Documentation Verification Report

Opposite

📁 Source: Mathlib/Algebra/Group/Opposite.lean

Statistics

MetricCount
DefinitionsinstAddCancelCommMonoid, instAddCancelMonoid, instAddCommGroup, instAddCommMonoid, instAddCommSemigroup, instAddGroup, instAddLeftCancelMonoid, instAddLeftCancelSemigroup, instAddMonoid, instAddRightCancelMonoid, instAddRightCancelSemigroup, instAddSemigroup, instAddZero, instAddZeroClass, instCommGroup, instCommMonoid, instCommSemigroup, instDivInvMonoid, instGroup, instLeftCancelSemigroup, instMonoid, instMulOneClass, instRightCancelSemigroup, instSemigroup, instSubNegMonoid, instSubtractionCommMonoid, instSubtractionMonoid, pow, instAddCommGroup, instAddCommMagma, instAddCommMonoid, instAddCommSemigroup, instAddGroup, instAddLeftCancelSemigroup, instAddMonoid, instAddRightCancelSemigroup, instAddSemigroup, instAddZeroClass, instCancelCommMonoid, instCancelMonoid, instCommGroup, instCommMonoid, instCommSemigroup, instDivInvMonoid, instDivisionCommMonoid, instDivisionMonoid, instGroup, instLeftCancelMonoid, instLeftCancelSemigroup, instMonoid, instMulOne, instMulOneClass, instRightCancelMonoid, instRightCancelSemigroup, instSemigroup, instSubNegMonoid
56
Theoremsop, unop, addCommute_op, addCommute_unop, addSemiconjBy_op, addSemiconjBy_unop, instAddTorsionFree, instIsCancelAdd, instIsDedekindFiniteAddMonoid, instIsLeftCancelAdd, instIsRightCancelAdd, instMulTorsionFree, isDedekindFiniteAddMonoid_iff, op_nsmul, op_pow, op_sub, op_zsmul, unop_nsmul, unop_pow, unop_sub, unop_zsmul, op, unop, op, unop, commute_op, commute_unop, instIsCancelMul, instIsDedekindFiniteMonoid, instIsLeftCancelMul, instIsRightCancelMul, isDedekindFiniteMonoid_iff, op_div, op_pow, op_zpow, semiconjBy_op, semiconjBy_unop, unop_div, unop_pow, unop_zpow, op, unop
42
Total98

AddCommute

Theorems

NameKindAssumesProvesValidatesDepends On
op 📖mathematicalAddCommuteAddOpposite
AddOpposite.instAdd
AddOpposite.op
AddSemiconjBy.op
unop 📖mathematicalAddCommute
AddOpposite
AddOpposite.instAdd
AddOpposite.unopAddSemiconjBy.unop

