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Theorem List for Metamath Proof Explorer - 20801-20900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremeqid1 20801 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 20802 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
 )  =  (/)
 
14.4  (Future - to be reviewed and classified)
 
14.4.1  Planar incidence geometry
 
Syntaxcplig 20803 Extend class notation with the class of all planar incidence geometries.
 class  Plig
 
Definitiondf-plig 20804* 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 20805* 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 20806* 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 20807* 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 )
 
14.4.2  Algebra preliminaries
 
Syntaxcrpm 20808 Ring primes.
 class RPrime
 
Definitiondf-rprm 20809* 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 12750. (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 ) ) } )
 
14.4.3  Transitive closure
 
Syntaxctcl 20810 Extend class notation to include the transitive closure symbol.
 class 
 t +
 
Syntaxcrtcl 20811 Extend class notation with transitive closure.
 class 
 t *
 
Definitiondf-trcl 20812* Transitive closure of a relation. Experimental. (Contributed by FL, 27-Jun-2011.)
 |-  t +  =  ( x  e.  _V  |->  |^| { z  |  ( x 
 C_  z  /\  (
 z  o.  z ) 
 C_  z ) }
 )
 
Definitiondf-rtrcl 20813* Reflexive-transitive closure of a relation. Experimental. (Contributed by FL, 27-Jun-2011.)
 |-  t *  =  ( x  e.  _V  |->  |^| { z  |  ( (  _I  |`  ( dom  x  u.  ran  x )
 )  C_  z  /\  x  C_  z  /\  (
 z  o.  z ) 
 C_  z ) }
 )
 
PART 15  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 20862 shows the relationship between the older group definition and the extensible structure definition.

 
15.1  Additional material on Group theory
 
15.1.1  Definitions and basic properties for groups
 
Syntaxcgr 20814 Extend class notation with the class of all group operations.
 class  GrpOp
 
Syntaxcgi 20815 Extend class notation with a function mapping a group operation to the group's identity element.
 class GId
 
Syntaxcgn 20816 Extend class notation with a function mapping a group operation to the inverse function for the group.
 class  inv
 
Syntaxcgs 20817 Extend class notation with a function mapping a group operation to the division (or subtraction) operation for the group.
 class  /g
 
Syntaxcgx 20818 Extend class notation with a function mapping a group operation to the power operation for the group.
 class  ^g
 
Definitiondf-grpo 20819* 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 20820* 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 20821* 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 20822* 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 20823* 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 20824* 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 20825* 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 20826* 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 20827 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 20828 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 20829* 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 20830 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 20831 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 20832 Lemma for grpoidinv 20836. (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 20833 Lemma for grpoidinv 20836. (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 20834* Lemma for grpoidinv 20836. (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 20835* Lemma for grpoidinv 20836. (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 20836* 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 20837* 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 20838 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 20839 The empty set is not a group. (Contributed by NM, 25-Apr-2007.) (New usage is discouraged.)
 |- 
 -.  (/)  e.  GrpOp
 
Theoremgrporn 20840 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 20841* 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 20842 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 20843 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 20844* Lemma for grpoidcl 20845 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 20845 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 20846* 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 20847 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 20848 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 20849 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 20850* 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 20851 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 20852* 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 20853* 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 20854 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 20855 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 20856 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 20857 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 20858 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 20859 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 20860 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 20861 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 20862 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 20863* 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 20864* 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 20865 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 20866 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 20867 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 20868 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 20869 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 20870* 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 20871 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 20872 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 20873 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 20874 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 20875 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 20876 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 20877 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 20878 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 20879 Cancellation law for group division. (pncan 9025 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 20880 Cancellation law for group division. (npcan 9028 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 20881 Cancellation law for mixed addition and group division. (pnpcan2 9055 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 20882 Cancellation law for group division. (nnncan2 9052 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 20883 Cancellation law for group division. (npncan 9037 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 20884 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 20885* 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 20886 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 20887 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 20888 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 20889 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 20890 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 20891 A negative group power is the inverse of the positive power (lemma with nonnegative exponent - use gxneg 20894 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 20892 Induction on group power (lemma with nonnegative exponent - use gxsuc 20900 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 20893 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 20894 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 20895 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 20896 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 20897 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 20898 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 20899 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 20900 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 ) )
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