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Theorem List for Metamath Proof Explorer - 25901-26000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem1p1e2apr1 25901 One plus one equals two. Using proof-shortening techniques pioneered by Mr. Mel L. O'Cat, along with the latest supercomputer technology, Prof. Loof Lirpa and colleagues were able to shorten Whitehead and Russell's 360-page proof that 1+1=2 in Principia Mathematica to this remarkable proof only two steps long, thus establishing a new world's record for this famous theorem. (Contributed by Prof. Loof Lirpa, 1-Apr-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( 1  +  1 )  =  2
 
Theoremeqid1 25902 Law of identity (reflexivity of class equality). Theorem 6.4 of [Quine] p. 41.

This law is thought to have originated with Aristotle (Metaphysics, Book VII, Part 17). It is one of the three axioms of Ayn Rand's philosophy (Atlas Shrugged, Part Three, Chapter VII). While some have proposed extending Rand's axiomatization to include Compassion and Kindness, others fear that such an extension may flirt with logical inconsistency. (Contributed by Stefan Allan, 1-Apr-2009.) (Proof modification is discouraged.) (New usage is discouraged.)

 |-  A  =  A
 
Theorem1div0apr 25903 Division by zero is forbidden! If we try, we encounter the DO NOT ENTER sign, which in mathematics means it is foolhardy to venture any further, possibly putting the underlying fabric of reality at risk. Based on a dare by David A. Wheeler. (Contributed by Mario Carneiro, 1-Apr-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( 1  /  0
 )  =  (/)
 
Theoremtopnfbey 25904 Nothing seems to be impossible to Prof. Lirpa. After years of intensive research, he managed to find a proof that when given a chance to reach infinity, one could indeed go beyond, thus giving formal soundness to Buzz Lightyear's motto "To infinity... and beyond!" (Contributed by Prof. Loof Lirpa, 1-Apr-2020.) (Modified by Thierry Arnoux, 2-Aug-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( B  e.  (
 0 ... +oo )  -> +oo  <  B )
 
17.3  (Future - to be reviewed and classified)
 
17.3.1  Planar incidence geometry
 
Syntaxcplig 25905 Extend class notation with the class of all planar incidence geometries.
 class  Plig
 
Definitiondf-plig 25906* Planar incidence geometry. I use Hilbert's "axioms" adapted to planar geometry.  e. is the incidence relation. I could take a generic incidence relation but I'm lazy and I'm not sure the gain is worth the extra work. Much of what follows is directly borrowed from Aitken. http://public.csusm.edu/aitken_html/m410/betweenness.08.pdf (Contributed by FL, 2-Aug-2009.)
 |- 
 Plig  =  { x  |  ( A. a  e. 
 U. x A. b  e.  U. x ( a  =/=  b  ->  E! l  e.  x  (
 a  e.  l  /\  b  e.  l )
 )  /\  A. l  e.  x  E. a  e. 
 U. x E. b  e.  U. x ( a  =/=  b  /\  a  e.  l  /\  b  e.  l )  /\  E. a  e.  U. x E. b  e.  U. x E. c  e.  U. x A. l  e.  x  -.  ( a  e.  l  /\  b  e.  l  /\  c  e.  l
 ) ) }
 
Theoremisplig 25907* The predicate "is a planar incidence geometry". (Contributed by FL, 2-Aug-2009.)
 |-  P  =  U. L   =>    |-  ( L  e.  A  ->  ( L  e.  Plig  <->  ( A. a  e.  P  A. b  e.  P  ( a  =/=  b  ->  E! l  e.  L  ( a  e.  l  /\  b  e.  l ) )  /\  A. l  e.  L  E. a  e.  P  E. b  e.  P  ( a  =/=  b  /\  a  e.  l  /\  b  e.  l )  /\  E. a  e.  P  E. b  e.  P  E. c  e.  P  A. l  e.  L  -.  ( a  e.  l  /\  b  e.  l  /\  c  e.  l ) ) ) )
 
Theoremtncp 25908* There exist three non colinear points. (Contributed by FL, 3-Aug-2009.)
 |-  P  =  U. L   =>    |-  ( L  e.  Plig  ->  E. a  e.  P  E. b  e.  P  E. c  e.  P  A. l  e.  L  -.  ( a  e.  l  /\  b  e.  l  /\  c  e.  l ) )
 
