Documentation Verification Report

Defs

πŸ“ Source: Mathlib/GroupTheory/Coset/Defs.lean

Statistics

MetricCount
DefinitionsquotientEquivOfEq, instCoeQuotientAddSubgroup, instDecidableEqQuotientAddSubgroupOfDecidablePredMem, instHasQuotientAddSubgroup, instInhabitedQuotientAddSubgroup, leftRel, leftRelDecidable, mk, quotientRightRelEquivQuotientLeftRel, rightRel, rightRelDecidable, instCoeQuotientSubgroup, instDecidableEqQuotientSubgroupOfDecidablePredMem, instHasQuotientSubgroup, instInhabitedQuotientSubgroup, leftRel, leftRelDecidable, mk, quotientRightRelEquivQuotientLeftRel, rightRel, rightRelDecidable, quotientEquivOfEq
22
Theoremseq, exists_mk, forall_mk, induction_on, induction_on', leftRel_apply, leftRel_eq, mk_add_of_mem, mk_out_eq_mul, mk_surjective, out_eq', preimage_image_mk, preimage_image_mk_eq_add, preimage_image_mk_eq_iUnion_image, preimage_mk_zero, quotient_liftOn_mk, range_mk, rightRel_apply, rightRel_eq, eq, exists_mk, forall_mk, induction_on, induction_on', leftRel_apply, leftRel_eq, mk_mul_of_mem, mk_out_eq_mul, mk_surjective, out_eq', preimage_image_mk, preimage_image_mk_eq_iUnion_image, preimage_image_mk_eq_mul, preimage_mk_one, quotient_liftOn_mk, range_mk, rightRel_apply, rightRel_eq, quotientEquivOfEq_mk
39
Total61

AddSubgroup

Definitions

NameCategoryTheorems
quotientEquivOfEq πŸ“–CompOpβ€”

