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Theorem List for Metamath Proof Explorer - 17001-17100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremisabl 17001 The predicate "is an Abelian (commutative) group." (Contributed by NM, 17-Oct-2011.)
 |-  ( G  e.  Abel  <->  ( G  e.  Grp  /\  G  e. CMnd ) )
 
Theoremablgrp 17002 An Abelian group is a group. (Contributed by NM, 26-Aug-2011.)
 |-  ( G  e.  Abel  ->  G  e.  Grp )
 
Theoremablcmn 17003 An Abelian group is a commutative monoid. (Contributed by Mario Carneiro, 6-Jan-2015.)
 |-  ( G  e.  Abel  ->  G  e. CMnd )
 
Theoremiscmn 17004* The predicate "is a commutative monoid." (Contributed by Mario Carneiro, 6-Jan-2015.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( G  e. CMnd  <->  ( G  e.  Mnd  /\  A. x  e.  B  A. y  e.  B  ( x  .+  y )  =  (
 y  .+  x )
 ) )
 
Theoremisabl2 17005* The predicate "is an Abelian (commutative) group." (Contributed by NM, 17-Oct-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( G  e.  Abel  <->  ( G  e.  Grp  /\  A. x  e.  B  A. y  e.  B  ( x  .+  y )  =  ( y  .+  x ) ) )
 
Theoremcmnpropd 17006* If two structures have the same group components (properties), one is a commutative monoid iff the other one is. (Contributed by Mario Carneiro, 6-Jan-2015.)
 |-  ( ph  ->  B  =  ( Base `  K )
 )   &    |-  ( ph  ->  B  =  ( Base `  L )
 )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B )
 )  ->  ( x ( +g  `  K )
 y )  =  ( x ( +g  `  L ) y ) )   =>    |-  ( ph  ->  ( K  e. CMnd  <->  L  e. CMnd ) )
 
Theoremablpropd 17007* If two structures have the same group components (properties), one is an Abelian group iff the other one is. (Contributed by NM, 6-Dec-2014.)
 |-  ( ph  ->  B  =  ( Base `  K )
 )   &    |-  ( ph  ->  B  =  ( Base `  L )
 )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B )
 )  ->  ( x ( +g  `  K )
 y )  =  ( x ( +g  `  L ) y ) )   =>    |-  ( ph  ->  ( K  e.  Abel 
 <->  L  e.  Abel )
 )
 
Theoremablprop 17008 If two structures have the same group components (properties), one is an Abelian group iff the other one is. (Contributed by NM, 11-Oct-2013.)
 |-  ( Base `  K )  =  ( Base `  L )   &    |-  ( +g  `  K )  =  ( +g  `  L )   =>    |-  ( K  e.  Abel  <->  L  e.  Abel )
 
Theoremiscmnd 17009* Properties that determine a commutative monoid. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  ( ph  ->  B  =  ( Base `  G )
 )   &    |-  ( ph  ->  .+  =  ( +g  `  G )
 )   &    |-  ( ph  ->  G  e.  Mnd )   &    |-  ( ( ph  /\  x  e.  B  /\  y  e.  B )  ->  ( x  .+  y
 )  =  ( y 
 .+  x ) )   =>    |-  ( ph  ->  G  e. CMnd )
 
Theoremisabld 17010* Properties that determine an Abelian group. (Contributed by NM, 6-Aug-2013.)
 |-  ( ph  ->  B  =  ( Base `  G )
 )   &    |-  ( ph  ->  .+  =  ( +g  `  G )
 )   &    |-  ( ph  ->  G  e.  Grp )   &    |-  ( ( ph  /\  x  e.  B  /\  y  e.  B )  ->  ( x  .+  y
 )  =  ( y 
 .+  x ) )   =>    |-  ( ph  ->  G  e.  Abel
 )
 
Theoremisabli 17011* Properties that determine an Abelian group. (Contributed by NM, 4-Sep-2011.)
 |-  G  e.  Grp   &    |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  (
 ( x  e.  B  /\  y  e.  B )  ->  ( x  .+  y )  =  (
 y  .+  x )
 )   =>    |-  G  e.  Abel
 
Theoremcmnmnd 17012 A commutative monoid is a monoid. (Contributed by Mario Carneiro, 6-Jan-2015.)
 |-  ( G  e. CMnd  ->  G  e.  Mnd )
 
Theoremcmncom 17013 A commutative monoid is commutative. (Contributed by Mario Carneiro, 6-Jan-2015.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( ( G  e. CMnd  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .+  Y )  =  ( Y  .+  X ) )
 
Theoremablcom 17014 An Abelian group operation is commutative. (Contributed by NM, 26-Aug-2011.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( ( G  e.  Abel  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .+  Y )  =  ( Y  .+  X ) )
 