Theoremlpni 25909* For any line, there exists a point not on the line. (Contributed by Jeff Hankins, 15-Aug-2009.)
 |-  P  =  U. G   =>    |-  (
 ( G  e.  Plig  /\  L  e.  G ) 
 ->  E. a  e.  P  a  e/  L )
 
17.3.2  Algebra preliminaries
 
Syntaxcrpm 25910 Ring primes.
 class RPrime
 
Definitiondf-rprm 25911* Define the set of prime elements in a ring. A prime element is a nonzero non-unit that satisfies an equivalent of Euclid's lemma euclemma 14664. (Contributed by Mario Carneiro, 17-Feb-2015.)
 |- RPrime  =  ( w  e.  _V  |->  [_ ( Base `  w )  /  b ]_ { p  e.  ( b  \  (
 (Unit `  w )  u.  { ( 0g `  w ) } )
 )  |  A. x  e.  b  A. y  e.  b  [. ( ||r `  w )  /  d ]. ( p d ( x ( .r `  w ) y )  ->  ( p d x  \/  p d y ) ) } )
 
PART 18  ADDITIONAL MATERIAL ON GROUPS, RINGS, AND FIELDS (DEPRECATED)

This part contains an earlier development of groups, rings, and fields that was defined before extensible structures were introduced.

Theorem grpo2grp 25960 shows the relationship between the older group definition and the extensible structure definition.

The intent is for this deprecated section to be deleted once its theorems have extensible structure versions (or are not useful). You can make a list of "terminal" theorems (i.e. theorems not referenced by anything else) and for each theorem see if there exists an extensible structure version (or decide it's not useful), and if so, delete it. Then repeat this recursively. One way to search for terminal theorems, for example in deprecated group theory, is to log the output ("open log x.txt") of "show usage cgr~circgrp" in metamath.exe and search for "(None)".

 
18.1  Additional material on group theory
 
18.1.1  Definitions and basic properties for groups
 
Syntaxcgr 25912 Extend class notation with the class of all group operations.
 class  GrpOp
 
Syntaxcgi 25913 Extend class notation with a function mapping a group operation to the group's identity element.
 class GId
 
Syntaxcgn 25914 Extend class notation with a function mapping a group operation to the inverse function for the group.
 class  inv
 
Syntaxcgs 25915 Extend class notation with a function mapping a group operation to the division (or subtraction) operation for the group.
 class  /g
 
Syntaxcgx 25916 Extend class notation with a function mapping a group operation to the power operation for the group.
 class  ^g
 
Definitiondf-grpo 25917* Define the class of all group operations. The base set for a group can be determined from its group operation. Based on the definition in Exercise 28 of [Herstein] p. 54. (Contributed by NM, 10-Oct-2006.) (New usage is discouraged.)
 |- 
 GrpOp  =  { g  |  E. t ( g : ( t  X.  t ) --> t  /\  A. x  e.  t  A. y  e.  t  A. z  e.  t  (
 ( x g y ) g z )  =  ( x g ( y g z ) )  /\  E. u  e.  t  A. x  e.  t  (
 ( u g x )  =  x  /\  E. y  e.  t  ( y g x )  =  u ) ) }
 
Definitiondf-gid 25918* Define a function that maps a group operation to the group's identity element. (Contributed by FL, 5-Feb-2010.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
 |- GId 
 =  ( g  e. 
 _V  |->  ( iota_ u  e. 
 ran  g A. x  e.  ran  g ( ( u g x )  =  x  /\  ( x g u )  =  x ) ) )
 
Definitiondf-ginv 25919* Define a function that maps a group operation to the group's inverse function. (Contributed by NM, 26-Oct-2006.) (New usage is discouraged.)
 |- 
 inv  =  ( g  e.  GrpOp  |->  ( x  e. 
 ran  g  |->  ( iota_ z  e.  ran  g (
 z g x )  =  (GId `  g
 ) ) ) )
 
Definitiondf-gdiv 25920* Define a function that maps a group operation to the group's division (or subtraction) operation. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
 |- 
 /g  =  ( g  e.  GrpOp  |->  ( x  e. 
 ran  g ,  y  e.  ran  g  |->  ( x g ( ( inv `  g ) `  y
 ) ) ) )
 