AddOpposite

Definitions

NameCategoryTheorems
instAddCancelCommMonoid 📖CompOp
instAddCancelMonoid 📖CompOp
instAddCommGroup 📖CompOp
instAddCommMonoid 📖CompOp
1 mathmath: instIsOrderedAddMonoid
instAddCommSemigroup 📖CompOp
instAddGroup 📖CompOp
71 mathmath: AddSubgroup.centerToAddOpposite_apply_coe, comap_op_leftUniformSpace, AddAction.stabilizer_addSubgroup_op, AddSubgroup.normal_op, AddSubgroup.op_inf, AddSubgroup.coe_unop, Topology.IsQuotientMap.isAddQuotientCoveringMap_of_addSubgroupOp, AddSubgroup.unop_normalizer, AddSubgroup.unop_sInf, AddSubgroup.op_le_iff, AddSubgroup.unop_closure, AddSubgroup.op_sInf, AddSubgroup.op_normalizer, AddSubgroup.unop_inf, AddCircle.instHasAddFundamentalDomainSubtypeAddOppositeRealMemAddSubgroupOpZmultiples, AddAction.stabilizer_op_addSubgroup, AddSubgroup.op.instNormal, AddSubgroup.le_op_iff, AddSubgroup.unop_iSup, AddSubgroup.op_top, AddSubgroup.op_toAddSubmonoid, AddSubgroup.op_le_op_iff, AddSubgroup.op_bot, AddSubgroup.unop_injective, AddSubgroup.op_toSubsemigroup, AddSubgroup.vadd_opposite_add, AddSubgroup.op_sup, AddSubgroup.op_closure, AddSubgroup.unop_toSubsemigroup, QuotientAddGroup.sound, AddSubgroup.unop_sup, AddSubgroup.coe_op, AddSubgroup.mem_op, AddSubgroup.isAddQuotientCoveringMap, AddSubgroup.opEquiv_symm_apply, AddSubgroup.op_sSup, instProperVAddAddOppositeOfIsTopologicalAddGroup, AddSubgroup.unop_eq_top, AddSubgroup.Normal.op, AddSubgroup.unop_eq_bot, instIsTopologicalAddGroupAddOpposite, AddSubgroup.unop_bot, AddSubgroup.op_iInf, AddSubgroup.unop_iInf, instIsUniformAddGroup, AddSubgroup.opEquiv_apply, AddSubgroup.properlyDiscontinuousVAdd_opposite_of_tendsto_cofinite, AddSubgroup.op_injective, AddSubgroup.centerToAddOpposite_symm_apply_coe, AddCircle.instAddQuotientMeasureEqMeasurePreimageSubtypeAddOppositeRealMemAddSubgroupOpZmultiplesVolume, QuotientAddGroup.orbit_eq_out_vadd, AddSubgroup.Normal.of_unop, AddAction.right_quotientAction, AddSubgroup.op_iSup, AddCircle.instProperlyDiscontinuousVAddSubtypeAddOppositeRealMemAddSubgroupOpZmultiples, AddSubgroup.unop_toAddSubmonoid, AddSubgroup.unop_top, AddSubgroup.mem_unop, AddSubgroup.equivOp_apply_coe, AddSubgroup.op_eq_bot, AddSubgroup.unop_sSup, AddSubgroup.vadd_opposite_image_add_preimage, AddSubgroup.normal_unop, instProperVAddSubtypeAddOppositeMemAddSubgroupOpOfIsTopologicalAddGroupOfIsClosedCoe, AddSubgroup.unop_le_unop_iff, isAddFundamentalDomain_Ioc', AddSubgroup.op_eq_top, comap_op_rightUniformSpace, QuotientAddGroup.orbit_mk_eq_vadd, AddSubgroup.instCountableSubtypeAddOppositeMemOp, AddSubgroup.equivOp_symm_apply_coe
instAddLeftCancelMonoid 📖CompOp
instAddLeftCancelSemigroup 📖CompOp
instAddMonoid 📖CompOp
48 mathmath: AddSubgroup.vadd_opposite_image_add_preimage', rightAddCoset_zero, isAddUnit_unop, Finset.neg_op_vadd_finset_distrib, HahnSeries.addOppositeEquiv_symm_support, rightAddCoset_mem_rightAddCoset, AddUnits.coe_unop_opEquiv, Finset.op_vadd_stabilizer_of_no_doubling, Set.neg_op_vadd_set_distrib, HahnSeries.addOppositeEquiv_support, op_vadd_add, instErgodicVAddAddOppositeOfIsAddRightInvariant, op_vadd_coe_set, op_vadd_op_vadd, instAddTorsionFree, mem_rightAddCoset_iff, NormedAddGroup.to_isIsometricVAdd_right, isAddUnit_op, HahnSeries.addOppositeEquiv_orderTop, rightAddCoset_eq_iff, AddAction.op_vadd_set_stabilizer_subset, eq_addCosets_of_normal, Set.op_vadd_inter_nonempty_iff, Finset.op_vadd_addConvolution_eq_addConvolution_vadd, DomAddAct.mk_nsmul, exponent, HahnSeries.addOppositeEquiv_symm_orderTop, mem_own_rightAddCoset, HahnSeries.addOppositeEquiv_symm_leadingCoeff, HahnSeries.addOppositeEquiv_leadingCoeff, unop_nsmul, DomAddAct.symm_mk_nsmul, Finset.addETransformLeft_fst, Set.neg_vadd_set_distrib, HahnSeries.addOppositeEquiv_symm_apply_coeff, orbit_addSubgroup_eq_rightCoset, normal_iff_eq_addCosets, HahnSeries.addOppositeEquiv_apply, lipschitzAdd, AddUnits.coe_opEquiv_symm, Finset.addConvolution_op_vadd_eq_addConvolution_add_neg, op_nsmul, IsOpen.right_addCoset, IsClosed.right_addCoset, AddAction.right_quotientAction', Finset.addETransformRight_fst, Finset.neg_vadd_finset_distrib, IsAddUnit.op
instAddRightCancelMonoid 📖CompOp
instAddRightCancelSemigroup 📖CompOp
instAddSemigroup 📖CompOp
instAddZero 📖CompOp
6 mathmath: AddMonoidHom.fromOpposite_apply, isDedekindFiniteAddMonoid_iff, AddMonoidHom.op_symm_apply_apply, AddMonoidHom.op_apply_apply, instIsDedekindFiniteAddMonoid, AddMonoidHom.toOpposite_apply
instAddZeroClass 📖CompOp
47 mathmath: AddUnits.embedProduct_injective, AddSubmonoid.unop_le_unop_iff, AddSubmonoid.unop_inf, AddSubmonoid.op_sup, AddSubmonoid.unop_eq_bot, AddSubmonoid.unop_iInf, AddSubmonoid.centerToAddOpposite_symm_apply_coe, AddSubmonoid.op_eq_top, AddSubmonoid.unop_bot, AddSubmonoid.equivOp_apply_coe, AddSubmonoid.op_eq_bot, AddSubmonoid.unop_top, AddSubmonoid.unop_iSup, AddSubmonoid.mem_op, AddSubmonoid.unop_sInf, AddSubmonoid.op_top, AddSubmonoid.unop_closure, AddSubmonoid.op_le_op_iff, AddSubmonoid.unop_sSup, AddSubmonoid.op_inf, AddSubmonoid.mem_unop, AddSubmonoid.coe_unop, AddSubmonoid.op_sSup, AddUnits.embedProduct_apply, AddSubmonoid.op_iSup, AddSubmonoid.op_sInf, AddUnits.range_embedProduct, AddSubmonoid.op_le_iff, AddSubmonoid.unop_injective, AddSubmonoid.unop_eq_top, AddSubmonoid.op_closure, AddSubmonoid.unop_toSubsemigroup, AddSubmonoid.op_injective, AddSubmonoid.coe_op, AddSubmonoid.op_iInf, AddUnits.continuous_embedProduct, AddSubmonoid.opEquiv_symm_apply, AddUnits.isClosedEmbedding_embedProduct, AddSubmonoid.unop_sup, AddSubmonoid.opEquiv_apply, AddSubmonoid.op_toSubsemigroup, AddUnits.isEmbedding_embedProduct, AddUnits.isInducing_embedProduct, AddSubmonoid.equivOp_symm_apply_coe, AddSubmonoid.centerToAddOpposite_apply_coe, AddSubmonoid.le_op_iff, AddSubmonoid.op_bot
instCommGroup 📖CompOp
instCommMonoid 📖CompOp
5 mathmath: Finset.unop_prod, unop_finsuppProd, op_finsuppProd, Finset.op_prod, instIsOrderedMonoid
instCommSemigroup 📖CompOp
instDivInvMonoid 📖CompOp
instGroup 📖CompOp
instLeftCancelSemigroup 📖CompOp
instMonoid 📖CompOp
instMulOneClass 📖CompOp
instRightCancelSemigroup 📖CompOp
instSemigroup 📖CompOp
instSubNegMonoid 📖CompOp
5 mathmath: op_zsmul, DomAddAct.mk_zsmul, DomAddAct.symm_mk_zsmul, unop_zsmul, unop_sub
instSubtractionCommMonoid 📖CompOp
instSubtractionMonoid 📖CompOp
pow 📖CompOp
2 mathmath: unop_pow, op_pow