Theoremcmn32 17015 Commutative/associative law for Abelian groups. (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 21-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( ( G  e. CMnd  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B ) )  ->  ( ( X  .+  Y ) 
 .+  Z )  =  ( ( X  .+  Z )  .+  Y ) )
 
Theoremcmn4 17016 Commutative/associative law for Abelian groups. (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 21-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( ( G  e. CMnd  /\  ( X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B ) )  ->  ( ( X  .+  Y )  .+  ( Z 
 .+  W ) )  =  ( ( X 
 .+  Z )  .+  ( Y  .+  W ) ) )
 
Theoremcmn12 17017 Commutative/associative law for Abelian monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( ( G  e. CMnd  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B ) )  ->  ( X 
 .+  ( Y  .+  Z ) )  =  ( Y  .+  ( X  .+  Z ) ) )
 
Theoremabl32 17018 Commutative/associative law for Abelian groups. (Contributed by Stefan O'Rear, 10-Apr-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   =>    |-  ( ph  ->  (
 ( X  .+  Y )  .+  Z )  =  ( ( X  .+  Z )  .+  Y ) )
 
Theoremablinvadd 17019 The inverse of an Abelian group operation. (Contributed by NM, 31-Mar-2014.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  N  =  ( invg `  G )   =>    |-  ( ( G  e.  Abel  /\  X  e.  B  /\  Y  e.  B )  ->  ( N `  ( X  .+  Y ) )  =  ( ( N `
  X )  .+  ( N `  Y ) ) )
 
Theoremablsub2inv 17020 Abelian group subtraction of two inverses. (Contributed by Stefan O'Rear, 24-May-2015.)
 |-  B  =  ( Base `  G )   &    |-  .-  =  ( -g `  G )   &    |-  N  =  ( invg `  G )   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  (
 ( N `  X )  .-  ( N `  Y ) )  =  ( Y  .-  X ) )
 
Theoremablsubadd 17021 Relationship between Abelian group subtraction and addition. (Contributed by NM, 31-Mar-2014.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  ( ( G  e.  Abel  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
 )  ->  ( ( X  .-  Y )  =  Z  <->  ( Y  .+  Z )  =  X ) )
 
Theoremablsub4 17022 Commutative/associative subtraction law for Abelian groups. (Contributed by NM, 31-Mar-2014.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  ( ( G  e.  Abel  /\  ( X  e.  B  /\  Y  e.  B ) 
 /\  ( Z  e.  B  /\  W  e.  B ) )  ->  ( ( X  .+  Y ) 
 .-  ( Z  .+  W ) )  =  ( ( X  .-  Z )  .+  ( Y 
 .-  W ) ) )
 
Theoremabladdsub4 17023 Abelian group addition/subtraction law. (Contributed by NM, 31-Mar-2014.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  ( ( G  e.  Abel  /\  ( X  e.  B  /\  Y  e.  B ) 
 /\  ( Z  e.  B  /\  W  e.  B ) )  ->  ( ( X  .+  Y )  =  ( Z  .+  W )  <->  ( X  .-  Z )  =  ( W  .-  Y ) ) )
 
Theoremabladdsub 17024 Associative-type law for group subtraction and addition. (Contributed by NM, 19-Apr-2014.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  ( ( G  e.  Abel  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
 )  ->  ( ( X  .+  Y )  .-  Z )  =  (
 ( X  .-  Z )  .+  Y ) )
 
Theoremablpncan2 17025 Cancellation law for subtraction. (Contributed by NM, 2-Oct-2014.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  ( ( G  e.  Abel  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .+  Y )  .-  X )  =  Y )
 
Theoremablpncan3 17026 A cancellation law for commutative groups. (Contributed by NM, 23-Mar-2015.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  ( ( G  e.  Abel  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  ( X  .+  ( Y  .-  X ) )  =  Y )
 
Theoremablsubsub 17027 Law for double subtraction. (Contributed by NM, 7-Apr-2015.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .-  =  ( -g `  G )   &    |-  ( ph  ->  G  e.  Abel
 )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   =>    |-  ( ph  ->  ( X  .-  ( Y  .-  Z ) )  =  ( ( X  .-  Y )  .+  Z ) )
 
Theoremablsubsub4 17028 Law for double subtraction. (Contributed by NM, 7-Apr-2015.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .-  =  ( -g `  G )   &    |-  ( ph  ->  G  e.  Abel
 )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   =>    |-  ( ph  ->  (
 ( X  .-  Y )  .-  Z )  =  ( X  .-  ( Y  .+  Z ) ) )
 
Theoremablpnpcan 17029 Cancellation law for mixed addition and subtraction. (pnpcan 9849 analog.) (Contributed by NM, 29-May-2015.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .-  =  ( -g `  G )   &    |-  ( ph  ->  G  e.  Abel
 )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   =>    |-  ( ph  ->  (
 ( X  .+  Y )  .-  ( X  .+  Z ) )  =  ( Y  .-  Z ) )
 