QuotientAddGroup

Definitions

NameCategoryTheorems
instCoeQuotientAddSubgroup πŸ“–CompOpβ€”
instDecidableEqQuotientAddSubgroupOfDecidablePredMem πŸ“–CompOp
9 mathmath: HurwitzZeta.isBigO_atTop_evenKernel_sub, HurwitzZeta.hurwitzZetaEven_apply_zero, HurwitzZeta.completedHurwitzZetaEven_eq, Finset.card_nsmul_quotient_add_nsmul_inter_addSubgroup_le, Finset.le_card_quotient_add_sq_inter_addSubgroup, HurwitzZeta.completedCosZeta_eq, HurwitzZeta.completedHurwitzZetaEven_residue_zero, HurwitzKernelBounds.isBigO_atTop_F_int_zero_sub, HurwitzZeta.hasSum_int_evenKernelβ‚€
instHasQuotientAddSubgroup πŸ“–CompOp
247 mathmath: kerLift_mk, AddCircle.not_isOfFinAddOrder_iff_forall_rat_ne_div, AddSubgroup.quotientEquivProdOfLE_symm_apply, exists_norm_mk_lt, instT3Space, AddSubgroup.quotientiInfAddSubgroupOfEmbedding_apply_mk, AddCircle.coe_eq_zero_iff, completeSpace_right', AddSubgroup.norm_normedMk_le, AddSubgroup.index_eq_zero_iff_infinite, AddCircle.coe_real_preimage_closedBall_eq_iUnion, AddSubgroup.IsComplement.quotientAddGroupMk_leftQuotientEquiv, univ_eq_iUnion_vadd, AddCircle.coe_neg, le_norm_iff, norm_mk_eq_zero_iff_mem_closure, equivQuotientZSMulOfEquiv_symm, mk'_eq_mk', NormedAddGroupHom.lift_mk, mk_surjective, isOpenQuotientMap_mk, ker_map, subsingleton_quotient_top, coe_mk', completeSpace_left, equivIocMod_zero, instSecondCountableTopology, measurableVAdd, AddAction.Quotient.vadd_coe, MeasureTheory.AddQuotientMeasureEqMeasurePreimage.addInvariantMeasure_quotient, AddAction.injective_ofQuotientStabilizer, liftEquiv_coe, AddCircle.coe_zsmul, AddSubgroup.quotient_finite_of_isOpen', AddAction.Quotient.coe_vadd_out, AddSubgroup.card_eq_card_quotient_mul_card_addSubgroup, ker_mk', AddCircle.isOfFinAddOrder_iff_exists_rat_eq_div, AddCircle.isAddQuotientCoveringMap_coe, AddCircle.isLocalHomeomorph_coe, AdjoinRoot.evalEval_apply, instContinuousVAdd, AddSubgroup.norm_trivial_quotient_mk, AddCircle.norm_div_natCast, isOpenMap_coe, AddSubgroup.instFiniteQuotientOfContinuousAddOfCompactSpace, equivQuotientZSMulOfEquiv_trans, NormedAddGroupHom.lift_normNoninc, completeSpace_right, norm_lt_iff, AddCircle.add_projection_respects_measure, CategoryTheory.ShortComplex.abLeftHomologyData_Ο€, preimage_image_mk, norm_eq_infDist, liftEquiv_mk, finite, nontrivial_iff, AddCircle.norm_coe_mul, AddCircle.norm_half_period_eq, AddGroup.exponent_quotient_dvd, finite_iff_addSubgroup_quotient, norm_mk, AddSubgroup.isAddQuotientCoveringMap_of_comm, AddCircle.addOrderOf_div_of_gcd_eq_one, AddSubgroup.card_add_eq_card_addSubgroup_add_card_quotient, strictMono_comap_prod_image, AddCircle.coe_sub, discreteTopology_iff, AddSubgroup.goursat_surjective, AddAction.Quotient.mk_vadd_out, AddSubgroup.quotientiInfEmbedding_apply_mk, lift_mk', AddAction.zmultiplesQuotientStabilizerEquiv_symm_apply, subsingleton_iff, map_comp_map, AddCircle.coe_eq_zero_iff_of_mem_Ico, t1Space_iff, instIsTopologicalAddGroup, instT2Space, AddSubgroup.isComplement_range_left, measurable_from_quotient, fg, quotient_norm_add_le, ker_le_range_iff, mk'_comp_subtype, AddSubgroup.discreteTopology, AddCircle.coe_nsmul, strictMono_comap_prod_map, mk_add, AddSubgroup.Characteristic.comap_quotient_mk, AddSubgroup.isComplement_addSubgroup_right_iff_bijective, AddSubgroup.isComplement_addSubgroup_right_iff_existsUnique_quotientAddGroupMk, equivIcoMod_symm_apply, image_coe_inj, comap_map_mk', instCompactSpaceQuotientAddSubgroup, sound, Finset.card_nsmul_quotient_add_nsmul_inter_addSubgroup_le, homQuotientZSMulOfHom_comp_of_rightInverse, AddCircle.coe_image_Icc_eq, preimage_image_mk_eq_add, AddSubgroup.index_ne_zero_iff_finite, nhds_zero_hasBasis, AddCircle.coe_real_preimage_closedBall_inter_eq, AddSubgroup.vadd_leftQuotientEquiv, AddCircle.coe_image_Ioc_eq, quotientAddEquivOfEq_mk, ker_lift, lift_comp_mk', AddSubgroup.isAddQuotientCoveringMap, lift_mk, mk_sub, quotientKerEquivOfRightInverse_symm_apply, quotient_norm_mk_eq, AddSubgroup.quotientiInfEmbedding_apply, AddCircle.norm_coe_eq_abs_iff, AddSubgroup.leftCoset_cover_const_iff_surjOn, measurable_coe, quotientMapAddSubgroupOfOfLe_mk, btw_coe_iff, dense_image_mk, mk_nat_mul, AddCircle.denseRange_zsmul_coe_iff, norm_lift_apply_le, instDiscreteMeasurableSpace, AddCircle.coe_eq_zero_of_pos_iff, AddSubgroup.finite_quotient_of_finiteIndex, AddSubgroup.instDiscreteTopologyQuotientOfContinuousAdd, mk_zero, instLocallyCompactSpace, norm_mk_eq_zero, AddCircle.norm_neg_