Theorems

NameKindAssumesProvesValidatesDepends On
addCommute_op 📖mathematicalAddCommute
AddOpposite
instAdd
op
addSemiconjBy_op
addCommute_unop 📖mathematicalAddCommute
unop
AddOpposite
instAdd
addSemiconjBy_unop
addSemiconjBy_op 📖mathematicalAddSemiconjBy
AddOpposite
instAdd
op
addSemiconjBy_unop 📖mathematicalAddSemiconjBy
unop
AddOpposite
instAdd
op_unop
addSemiconjBy_op
instAddTorsionFree 📖mathematicalIsAddTorsionFree
AddOpposite
instAddMonoid
op_injective
nsmul_right_injective
unop_injective
instIsCancelAdd 📖mathematicalIsCancelAdd
AddOpposite
instAdd
instIsLeftCancelAdd
IsCancelAdd.toIsRightCancelAdd
instIsRightCancelAdd
IsCancelAdd.toIsLeftCancelAdd
instIsDedekindFiniteAddMonoid 📖mathematicalIsDedekindFiniteAddMonoid
AddOpposite
instAddZero
isDedekindFiniteAddMonoid_iff
instIsLeftCancelAdd 📖mathematicalIsLeftCancelAdd
AddOpposite
instAdd
unop_injective
add_right_cancel
op_injective
instIsRightCancelAdd 📖mathematicalIsRightCancelAdd
AddOpposite
instAdd
unop_injective
add_left_cancel
op_injective
instMulTorsionFree 📖mathematicalIsMulTorsionFree
MulOpposite
MulOpposite.instMonoid
op_injective
pow_left_injective
unop_injective
isDedekindFiniteAddMonoid_iff 📖mathematicalIsDedekindFiniteAddMonoid
AddOpposite
instAddZero
isDedekindFiniteAddMonoid_iff
Equiv.forall_congr_right
forall_swap
op_nsmul 📖mathematicalop
AddMonoid.toNatSMul
AddOpposite
instAddMonoid
op_pow 📖mathematicalop
AddOpposite
pow
op_sub 📖mathematicalop
SubNegMonoid.toSub
AddOpposite
instAdd
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
instNeg
SubNegMonoid.toNeg
sub_eq_add_neg
op_zsmul 📖mathematicalop
SubNegMonoid.toZSMul
AddOpposite
instSubNegMonoid
unop_nsmul 📖mathematicalunop
AddOpposite
AddMonoid.toNatSMul
instAddMonoid
unop_pow 📖mathematicalunop
AddOpposite
pow
unop_sub 📖mathematicalunop
AddOpposite
SubNegMonoid.toSub
instSubNegMonoid
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
SubNegMonoid.toNeg
unop_zsmul 📖mathematicalunop
AddOpposite
SubNegMonoid.toZSMul
instSubNegMonoid