Theoremablnncan 17030 Cancellation law for group division. (nncan 9839 analog.) (Contributed by NM, 7-Apr-2015.)
 |-  B  =  ( Base `  G )   &    |-  .-  =  ( -g `  G )   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( X  .-  ( X  .-  Y ) )  =  Y )
 
Theoremablsub32 17031 Swap the second and third terms in a double subtraction. (Contributed by NM, 7-Apr-2015.)
 |-  B  =  ( Base `  G )   &    |-  .-  =  ( -g `  G )   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   =>    |-  ( ph  ->  (
 ( X  .-  Y )  .-  Z )  =  ( ( X  .-  Z )  .-  Y ) )
 
Theoremablnnncan1 17032 Cancellation law for subtraction. (nnncan1 9846 analog.) (Contributed by NM, 7-Apr-2015.)
 |-  B  =  ( Base `  G )   &    |-  .-  =  ( -g `  G )   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   =>    |-  ( ph  ->  (
 ( X  .-  Y )  .-  ( X  .-  Z ) )  =  ( Z  .-  Y ) )
 
Theoremmulgnn0di 17033 Group multiple of a sum, for nonnegative multiples. (Contributed by Mario Carneiro, 13-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( ( G  e. CMnd  /\  ( M  e.  NN0  /\  X  e.  B  /\  Y  e.  B ) )  ->  ( M 
 .x.  ( X  .+  Y ) )  =  ( ( M  .x.  X )  .+  ( M 
 .x.  Y ) ) )
 
Theoremmulgdi 17034 Group multiple of a sum. (Contributed by Mario Carneiro, 13-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( ( G  e.  Abel  /\  ( M  e.  ZZ  /\  X  e.  B  /\  Y  e.  B ) )  ->  ( M  .x.  ( X 
 .+  Y ) )  =  ( ( M 
 .x.  X )  .+  ( M  .x.  Y ) ) )
 
Theoremmulgmhm 17035* The map from  x to  n x for a fixed positive integer  n is a monoid homomorphism if the monoid is commutative. (Contributed by Mario Carneiro, 4-May-2015.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   =>    |-  ( ( G  e. CMnd  /\  M  e.  NN0 )  ->  ( x  e.  B  |->  ( M  .x.  x ) )  e.  ( G MndHom  G ) )
 
Theoremmulgghm 17036* The map from  x to  n x for a fixed integer  n is a group homomorphism if the group is commutative. (Contributed by Mario Carneiro, 4-May-2015.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   =>    |-  ( ( G  e.  Abel  /\  M  e.  ZZ )  ->  ( x  e.  B  |->  ( M  .x.  x ) )  e.  ( G 
 GrpHom  G ) )
 
Theoremmulgsubdi 17037 Group multiple of a difference. (Contributed by Mario Carneiro, 13-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  ( ( G  e.  Abel  /\  ( M  e.  ZZ  /\  X  e.  B  /\  Y  e.  B ) )  ->  ( M  .x.  ( X 
 .-  Y ) )  =  ( ( M 
 .x.  X )  .-  ( M  .x.  Y ) ) )
 
Theoremghmfghm 17038* The function fulfilling the conditions of ghmgrp 16393 is a group homomorphism. (Contributed by Thierry Arnoux, 26-Jan-2020.)
 |-  X  =  ( Base `  G )   &    |-  Y  =  (
 Base `  H )   &    |-  .+  =  ( +g  `  G )   &    |-  .+^  =  (
 +g  `  H )   &    |-  (
 ( ph  /\  x  e.  X  /\  y  e.  X )  ->  ( F `  ( x  .+  y ) )  =  ( ( F `  x )  .+^  ( F `
  y ) ) )   &    |-  ( ph  ->  F : X -onto-> Y )   &    |-  ( ph  ->  G  e.  Grp )   =>    |-  ( ph  ->  F  e.  ( G  GrpHom  H ) )
 
Theoremghmcmn 17039* The image of a commutative monoid 
G under a group homomorphism  F is a commutative monoid. (Contributed by Thierry Arnoux, 26-Jan-2020.)
 |-  X  =  ( Base `  G )   &    |-  Y  =  (
 Base `  H )   &    |-  .+  =  ( +g  `  G )   &    |-  .+^  =  (
 +g  `  H )   &    |-  (
 ( ph  /\  x  e.  X  /\  y  e.  X )  ->  ( F `  ( x  .+  y ) )  =  ( ( F `  x )  .+^  ( F `
  y ) ) )   &    |-  ( ph  ->  F : X -onto-> Y )   &    |-  ( ph  ->  G  e. CMnd )   =>    |-  ( ph  ->  H  e. CMnd )
 
