateField.coe_toSubfield                        |0
016 fixingSubmonoid_fixedPoints_gc                          |0
017 IntermediateField.sInf_toSubfield                       |0
018 IntermediateField.coe_toSubalgebra                      |0
019 IntermediateField.subsingleton_of_rank_adjoin_eq_one    |0
020 IntermediateField.iInf_toSubalgebra                     |0
021 IntermediateField.mem_fixingSubgroup_iff                |0
022 IsGalois.is_separable_splitting_field                   |0
023 mem_subalgebraEquivIntermediateField                    |0
024 IntermediateField.top_toSubfield                        |0
025 IntermediateField.sup_toSubalgebra                      |0
026 IntermediateField.eq_adjoin_of_eq_algebra_adjoin        |0
027 IntermediateField.coe_iInf                              |0
028 IntermediateField.mem_top                               |0
029 IntermediateField.subsingleton_of_finrank_adjoin_eq_one |0
030 Polynomial.Gal.restrictDvd_def                          |0
031 fixingSubgroup_iUnion                                   |0
032 mem_subalgebraEquivIntermediateField_symm               |0
033 Polynomial.Gal.card_of_separable                        |0
034 GaloisField.card                                        |0
035 IntermediateField.coe_inv                               |0
036 IntermediateField.finrank_eq_finrank_subalgebra         |0
037 IntermediateField.subset_adjoin                         |0
038 IntermediateField.AdjoinSimple.algebraMap_gen           |0
039 IntermediateField.coe_top                               |0
040 IntermediateField.normalClosure_def'                    |0
041 toIntermediateField'_toSubalgebra                       |0
042 fixingSubgroup_union                                    |0
043 IntermediateField.range_val                             |0
044 IntermediateField.coe_map                               |0
045 gal_X_pow_sub_C_isSolvable                              |0
046 IntermediateField.coe_zero                              |0
047 IntermediateField.exists_finset_of_mem_supr'            |0
048 IntermediateField.exists_finset_of_mem_iSup             |0
049 IntermediateField.adjoin_contains_field_as_subfield     |0
050 IntermediateField.mem_toSubalgebra                      |0



#Query
any set is not an element of itself in ZFC
$x \notin x$
`¬ x ∈ x`
#Retrieved theorem
001 ZFSet.mem_irrefl                                             |2
002 PSet.mem_irrefl                                              |1
003 AddGroupFilterBasis.instInhabitedAddGroupFilterBasis.proof_1 |0
004 ZFSet.not_mem_empty                                          |0
005 Set.Nontrivial.not_subset_singleton                          |0
006 Set.Intersecting.not_mem                                     |0
007 AddSubgroup.not_mem_of_not_mem_closure                       |0
008 Set.Nontrivial.not_subsingleton                              |0
009 PSet.not_mem_empty                                           |0
010 ZFSet.not_mem_sInter_of_not_mem                              |0
011 Set.Nontrivial.ne_singleton                                  |0
012 Finset.not_mem_erase                                         |0
013 Set.not_mem_empty                                            |0
014 Part.not_mem_none                                            |0
015 YoungDiagram.not_mem_bot                                     |0
016 Set.not_nontrivial_empty                                     |0
017 Set.Nontrivial.not_subset_empty                              |0
018 AList.not_mem_empty                                          |0
019 AddSubmonoid.not_mem_of_not_mem_closure                      |0
020 Set.subset_compl_singleton_iff                               |0
021 Finset.not_mem_mono                                          |0
022 Finset.not_mem_empty                                         |0
023 Set.Nontrivial.ne_empty                                      |0
024 Setoid.empty_not_mem_classes                                 |0
025 Subfield.not_mem_of_not_mem_closure                          |0
026 Subsemiring.not_mem_of_not_mem_closure                       |0
027 Disjoint.zero_not_mem_sub_set                                |0
028 Finset.zero_mem_sub_iff                                      |0
029 Class.univ_not_mem_univ                                      |0
030 Finset.not_mem_of_mem_powerset_of_not_mem                    |0
031 Set.nmem_setOf_iff                                           |0
032 NonUnitalSubring.not_mem_of_not_mem_closure                  |0
033 Finset.not_mem_sigmaLift_of_ne_right                         |0
034 Disjoint.one_not_mem_div_set                                 |0
035 Set.zero_nonempty                                            |0
036 Set.not_mem_of_not_mem_sUnion                                |0
037 Finset.not_ssubset_empty                                     |0
038 Set.Nontrivial.exists_ne                                     |0
039 ssubset_asymm                                                |0
040 Filter.nmem_hyperfilter_of_finite                            |0
041 Counterexample.exist_ne_and_fst_eq_fst_and_snd_eq_snd        |0
042 Ne.irrefl                                                    |0
043 ZFSet.mem_asymm                                              |1
044 Associates.reducible_not_mem_factorSet                       |0
045 Class.not_mem_empty                                          |0
046 Set.Finite.nmem_hyperfilter                                  |0
047 Finset.Nonempty.ne_empty                                     |0
048 Finset.not_mem_union                                         |0
049 not_forall_of_exists_not                                     |0
050 PSet.not_nonempty_empty                                      |0
000 ZFSet.regularity |1





