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Theorem List for Metamath Proof Explorer - 15901-16000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremgrpinvinv 15901 Double inverse law for groups. Lemma 2.2.1(c) of [Herstein] p. 55. (Contributed by NM, 31-Mar-2014.)
 |-  B  =  ( Base `  G )   &    |-  N  =  ( invg `  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  B ) 
 ->  ( N `  ( N `  X ) )  =  X )
 
Theoremgrpinvcnv 15902 The group inverse is its own inverse function. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  B  =  ( Base `  G )   &    |-  N  =  ( invg `  G )   =>    |-  ( G  e.  Grp  ->  `' N  =  N )
 
Theoremgrpinv11 15903 The group inverse is one-to-one. (Contributed by NM, 22-Mar-2015.)
 |-  B  =  ( Base `  G )   &    |-  N  =  ( invg `  G )   &    |-  ( ph  ->  G  e.  Grp )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  (
 ( N `  X )  =  ( N `  Y )  <->  X  =  Y ) )
 
Theoremgrpinvf1o 15904 The group inverse is a one-to-one onto function. (Contributed by NM, 22-Oct-2014.) (Proof shortened by Mario Carneiro, 14-Aug-2015.)
 |-  B  =  ( Base `  G )   &    |-  N  =  ( invg `  G )   &    |-  ( ph  ->  G  e.  Grp )   =>    |-  ( ph  ->  N : B -1-1-onto-> B )
 
Theoremgrpinvnz 15905 The inverse of a nonzero group element is not zero. (Contributed by Stefan O'Rear, 27-Feb-2015.)
 |-  B  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  N  =  ( invg `
  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  B  /\  X  =/=  .0.  )  ->  ( N `  X )  =/=  .0.  )
 
Theoremgrpinvnzcl 15906 The inverse of a nonzero group element is a nonzero group element. (Contributed by Stefan O'Rear, 27-Feb-2015.)
 |-  B  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  N  =  ( invg `
  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  ( B  \  {  .0.  } ) )  ->  ( N `  X )  e.  ( B  \  {  .0.  } ) )
 
Theoremgrpsubinv 15907 Subtraction of an inverse. (Contributed by NM, 7-Apr-2015.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .-  =  ( -g `  G )   &    |-  N  =  ( invg `
  G )   &    |-  ( ph  ->  G  e.  Grp )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( X  .-  ( N `  Y ) )  =  ( X  .+  Y ) )
 
Theoremgrplmulf1o 15908* Left multiplication by a group element is a bijection on any group. (Contributed by Mario Carneiro, 17-Jan-2015.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  F  =  ( x  e.  B  |->  ( X  .+  x ) )   =>    |-  ( ( G  e.  Grp  /\  X  e.  B ) 
 ->  F : B -1-1-onto-> B )
 
Theoremgrpinvpropd 15909* If two structures have the same group components (properties), they have the same group inversion function. (Contributed by Mario Carneiro, 27-Nov-2014.) (Revised by Stefan O'Rear, 21-Mar-2015.)
 |-  ( ph  ->  B  =  ( Base `  K )
 )   &    |-  ( ph  ->  B  =  ( Base `  L )
 )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B )
 )  ->  ( x ( +g  `  K )
 y )  =  ( x ( +g  `  L ) y ) )   =>    |-  ( ph  ->  ( invg `
  K )  =  ( invg `  L ) )
 
Theoremgrpidssd 15910* If the base set of a group is contained in the base set of another group, and the group operation of the group is the restriction of the group operation of the other group to its base set, then both groups have the same identity element. (Contributed by AV, 15-Mar-2019.)
 |-  ( ph  ->  M  e.  Grp )   &    |-  ( ph  ->  S  e.  Grp )   &    |-  B  =  ( Base `  S )   &    |-  ( ph  ->  B  C_  ( Base `  M ) )   &    |-  ( ph  ->  A. x  e.  B  A. y  e.  B  ( x (
 +g  `  M )
 y )  =  ( x ( +g  `  S ) y ) )   =>    |-  ( ph  ->  ( 0g `  M )  =  ( 0g `  S ) )
 