Theoremghmabl 17040* The image of an abelian group  G under a group homomorphism  F is an abelian group. (Contributed by Mario Carneiro, 12-May-2014.) (Revised by Thierry Arnoux, 26-Jan-2020.)
 |-  X  =  ( Base `  G )   &    |-  Y  =  (
 Base `  H )   &    |-  .+  =  ( +g  `  G )   &    |-  .+^  =  (
 +g  `  H )   &    |-  (
 ( ph  /\  x  e.  X  /\  y  e.  X )  ->  ( F `  ( x  .+  y ) )  =  ( ( F `  x )  .+^  ( F `
  y ) ) )   &    |-  ( ph  ->  F : X -onto-> Y )   &    |-  ( ph  ->  G  e.  Abel
 )   =>    |-  ( ph  ->  H  e.  Abel )
 
Theoreminvghm 17041 The inversion map is a group automorphism if and only if the group is abelian. (In general it is only a group homomorphism into the opposite group, but in an abelian group the opposite group coincides with the group itself.) (Contributed by Mario Carneiro, 4-May-2015.)
 |-  B  =  ( Base `  G )   &    |-  I  =  ( invg `  G )   =>    |-  ( G  e.  Abel  <->  I  e.  ( G  GrpHom  G ) )
 
Theoremeqgabl 17042 Value of the subgroup coset equivalence relation on an abelian group. (Contributed by Mario Carneiro, 14-Jun-2015.)
 |-  X  =  ( Base `  G )   &    |-  .-  =  ( -g `  G )   &    |-  .~  =  ( G ~QG  S )   =>    |-  ( ( G  e.  Abel  /\  S  C_  X )  ->  ( A  .~  B  <->  ( A  e.  X  /\  B  e.  X  /\  ( B  .-  A )  e.  S ) ) )
 
Theoremsubgabl 17043 A subgroup of an abelian group is also abelian. (Contributed by Mario Carneiro, 3-Dec-2014.)
 |-  H  =  ( Gs  S )   =>    |-  ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G ) )  ->  H  e.  Abel
 )
 
Theoremsubcmn 17044 A submonoid of a commutative monoid is also commutative. (Contributed by Mario Carneiro, 10-Jan-2015.)
 |-  H  =  ( Gs  S )   =>    |-  ( ( G  e. CMnd  /\  H  e.  Mnd )  ->  H  e. CMnd )
 
Theoremsubmcmn 17045 A submonoid of a commutative monoid is also commutative. (Contributed by Mario Carneiro, 24-Apr-2016.)
 |-  H  =  ( Gs  S )   =>    |-  ( ( G  e. CMnd  /\  S  e.  (SubMnd `  G ) )  ->  H  e. CMnd )
 
Theoremsubmcmn2 17046 A submonoid is commutative iff it is a subset of its own centralizer. (Contributed by Mario Carneiro, 24-Apr-2016.)
 |-  H  =  ( Gs  S )   &    |-  Z  =  (Cntz `  G )   =>    |-  ( S  e.  (SubMnd `  G )  ->  ( H  e. CMnd  <->  S  C_  ( Z `
  S ) ) )
 
Theoremcntzcmn 17047 The centralizer of any subset in a commutative monoid is the whole monoid. (Contributed by Mario Carneiro, 3-Oct-2015.)
 |-  B  =  ( Base `  G )   &    |-  Z  =  (Cntz `  G )   =>    |-  ( ( G  e. CMnd  /\  S  C_  B )  ->  ( Z `  S )  =  B )
 
Theoremcntzcmnss 17048 Any subset in a commutative monoid is a subset of its centralizer. (Contributed by AV, 12-Jan-2019.)
 |-  B  =  ( Base `  G )   &    |-  Z  =  (Cntz `  G )   =>    |-  ( ( G  e. CMnd  /\  S  C_  B )  ->  S  C_  ( Z `  S ) )
 
Theoremcntzspan 17049 If the generators commute, the generated monoid is commutative. (Contributed by Mario Carneiro, 25-Apr-2016.)
 |-  Z  =  (Cntz `  G )   &    |-  K  =  (mrCls `  (SubMnd `  G )
 )   &    |-  H  =  ( Gs  ( K `  S ) )   =>    |-  ( ( G  e.  Mnd  /\  S  C_  ( Z `  S ) )  ->  H  e. CMnd )
 
Theoremcntzcmnf 17050 Discharge the centralizer assumption in a commutative monoid. (Contributed by Mario Carneiro, 24-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  Z  =  (Cntz `  G )   &    |-  ( ph  ->  G  e. CMnd )   &    |-  ( ph  ->  F : A --> B )   =>    |-  ( ph  ->  ran  F  C_  ( Z `  ran  F ) )
 