AddSemiconjBy

Theorems

NameKindAssumesProvesValidatesDepends On
op 📖mathematicalAddSemiconjByAddOpposite
AddOpposite.instAdd
AddOpposite.op
AddOpposite.addSemiconjBy_op
unop 📖mathematicalAddSemiconjBy
AddOpposite
AddOpposite.instAdd
AddOpposite.unopAddOpposite.addSemiconjBy_unop

Commute

Theorems

NameKindAssumesProvesValidatesDepends On
op 📖mathematicalCommuteMulOpposite
MulOpposite.instMul
MulOpposite.op
SemiconjBy.op
unop 📖mathematicalCommute
MulOpposite
MulOpposite.instMul
MulOpposite.unopSemiconjBy.unop

MulOpposite

Definitions

NameCategoryTheorems
instAddCommGroup 📖CompOp
1 mathmath: instFiniteDimensional
instAddCommMagma 📖CompOp
instAddCommMonoid 📖CompOp
85 mathmath: op_finsuppSum, coe_opLinearEquiv_symm, Submodule.comap_op_mul, Submodule.comap_op_pow, CategoryTheory.Preadditive.homSelfLinearEquivEndMulOpposite_apply, Submodule.LinearDisjoint.op, RingEquiv.mopMatrix_symm_apply, AlgHom.mulLeftRightMatrix.inv_comp, coe_opLinearEquiv_symm_addEquiv, RingEquiv.mopMatrix_apply, MonoidAlgebra.opRingEquiv_symm_apply, Subalgebra.LinearDisjoint.linearIndependent_left_op_of_flat, Module.Basis.mulOpposite_apply, AddMonoidAlgebra.opRingEquiv_symm_apply, IsSimpleRing.exists_ringEquiv_matrix_end_mulOpposite, Module.Basis.mulOpposite_is_orthonormal_iff, AlgHom.toLinearMap_toOpposite, finrank, Submodule.map_unop_mul, coe_opLinearEquiv_addEquiv, instFree, summable_unop, isometry_opLinearEquiv, Algebra.TensorProduct.opAlgEquiv_apply, IsAzumaya.mulLeftRight_comp_congr, AlgHom.mulLeftRight_apply, IsAzumaya.bij, Module.Basis.mulOpposite_repr_op, opContinuousLinearEquiv_symm_apply, Submodule.map_unop_pow, hasSum_unop, Submodule.equivOpposite_apply, counit_def, coe_opLinearEquiv_toLinearMap, Submodule.map_op_pow, instIsStablyFiniteRingMulOpposite, HasSum.op, opLinearEquiv_toAddEquiv, hasSum_op, Module.Finite.instMulOpposite, IsAzumaya.coe_tensorEquivEnd, Algebra.TensorProduct.opAlgEquiv_symm_apply, Submodule.linearDisjoint_op, CliffordAlgebra.toBaseChange_comp_reverseOp, instIsOrderedAddMonoid, Submodule.map_op_mul, tsum_unop, AlgHom.toLinearMap_fromOpposite, AddMonoidAlgebra.opRingEquiv_apply, Submodule.map_unop_one, Module.Basis.mulOpposite_repr_eq, CategoryTheory.Preadditive.homSelfLinearEquivEndMulOpposite_symm_apply, IsSemisimpleRing.exists_ringEquiv_pi_matrix_end_mulOpposite, op_geom_sum, isStablyFiniteRing_iff, Submodule.comap_unop_pow, rank, Algebra.TensorProduct.opAlgEquiv_tmul, TwoSidedIdeal.subtypeMop_apply, coe_opLinearEquiv, comul_def, summable_op, opLinearEquiv_symm_toAddEquiv, opContinuousLinearEquiv_apply, Submodule.equivOpposite_symm_apply, op_geom_sum₂, Submodule.comap_unop_one, Submodule.mulMap_op, TwoSidedIdeal.subtypeMop_injective, coe_opLinearEquiv_symm_toLinearMap, Submodule.comap_op_one, instModuleIsReflexive, Submodule.map_op_one, Finset.unop_sum, Algebra.TensorProduct.opAlgEquiv_symm_tmul, tsum_op, Subalgebra.LinearDisjoint.mulLeftMap_ker_eq_bot_iff_linearIndependent_op, MonoidAlgebra.opRingEquiv_apply, unop_finsuppSum, Finset.op_sum, Submodule.comap_unop_mul, Module.Basis.repr_unop_eq_mulOpposite_repr, AlgHom.mulLeftRightMatrix.comp_inv, IsAzumaya.AlgHom.mulLeftRight_bij, Summable.op
instAddCommSemigroup 📖CompOp
instAddGroup 📖CompOp
instAddLeftCancelSemigroup 📖CompOp
instAddMonoid 📖CompOp
instAddRightCancelSemigroup 📖CompOp
instAddSemigroup 📖CompOp
instAddZeroClass 📖CompOp
5 mathmath: AddMonoidHom.mulOp_apply_apply, NonUnitalRingHom.op_symm_apply_apply, AddMonoidHom.mul_op_ext_iff, NonUnitalRingHom.op_apply_apply, AddMonoidHom.mulOp_symm_apply_apply
instCancelCommMonoid 📖CompOp
instCancelMonoid 📖CompOp