period, quotientBot_apply, equivIocMod_symm_apply, quotientKerEquivOfRightInverse_apply, UnitAddCircle.norm_eq, Finset.le_card_quotient_add_sq_inter_addSubgroup, preimage_image_coe, AddAction.orbitEquivQuotientStabilizer_symm_apply, AddSubgroup.surjective_normedMk, mk_zsmul, mk_nsmul, btw_coe_iff', image_coe, instFirstCountableTopology, AddCircle.continuous_mk', AddCircle.coe_real_preimage_closedBall_period_zero, dense_preimage_mk, comap_comap_center, instContinuousConstVAdd, AddSubgroup.ker_normedMk, isQuotientMap_mk, mk_sum, map_mk', zmultiples_zsmul_eq_zsmul_iff, NormedAddGroupHom.lift_norm_le, quotientBot_symm_apply, AddSubgroup.quotientiInfAddSubgroupOfEmbedding_apply, addMonoidHom_ext_iff, isClosedMap_coe, NormedAddGroupHom.norm_lift_le, range_mk', map_mk, lift_surjective_of_surjective, AddAction.Quotient.vadd_mk, equivQuotientZSMulOfEquiv_refl, zmultiples_nsmul_eq_nsmul_iff, norm_eq_addGroupSeminorm, continuous_mk, map_normal, AddCircle.coe_period, map_id, map_map, lift_quot_mk, nhds_eq, NormedAddGroupHom.isQuotientQuotient, norm_mk_le_norm, AddCircle.addOrderOf_period_div, measurePreserving_quotientAddGroup_mk_of_AddQuotientMeasureEqMeasurePreimage, AddAction.stabilizer_image_coe_quotient, preimage_image_mk_eq_iUnion_image, kerLift_mk', AddSubgroup.index_eq_card, equivIcoMod_zero, AddCircle.norm_eq, fourier_coe_apply', automorphize_smul_left, AddSubgroup.vadd_apply_eq_vadd_apply_neg_vadd, equivIcoMod_coe, quotientQuotientEquivQuotientAux_mk, eq_zero_iff, AddCircle.addOrderOf_div_of_gcd_eq_one', mk_int_mul, AddCircle.coe_image_Ico_eq, mk'_surjective, AddSubgroup.quotientEquivProdOfLE_apply, discreteTopology, kerLift_injective, eq_class_eq_leftCoset, mk_neg, map_mk'_self, AddSubgroup.finiteIndex_iff_finite_quotient, AddSubgroup.norm_normedMk, exists_norm_add_lt, AddCircle.intCast_div_mul_eq_zsmul, AddCircle.addOrderOf_coe_rat, AddSubgroup.quotient_finite_of_isOpen, borelSpace, AddCircle.norm_eq', MeasureTheory.AddQuotientMeasureEqMeasurePreimage.addHaarMeasure_quotient, instT1Space, AddSubgroup.quotientAddSubgroupOfEmbeddingOfLE_apply_mk, AddSubgroup.goursat, AddCircle.coe_add, MeasureTheory.IsAddFundamentalDomain.absolutelyContinuous_map, AddSubgroup.normedMk.apply, AddCircle.isCoveringMap_coe, CategoryTheory.ShortComplex.abLeftHomologyData_H_coe, card_quotient_rightRel, rangeKerLift_injective, AddSubgroup.card_quotient_dvd_card, MeasureTheory.AddQuotientMeasureEqMeasurePreimage.vaddInvariantMeasure_quotient, KaehlerDifferential.quotKerTotalEquiv_apply, AddAction.ofQuotientStabilizer_vadd, instIsAddTorsionFree, AddCircle.norm_eq_of_zero, range_mk, rangeKerLift_surjective, quotientQuotientEquivQuotientAux_mk_mk, mk'_apply, AddCosetSpace.borelSpace, AddSubgroup.instFiniteQuotientSubtypeMemOpenAddSubgroupOfIsTopologicalAddGroupOfCompactSpace, AddSubgroup.IsComplement.leftQuotientEquiv_apply, preimage_mk_zero, homQuotientZSMulOfHom_comp, completeSpace_left', AddCircle.gcd_mul_addOrderOf_div_eq, homQuotientZSMulOfHom_id, equivIocMod_coe, AddAction.isPretransitive_quotient, le_comap_mk', HurwitzZeta.hasSum_int_evenKernelβ‚€, AddAction.stabilizer_quotient, Rep.FiniteCyclicGroup.coinvariantsTensorResolutionIso_hom_f_hom_apply, AddCircle.natCast_div_mul_eq_nsmul, IsFundamentalDomain.AddQuotientMeasureEqMeasurePreimage_vaddAddHaarMeasure, map_id_apply
instInhabitedQuotientAddSubgroup πŸ“–CompOpβ€”
leftRel πŸ“–CompOp
21 mathmath: AddSubgroup.quotientEquivProdOfLE_symm_apply, leftRel_apply, AddSubgroup.IsComplement.quotientAddGroupMk_leftQuotientEquiv, AddCircle.equivIccQuot_comp_mk_eq_toIcoMod, univ_eq_iUnion_vadd, leftRel_eq_top, AddAction.Quotient.coe_vadd_out, quotient_liftOn_mk, leftRel_eq, AddAction.Quotient.mk_vadd_out, AddCircle.equivIccQuot_comp_mk_eq_toIocMod, leftRel_prod, mk_out_eq_mul, out_eq', orbit_eq_out_vadd, AddSubgroup.quotientEquivProdOfLE'_apply, lift_quot_mk, AddSubgroup.quotientEquivProdOfLE'_symm_apply, automorphize_smul_left, AddSubgroup.quotientEquivProdOfLE_apply, leftRel_pi
leftRelDecidable πŸ“–CompOpβ€”
mk πŸ“–CompOpβ€”
quotientRightRelEquivQuotientLeftRel πŸ“–CompOpβ€”
rightRel πŸ“–CompOp
9 mathmath: AddSubgroup.isComplement_addSubgroup_left_iff_bijective, AddSubgroup.IsComplement.mk''_rightQuotientEquiv, rightRel_eq_top, rightRel_pi, rightRel_apply, AddSubgroup.isComplement_addSubgroup_left_iff_existsUnique_quotientMk'', rightRel_eq, card_quotient_rightRel, rightRel_prod
rightRelDecidable πŸ“–CompOpβ€”