Definitiondf-gx 25921* Define a function that maps a group operation to the group's power operation. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)
 |- 
 ^g  =  ( g  e.  GrpOp  |->  ( x  e. 
 ran  g ,  y  e.  ZZ  |->  if ( y  =  0 ,  (GId `  g ) ,  if ( 0  <  y ,  (  seq 1
 ( g ,  ( NN  X.  { x }
 ) ) `  y
 ) ,  ( ( inv `  g ) `  (  seq 1
 ( g ,  ( NN  X.  { x }
 ) ) `  -u y
 ) ) ) ) ) )
 
Theoremisgrpo 25922* The predicate "is a group operation." Note that  X is the base set of the group. (Contributed by NM, 10-Oct-2006.) (New usage is discouraged.)
 |-  X  =  ran  G   =>    |-  ( G  e.  A  ->  ( G  e.  GrpOp  <->  ( G :
 ( X  X.  X )
 --> X  /\  A. x  e.  X  A. y  e.  X  A. z  e.  X  ( ( x G y ) G z )  =  ( x G ( y G z ) ) 
 /\  E. u  e.  X  A. x  e.  X  ( ( u G x )  =  x  /\  E. y  e.  X  ( y G x )  =  u ) ) ) )
 
Theoremisgrpo2 25923* The predicate "is a group operation." (Contributed by NM, 23-Oct-2012.) (New usage is discouraged.)
 |-  X  =  ran  G   =>    |-  ( G  e.  A  ->  ( G  e.  GrpOp  <->  ( G :
 ( X  X.  X )
 --> X  /\  A. x  e.  X  A. y  e.  X  A. z  e.  X  ( ( x G y )  e.  X  /\  ( ( x G y ) G z )  =  ( x G ( y G z ) ) )  /\  E. u  e.  X  A. x  e.  X  ( ( u G x )  =  x  /\  E. n  e.  X  ( n G x )  =  u ) ) ) )
 
Theoremisgrpoi 25924* Properties that determine a group operation. Read  N as  N ( x ). (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)
 |-  X  e.  _V   &    |-  G : ( X  X.  X ) --> X   &    |-  (
 ( x  e.  X  /\  y  e.  X  /\  z  e.  X )  ->  ( ( x G y ) G z )  =  ( x G ( y G z ) ) )   &    |-  U  e.  X   &    |-  ( x  e.  X  ->  ( U G x )  =  x )   &    |-  ( x  e.  X  ->  N  e.  X )   &    |-  ( x  e.  X  ->  ( N G x )  =  U )   =>    |-  G  e.  GrpOp
 
Theoremgrpofo 25925 A group operation maps onto the group's underlying set. (Contributed by NM, 30-Oct-2006.) (New usage is discouraged.)
 |-  X  =  ran  G   =>    |-  ( G  e.  GrpOp  ->  G : ( X  X.  X ) -onto-> X )
 
Theoremgrpocl 25926 Closure law for a group operation. (Contributed by NM, 10-Oct-2006.) (New usage is discouraged.)
 |-  X  =  ran  G   =>    |-  (
 ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  e.  X )
 
Theoremgrpolidinv 25927* A group has a left identity element, and every member has a left inverse. (Contributed by NM, 2-Nov-2006.) (New usage is discouraged.)
 |-  X  =  ran  G   =>    |-  ( G  e.  GrpOp  ->  E. u  e.  X  A. x  e.  X  ( ( u G x )  =  x  /\  E. y  e.  X  ( y G x )  =  u ) )
 
Theoremgrpon0 25928 The base set of a group is not empty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.) (New usage is discouraged.)
 |-  X  =  ran  G   =>    |-  ( G  e.  GrpOp  ->  X  =/= 
 (/) )
 
Theoremgrpoass 25929 A group operation is associative. (Contributed by NM, 10-Oct-2006.) (New usage is discouraged.)
 |-  X  =  ran  G   =>    |-  (
 ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A G B ) G C )  =  ( A G ( B G C ) ) )
 