Theoremgrpinvssd 15911* If the base set of a group is contained in the base set of another group, and the group operation of the group is the restriction of the group operation of the other group to its base set, then the elements of the first group have the same inverses in both groups. (Contributed by AV, 15-Mar-2019.)
 |-  ( ph  ->  M  e.  Grp )   &    |-  ( ph  ->  S  e.  Grp )   &    |-  B  =  ( Base `  S )   &    |-  ( ph  ->  B  C_  ( Base `  M ) )   &    |-  ( ph  ->  A. x  e.  B  A. y  e.  B  ( x (
 +g  `  M )
 y )  =  ( x ( +g  `  S ) y ) )   =>    |-  ( ph  ->  ( X  e.  B  ->  ( ( invg `  S ) `
  X )  =  ( ( invg `  M ) `  X ) ) )
 
Theoremgrpinvadd 15912 The inverse of the group operation reverses the arguments. Lemma 2.2.1(d) of [Herstein] p. 55. (Contributed by NM, 27-Oct-2006.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  N  =  ( invg `  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( N `  ( X  .+  Y ) )  =  ( ( N `
  Y )  .+  ( N `  X ) ) )
 
Theoremgrpsubf 15913 Functionality of group subtraction. (Contributed by Mario Carneiro, 9-Sep-2014.)
 |-  B  =  ( Base `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  ( G  e.  Grp 
 ->  .-  : ( B  X.  B ) --> B )
 
Theoremgrpsubcl 15914 Closure of group subtraction. (Contributed by NM, 31-Mar-2014.)
 |-  B  =  ( Base `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .-  Y )  e.  B )
 
Theoremgrpsubrcan 15915 Right cancellation law for group subtraction. (Contributed by NM, 31-Mar-2014.)
 |-  B  =  ( Base `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B ) )  ->  ( ( X  .-  Z )  =  ( Y  .-  Z )  <->  X  =  Y ) )
 
Theoremgrpinvsub 15916 Inverse of a group subtraction. (Contributed by NM, 9-Sep-2014.)
 |-  B  =  ( Base `  G )   &    |-  .-  =  ( -g `  G )   &    |-  N  =  ( invg `  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( N `  ( X  .-  Y ) )  =  ( Y  .-  X ) )
 
Theoremgrpinvval2 15917 A df-neg 9799-like equation for inverse in terms of group subtraction. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  B  =  ( Base `  G )   &    |-  .-  =  ( -g `  G )   &    |-  N  =  ( invg `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  B ) 
 ->  ( N `  X )  =  (  .0.  .-  X ) )
 
Theoremgrpsubid 15918 Subtraction of a group element from itself. (Contributed by NM, 31-Mar-2014.)
 |-  B  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  B ) 
 ->  ( X  .-  X )  =  .0.  )
 
Theoremgrpsubid1 15919 Subtraction of the identity from a group element. (Contributed by Mario Carneiro, 14-Jan-2015.)
 |-  B  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  B ) 
 ->  ( X  .-  .0.  )  =  X )
 
Theoremgrpsubeq0 15920 If the difference between two group elements is zero, they are equal. (subeq0 9836 analog.) (Contributed by NM, 31-Mar-2014.)
 |-  B  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .-  Y )  =  .0.  <->  X  =  Y ) )
 
Theoremgrpsubadd0sub 15921 Subtraction expressed as addition of the difference of the identity element and the subtrahend. (Contributed by AV, 9-Nov-2019.)
 |-  B  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  .-  =  ( -g `  G )   &    |- 
 .+  =  ( +g  `  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .-  Y )  =  ( X  .+  (  .0.  .-  Y ) ) )
 
Theoremgrpsubadd 15922 Relationship between group subtraction and addition. (Contributed by NM, 31-Mar-2014.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
 )  ->  ( ( X  .-  Y )  =  Z  <->  ( Z  .+  Y )  =  X ) )
 
Theoremgrpsubsub 15923 Double group subtraction. (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 2-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
 )  ->  ( X  .-  ( Y  .-  Z ) )  =  ( X  .+  ( Z  .-  Y ) ) )
 
