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Theorem List for Metamath Proof Explorer - 20901-21000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremgrpoinvid 20901 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 20902 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 20903 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 20904* 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 20905* 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 20906 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 20907 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 20908 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 20909 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 20910 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 20911* 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 20912 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 20913 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 20914 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 20915 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 20916 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 20917 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 20918 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 20919 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 20920 Cancellation law for group division. (pncan 9059 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 20921 Cancellation law for group division. (npcan 9062 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 20922 Cancellation law for mixed addition and group division. (pnpcan2 9089 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 20923 Cancellation law for group division. (nnncan2 9086 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 20924 Cancellation law for group division. (npncan 9071 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 20925 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 20926* 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 20927 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 20928 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 20929 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 20930 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 20931 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 20932 A negative group power is the inverse of the positive power (lemma with nonnegative exponent - use gxneg 20935 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 20933 Induction on group power (lemma with nonnegative exponent - use gxsuc 20941 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 20934 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 20935 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 20936 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 20937 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 20938 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 20939 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 20940 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 20941 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 20942 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 20943 The group power of a sum is the group product of the powers (lemma with nonnegative exponent - use gxadd 20944 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 ) ) )
 
Theoremgxadd 20944 The group power of a sum is the group product of the powers. (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.  ZZ )
 )  ->  ( A P ( J  +  K ) )  =  ( ( A P J ) G ( A P K ) ) )
 
Theoremgxsub 20945 The group power of a difference is the group quotient of the powers. (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  /\  ( J  e.  ZZ  /\  K  e.  ZZ )
 )  ->  ( A P ( J  -  K ) )  =  ( ( A P J ) G ( N `  ( A P K ) ) ) )
 
Theoremgxnn0mul 20946 The group power of a product is the composition of the powers (lemma with nonnegative exponent - use gxmul 20947 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  x.  K ) )  =  ( ( A P J ) P K ) )
 
Theoremgxmul 20947 The group power of a product is the composition of the powers. (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.  ZZ )
 )  ->  ( A P ( J  x.  K ) )  =  ( ( A P J ) P K ) )
 
Theoremgxmodid 20948 Casting out powers of the identity element leaves the group power unchanged. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  U  =  (GId `  G )   &    |-  P  =  ( ^g `  G )   =>    |-  ( ( G  e.  GrpOp  /\  ( K  e.  ZZ  /\  M  e.  NN )  /\  ( A  e.  X  /\  ( A P M )  =  U )
 )  ->  ( A P ( K  mod  M ) )  =  ( A P K ) )
 
Theoremresgrprn 20949 The underlying set of a group operation which is a restriction of a mapping. (Contributed by Paul Chapman, 25-Mar-2008.) (New usage is discouraged.)
 |-  H  =  ( G  |`  ( Y  X.  Y ) )   =>    |-  ( ( dom  G  =  ( X  X.  X )  /\  H  e.  GrpOp  /\  Y  C_  X )  ->  Y  =  ran  H )
 
15.1.2  Definition and basic properties of Abelian groups
 
Syntaxcablo 20950 Extend class notation with the class of all Abelian group operations.
 class  AbelOp
 
Definitiondf-ablo 20951* Define the class of all Abelian group operations. (Contributed by NM, 2-Nov-2006.) (New usage is discouraged.)
 |-  AbelOp  =  { g  e. 
 GrpOp  |  A. x  e. 
 ran  g A. y  e.  ran  g ( x g y )  =  ( y g x ) }
 
Theoremisablo 20952* The predicate "is an Abelian (commutative) group operation." (Contributed by NM, 2-Nov-2006.) (New usage is discouraged.)
 |-  X  =  ran  G   =>    |-  ( G  e.  AbelOp  <->  ( G  e.  GrpOp  /\  A. x  e.  X  A. y  e.  X  ( x G y )  =  ( y G x ) ) )
 