#Query
if there exist injective maps of sets from $A$ to $B$ and from $B$ to $A$, then there exist bijective map between $A$ and $B$
If there exist $f : A \to B$ injective, $g : A \to B$ injective, then there exist $h : A \to B$ bijective.
Schroeder Bernstein theorem
`{f : A → B} {g : B → A} (hf : f.Injective) (hg : g.Injective) : ∃ h, h.Bijective`
#Retrieved theorem
001 Function.Injective.of_comp_iff'                     |0
002 Function.Embedding.schroeder_bernstein              |2
003 Function.RightInverse.injective                     |0
004 Function.Bijective.comp                             |0
005 Function.Bijective.of_comp_iff                      |0
006 Equiv.injective                                     |0
007 Function.Injective.of_comp                          |0
008 Function.Bijective.comp_left                        |0
009 Function.Bijective.of_comp_iff'                     |0
010 Function.Bijective.injective                        |0
011 Function.Surjective.of_comp_iff'                    |0
012 Function.injective_of_isPartialInv                  |0
013 EquivLike.bijective_comp                            |0
014 Function.Injective.of_comp_iff                      |0
015 Function.LeftInverse.rightInverse_of_injective      |0
016 Function.Injective.injOn_range                      |0
017 Function.Injective.surjective_comp_right'           |0
018 Function.Surjective.iUnion_comp                     |0
019 Function.Injective.comp                             |0
020 Function.Surjective.of_comp                         |0
021 Set.InjOn.rightInvOn_of_leftInvOn                   |0
022 Function.Bijective.comp_right                       |0
023 Function.Injective.factorsThrough                   |0
024 Function.Injective.comp_injOn                       |0
025 Function.LeftInverse.injective                      |0
026 Function.Injective.bijective_of_finite              |0
027 EquivLike.comp_injective                            |0
028 Function.Injective.image_injective                  |0
029 Finset.image_injective                              |0
030 Set.InjOn.comp                                      |0
031 Set.BijOn.injOn                                     |0
032 Set.BijOn.symm                                      |0
033 Set.BijOn.comp                                      |0
034 Unique.bijective                                    |0
035 Function.Bijective.prod_comp                        |0
036 Set.bijOn_comm                                      |0
037 Subsemigroup.comap_injective_of_surjective          |0
038 EquivLike.comp_bijective                            |0
039 Set.SurjOn.comp                                     |0
040 Cardinal.extend_function                            |0
041 finprod_mem_range                                   |0
042 Function.Surjective.comp_left                       |0
043 Function.RightInverse.leftInverse_of_injective      |0
044 Cardinal.extend_function_finite                     |0
045 Function.Injective.sum_map                          |0
046 Function.Surjective.injective_comp_right            |0
047 Subgroup.map_injective                              |0
048 Function.Surjective.of_comp_iff                     |0
049 Ideal.comap_injective_of_surjective                 |0
050 Equiv.bijective                                     |0
000 Function.Embedding.antisymm |2
000 Function.Embedding.total |1
000 Function.Embedding.min_injective |1


#Query
if $p$ implies $q$, then not $q$ implies not $p$
modus tollens
$(p \to q) \to (\not q \to \not p)$
`(p → q) → (¬ q → ¬ p)`
#Retrieved theorem
001 imp_and_neg_imp_iff                               |0
002 by_cases                                          |0
003 Classical.byCases                                 |0
004 Classical.by_cases                                |0
005 imp_not_comm                                      |1
006 imp_not_self                                      |0
007 not_of_not_imp                                    |0
008 ne_true_of_not                                    |0
009 not_not_of_not_imp                                |0
010 Decidable.not_and_not_right                       |0
011 mt                                                |2
012 not_imp_not                                       |2
013 DvdNotUnit.ne                                     |0
014 Function.mt                                       |2
015 DvdNotUnit.not_unit                               |0
016 Function.mtr                                      |1
017 ne_of_not_subset                                  |0
018 AddCommGroup.not_modEq_iff_ne_mod_zmultiples      |0
019 padicValRat.inv                                   |0
020 ne_of_not_superset                                |0
021 neq_of_not_iff                                    |0
022 not_ssubset_of_subset                             |0
023 HasSubset.Subset.not_ssubset                      |0
024 not_or_of_imp                                     |0
025 Subsemiring.not_mem_of_not_mem_closure            |0
026 Subfield.not_mem_of_not_mem_closure               |0
027 Subsemigroup.not_mem_of_not_mem_closure           |0
028 not_lt_of_lt                                      |0
029 false_iff_true                                    |0
030 not_subset_of_ssubset                             |0
031 HasSSubset.SSubset.not_subset                     |0
032 not_mem_of_not_mem_closure                        |0
033 Subring.not_mem_of_not_mem_closure                |0
034 NonUnitalSubring.not_mem_of_not_mem_closure       |0
035 IsSquare.not_prime                                |0
036 Set.Subsingleton.induction_on                     |0
037 Set.not_mem_of_mem_compl                          |0
038 SimpleGraph.IsNClique.not_cliqueFree              |0
039 not_isEmpty_of_nonempty                           |0
040 Sbtw.not_rotate                                   |0
041 Submonoid.closure_induction'                      |0
042 multiplicity.pow_prime_sub_pow_prime              |0
043 ne_of_gt                                          |0
044 Real.IsConjugateExponent.inv_add_inv_conj_ennreal |0
045 Real.IsConjugateExponent.inv_add_inv_conj_nnreal  |0
046 Real.IsConjugateExponent.div_conj_eq_sub_one      |0
047 LT.lt.ne                                          |0
048 SetCoe.exists'                                    |0
049 Wbtw.left_ne_right_of_ne_right                    |0
050 List.not_mem_of_not_mem_cons                      |0