Theoremgrpoidinvlem1 25930 Lemma for grpoidinv 25934. (Contributed by NM, 10-Oct-2006.) (New usage is discouraged.)
 |-  X  =  ran  G   =>    |-  (
 ( ( G  e.  GrpOp  /\  ( Y  e.  X  /\  A  e.  X ) )  /\  ( ( Y G A )  =  U  /\  ( A G A )  =  A ) )  ->  ( U G A )  =  U )
 
Theoremgrpoidinvlem2 25931 Lemma for grpoidinv 25934. (Contributed by NM, 10-Oct-2006.) (New usage is discouraged.)
 |-  X  =  ran  G   =>    |-  (
 ( ( G  e.  GrpOp  /\  ( Y  e.  X  /\  A  e.  X ) )  /\  ( ( U G Y )  =  Y  /\  ( Y G A )  =  U ) )  ->  ( ( A G Y ) G ( A G Y ) )  =  ( A G Y ) )
 
Theoremgrpoidinvlem3 25932* Lemma for grpoidinv 25934. (Contributed by NM, 11-Oct-2006.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  ( ph 
 <-> 
 A. x  e.  X  ( U G x )  =  x )   &    |-  ( ps 
 <-> 
 A. x  e.  X  E. z  e.  X  ( z G x )  =  U )   =>    |-  ( ( ( ( G  e.  GrpOp  /\  U  e.  X )  /\  ( ph  /\  ps ) ) 
 /\  A  e.  X )  ->  E. y  e.  X  ( ( y G A )  =  U  /\  ( A G y )  =  U ) )
 
Theoremgrpoidinvlem4 25933* Lemma for grpoidinv 25934. (Contributed by NM, 14-Oct-2006.) (New usage is discouraged.)
 |-  X  =  ran  G   =>    |-  (
 ( ( G  e.  GrpOp  /\  A  e.  X ) 
 /\  E. y  e.  X  ( ( y G A )  =  U  /\  ( A G y )  =  U ) )  ->  ( A G U )  =  ( U G A ) )
 
Theoremgrpoidinv 25934* A group has a left and right identity element, and every member has a left and right inverse. (Contributed by NM, 14-Oct-2006.) (New usage is discouraged.)
 |-  X  =  ran  G   =>    |-  ( G  e.  GrpOp  ->  E. u  e.  X  A. x  e.  X  ( ( ( u G x )  =  x  /\  ( x G u )  =  x )  /\  E. y  e.  X  (
 ( y G x )  =  u  /\  ( x G y )  =  u ) ) )
 
Theoremgrpoideu 25935* The left identity element of a group is unique. Lemma 2.2.1(a) of [Herstein] p. 55. (Contributed by NM, 14-Oct-2006.) (New usage is discouraged.)
 |-  X  =  ran  G   =>    |-  ( G  e.  GrpOp  ->  E! u  e.  X  A. x  e.  X  ( u G x )  =  x )
 
Theoremgrporndm 25936 A group's range in terms of its domain. (Contributed by NM, 6-Apr-2008.) (New usage is discouraged.)
 |-  ( G  e.  GrpOp  ->  ran  G  =  dom  dom  G )
 
Theorem0ngrp 25937 The empty set is not a group. (Contributed by NM, 25-Apr-2007.) (New usage is discouraged.)
 |- 
 -.  (/)  e.  GrpOp
 
Theoremgrporn 25938 The range of a group operation. Useful for satisfying group base set hypotheses of the form  X  =  ran  G. (Contributed by NM, 5-Nov-2006.) (New usage is discouraged.)
 |-  G  e.  GrpOp   &    |- 
 dom  G  =  ( X  X.  X )   =>    |-  X  =  ran  G
 
Theoremgidval 25939* The value of the identity element of a group. (Contributed by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
 |-  X  =  ran  G   =>    |-  ( G  e.  V  ->  (GId `  G )  =  (
 iota_ u  e.  X  A. x  e.  X  ( ( u G x )  =  x  /\  ( x G u )  =  x ) ) )
 
Theoremfngid 25940 GId is a function. (Contributed by FL, 5-Feb-2010.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
 |- GId  Fn  _V
 
Theoremgrposn 25941 The group operation for the singleton group. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)
 |-  A  e.  _V   =>    |-  { <. <. A ,  A >. ,  A >. }  e.  GrpOp
 