instCommGroup 📖CompOp
instCommMonoid 📖CompOp
1 mathmath: instIsOrderedMonoid
instCommSemigroup 📖CompOp
instDivInvMonoid 📖CompOp
5 mathmath: unop_zpow, DomMulAct.mk_zpow, op_zpow, DomMulAct.symm_mk_zpow, unop_div
instDivisionCommMonoid 📖CompOp
instDivisionMonoid 📖CompOp
instGroup 📖CompOp
74 mathmath: Subgroup.unop_inf, Subgroup.instCountableSubtypeMulOppositeMemOp, Subgroup.equivOp_symm_apply_coe, Subgroup.op_le_op_iff, Subgroup.op_iSup, Function.Embedding.instIsCentralScalar, RootPairing.range_weylGroup_coweightHom, Subgroup.op_bot, DomMulAct.mem_stabilizer_iff, Subgroup.op_iInf, Subgroup.unop_closure, Subgroup.unop_eq_bot, MulAction.right_quotientAction, Subgroup.mem_op, Subgroup.unop_normalizer, Subgroup.op_top, Subgroup.smul_diff_smul', Subgroup.centerToMulOpposite_apply_coe, Subgroup.Normal.of_unop, Subgroup.op_inf, Subgroup.smul_opposite_image_mul_preimage, Subgroup.normal_op, Subgroup.op_normalizer, Subgroup.op_toSubsemigroup, instIsUniformGroup, Topology.IsQuotientMap.isQuotientCoveringMap_of_subgroupOp, instIsTopologicalGroupMulOpposite, Subgroup.properlyDiscontinuousSMul_opposite_of_tendsto_cofinite, ValuationSubring.pointwise_central_scalar, MulAction.stabilizer_subgroup_op, Subgroup.Normal.op, Subgroup.unop_sInf, MulAction.stabilizer_op_subgroup, QuotientGroup.orbit_eq_out_smul, Subgroup.op_eq_top, Subgroup.op_sInf, Subgroup.unop_toSubsemigroup, Subgroup.coe_unop, Subgroup.equivOp_apply_coe, Subgroup.smul_opposite_mul, Subgroup.opEquiv_symm_apply, Subgroup.unop_eq_top, Subgroup.unop_iSup, DomMulAct.stabilizerMulEquiv_apply, comap_op_leftUniformSpace, Subgroup.unop_injective, instProperSMulSubtypeMulOppositeMemSubgroupOpOfIsTopologicalGroupOfIsClosedCoe, Subgroup.op_closure, Subgroup.op_injective, Subgroup.op_sup, instProperSMulMulOppositeOfIsTopologicalGroup, Subgroup.unop_sSup, Subgroup.unop_bot, QuotientGroup.orbit_mk_eq_smul, Subgroup.op_toSubmonoid, Subgroup.coe_op, comap_op_rightUniformSpace, Subgroup.op_eq_bot, Subgroup.le_op_iff, Subgroup.unop_le_unop_iff, Subgroup.op.instNormal, Subgroup.op_sSup, Subgroup.mem_unop, Subgroup.centerToMulOpposite_symm_apply_coe, Subgroup.unop_sup, Subgroup.op_le_iff, Subgroup.normal_unop, Subgroup.smul_diff', QuotientGroup.sound, Subgroup.unop_toSubmonoid, Subgroup.unop_iInf, Subgroup.isQuotientCoveringMap, Subgroup.opEquiv_apply, Subgroup.unop_top
instLeftCancelMonoid 📖CompOp
instLeftCancelSemigroup 📖CompOp
instMonoid 📖CompOp
186 mathmath: DoubleCoset.doubleCoset_union_rightCoset, AddOpposite.instMulTorsionFree, HasStrictFDerivAt.fun_mul', NormedGroup.to_isIsometricSMul_right, hasStrictFDerivAt_list_prod_finRange', TrivSqZeroExt.lift_inlAlgHom_inrHom, mem_own_rightCoset, TrivSqZeroExt.snd_pow_eq_sum, MeasureTheory.integral_domSMul, MeasureTheory.addHaarScalarFactor_smul_inv_eq_distribHaarChar, fderivWithin_fun_pow', MulAction.right_quotientAction', DomMulAct.symm_mk_pow, Finset.inv_smul_finset_distrib, Finset.doubling_lt_three_halves, TrivSqZeroExt.inv_inl, MeasureTheory.Measure.domSMul_apply, TrivSqZeroExt.inv_one, CategoryTheory.Mat.instFaithfulMat_SingleObjMulOppositeEquivalenceSingleObjInverse, HasStrictFDerivAt.mul', MeasureTheory.addHaarScalarFactor_smul_eq_distribHaarChar_inv, TrivSqZeroExt.snd_list_prod, TrivSqZeroExt.fstHom_comp_map, fderiv_fun_pow', MeasureTheory.Measure.IsAddHaarMeasure.domSMul, Finset.inv_op_smul_finset_distrib, rightCoset_eq_iff, Monoid.CoprodI.inv_def, DualNumber.inv_eps, HasStrictFDerivAt.list_prod', MulAction.op_smul_set_stabilizer_subset, CategoryTheory.End.smul_left, NormMulClass.toNormSMulClass_op, TrivSqZeroExt.fst_invOf, Finset