Theoremghmplusg 17051 The pointwise sum of two linear functions is linear. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |- 
 .+  =  ( +g  `  N )   =>    |-  ( ( N  e.  Abel  /\  F  e.  ( M 
 GrpHom  N )  /\  G  e.  ( M  GrpHom  N ) )  ->  ( F  oF  .+  G )  e.  ( M  GrpHom  N ) )
 
Theoremablnsg 17052 Every subgroup of an abelian group is normal. (Contributed by Mario Carneiro, 14-Jun-2015.)
 |-  ( G  e.  Abel  ->  (NrmSGrp `  G )  =  (SubGrp `  G )
 )
 
Theoremodadd1 17053 The order of a product in an abelian group divides the LCM of the orders of the factors. (Contributed by Mario Carneiro, 20-Oct-2015.)
 |-  O  =  ( od
 `  G )   &    |-  X  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  (
 ( G  e.  Abel  /\  A  e.  X  /\  B  e.  X )  ->  ( ( O `  ( A  .+  B ) )  x.  ( ( O `  A ) 
 gcd  ( O `  B ) ) ) 
 ||  ( ( O `
  A )  x.  ( O `  B ) ) )
 
Theoremodadd2 17054 The order of a product in an abelian group is divisible by the LCM of the orders of the factors divided by the GCD. (Contributed by Mario Carneiro, 20-Oct-2015.)
 |-  O  =  ( od
 `  G )   &    |-  X  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  (
 ( G  e.  Abel  /\  A  e.  X  /\  B  e.  X )  ->  ( ( O `  A )  x.  ( O `  B ) ) 
 ||  ( ( O `
  ( A  .+  B ) )  x.  ( ( ( O `
  A )  gcd  ( O `  B ) ) ^ 2 ) ) )
 
Theoremodadd 17055 The order of a product is the product of the orders, if the factors have coprime order. (Contributed by Mario Carneiro, 20-Oct-2015.)
 |-  O  =  ( od
 `  G )   &    |-  X  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  (
 ( ( G  e.  Abel  /\  A  e.  X  /\  B  e.  X )  /\  ( ( O `  A )  gcd  ( O `
  B ) )  =  1 )  ->  ( O `  ( A 
 .+  B ) )  =  ( ( O `
  A )  x.  ( O `  B ) ) )
 
Theoremgex2abl 17056 A group with exponent 2 (or 1) is abelian. (Contributed by Mario Carneiro, 24-Apr-2016.)
 |-  X  =  ( Base `  G )   &    |-  E  =  (gEx `  G )   =>    |-  ( ( G  e.  Grp  /\  E  ||  2 )  ->  G  e.  Abel )
 
Theoremgexexlem 17057* Lemma for gexex 17058. (Contributed by Mario Carneiro, 24-Apr-2016.)
 |-  X  =  ( Base `  G )   &    |-  E  =  (gEx `  G )   &    |-  O  =  ( od `  G )   &    |-  ( ph  ->  G  e.  Abel
 )   &    |-  ( ph  ->  E  e.  NN )   &    |-  ( ph  ->  A  e.  X )   &    |-  (
 ( ph  /\  y  e.  X )  ->  ( O `  y )  <_  ( O `  A ) )   =>    |-  ( ph  ->  ( O `  A )  =  E )
 
Theoremgexex 17058* In an abelian group with finite exponent, there is an element in the group with order equal to the exponent. In other words, all orders of elements divide the largest order of an element of the group. This fails if  E  =  0, for example in an infinite p-group, where there are elements of arbitrarily large orders (so  E is zero) but no elements of infinite order. (Contributed by Mario Carneiro, 24-Apr-2016.)
 |-  X  =  ( Base `  G )   &    |-  E  =  (gEx `  G )   &    |-  O  =  ( od `  G )   =>    |-  ( ( G  e.  Abel  /\  E  e.  NN )  ->  E. x  e.  X  ( O `  x )  =  E )
 
Theoremtorsubg 17059 The set of all elements of finite order forms a subgroup of any abelian group, called the torsion subgroup. (Contributed by Mario Carneiro, 20-Oct-2015.)
 |-  O  =  ( od
 `  G )   =>    |-  ( G  e.  Abel 
 ->  ( `' O " NN )  e.  (SubGrp `  G ) )
 
Theoremoddvdssubg 17060* The set of all elements whose order divides a fixed integer is a subgroup of any abelian group. (Contributed by Mario Carneiro, 19-Apr-2016.)
 |-  O  =  ( od
 `  G )   &    |-  B  =  ( Base `  G )   =>    |-  (
 ( G  e.  Abel  /\  N  e.  ZZ )  ->  { x  e.  B  |  ( O `  x )  ||  N }  e.  (SubGrp `  G ) )
 