Theoremablogrpo 20953 An Abelian group operation is a group operation. (Contributed by NM, 2-Nov-2006.) (New usage is discouraged.)
 |-  ( G  e.  AbelOp  ->  G  e.  GrpOp )
 
Theoremablocom 20954 An Abelian group operation is commutative. (Contributed by NM, 2-Nov-2006.) (New usage is discouraged.)
 |-  X  =  ran  G   =>    |-  (
 ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  =  ( B G A ) )
 
Theoremablo32 20955 Commutative/associative law for Abelian groups. (Contributed by NM, 26-Apr-2007.) (New usage is discouraged.)
 |-  X  =  ran  G   =>    |-  (
 ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A G B ) G C )  =  ( ( A G C ) G B ) )
 
Theoremablo4 20956 Commutative/associative law for Abelian groups. (Contributed by NM, 26-Apr-2007.) (New usage is discouraged.)
 |-  X  =  ran  G   =>    |-  (
 ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X ) 
 /\  ( C  e.  X  /\  D  e.  X ) )  ->  ( ( A G B ) G ( C G D ) )  =  ( ( A G C ) G ( B G D ) ) )
 
Theoremisabloi 20957* Properties that determine an Abelian group operation. (Contributed by NM, 5-Nov-2006.) (New usage is discouraged.)
 |-  G  e.  GrpOp   &    |- 
 dom  G  =  ( X  X.  X )   &    |-  (
 ( x  e.  X  /\  y  e.  X )  ->  ( x G y )  =  ( y G x ) )   =>    |-  G  e.  AbelOp
 
Theoremablomuldiv 20958 Law for group multiplication and division. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  D  =  (  /g  `  G )   =>    |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A G B ) D C )  =  ( ( A D C ) G B ) )
 
Theoremablodivdiv 20959 Law for double group division. (Contributed by NM, 29-Feb-2008.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  D  =  (  /g  `  G )   =>    |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( A D ( B D C ) )  =  ( ( A D B ) G C ) )
 
Theoremablodivdiv4 20960 Law for double group division. (Contributed by NM, 29-Feb-2008.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  D  =  (  /g  `  G )   =>    |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A D B ) D C )  =  ( A D ( B G C ) ) )
 
Theoremablodiv32 20961 Swap the second and third terms in a double division. (Contributed by NM, 29-Feb-2008.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  D  =  (  /g  `  G )   =>    |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A D B ) D C )  =  ( ( A D C ) D B ) )
 
Theoremablonnncan 20962 Cancellation law for group division. (nnncan 9084 analog.) (Contributed by NM, 29-Feb-2008.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  D  =  (  /g  `  G )   =>    |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A D ( B D C ) ) D C )  =  ( A D B ) )
 
Theoremablonncan 20963 Cancellation law for group division. (nncan 9078 analog.) (Contributed by NM, 7-Mar-2008.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  D  =  (  /g  `  G )   =>    |-  ( ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A D ( A D B ) )  =  B )
 
Theoremablonnncan1 20964 Cancellation law for group division. (nnncan1 9085 analog.) (Contributed by NM, 7-Mar-2008.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  D  =  (  /g  `  G )   =>    |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A D B ) D ( A D C ) )  =  ( C D B ) )
 
Theoremgxdi 20965 Distribution of group power over group operation for abelian groups. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  P  =  ( ^g `  G )   =>    |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X ) 
 /\  K  e.  ZZ )  ->  ( ( A G B ) P K )  =  ( ( A P K ) G ( B P K ) ) )
 
Theoremisgrpda 20966* Properties that determine a group operation. (Contributed by Jeff Madsen, 1-Dec-2009.) (New usage is discouraged.)
 |-  ( ph  ->  X  e.  _V )   &    |-  ( 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  ->  U  e.  X )   &    |-  (
 ( ph  /\  x  e.  X )  ->  ( U G x )  =  x )   &    |-  ( ( ph  /\  x  e.  X ) 
 ->  E. n  e.  X  ( n G x )  =  U )   =>    |-  ( ph  ->  G  e.  GrpOp )
 