.smul_inv_mul_eq_inv_mul_opSMul, exponent, MeasureTheory.addHaarScalarFactor_smul_eq_distribHaarChar, TrivSqZeroExt.map_comp_inlAlgHom, MeasureTheory.Measure.addHaarScalarFactor_smul_congr, fderiv_mul_const', TrivSqZeroExt.invertibleEquivInvertibleFst_symm_apply_invOf, TrivSqZeroExt.inl_mul_inr, Subgroup.smul_diff_smul', TrivSqZeroExt.fst_list_prod, isUnit_op, MeasureTheory.distribHaarChar_apply, TrivSqZeroExt.mul_left_eq_one, CategoryTheory.Mat.instEssSurjMat_SingleObjMulOppositeEquivalenceSingleObjInverse, fderiv_pow', CategoryTheory.Mat.equivalenceSingleObjInverse_obj_carrier, AddSubmonoid.pointwise_isCentralScalar, op_smul_coe_set, op_pow, TrivSqZeroExt.inr_mul_inr, Subgroup.pointwise_isCentralScalar, fderivWithin_fun_mul', Subsemiring.pointwise_central_scalar, hasFDerivAt_list_prod', fderiv_mul', DualNumber.exists_mul_left_or_mul_right, unop_pow, TrivSqZeroExt.invertibleEquivInvertibleFst_apply_invOf, hasFDerivWithinAt_pow', TrivSqZeroExt.mul_inv_cancel, AffineMap.isCentralScalar, RingEquiv.moduleEndSelf_apply, TrivSqZeroExt.inv_mul_cancel, fderivWithin_mul_const', hasFDerivAt_list_prod_finRange', TrivSqZeroExt.invOf_eq_inv, normal_iff_eq_cosets, MeasureTheory.Measure.Regular.domSMul, TrivSqZeroExt.inl_mul, op_smul_op_smul, MeasureTheory.Measure.instSMulCommClassDomMulActNNReal, Set.op_smul_inter_nonempty_iff, HasFDerivWithinAt.mul_const', AddMonoid.End.isCentralScalar, Units.coe_unop_opEquiv, IsTopologicalSemiring.toOppositeIsModuleTopology, fderiv_pow_ring', lipschitzMul, DoubleCentralizer.instIsCentralScalar, Matrix.instSMulCommClassMulOppositeForallOfIsScalarTower_1, TrivSqZeroExt.inv_inv, HasFDerivAt.fun_pow', HasMFDerivWithinAt.mul', isUnit_unop, TrivSqZeroExt.inr_mul_inl, TrivSqZeroExt.snd_invOf, TrivSqZeroExt.inl_mul_inl, MeasureTheory.Measure.instSMulCommClassNNRealDomMulAct, hasStrictFDerivAt_pow', LieSubmodule.Quotient.isCentralScalar, fderiv_fun_mul', Derivation.instIsCentralScalar, Set.inv_smul_set_distrib, IsOpen.rightCoset, TrivSqZeroExt.isUnit_inv_iff, Subring.pointwise_central_scalar, rightCoset_mem_rightCoset, TrivSqZeroExt.inv_neg, Submonoid.pointwise_isCentralScalar, hasStrictFDerivAt_list_prod_attach', MeasureTheory.Lp.instIsCentralScalar, Matrix.scalar_comm_iff, Finset.op_smul_convolution_eq_convolution_smul, HasFDerivWithinAt.pow', HasFDerivAt.list_prod', hasFDerivAt_pow', DualNumber.commute_eps_left, Subgroup.smul_opposite_image_mul_preimage', CategoryTheory.Mat.instIsEquivalenceMat_SingleObjMulOppositeEquivalenceSingleObjInverse, StarSemigroup.toOpposite_starModule, Finset.mulETransformRight_fst, orbit_subgroup_eq_rightCoset, op_geom_sum, DualNumber.commute_eps_right, Set.inv_op_smul_set_distrib, HasFDerivAt.pow', HasStrictFDerivAt.pow', CentroidHom.instIsCentralScalar, HasStrictFDerivAt.mul_const', Units.coe_opEquiv_symm, DomMulAct.mk_pow, Ideal.pointwise_central_scalar, fderivWithin_list_prod', FreeLieAlgebra.instIsCentralScalar, HasFDerivWithinAt.fun_mul', ContinuousAlternatingMap.instIsCentralScalar, Matrix.instSMulCommClassMulOppositeForallOfIsScalarTower, instErgodicSMulMulOppositeOfIsMulRightInvariant, fderivWithin_pow_ring', eq_cosets_of_normal, HasFDerivWithinAt.fun_pow', CategoryTheory.Mat.instFullMat_SingleObjMulOppositeEquivalenceSingleObjInverse, MeasureTheory.Measure.addHaarScalarFactor_domSMul, Matrix.instIsScalarTowerMulOppositeForallOfSMulCommClass, HasFDerivAt.mul_const', HasStrictFDerivAt.fun_pow', Finset.op_smul_stabilizer_of_no_doubling, DualNumber.lift_op_smul, TrivSqZeroExt.mul_right_eq_one, CategoryTheory.Mat.equivalenceSingleObjInverse_map, op_geom_sum₂, MeasureTheory.Measure.addHaarScalarFactor_smul_congr', TrivSqZeroExt.inl_mul_eq_smul, hasStrictFDerivAt_list_prod', TrivSqZeroExt.inv_zero, Ring.uniformContinuousConstSMul_op, DualNumber.eps_mul_eps, IsUnit.op, ContinuousLinearMap.isCentralScalar, rightCoset_one, CategoryTheory.Mat.equivalenceSingleObjInverse_obj_str, HasFDerivWithinAt.mul', DualNumber.snd_mul, fderivWithin_mul', RestrictScalars.isCentralScalar, mem_rightCoset_iff, TrivSqZeroExt.mul_inv_rev, Matrix.scalar_commute_iff, fderiv_list_prod', HasMFDerivAt.mul', RootPairing.Equiv.instSMulCommClassMulOppositeAut, AddSubgroup.pointwise_isCentralScalar, HasFDerivAt.fun_mul', LieSubalgebra.instIsCentralScalarSubtypeMem, HasFDerivAt.mul', hasFDerivAt_list_prod_attach', Subgroup.smul_diff', Matrix.op_smul_eq_vecMul, Finset.mulETransformLeft_fst, TrivSqZeroExt.mul_inl_eq_op_smul, op_smul_mul, ContinuousAffineMap.instIsCentralScalar, TrivSqZeroExt.inv_inr, Finset.smul_inv_mul_opSMul_eq_mul_of_doubling_lt_three_halves, TensorProduct.instIsCentralScalar, AlternatingMap.instIsCentralScalar, Submodule.pointwiseCentralScalar, IsClosed.rightCoset, HasFDerivWithinAt.list_prod', TwoSidedIdeal.coe_mop_smul, Finset.convolution_op_smul_eq_convolution_mul_inv, fderivWithin_pow'
instMulOne 📖CompOp
19 mathmath: Monoid.Foldl.ofFreeMonoid_apply, Traversable.foldlm.ofFreeMonoid_comp_of, Monoid.CoprodI.inv_def, RootPairing.Equiv.coweightHom_injective, RootPairing.Hom.coweightHom_injective, instIsStablyFiniteRingMulOpposite, MonoidHom.op_symm_apply_apply, Traversable.foldl.unop_ofFreeMonoid, Monoid.foldlM.ofFreeMonoid_apply, RootPairing.Equiv.coweightHom_op, isStablyFiniteRing_iff, Traversable.foldl.ofFreeMonoid_comp_of, isDedekindFiniteMonoid_iff, MonoidHom.op_apply_apply, instIsDedekindFiniteMonoid, RootPairing.Equiv.coweightHom_toLinearMap, MonoidHom.toOpposite_apply, RootPairing.Equiv.coweightHom_apply, MonoidHom.fromOpposite_apply
instMulOneClass 📖CompOp
58 mathmath: Submonoid.le_op_iff, Submonoid.equivOp_symm_apply_coe, Submonoid.op_iInf, Submonoid.unop_eq_top, Subring.addEquivOp_apply_coe, Submonoid.unop_iInf, Submonoid.unop_iSup, Submonoid.unop_inf, RootPairing.range_weylGroup_coweightHom, Monoid.CoprodI.inv_def, Subsemiring.addEquivOp_apply_coe, Submonoid.op_eq_bot, Submonoid.centerToMulOpposite_apply_coe, Submonoid.op_sSup, Submonoid.op_sInf, Submonoid.unop_top, Subsemiring.addEquivOp_symm_apply_coe, Submonoid.op_iSup, Submonoid.coe_op, Submonoid.centerToMulOpposite_symm_apply_coe, Submonoid.op_le_iff, Units.isEmbedding_embedProduct, Submonoid.op_sup, Submonoid.unop_eq_bot, Submonoid.unop_le_unop_iff, Submonoid.op_le_op_iff, Submonoid.unop_sup, Submonoid.equivOp_apply_coe, Submonoid.op_inf, Submonoid.unop_sInf, Subring.ringEquivOpMop_symm_apply_coe, Submonoid.opEquiv_apply, Subsemiring.ringEquivOpMop_symm_apply_coe, Submonoid.unop_sSup, Submonoid.op_eq_top, Submonoid.unop_bot, Units.isClosedEmbedding_embedProduct, Submonoid.op_closure, Units.continuous_embedProduct, Submonoid.op_injective, Submonoid.unop_injective, Submonoid.mem_op, Subalgebra.linearEquivOp_symm_apply_coe, Submonoid.mem_unop, Units.embed_product_star, Submonoid.unop_closure, Submonoid.coe_unop, Subalgebra.algEquivOpMop_symm_apply_coe, Submonoid.op_top, Units.range_embedProduct, Units.embedProduct_injective, Submonoid.op_bot, Submonoid.opEquiv_symm_apply, Units.isInducing_embedProduct, Subring.addEquivOp_symm_apply_coe, Submonoid.unop_toSubsemigroup, Units.embedProduct_apply, Submonoid.op_toSubsemigroup
instRightCancelMonoid 📖CompOp
instRightCancelSemigroup 📖CompOp
instSemigroup 📖CompOp
instSubNegMonoid 📖CompOp