Theoremlsmcomx 17061 Subgroup sum commutes (extended domain version). (Contributed by NM, 25-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  .(+)  =  ( LSSum `  G )   =>    |-  ( ( G  e.  Abel  /\  T  C_  B  /\  U  C_  B )  ->  ( T  .(+)  U )  =  ( U  .(+)  T ) )
 
Theoremablcntzd 17062 All subgroups in an abelian group commute. (Contributed by Mario Carneiro, 19-Apr-2016.)
 |-  Z  =  (Cntz `  G )   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  T  e.  (SubGrp `  G ) )   &    |-  ( ph  ->  U  e.  (SubGrp `  G ) )   =>    |-  ( ph  ->  T  C_  ( Z `  U ) )
 
Theoremlsmcom 17063 Subgroup sum commutes. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 21-Jun-2014.)
 |-  .(+)  =  ( LSSum `  G )   =>    |-  ( ( G  e.  Abel  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  ->  ( T  .(+)  U )  =  ( U  .(+)  T ) )
 
Theoremlsmsubg2 17064 The sum of two subgroups is a subgroup. (Contributed by NM, 4-Feb-2014.) (Proof shortened by Mario Carneiro, 19-Apr-2016.)
 |-  .(+)  =  ( LSSum `  G )   =>    |-  ( ( G  e.  Abel  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  ->  ( T  .(+)  U )  e.  (SubGrp `  G ) )
 
Theoremlsm4 17065 Commutative/associative law for subgroup sum. (Contributed by NM, 26-Sep-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
 |-  .(+)  =  ( LSSum `  G )   =>    |-  ( ( G  e.  Abel  /\  ( Q  e.  (SubGrp `  G )  /\  R  e.  (SubGrp `  G )
 )  /\  ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) ) )  ->  ( ( Q  .(+)  R )  .(+)  ( T  .(+)  U ) )  =  ( ( Q  .(+)  T )  .(+)  ( R  .(+)  U ) ) )
 
Theoremprdscmnd 17066 The product of a family of commutative monoids is commutative. (Contributed by Stefan O'Rear, 10-Jan-2015.)
 |-  Y  =  ( S
 X_s
 R )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  R : I -->CMnd )   =>    |-  ( ph  ->  Y  e. CMnd )
 
Theoremprdsabld 17067 The product of a family of Abelian groups is an Abelian group. (Contributed by Stefan O'Rear, 10-Jan-2015.)
 |-  Y  =  ( S
 X_s
 R )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  R : I --> Abel )   =>    |-  ( ph  ->  Y  e.  Abel )
 
Theorempwscmn 17068 The structure power on a commutative monoid is commutative. (Contributed by Mario Carneiro, 11-Jan-2015.)
 |-  Y  =  ( R 
 ^s  I )   =>    |-  ( ( R  e. CMnd  /\  I  e.  V ) 
 ->  Y  e. CMnd )
 
Theorempwsabl 17069 The structure power on an Abelian group is Abelian. (Contributed by Mario Carneiro, 21-Jan-2015.)
 |-  Y  =  ( R 
 ^s  I )   =>    |-  ( ( R  e.  Abel  /\  I  e.  V )  ->  Y  e.  Abel )
 
Theoremqusabl 17070 If  Y is a subgroup of the abelian group  G, then  H  =  G  /  Y is an abelian group. (Contributed by Mario Carneiro, 26-Apr-2016.)
 |-  H  =  ( G 
 /.s 
 ( G ~QG  S ) )   =>    |-  ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G ) ) 
 ->  H  e.  Abel )
 
Theoremabl1 17071 The (smallest) structure representing a trivial abelian group. (Contributed by AV, 28-Apr-2019.)
 |-  M  =  { <. (
 Base `  ndx ) ,  { I } >. , 
 <. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. }
 >. }   =>    |-  ( I  e.  V  ->  M  e.  Abel )
 
Theoremabln0 17072 Abelian groups (and therefore also groups and monoids) exist. (Contributed by AV, 29-Apr-2019.)
 |- 
 Abel  =/=  (/)
 
Theoremcnaddablx 17073 The complex numbers are an Abelian group under addition. This version of cnaddabl 17074 shows the explicit structure "scaffold" we chose for the definition for Abelian groups. Note: This theorem has hard-coded structure indices for demonstration purposes. It is not intended for general use; use cnaddabl 17074 instead. (New usage is discouraged.) (Contributed by NM, 18-Oct-2012.)
 |-  G  =  { <. 1 ,  CC >. ,  <. 2 ,  +  >. }   =>    |-  G  e.  Abel
 