Theoremisgrpod 20967* Properties that determine a group operation. (Renamed from isgrpd 14509 to isgrpod 20967 to prevent naming conflict. -NM 5-Jun-2013) (Contributed by Jeff Madsen, 1-Dec-2009.) (New usage is discouraged.)
 |-  ( ph  ->  X  e.  _V )   &    |-  ( 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  ->  U  e.  X )   &    |-  (
 ( ph  /\  x  e.  X )  ->  ( U G x )  =  x )   &    |-  ( ( ph  /\  x  e.  X ) 
 ->  N  e.  X )   &    |-  ( ( ph  /\  x  e.  X )  ->  ( N G x )  =  U )   =>    |-  ( ph  ->  G  e.  GrpOp )
 
Theoremisabloda 20968* Properties that determine an Abelian group operation. (Contributed by Jeff Madsen, 11-Jun-2010.) (New usage is discouraged.)
 |-  ( ph  ->  X  e.  _V )   &    |-  ( 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  ->  U  e.  X )   &    |-  (
 ( ph  /\  x  e.  X )  ->  ( U G x )  =  x )   &    |-  ( ( ph  /\  x  e.  X ) 
 ->  E. n  e.  X  ( n G x )  =  U )   &    |-  (
 ( ph  /\  ( x  e.  X  /\  y  e.  X ) )  ->  ( x G y )  =  ( y G x ) )   =>    |-  ( ph  ->  G  e.  AbelOp )
 
Theoremisablod 20969* Properties that determine an Abelian group operation. (Changed label from isabld 15104 to isablod 20969-NM 6-Aug-2013.) (Contributed by Jeff Madsen, 5-Dec-2009.) (New usage is discouraged.)
 |-  ( ph  ->  X  e.  _V )   &    |-  ( 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  ->  U  e.  X )   &    |-  (
 ( ph  /\  x  e.  X )  ->  ( U G x )  =  x )   &    |-  ( ( ph  /\  x  e.  X ) 
 ->  N  e.  X )   &    |-  ( ( ph  /\  x  e.  X )  ->  ( N G x )  =  U )   &    |-  ( ( ph  /\  ( x  e.  X  /\  y  e.  X ) )  ->  ( x G y )  =  ( y G x ) )   =>    |-  ( ph  ->  G  e.  AbelOp )
 
15.1.3  Subgroups
 
Syntaxcsubgo 20970 Extend class notation to include the class of subgroups.
 class  SubGrpOp
 
Definitiondf-subgo 20971 Define the set of subgroups of  g. (Contributed by Paul Chapman, 3-Mar-2008.) (New usage is discouraged.)
 |-  SubGrpOp 
 =  ( g  e. 
 GrpOp  |->  ( GrpOp  i^i  ~P g ) )
 
Theoremissubgo 20972 The predicate "is a subgroup of  G." (Contributed by Paul Chapman, 3-Mar-2008.) (Revised by Mario Carneiro, 12-Jul-2014.) (New usage is discouraged.)
 |-  ( H  e.  ( SubGrpOp `  G )  <->  ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  H  C_  G ) )
 
Theoremsubgores 20973 A subgroup operation is the restriction of its parent group operation to its underlying set. (Contributed by Paul Chapman, 3-Mar-2008.) (New usage is discouraged.)
 |-  W  =  ran  H   =>    |-  ( H  e.  ( SubGrpOp `  G )  ->  H  =  ( G  |`  ( W  X.  W ) ) )
 
Theoremsubgoov 20974 The result of a subgroup operation is the same as the result of its parent operation. (Contributed by Paul Chapman, 3-Mar-2008.) (Revised by Mario Carneiro, 8-Jul-2014.) (New usage is discouraged.)
 |-  W  =  ran  H   =>    |-  (
 ( H  e.  ( SubGrpOp `  G )  /\  ( A  e.  W  /\  B  e.  W )
 )  ->  ( A H B )  =  ( A G B ) )
 