Theorems

NameKindAssumesProvesValidatesDepends On
eq πŸ“–mathematicalβ€”AddSubgroup
SetLike.instMembership
AddSubgroup.instSetLike
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
NegZeroClass.toNeg
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
AddGroup.toSubtractionMonoid
β€”Quotient.eq''
leftRel_apply
exists_mk πŸ“–β€”β€”β€”β€”Function.Surjective.exists
mk_surjective
forall_mk πŸ“–β€”β€”β€”β€”Function.Surjective.forall
mk_surjective
induction_on πŸ“–β€”β€”β€”β€”Quotient.inductionOn'
induction_on' πŸ“–β€”β€”β€”β€”induction_on
leftRel_apply πŸ“–mathematicalβ€”leftRel
AddSubgroup
SetLike.instMembership
AddSubgroup.instSetLike
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
NegZeroClass.toNeg
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
AddGroup.toSubtractionMonoid
β€”Equiv.exists_congr_left
neg_add_eq_iff_eq_add
eq_add_neg_iff_add_eq
exists_neg_mem_iff_exists_mem
AddSubgroup.instAddSubgroupClass
leftRel_eq πŸ“–mathematicalβ€”leftRel
AddSubgroup
SetLike.instMembership
AddSubgroup.instSetLike
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
NegZeroClass.toNeg
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
AddGroup.toSubtractionMonoid
β€”leftRel_apply
mk_add_of_mem πŸ“–mathematicalAddSubgroup
SetLike.instMembership
AddSubgroup.instSetLike
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
β€”eq
neg_add_rev
neg_add_cancel_right
AddSubgroup.neg_mem_iff
mk_out_eq_mul πŸ“–mathematicalβ€”Quotient.out
leftRel
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
AddSubgroup
SetLike.instMembership
AddSubgroup.instSetLike
β€”eq
Quotient.out_eq'
add_neg_cancel_left
mk_surjective πŸ“–mathematicalβ€”HasQuotient.Quotient
AddSubgroup
instHasQuotientAddSubgroup
β€”Quotient.mk''_surjective
out_eq' πŸ“–mathematicalβ€”Quotient.out
leftRel
β€”Quotient.out_eq'
preimage_image_mk πŸ“–mathematicalβ€”Set.preimage
HasQuotient.Quotient
AddSubgroup
instHasQuotientAddSubgroup
Set.image
Set.iUnion
SetLike.instMembership
AddSubgroup.instSetLike
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
β€”Set.ext
Set.mem_preimage
Set.mem_image
eq
Set.mem_iUnion
SetLike.exists
AddSubgroup.neg_mem
neg_add_rev
neg_neg
add_neg_cancel_left
neg_add_cancel_right
neg_mem_iff
AddSubgroupClass.toNegMemClass
AddSubgroup.instAddSubgroupClass
preimage_image_mk_eq_add πŸ“–mathematicalβ€”Set.preimage
HasQuotient.Quotient
AddSubgroup
instHasQuotientAddSubgroup
Set.image
Set
Set.add
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
SetLike.coe
AddSubgroup.instSetLike
β€”preimage_image_mk_eq_iUnion_image
Set.iUnion_subtype
Set.image2_add
Set.iUnion_image_right
Set.iUnion_congr_Prop
Set.image_congr
SetLike.mem_coe
preimage_image_mk_eq_iUnion_image πŸ“–mathematicalβ€”Set.preimage
HasQuotient.Quotient
AddSubgroup
instHasQuotientAddSubgroup
Set.image
Set.iUnion
SetLike.instMembership
AddSubgroup.instSetLike
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
β€”Set.iUnion_congr_of_surjective
neg_surjective
Set.image_add_right'
preimage_mk_zero πŸ“–mathematicalβ€”Set.preimage
HasQuotient.Quotient
AddSubgroup
instHasQuotientAddSubgroup
Set
Set.instSingletonSet
NegZeroClass.toZero
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
AddGroup.toSubtractionMonoid
SetLike.coe
AddSubgroup.instSetLike
β€”Set.image_singleton
preimage_image_mk_eq_add
Set.singleton_add
Set.image_congr
zero_add
Set.image_id'
quotient_liftOn_mk πŸ“–mathematicalβ€”Quotient.liftOn'
leftRel
β€”β€”
range_mk πŸ“–mathematicalβ€”Set.range
HasQuotient.Quotient
AddSubgroup
instHasQuotientAddSubgroup
Set.univ
β€”Set.range_eq_univ
mk_surjective
rightRel_apply πŸ“–mathematicalβ€”rightRel
AddSubgroup
SetLike.instMembership
AddSubgroup.instSetLike
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
NegZeroClass.toNeg
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
AddGroup.toSubtractionMonoid
β€”add_neg_eq_iff_eq_add
eq_neg_add_iff_add_eq
exists_neg_mem_iff_exists_mem
AddSubgroup.instAddSubgroupClass
rightRel_eq πŸ“–mathematicalβ€”rightRel
AddSubgroup
SetLike.instMembership
AddSubgroup.instSetLike
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
NegZeroClass.toNeg
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
AddGroup.toSubtractionMonoid
β€”rightRel_apply