Theoremgrpoidval 25942* Lemma for grpoidcl 25943 and others. (Contributed by NM, 5-Feb-2010.) (Proof shortened by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  U  =  (GId `  G )   =>    |-  ( G  e.  GrpOp  ->  U  =  ( iota_ u  e.  X  A. x  e.  X  ( u G x )  =  x ) )
 
Theoremgrpoidcl 25943 The identity element of a group belongs to the group. (Contributed by NM, 24-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  U  =  (GId `  G )   =>    |-  ( G  e.  GrpOp  ->  U  e.  X )
 
Theoremgrpoidinv2 25944* A group's properties using the explicit identity element. (Contributed by NM, 5-Feb-2010.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  U  =  (GId `  G )   =>    |-  (
 ( G  e.  GrpOp  /\  A  e.  X ) 
 ->  ( ( ( U G A )  =  A  /\  ( A G U )  =  A )  /\  E. y  e.  X  (
 ( y G A )  =  U  /\  ( A G y )  =  U ) ) )
 
Theoremgrpolid 25945 The identity element of a group is a left identity. (Contributed by NM, 24-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  U  =  (GId `  G )   =>    |-  (
 ( G  e.  GrpOp  /\  A  e.  X ) 
 ->  ( U G A )  =  A )
 
Theoremgrporid 25946 The identity element of a group is a right identity. (Contributed by NM, 24-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  U  =  (GId `  G )   =>    |-  (
 ( G  e.  GrpOp  /\  A  e.  X ) 
 ->  ( A G U )  =  A )
 
Theoremgrporcan 25947 Right cancellation law for groups. (Contributed by NM, 26-Oct-2006.) (New usage is discouraged.)
 |-  X  =  ran  G   =>    |-  (
 ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A G C )  =  ( B G C ) 
 <->  A  =  B ) )
 
Theoremgrpoinveu 25948* The left inverse element of a group is unique. Lemma 2.2.1(b) of [Herstein] p. 55. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  U  =  (GId `  G )   =>    |-  (
 ( G  e.  GrpOp  /\  A  e.  X ) 
 ->  E! y  e.  X  ( y G A )  =  U )
 
Theoremgrpoid 25949 Two ways of saying that an element of a group is the identity element. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  U  =  (GId `  G )   =>    |-  (
 ( G  e.  GrpOp  /\  A  e.  X ) 
 ->  ( A  =  U  <->  ( A G A )  =  A ) )
 
Theoremgrpoinvfval 25950* The inverse function of a group. (Contributed by NM, 26-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  U  =  (GId `  G )   &    |-  N  =  ( inv `  G )   =>    |-  ( G  e.  GrpOp  ->  N  =  ( x  e.  X  |->  ( iota_ y  e.  X  ( y G x )  =  U ) ) )
 
Theoremgrpoinvval 25951* The inverse of a group element. (Contributed by NM, 26-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  U  =  (GId `  G )   &    |-  N  =  ( inv `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X ) 
 ->  ( N `  A )  =  ( iota_ y  e.  X  ( y G A )  =  U ) )
 
Theoremgrpoinvcl 25952 A group element's inverse is a group element. (Contributed by NM, 27-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  N  =  ( inv `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X ) 
 ->  ( N `  A )  e.  X )
 
Theoremgrpoinv 25953 The properties of a group element's inverse. (Contributed by NM, 27-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  U  =  (GId `  G )   &    |-  N  =  ( inv `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X ) 
 ->  ( ( ( N `
  A ) G A )  =  U  /\  ( A G ( N `  A ) )  =  U ) )
 
Theoremgrpolinv 25954 The left inverse of a group element. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  U  =  (GId `  G )   &    |-  N  =  ( inv `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X ) 
 ->  ( ( N `  A ) G A )  =  U )
 
Theoremgrporinv 25955 The right inverse of a group element. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  U  =  (GId `  G )   &    |-  N  =  ( inv `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X ) 
 ->  ( A G ( N `  A ) )  =  U )
 
Theoremgrpoinvid1 25956 The inverse of a group element expressed in terms of the identity element. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  U  =  (GId `  G )   &    |-  N  =  ( inv `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( ( N `  A )  =  B  <->  ( A G B )  =  U ) )
 