Theorems

NameKindAssumesProvesValidatesDepends On
commute_op 📖mathematicalCommute
MulOpposite
instMul
op
semiconjBy_op
commute_unop 📖mathematicalCommute
unop
MulOpposite
instMul
semiconjBy_unop
instIsCancelMul 📖mathematicalIsCancelMul
MulOpposite
instMul
instIsLeftCancelMul
IsCancelMul.toIsRightCancelMul
instIsRightCancelMul
IsCancelMul.toIsLeftCancelMul
instIsDedekindFiniteMonoid 📖mathematicalIsDedekindFiniteMonoid
MulOpposite
instMulOne
isDedekindFiniteMonoid_iff
instIsLeftCancelMul 📖mathematicalIsLeftCancelMul
MulOpposite
instMul
unop_injective
mul_right_cancel
op_injective
instIsRightCancelMul 📖mathematicalIsRightCancelMul
MulOpposite
instMul
unop_injective
mul_left_cancel
op_injective
isDedekindFiniteMonoid_iff 📖mathematicalIsDedekindFiniteMonoid
MulOpposite
instMulOne
Equiv.forall_congr_right
forall_swap
op_div 📖mathematicalop
DivInvMonoid.toDiv
MulOpposite
instMul
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
instInv
DivInvMonoid.toInv
div_eq_mul_inv
op_pow 📖mathematicalop
Monoid.toNatPow
MulOpposite
instMonoid
op_zpow 📖mathematicalop
DivInvMonoid.toZPow
MulOpposite
instDivInvMonoid
semiconjBy_op 📖mathematicalSemiconjBy
MulOpposite
instMul
op
semiconjBy_unop 📖mathematicalSemiconjBy
unop
MulOpposite
instMul
op_unop
semiconjBy_op
unop_div 📖mathematicalunop
MulOpposite
DivInvMonoid.toDiv
instDivInvMonoid
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
DivInvMonoid.toMonoid
DivInvMonoid.toInv
unop_pow 📖mathematicalunop
MulOpposite
Monoid.toNatPow
instMonoid
unop_zpow 📖mathematicalunop
MulOpposite
DivInvMonoid.toZPow
instDivInvMonoid

SemiconjBy

Theorems

NameKindAssumesProvesValidatesDepends On
op 📖mathematicalSemiconjByMulOpposite
MulOpposite.instMul
MulOpposite.op
MulOpposite.semiconjBy_op
unop 📖mathematicalSemiconjBy
MulOpposite
MulOpposite.instMul
MulOpposite.unopMulOpposite.semiconjBy_unop

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