Theoremsubgornss 20975 The underlying set of a subgroup is a subset of its parent group's underlying set. (Contributed by Paul Chapman, 3-Mar-2008.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  W  =  ran  H   =>    |-  ( H  e.  ( SubGrpOp `  G )  ->  W  C_  X )
 
Theoremsubgoid 20976 The identity element of a subgroup is the same as its parent's. (Contributed by Paul Chapman, 3-Mar-2008.) (New usage is discouraged.)
 |-  U  =  (GId `  G )   &    |-  T  =  (GId `  H )   =>    |-  ( H  e.  ( SubGrpOp `  G )  ->  T  =  U )
 
Theoremsubgoinv 20977 The inverse of a subgroup element is the same as its inverse in the parent group. (Contributed by Mario Carneiro, 8-Jul-2014.) (New usage is discouraged.)
 |-  W  =  ran  H   &    |-  M  =  ( inv `  G )   &    |-  N  =  ( inv `  H )   =>    |-  ( ( H  e.  ( SubGrpOp `  G )  /\  A  e.  W ) 
 ->  ( N `  A )  =  ( M `  A ) )
 
Theoremissubgoilem 20978* Lemma for issubgoi 20979. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.)
 |-  ( ( x  e.  Y  /\  y  e.  Y )  ->  ( x H y )  =  ( x G y ) )   =>    |-  ( ( A  e.  Y  /\  B  e.  Y )  ->  ( A H B )  =  ( A G B ) )
 
Theoremissubgoi 20979* Properties that determine a subgroup. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.)
 |-  G  e.  GrpOp   &    |-  X  =  ran  G   &    |-  U  =  (GId `  G )   &    |-  N  =  ( inv `  G )   &    |-  Y  C_  X   &    |-  H  =  ( G  |`  ( Y  X.  Y ) )   &    |-  (
 ( x  e.  Y  /\  y  e.  Y )  ->  ( x G y )  e.  Y )   &    |-  U  e.  Y   &    |-  ( x  e.  Y  ->  ( N `  x )  e.  Y )   =>    |-  H  e.  ( SubGrpOp `  G )
 
Theoremsubgoablo 20980 A subgroup of an Abelian group is Abelian. (Contributed by Paul Chapman, 25-Apr-2008.) (New usage is discouraged.)
 |-  ( ( G  e.  AbelOp  /\  H  e.  ( SubGrpOp `  G ) )  ->  H  e.  AbelOp )
 
15.1.4  Operation properties
 
Syntaxcass 20981 Extend class notation with a device to add associativity to internal operations.
 class  Ass
 
Definitiondf-ass 20982* A device to add associativity to various sorts of internal operations. The definition is meaningful when  g is a magma at least. (Contributed by FL, 1-Nov-2009.) (New usage is discouraged.)
 |- 
 Ass  =  { g  |  A. x  e.  dom  dom  g A. y  e. 
 dom  dom  g A. z  e.  dom  dom  g (
 ( x g y ) g z )  =  ( x g ( y g z ) ) }
 
Syntaxcexid 20983 Extend class notation with the class of all the internal operations with an identity element.
 class  ExId
 
Definitiondf-exid 20984* A device to add an identity element to various sorts of internal operations. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
 |- 
 ExId  =  { g  |  E. x  e.  dom  dom  g A. y  e. 
 dom  dom  g ( ( x g y )  =  y  /\  (
 y g x )  =  y ) }
 
Theoremisass 20985* The predicate "is an associative operation". (Contributed by FL, 1-Nov-2009.) (New usage is discouraged.)
 |-  X  =  dom  dom  G   =>    |-  ( G  e.  A  ->  ( G  e.  Ass  <->  A. x  e.  X  A. y  e.  X  A. z  e.  X  ( ( x G y ) G z )  =  ( x G ( y G z ) ) ) )
 