Theoremcnaddabl 17074 The complex numbers are an Abelian group under addition. This version of cnaddablx 17073 hides the explicit structure indices i.e. is "scaffold-independent". Note that the proof also does not reference explicit structure indices. The actual structure is dependent on how  Base and  +g is defined. This theorem should not be referenced in any proof. For the group/ring properties of the complex numbers, see cnring 18635. (Contributed by NM, 20-Oct-2012.) (New usage is discouraged.)
 |-  G  =  { <. (
 Base `  ndx ) ,  CC >. ,  <. ( +g  ` 
 ndx ) ,  +  >. }   =>    |-  G  e.  Abel
 
Theoremzaddablx 17075 The integers are an Abelian group under addition. Note: This theorem has hard-coded structure indices for demonstration purposes. It is not intended for general use. Use zsubrg 18666 instead. (New usage is discouraged.) (Contributed by NM, 4-Sep-2011.)
 |-  G  =  { <. 1 ,  ZZ >. ,  <. 2 ,  +  >. }   =>    |-  G  e.  Abel
 
Theoremfrgpnabllem1 17076* Lemma for frgpnabl 17078. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  G  =  (freeGrp `  I
 )   &    |-  W  =  (  _I  ` Word  ( I  X.  2o ) )   &    |-  .~  =  ( ~FG  `  I )   &    |-  .+  =  ( +g  `  G )   &    |-  M  =  ( y  e.  I ,  z  e.  2o  |->  <.
 y ,  ( 1o  \  z ) >. )   &    |-  T  =  ( v  e.  W  |->  ( n  e.  (
 0 ... ( # `  v
 ) ) ,  w  e.  ( I  X.  2o )  |->  ( v splice  <. n ,  n ,  <" w ( M `  w ) "> >. ) ) )   &    |-  D  =  ( W  \  U_ x  e.  W  ran  ( T `
  x ) )   &    |-  U  =  (varFGrp `  I )   &    |-  ( ph  ->  I  e.  _V )   &    |-  ( ph  ->  A  e.  I
 )   &    |-  ( ph  ->  B  e.  I )   =>    |-  ( ph  ->  <" <. A ,  (/) >. <. B ,  (/) >. ">  e.  ( D  i^i  (
 ( U `  A )  .+  ( U `  B ) ) ) )
 
Theoremfrgpnabllem2 17077* Lemma for frgpnabl 17078. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  G  =  (freeGrp `  I
 )   &    |-  W  =  (  _I  ` Word  ( I  X.  2o ) )   &    |-  .~  =  ( ~FG  `  I )   &    |-  .+  =  ( +g  `  G )   &    |-  M  =  ( y  e.  I ,  z  e.  2o  |->  <.
 y ,  ( 1o  \  z ) >. )   &    |-  T  =  ( v  e.  W  |->  ( n  e.  (
 0 ... ( # `  v
 ) ) ,  w  e.  ( I  X.  2o )  |->  ( v splice  <. n ,  n ,  <" w ( M `  w ) "> >. ) ) )   &    |-  D  =  ( W  \  U_ x  e.  W  ran  ( T `
  x ) )   &    |-  U  =  (varFGrp `  I )   &    |-  ( ph  ->  I  e.  _V )   &    |-  ( ph  ->  A  e.  I
 )   &    |-  ( ph  ->  B  e.  I )   &    |-  ( ph  ->  ( ( U `  A )  .+  ( U `  B ) )  =  ( ( U `  B )  .+  ( U `
  A ) ) )   =>    |-  ( ph  ->  A  =  B )
 
Theoremfrgpnabl 17078 The free group on two or more generators is not abelian. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  G  =  (freeGrp `  I
 )   =>    |-  ( 1o  ~<  I  ->  -.  G  e.  Abel )
 
10.3.2  Cyclic groups
 
Syntaxccyg 17079 Cyclic group.
 class CycGrp
 
Definitiondf-cyg 17080* Define a cyclic group, which is a group with an element  x, called the generator of the group, such that all elements in the group are multiples of  x. A generator is usually not unique. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |- CycGrp  =  { g  e.  Grp  | 
 E. x  e.  ( Base `  g ) ran  ( n  e.  ZZ  |->  ( n (.g `  g ) x ) )  =  (
 Base `  g ) }
 
Theoremiscyg 17081* Definition of a cyclic group. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   =>    |-  ( G  e. CycGrp  <->  ( G  e.  Grp  /\  E. x  e.  B  ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  B ) )
 
Theoremiscyggen 17082* The property of being a cyclic generator for a group. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  E  =  { x  e.  B  |  ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  B }   =>    |-  ( X  e.  E  <->  ( X  e.  B  /\  ran  ( n  e.  ZZ  |->  ( n  .x.  X ) )  =  B ) )
 
Theoremiscyggen2 17083* The property of being a cyclic generator for a group. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  E  =  { x  e.  B  |  ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  B }   =>    |-  ( G  e.  Grp  ->  ( X  e.  E  <->  ( X  e.  B  /\  A. y  e.  B  E. n  e. 
 ZZ  y  =  ( n  .x.  X )
 ) ) )
 