QuotientGroup

Definitions

NameCategoryTheorems
instCoeQuotientSubgroup πŸ“–CompOpβ€”
instDecidableEqQuotientSubgroupOfDecidablePredMem πŸ“–CompOp
2 mathmath: Finset.le_card_quotient_mul_sq_inter_subgroup, Finset.card_pow_quotient_mul_pow_inter_subgroup_le
instHasQuotientSubgroup πŸ“–CompOp
276 mathmath: MulAction.Quotient.coe_smul_out, homQuotientZPowOfHom_id, Subgroup.card_eq_card_quotient_mul_card_subgroup, mk_one, isQuotientMap_mk, Subgroup.quotientEquivOfEq_mk, norm_mk, image_coe_inj, groupHomology.map₁_quotientGroupMk'_epi, kerLift_mk, MeasureTheory.IsFundamentalDomain.absolutelyContinuous_map, lift_mk, groupHomology.coinfNatTrans_app, coe_mk', MonoidHom.restrictHomKerEquiv_apply_coe, NumberField.Units.fun_eq_repr, discreteTopology, Subgroup.transferTransversal_apply, instSecondCountableTopology, DiscreteTiling.PlacedTile.ext_iff_of_preimage, borelSpace, finite, norm_mk_eq_zero, nilpotencyClass_quotient_center, nhds_eq, preimage_image_mk, rangeKerLift_injective, InfiniteGalois.normalAutEquivQuotient_apply, completeSpace_left, card_quotient_rightRel, MulAction.quotient_out_smul_fixedPoints, groupHomology.H1CoresCoinf_X₃, ModularForm.norm_eq_zero_iff, kerLift_mk', Subgroup.quotientEquivProdOfLE_apply, instDiscreteMeasurableSpace, SlashInvariantForm.quotientFunc_smul, CuspForm.coe_trace, Subgroup.instDiscreteTopologyQuotientOfContinuousMul, norm_mk_eq_zero_iff_mem_closure, finite_iff_subgroup_quotient, Subgroup.Normal.quotient_commutative_iff_commutator_le, isClosedMap_coe, MulAction.Quotient.smul_mk, Subgroup.quotientCentralizerEmbedding_apply, liftEquiv_coe, IsPGroup.to_quotient, MeasureTheory.QuotientMeasureEqMeasurePreimage.mulInvariantMeasure_quotient, preimage_image_mk_eq_mul, DoubleCoset.left_bot_eq_left_quot, completeSpace_right, quotientKerEquivOfRightInverse_apply, Subgroup.smul_leftQuotientEquiv, Subgroup.quotientSubgroupOfEmbeddingOfLE_apply_mk, MeasureTheory.QuotientMeasureEqMeasurePreimage.haarMeasure_quotient, Subgroup.discreteTopology, Subgroup.finite_quotient_of_finiteIndex, NumberField.Units.logEmbeddingQuot_apply, completeSpace_right', DiscreteTiling.PlacedTile.ext_iff, Subgroup.leftCoset_cover_const_iff_surjOn, liftEquiv_mk, eq_class_eq_leftCoset, congr_refl, groupCohomology.infNatTrans_app, nontrivial_iff, nhds_one_hasBasis, IsDedekindDomain.selmerGroup.monotone, congr_symm, instIsMulTorsionFree, Representation.ofQuotient_coe_apply, univ_eq_iUnion_smul, MonoidHom.restrictHomKerEquiv_symm_coe_apply, quotientQuotientEquivQuotientAux_mk, lift_comp_mk', NumberField.Units.fundSystem_mk, instT2Space, groupHomology.H1CoresCoinf_g, Subgroup.smul_apply_eq_smul_apply_inv_smul, NumberField.Units.rank_modTorsion, Action.FintypeCat.toEndHom_apply, mk'_apply, IsGalois.normalAutEquivQuotient_apply, rangeKerLift_surjective, Subgroup.index_eq_zero_iff_infinite, quotientKerEquivOfRightInverse_symm_apply, Subgroup.quotientEquivProdOfLE_symm_apply, map_id, MulAction.coe_quotient_smul_fixedPoints, dense_preimage_mk, mk'_comp_subtype, mk'_eq_mk', map_mk', MulAction.isPretransitive_quotient, map_mk, measurableSMul, quotientMapSubgroupOfOfLe_mk, IsPGroup.cyclic_center_quotient_of_card_eq_prime_sq, ker_map, MeasureTheory.QuotientMeasureEqMeasurePreimage.smulInvariantMeasure_quotient, Rep.quotientToInvariantsFunctor_map_hom, Ideal.Quotient.stabilizerQuotientInertiaEquiv_mk, Rep.quotientToCoinvariantsFunctor_map_hom, lift_mk', discreteTopology_iff, Subgroup.quotientiInfSubgroupOfEmbedding_apply_mk, kerLift_injective, instIsInvariantSubtypeMemSubalgebraSubalgebraSubgroupQuotient, instT3Space, Subgroup.quotientEquivSigmaZMod_apply, surjective_cosetToCuspOrbit, Action.FintypeCat.quotientToEndHom_mk, quotientQuotientEquivQuotientAux_mk_mk, congr_apply, preimage_image_mk_eq_iUnion_image, mk_pow, Subgroup.instFiniteQuotientSubtypeMemOpenSubgroupOfIsTopologicalGroupOfCompactSpace, continuous_mk, Submodule.unitsQuotEquivRelPic_symm_apply, range_mk', range_mk, strictMono_comap_prod_image, homQuotientZPowOfHom_comp_of_rightInverse, MulAction.stabilizer_quotient, ker_lift, comap_upperCentralSeries_quotient_center, MulAction.Quotient.mk_smul_out, fg, Subgroup.index_ne_zero_iff_finite, Subgroup.transferTransversal_apply', map_normal, instContinuousSMul, MulAction.orbitEquivQuotientStabilizer_symm_apply, MulActionHom.toQuotient_apply, homQuotientZPowOfHom_comp, groupCohomology.H1InfRes_X₁, subsingleton_quotient_top, Rep.quotientToInvariantsFunctor_obj_V, Subgroup.quotientiInfEmbedding_apply_mk, exists_norm_mk_lt, Subgroup.finiteIndex_iff_finite_quotient, completeSpace_left', Subgroup.quotientiInfEmbedding_apply, SlashInvariantForm.coe_trace, lift_surjective_of_surjective, Subgroup.isComplement_range_left, ClassGroup.equivPic_symm_apply, Subgroup.card_quotient_dvd_card, Finset.le_card_quotient_mul_sq_inter_subgroup, congr_mk', mk_inv, MulAction.zpowersQuotientStabilizerEquiv_symm_apply, Subgroup.quotient_finite_of_isOpen', MulAction.injective_ofQuotientStabilizer, IsDedekindDomain.selmerGroup.fromUnit_ker, MulAction.stabilizer_image_coe_quotient, Subgroup.goursat_surjective, isOpenMap_coe, solvable_quotient_of_solvable, exists_norm_mul_lt, equivQuotientZPowOfEquiv_trans, comap_map_mk', CategoryTheory.PreGaloisCategory.has_decomp_quotients, ValuationSubring.unitsModPrincipalUnitsEquivResidueFieldUnits_comp_quotientGroup_mk_apply, Subgroup.index_eq_card, IsDedekindDomain.selmerGroup.fromUnitLift_injective, NumberField.Units.logEmbeddingQuot_injective, map_id_apply, IsFundamentalDomain.QuotientMeasureEqMeasurePreimage_smulHaarMeasure, Action.FintypeCat.quotientToQuotientOfLE_hom_mk, Subgroup.quotientCenterEmbedding_apply, measurable_coe, Subgroup.isQuotientCoveringMap_of_comm, Finset.card_pow_quotient_mul_pow_inter_subgroup_le, ModularForm.coe_norm, nilpotencyClass_quotient_le, Sylow.card_quotient_normalizer_modEq_card_quotient, Subgroup.normalCore_eq_ker, groupHomology.H1CoresCoinfOfTrivial_g, Subgroup.quotientiInfSubgroupOfEmbedding_apply, map_comp_map, automorphize_smul_left, norm_eq_infDist, le_norm_iff, mk_zpow, CosetSpace.borelSpace, measurable_from_quotient, instContinuousConstSMul, Subgroup.transferFunction_apply, card_preimage_mk, preimage_image_coe, Subgroup.IsComplement'.QuotientMulEquiv_symm_apply, Subgroup.instFiniteQuotientOfContinuousMulOfCompactSpace, instT1Space, NumberField.Units.logEmbeddingEquiv_apply, monoidHom_ext_iff, mk_surjective, quotientBot_apply, isOpenQuotientMap_mk, congr_mk, Subgroup.transferTransversal_apply'', preimage_mk_one, instFirstCountableTopology, upperCentralSeriesStep_eq_comap_center, nilpotencyClass_eq_quotient_center_plus_one, map_mk'_self, ClassGroup.equiv_mk0, Subgroup.isComplement_subgroup_right_iff_existsUnique_quotientGroupMk, Subgroup.card_mul_eq_card_subgroup_mul_card_quotient, Sylow.prime_dvd_card_quotient_normalizer, groupHomology.H1CoresCoinfOfTrivial_X₃, Subgroup.quotientEquivSigmaZMod_symm_apply, ker_le_range_iff, mk'_surjective, NumberField.Units.instFiniteIntAdditiveQuotientUnitsRingOfIntegersSubgroupTorsion, equivQuotientZPowOfEquiv_refl, mk_mul, CategoryTheory.PreGaloisCategory.exists_lift_of_quotient_openSubgroup, image_coe, norm_eq_groupSeminorm, Subgroup.IsComplement'.QuotientMulEquiv_apply, Sylow.instFiniteQuotientSubgroupNormalizerOfFactPrime, measurePreserving_quotientGroup_mk_of_QuotientMeasureEqMeasurePreimage, norm_lt_iff, equivQuotientZPowOfEquiv_symm, mk_prod, comap_comap_center, Subgroup.IsComplement.leftQuotientEquiv_apply, instLocallyCompactSpace, map_map, ClassGroup.equivPic_apply, IsDedekindDomain.HeightOneSpectrum.valuation_of_unit_mod_eq, instIsTopologicalGroup, ker_mk', Sylow.card_eq_card_quotient_normalizer, ModularForm.coe_trace, dense_image_mk, eq_one_iff, lift_quot_mk, Subgroup.commProb_quotient_le, strictMono_comap_prod_map, groupCohomology.H1InfRes_f, quotientBot_symm_apply, mk_div, le_comap_mk', Subgroup.quotient_finite_of_isOpen, Subgroup.isComplement_subgroup_right_iff_bijective, nilpotent_quotient_of_nilpotent, MonoidHom.transfer_eq_prod_quotient_orbitRel_zpowers_quot, ValuationSubring.unitsModPrincipalUnitsEquivResidueFieldUnits_comp_quotientGroup_mk, instSMulCommClassQuotientSubgroupSubtypeMemSubalgebraSubalgebra, Sylow.mem_fixedPoints_mul_left_cosets_iff_mem_normalizer, MulAction.ofQuotientStabilizer_smul, Subgroup.Characteristic.comap_quotient_mk, ClassGroup.mk_def, subsingleton_iff, sound, SlashInvariantForm.coe_norm, groupHomology.mapCycles₁_quotientGroupMk'_epi, MulAction.Quotient.smul_coe, IsZGroup.instQuotientSubgroupOfFinite, Subgroup.goursat, instCompactSpaceQuotientSubgroup, norm_mk_le_norm, Subgroup.IsComplement.quotientGroupMk_leftQuotientEquiv, ClassGroup.equiv_mk, IsDedekindDomain.selmerGroup.valuation_ker_eq, NumberField.Units.instFreeIntAdditiveQuotientUnitsRingOfIntegersSubgroupTorsion, Group.exponent_quotient_dvd, Action.FintypeCat.toEndHom_trivial_of_mem, Subgroup.isQuotientCoveringMap, Submodule.unitsQuotEquivRelPic_apply_coe, quotientMulEquivOfEq_mk, t1Space_iff, out_conj_pow_minimalPeriod_mem, Rep.quotientToCoinvariantsFunctor_obj_V
instInhabitedQuotientSubgroup πŸ“–CompOpβ€”
leftRel πŸ“–CompOp
39 mathmath: MulAction.Quotient.coe_smul_out, AlternatingMap.domCoprod.summand_mk'', DiscreteTiling.PlacedTile.ext_iff_of_preimage, ClassGroup.Quot_mk_eq_mk, leftRel_r_eq_leftCosetEquivalence, SlashInvariantForm.quotientFunc_mk, MulAction.quotient_out_smul_fixedPoints, out_eq', Subgroup.quotientEquivProdOfLE_apply, Abelianization.mk_eq_of, DiscreteTiling.PlacedTile.ext_iff_of_exists, leftRel_prod, leftRel_eq, quotient_liftOn_mk, DiscreteTiling.PlacedTile.smul_mk_mk, univ_eq_iUnion_smul, Action.FintypeCat.toEndHom_apply, Subgroup.quotientEquivProdOfLE_symm_apply, DiscreteTiling.PlacedTile.coe_mk_mk, Subgroup.quotientEquivProdOfLE'_symm_apply, mk_out_eq_mul, Action.FintypeCat.quotientToEndHom_mk, orbit_eq_out_smul, MulAction.Quotient.mk_smul_out, Subgroup.transferTransversal_apply', cosetToCuspOrbit_apply_mk, Action.FintypeCat.quotientToQuotientOfLE_hom_mk, automorphize_smul_left, Subgroup.transferFunction_apply, Subgroup.quotientEquivProdOfLE'_apply, Subgroup.transferTransversal_apply'', DoubleCoset.bot_rel_eq_leftRel, leftRel_eq_top, lift_quot_mk, MonoidHom.transfer_eq_prod_quotient_orbitRel_zpowers_quot, leftRel_pi, Subgroup.IsComplement.quotientGroupMk_leftQuotientEquiv, out_conj_pow_minimalPeriod_mem, leftRel_apply
leftRelDecidable πŸ“–CompOp
2 mathmath: AlternatingMap.domCoprod_coe, AlternatingMap.domCoprod_apply
mk πŸ“–CompOpβ€”
quotientRightRelEquivQuotientLeftRel πŸ“–CompOpβ€”
rightRel πŸ“–CompOp
13 mathmath: rightRel_prod, card_quotient_rightRel, DoubleCoset.right_bot_eq_right_quot, Subgroup.isComplement_subgroup_left_iff_existsUnique_quotientMk'', rightRel_r_eq_rightCosetEquivalence, Rep.coindToInd_apply, DoubleCoset.rel_bot_eq_right_group_rel, rightRel_apply, Subgroup.isComplement_subgroup_left_iff_bijective, rightRel_eq_top, Subgroup.IsComplement.mk''_rightQuotientEquiv, rightRel_eq, rightRel_pi
rightRelDecidable πŸ“–CompOpβ€”