Theoremgrpoinvid2 25957 The inverse of a group element expressed in terms of the identity element. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  U  =  (GId `  G )   &    |-  N  =  ( inv `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( ( N `  A )  =  B  <->  ( B G A )  =  U ) )
 
Theoremgrpoinvid 25958 The inverse of the identity element of a group. (Contributed by NM, 4-Dec-2006.) (New usage is discouraged.)
 |-  U  =  (GId `  G )   &    |-  N  =  ( inv `  G )   =>    |-  ( G  e.  GrpOp  ->  ( N `  U )  =  U )
 
Theoremgrpolcan 25959 Left cancellation law for groups. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)
 |-  X  =  ran  G   =>    |-  (
 ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( C G A )  =  ( C G B ) 
 <->  A  =  B ) )
 
Theoremgrpo2grp 25960 Convert a group operation to a group structure. (Contributed by NM, 25-Oct-2012.) (Revised by Mario Carneiro, 6-Jan-2015.) (New usage is discouraged.)
 |-  ( Base `  K )  =  ran  .+   &    |-  ( +g  `  K )  =  .+   &    |-  .+  e.  GrpOp   =>    |-  K  e.  Grp
 
Theoremisgrp2d 25961* An alternate way to show a group operation. Exercise 1 of [Herstein] p. 57. (Contributed by Mario Carneiro, 12-May-2014.) (New usage is discouraged.)
 |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  X  =/=  (/) )   &    |-  ( ph  ->  G : ( X  X.  X ) --> X )   &    |-  ( ( ph  /\  ( x  e.  X  /\  y  e.  X  /\  z  e.  X )
 )  ->  ( ( x G y ) G z )  =  ( x G ( y G z ) ) )   &    |-  ( ( ph  /\  ( x  e.  X  /\  y  e.  X ) )  ->  E. z  e.  X  ( z G x )  =  y )   &    |-  ( ( ph  /\  ( x  e.  X  /\  y  e.  X ) )  ->  E. z  e.  X  ( x G z )  =  y )   =>    |-  ( ph  ->  G  e.  GrpOp )
 
Theoremisgrp2i 25962* An alternate way to show a group operation. Exercise 1 of [Herstein] p. 57. (Contributed by NM, 5-Dec-2006.) (Revised by Mario Carneiro, 12-May-2014.) (New usage is discouraged.)
 |-  X  e.  _V   &    |-  X  =/= 
 (/)   &    |-  G : ( X  X.  X ) --> X   &    |-  (
 ( x  e.  X  /\  y  e.  X  /\  z  e.  X )  ->  ( ( x G y ) G z )  =  ( x G ( y G z ) ) )   &    |-  ( ( x  e.  X  /\  y  e.  X )  ->  E. z  e.  X  ( z G x )  =  y )   &    |-  ( ( x  e.  X  /\  y  e.  X )  ->  E. z  e.  X  ( x G z )  =  y )   =>    |-  G  e.  GrpOp
 
Theoremgrpoasscan1 25963 An associative cancellation law for groups. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  N  =  ( inv `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A G ( ( N `  A ) G B ) )  =  B )
 
Theoremgrpoasscan2 25964 An associative cancellation law for groups. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  N  =  ( inv `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( ( A G ( N `  B ) ) G B )  =  A )
 
Theoremgrpo2inv 25965 Double inverse law for groups. Lemma 2.2.1(c) of [Herstein] p. 55. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  N  =  ( inv `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X ) 
 ->  ( N `  ( N `  A ) )  =  A )
 
Theoremgrpoinvf 25966 Mapping of the inverse function of a group. (Contributed by NM, 29-Mar-2008.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  N  =  ( inv `  G )   =>    |-  ( G  e.  GrpOp  ->  N : X -1-1-onto-> X )
 
Theoremgrpoinvop 25967 The inverse of the group operation reverses the arguments. Lemma 2.2.1(d) of [Herstein] p. 55. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  N  =  ( inv `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A G B ) )  =  ( ( N `
  B ) G ( N `  A ) ) )
 