Theoremiscyg2 17084* A cyclic group is a group which contains a generator. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  E  =  { x  e.  B  |  ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  B }   =>    |-  ( G  e. CycGrp  <->  ( G  e.  Grp  /\  E  =/=  (/) ) )
 
Theoremcyggeninv 17085* The inverse of a cyclic generator is a generator. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  E  =  { x  e.  B  |  ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  B }   &    |-  N  =  ( invg `  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  E ) 
 ->  ( N `  X )  e.  E )
 
Theoremcyggenod 17086* An element is the generator of a finite group iff the order of the generator equals the order of the group. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  E  =  { x  e.  B  |  ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  B }   &    |-  O  =  ( od `  G )   =>    |-  ( ( G  e.  Grp  /\  B  e.  Fin )  ->  ( X  e.  E  <->  ( X  e.  B  /\  ( O `  X )  =  ( # `  B ) ) ) )
 
Theoremcyggenod2 17087* In an infinite cyclic group, the generator must have infinite order, but this property no longer characterizes the generators. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  E  =  { x  e.  B  |  ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  B }   &    |-  O  =  ( od `  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  E ) 
 ->  ( O `  X )  =  if ( B  e.  Fin ,  ( # `
  B ) ,  0 ) )
 
Theoremiscyg3 17088* Definition of a cyclic group. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   =>    |-  ( G  e. CycGrp  <->  ( G  e.  Grp  /\  E. x  e.  B  A. y  e.  B  E. n  e.  ZZ  y  =  ( n  .x.  x ) ) )
 
Theoremiscygd 17089* Definition of a cyclic group. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  ( ph  ->  G  e.  Grp )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ( ph  /\  y  e.  B )  ->  E. n  e.  ZZ  y  =  ( n  .x.  X )
 )   =>    |-  ( ph  ->  G  e. CycGrp )
 
Theoremiscygodd 17090 Show that a group with an element the same order as the group is cyclic. (Contributed by Mario Carneiro, 27-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  ( ph  ->  G  e.  Grp )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  ( O `  X )  =  ( # `
  B ) )   =>    |-  ( ph  ->  G  e. CycGrp )
 
Theoremcyggrp 17091 A cyclic group is a group. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  ( G  e. CycGrp  ->  G  e.  Grp )
 
Theoremcygabl 17092 A cyclic group is abelian. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  ( G  e. CycGrp  ->  G  e.  Abel )
 
Theoremcygctb 17093 A cyclic group is countable. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  B  =  ( Base `  G )   =>    |-  ( G  e. CycGrp  ->  B  ~<_ 
 om )
 
Theorem0cyg 17094 The trivial group is cyclic. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  B  =  ( Base `  G )   =>    |-  ( ( G  e.  Grp  /\  B  ~~  1o )  ->  G  e. CycGrp )
 
Theoremprmcyg 17095 A group with prime order is cyclic. (Contributed by Mario Carneiro, 27-Apr-2016.)
 |-  B  =  ( Base `  G )   =>    |-  ( ( G  e.  Grp  /\  ( # `  B )  e.  Prime )  ->  G  e. CycGrp )
 
Theoremlt6abl 17096 A group with fewer than  6 elements is abelian. (Contributed by Mario Carneiro, 27-Apr-2016.)
 |-  B  =  ( Base `  G )   =>    |-  ( ( G  e.  Grp  /\  ( # `  B )  <  6 )  ->  G  e.  Abel )
 
Theoremghmcyg 17097 The image of a cyclic group under a surjective group homomorphism is cyclic. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  C  =  (
 Base `  H )   =>    |-  ( ( F  e.  ( G  GrpHom  H )  /\  F : B -onto-> C )  ->  ( G  e. CycGrp  ->  H  e. CycGrp ) )
 
Theoremcyggex2 17098 The exponent of a cyclic group is  0 if the group is infinite, otherwise it equals the order of the group. (Contributed by Mario Carneiro, 24-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  E  =  (gEx `  G )   =>    |-  ( G  e. CycGrp  ->  E  =  if ( B  e.  Fin ,  ( # `
  B ) ,  0 ) )
 
Theoremcyggex 17099 The exponent of a finite cyclic group is the order of the group. (Contributed by Mario Carneiro, 24-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  E  =  (gEx `  G )   =>    |-  ( ( G  e. CycGrp  /\  B  e.  Fin )  ->  E  =  ( # `  B ) )
 
Theoremcyggexb 17100 A finite abelian group is cyclic iff the exponent equals the order of the group. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  E  =  (gEx `  G )   =>    |-  ( ( G  e.  Abel  /\  B  e.  Fin )  ->  ( G  e. CycGrp  <->  E  =  ( # `
  B ) ) )
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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38473
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