Theoremisexid 20986* The predicate  G has a left and right identity element. (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
 |-  X  =  dom  dom  G   =>    |-  ( G  e.  A  ->  ( G  e.  ExId  <->  E. x  e.  X  A. y  e.  X  ( ( x G y )  =  y  /\  ( y G x )  =  y ) ) )
 
15.1.5  Group-like structures
 
Syntaxcmagm 20987 Extend class notation with the class of all magmas.
 class  Magma
 
Definitiondf-mgm 20988* A magma is a binary internal operation. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
 |- 
 Magma  =  { g  |  E. t  g : ( t  X.  t
 ) --> t }
 
Theoremismgm 20989 The predicate "is a magma". (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
 |-  X  =  dom  dom  G   =>    |-  ( G  e.  A  ->  ( G  e.  Magma  <->  G : ( X  X.  X ) --> X ) )
 
Theoremclmgm 20990 Closure of a magma. (Contributed by FL, 14-Sep-2010.) (New usage is discouraged.)
 |-  X  =  dom  dom  G   =>    |-  ( ( G  e.  Magma  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  e.  X )
 
Theoremopidon 20991 An operation with a left and right identity element is onto. (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
 |-  X  =  dom  dom  G   =>    |-  ( G  e.  ( Magma  i^i  ExId  )  ->  G : ( X  X.  X ) -onto-> X )
 
Theoremrngopid 20992 Range of an operation with a left and right identity element. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
 |-  ( G  e.  ( Magma  i^i  ExId  )  ->  ran 
 G  =  dom  dom  G )
 
Theoremopidon2 20993 An operation with a left and right identity element is onto. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
 |-  X  =  ran  G   =>    |-  ( G  e.  ( Magma  i^i  ExId  )  ->  G :
 ( X  X.  X ) -onto-> X )
 
Theoremisexid2 20994* If  G  e.  ( Magma  i^i  ExId  ), then it has a left and right identity element that belongs to the range of the operation. (Contributed by FL, 12-Dec-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
 |-  X  =  ran  G   =>    |-  ( G  e.  ( Magma  i^i  ExId  )  ->  E. u  e.  X  A. x  e.  X  ( ( u G x )  =  x  /\  ( x G u )  =  x ) )
 
Theoremexidu1 20995* Unicity of the left and right identity element of a magma when it exists. (Contributed by FL, 12-Dec-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
 |-  X  =  ran  G   =>    |-  ( G  e.  ( Magma  i^i  ExId  )  ->  E! u  e.  X  A. x  e.  X  ( ( u G x )  =  x  /\  ( x G u )  =  x ) )
 
Theoremidrval 20996* The value of the identity element. (Contributed by FL, 12-Dec-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  U  =  (GId `  G )   =>    |-  ( G  e.  A  ->  U  =  ( iota_ u  e.  X A. x  e.  X  ( ( u G x )  =  x  /\  ( x G u )  =  x ) ) )
 
Theoremiorlid 20997 A magma right and left identity belongs to the underlying set of the operation. (Contributed by FL, 12-Dec-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  U  =  (GId `  G )   =>    |-  ( G  e.  ( Magma  i^i  ExId  )  ->  U  e.  X )
 
Theoremcmpidelt 20998 A magma right and left identity element keeps the other elements unchanged. (Contributed by FL, 12-Dec-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  U  =  (GId `  G )   =>    |-  (
 ( G  e.  ( Magma  i^i  ExId  )  /\  A  e.  X )  ->  ( ( U G A )  =  A  /\  ( A G U )  =  A )
 )
 
Syntaxcsem 20999 Extend class notation with the class of all semi-groups.
 class  SemiGrp
 
Definitiondf-sgr 21000 A semi-group is an associative magma. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
 |-  SemiGrp  =  ( Magma  i^i  Ass )
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