Theoremgrpodivfval 25968* Group division (or subtraction) operation. (Contributed by NM, 15-Feb-2008.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  N  =  ( inv `  G )   &    |-  D  =  (  /g  `  G )   =>    |-  ( G  e.  GrpOp  ->  D  =  ( x  e.  X ,  y  e.  X  |->  ( x G ( N `  y
 ) ) ) )
 
Theoremgrpodivval 25969 Group division (or subtraction) operation value. (Contributed by NM, 15-Feb-2008.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  N  =  ( inv `  G )   &    |-  D  =  (  /g  `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  =  ( A G ( N `  B ) ) )
 
Theoremgrpodivinv 25970 Group division by an inverse. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  N  =  ( inv `  G )   &    |-  D  =  (  /g  `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A D ( N `  B ) )  =  ( A G B ) )
 
Theoremgrpoinvdiv 25971 Inverse of a group division. (Contributed by NM, 24-Feb-2008.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  N  =  ( inv `  G )   &    |-  D  =  (  /g  `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A D B ) )  =  ( B D A ) )
 
Theoremgrpodivf 25972 Mapping for group division. (Contributed by NM, 10-Apr-2008.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  D  =  (  /g  `  G )   =>    |-  ( G  e.  GrpOp  ->  D : ( X  X.  X ) --> X )
 
Theoremgrpodivcl 25973 Closure of group division (or subtraction) operation. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  D  =  (  /g  `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  e.  X )
 
Theoremgrpodivdiv 25974 Double group division. (Contributed by NM, 24-Feb-2008.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  D  =  (  /g  `  G )   =>    |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( A D ( B D C ) )  =  ( A G ( C D B ) ) )
 
Theoremgrpomuldivass 25975 Associative-type law for multiplication and division. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  D  =  (  /g  `  G )   =>    |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A G B ) D C )  =  ( A G ( B D C ) ) )
 
Theoremgrpodivid 25976 Division of a group member by itself. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  D  =  (  /g  `  G )   &    |-  U  =  (GId `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X ) 
 ->  ( A D A )  =  U )
 
Theoremgrpopncan 25977 Cancellation law for group division. (pncan 9888 analog.) (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  D  =  (  /g  `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( ( A G B ) D B )  =  A )
 
Theoremgrponpcan 25978 Cancellation law for group division. (npcan 9891 analog.) (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  D  =  (  /g  `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( ( A D B ) G B )  =  A )
 
Theoremgrpopnpcan2 25979 Cancellation law for mixed addition and group division. (pnpcan2 9921 analog.) (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  D  =  (  /g  `  G )   =>    |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A G C ) D ( B G C ) )  =  ( A D B ) )
 
Theoremgrponnncan2 25980 Cancellation law for group division. (nnncan2 9918 analog.) (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  D  =  (  /g  `  G )   =>    |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A D C ) D ( B D C ) )  =  ( A D B ) )
 
Theoremgrponpncan 25981 Cancellation law for group division. (npncan 9902 analog.) (Contributed by NM, 24-Feb-2008.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  D  =  (  /g  `  G )   =>    |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A D B ) G ( B D C ) )  =  ( A D C ) )
 
Theoremgrpodiveq 25982 Relationship between group division and group multiplication. (Contributed by Mario Carneiro, 11-Jul-2014.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  D  =  (  /g  `  G )   =>    |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A D B )  =  C  <->  ( C G B )  =  A ) )
 
Theoremgxfval 25983* The value of the group power operator function. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  U  =  (GId `  G )   &    |-  N  =  ( inv `  G )   &    |-  P  =  ( ^g `  G )   =>    |-  ( G  e.  GrpOp  ->  P  =  ( x  e.  X ,  y  e. 
 ZZ  |->  if ( y  =  0 ,  U ,  if ( 0  <  y ,  (  seq 1
 ( G ,  ( NN  X.  { x }
 ) ) `  y
 ) ,  ( N `
  (  seq 1
 ( G ,  ( NN  X.  { x }
 ) ) `  -u y
 ) ) ) ) ) )
 
Theoremgxval 25984 The result of the group power operator. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  U  =  (GId `  G )   &    |-  N  =  ( inv `  G )   &    |-  P  =  ( ^g `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  ZZ )  ->  ( A P K )  =  if ( K  =  0 ,  U ,  if (
 0  <  K ,  (  seq 1 ( G ,  ( NN  X.  { A } ) ) `
  K ) ,  ( N `  (  seq 1 ( G ,  ( NN  X.  { A } ) ) `  -u K ) ) ) ) )
 