Theorems

NameKindAssumesProvesValidatesDepends On
eq πŸ“–mathematicalβ€”Subgroup
SetLike.instMembership
Subgroup.instSetLike
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
Group.toDivisionMonoid
β€”Quotient.eq''
leftRel_apply
exists_mk πŸ“–β€”β€”β€”β€”Function.Surjective.exists
mk_surjective
forall_mk πŸ“–β€”β€”β€”β€”Function.Surjective.forall
mk_surjective
induction_on πŸ“–β€”β€”β€”β€”Quotient.inductionOn'
induction_on' πŸ“–β€”β€”β€”β€”induction_on
leftRel_apply πŸ“–mathematicalβ€”leftRel
Subgroup
SetLike.instMembership
Subgroup.instSetLike
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
Group.toDivisionMonoid
β€”Equiv.exists_congr_left
Subgroup.instSubgroupClass
leftRel_eq πŸ“–mathematicalβ€”leftRel
Subgroup
SetLike.instMembership
Subgroup.instSetLike
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
Group.toDivisionMonoid
β€”leftRel_apply
mk_mul_of_mem πŸ“–mathematicalSubgroup
SetLike.instMembership
Subgroup.instSetLike
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
β€”eq
mul_inv_rev
inv_mul_cancel_right
Subgroup.inv_mem_iff
mk_out_eq_mul πŸ“–mathematicalβ€”Quotient.out
leftRel
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
Subgroup
SetLike.instMembership
Subgroup.instSetLike
β€”eq
Quotient.out_eq'
mul_inv_cancel_left
mk_surjective πŸ“–mathematicalβ€”HasQuotient.Quotient
Subgroup
instHasQuotientSubgroup
β€”Quotient.mk''_surjective
out_eq' πŸ“–mathematicalβ€”Quotient.out
leftRel
β€”Quotient.out_eq'
preimage_image_mk πŸ“–mathematicalβ€”Set.preimage
HasQuotient.Quotient
Subgroup
instHasQuotientSubgroup
Set.image
Set.iUnion
SetLike.instMembership
Subgroup.instSetLike
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
β€”Set.ext
Subgroup.inv_mem
mul_inv_rev
inv_inv
mul_inv_cancel_left
inv_mul_cancel_right
SubgroupClass.toInvMemClass
Subgroup.instSubgroupClass
preimage_image_mk_eq_iUnion_image πŸ“–mathematicalβ€”Set.preimage
HasQuotient.Quotient
Subgroup
instHasQuotientSubgroup
Set.image
Set.iUnion
SetLike.instMembership
Subgroup.instSetLike
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
β€”Set.iUnion_congr_of_surjective
inv_surjective
Set.image_mul_right'
preimage_image_mk_eq_mul πŸ“–mathematicalβ€”Set.preimage
HasQuotient.Quotient
Subgroup
instHasQuotientSubgroup
Set.image
Set
Set.mul
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
SetLike.coe
Subgroup.instSetLike
β€”preimage_image_mk_eq_iUnion_image
Set.iUnion_subtype
Set.image2_mul
Set.iUnion_image_right
Set.iUnion_congr_Prop
Set.image_congr
preimage_mk_one πŸ“–mathematicalβ€”Set.preimage
HasQuotient.Quotient
Subgroup
instHasQuotientSubgroup
Set
Set.instSingletonSet
InvOneClass.toOne
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
Group.toDivisionMonoid
SetLike.coe
Subgroup.instSetLike
β€”Set.image_singleton
preimage_image_mk_eq_mul
Set.singleton_mul
Set.image_congr
one_mul
Set.image_id'
quotient_liftOn_mk πŸ“–mathematicalβ€”Quotient.liftOn'
leftRel
β€”β€”
range_mk πŸ“–mathematicalβ€”Set.range
HasQuotient.Quotient
Subgroup
instHasQuotientSubgroup
Set.univ
β€”Set.range_eq_univ
mk_surjective
rightRel_apply πŸ“–mathematicalβ€”rightRel
Subgroup
SetLike.instMembership
Subgroup.instSetLike
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
Group.toDivisionMonoid
β€”Subgroup.instSubgroupClass
rightRel_eq πŸ“–mathematicalβ€”rightRel
Subgroup
SetLike.instMembership
Subgroup.instSetLike
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
Group.toDivisionMonoid
β€”rightRel_apply

Subgroup

Definitions

NameCategoryTheorems
quotientEquivOfEq πŸ“–CompOp
1 mathmath: quotientEquivOfEq_mk

Theorems

NameKindAssumesProvesValidatesDepends On
quotientEquivOfEq_mk πŸ“–mathematicalβ€”DFunLike.coe
Equiv
HasQuotient.Quotient
Subgroup
QuotientGroup.instHasQuotientSubgroup
EquivLike.toFunLike
Equiv.instEquivLike
quotientEquivOfEq
β€”β€”

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