Theoremgxpval 25985 The result of the group power operator when the exponent is positive. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  P  =  ( ^g `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  NN )  ->  ( A P K )  =  (  seq 1 ( G ,  ( NN  X.  { A } ) ) `  K ) )
 
Theoremgxnval 25986 The result of the group power operator when the exponent is negative. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  P  =  ( ^g `  G )   &    |-  N  =  ( inv `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( K  e.  ZZ  /\  K  <  0 ) )  ->  ( A P K )  =  ( N `  (  seq 1 ( G ,  ( NN  X.  { A } ) ) `  -u K ) ) )
 
Theoremgx0 25987 The result of the group power operator when the exponent is zero. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  U  =  (GId `  G )   &    |-  P  =  ( ^g `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X ) 
 ->  ( A P 0 )  =  U )
 
Theoremgx1 25988 The result of the group power operator when the exponent is one. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  P  =  ( ^g `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X ) 
 ->  ( A P 1 )  =  A )
 
Theoremgxnn0neg 25989 A negative group power is the inverse of the positive power (lemma with nonnegative exponent - use gxneg 25992 instead). (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  N  =  ( inv `  G )   &    |-  P  =  ( ^g `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  NN0 )  ->  ( A P -u K )  =  ( N `  ( A P K ) ) )
 
Theoremgxnn0suc 25990 Induction on group power (lemma with nonnegative exponent - use gxsuc 25998 instead). (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  P  =  ( ^g `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  NN0 )  ->  ( A P ( K  +  1 ) )  =  ( ( A P K ) G A ) )
 
Theoremgxcl 25991 Closure of the group power operator. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  P  =  ( ^g `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  ZZ )  ->  ( A P K )  e.  X )
 
Theoremgxneg 25992 A negative group power is the inverse of the positive power. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  N  =  ( inv `  G )   &    |-  P  =  ( ^g `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  ZZ )  ->  ( A P -u K )  =  ( N `  ( A P K ) ) )
 
Theoremgxneg2 25993 The inverse of a negative group power is the positive power. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  N  =  ( inv `  G )   &    |-  P  =  ( ^g `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  ZZ )  ->  ( N `  ( A P -u K ) )  =  ( A P K ) )
 
Theoremgxm1 25994 The result of the group power operator when the exponent is minus one. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  N  =  ( inv `  G )   &    |-  P  =  ( ^g `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X ) 
 ->  ( A P -u 1 )  =  ( N `  A ) )
 
Theoremgxcom 25995 The group power operator commutes with the group operation. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  P  =  ( ^g `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  ZZ )  ->  ( ( A P K ) G A )  =  ( A G ( A P K ) ) )
 
Theoremgxinv 25996 The group power operator commutes with the group inverse function. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  N  =  ( inv `  G )   &    |-  P  =  ( ^g `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  ZZ )  ->  ( ( N `  A ) P K )  =  ( N `  ( A P K ) ) )
 
Theoremgxinv2 25997 The group power operator commutes with the group inverse function. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  N  =  ( inv `  G )   &    |-  P  =  ( ^g `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  ZZ )  ->  ( N `  (
 ( N `  A ) P K ) )  =  ( A P K ) )
 
Theoremgxsuc 25998 Induction on group power. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  P  =  ( ^g `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  ZZ )  ->  ( A P ( K  +  1 ) )  =  ( ( A P K ) G A ) )
 
Theoremgxid 25999 The identity element of a group to any power remains unchanged. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)
 |-  U  =  (GId `  G )   &    |-  P  =  ( ^g `  G )   =>    |-  ( ( G  e.  GrpOp  /\  K  e.  ZZ )  ->  ( U P K )  =  U )
 
Theoremgxnn0add 26000 The group power of a sum is the group product of the powers (lemma with nonnegative exponent - use gxadd 26001 instead). (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  P  =  ( ^g `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( J  e.  ZZ  /\  K  e.  NN0 )
 )  ->  ( A P ( J  +  K ) )  =  ( ( A P J ) G ( A P K ) ) )
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78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 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 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