Theoremgrpaddsubass 15924 Associative-type law for group subtraction and addition. (Contributed by NM, 16-Apr-2014.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
 )  ->  ( ( X  .+  Y )  .-  Z )  =  ( X  .+  ( Y  .-  Z ) ) )
 
Theoremgrppncan 15925 Cancellation law for subtraction (pncan 9817 analog). (Contributed by NM, 16-Apr-2014.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .+  Y )  .-  Y )  =  X )
 
Theoremgrpnpcan 15926 Cancellation law for subtraction (npcan 9820 analog). (Contributed by NM, 19-Apr-2014.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .-  Y )  .+  Y )  =  X )
 
Theoremgrpsubsub4 15927 Double group subtraction (subsub4 9843 analog). (Contributed by Mario Carneiro, 2-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
 )  ->  ( ( X  .-  Y )  .-  Z )  =  ( X  .-  ( Z  .+  Y ) ) )
 
Theoremgrppnpcan2 15928 Cancellation law for mixed addition and subtraction. (pnpcan2 9850 analog.) (Contributed by NM, 15-Feb-2008.) (Revised by Mario Carneiro, 2-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
 )  ->  ( ( X  .+  Z )  .-  ( Y  .+  Z ) )  =  ( X 
 .-  Y ) )
 
Theoremgrpnpncan 15929 Cancellation law for group subtraction. (npncan 9831 analog.) (Contributed by NM, 15-Feb-2008.) (Revised by Mario Carneiro, 2-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
 )  ->  ( ( X  .-  Y )  .+  ( Y  .-  Z ) )  =  ( X 
 .-  Z ) )
 
Theoremgrpnpncan0 15930 Cancellation law for group subtraction (npncan2 9837 analog). (Contributed by AV, 24-Nov-2019.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .-  =  ( -g `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  ( ( X  .-  Y )  .+  ( Y  .-  X ) )  =  .0.  )
 
Theoremgrpnnncan2 15931 Cancellation law for group subtraction. (nnncan2 9847 analog.) (Contributed by NM, 15-Feb-2008.) (Revised by Mario Carneiro, 2-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B ) )  ->  ( ( X  .-  Z )  .-  ( Y 
 .-  Z ) )  =  ( X  .-  Y ) )
 
Theoremgrplactfval 15932* The left group action of element  A of group  G. (Contributed by Paul Chapman, 18-Mar-2008.)
 |-  F  =  ( g  e.  X  |->  ( a  e.  X  |->  ( g 
 .+  a ) ) )   &    |-  X  =  (
 Base `  G )   =>    |-  ( A  e.  X  ->  ( F `  A )  =  (
 a  e.  X  |->  ( A  .+  a ) ) )
 
Theoremgrplactval 15933* The value of the left group action of element  A of group  G at  B. (Contributed by Paul Chapman, 18-Mar-2008.)
 |-  F  =  ( g  e.  X  |->  ( a  e.  X  |->  ( g 
 .+  a ) ) )   &    |-  X  =  (
 Base `  G )   =>    |-  ( ( A  e.  X  /\  B  e.  X )  ->  (
 ( F `  A ) `  B )  =  ( A  .+  B ) )
 
Theoremgrplactcnv 15934* The left group action of element  A of group  G maps the underlying set  X of  G one-to-one onto itself. (Contributed by Paul Chapman, 18-Mar-2008.) (Proof shortened by Mario Carneiro, 14-Aug-2015.)
 |-  F  =  ( g  e.  X  |->  ( a  e.  X  |->  ( g 
 .+  a ) ) )   &    |-  X  =  (
 Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  I  =  ( invg `  G )   =>    |-  ( ( G  e.  Grp  /\  A  e.  X ) 
 ->  ( ( F `  A ) : X -1-1-onto-> X  /\  `' ( F `  A )  =  ( F `  ( I `  A ) ) ) )
 
Theoremgrplactf1o 15935* The left group action of element  A of group  G maps the underlying set  X of  G one-to-one onto itself. (Contributed by Paul Chapman, 18-Mar-2008.) (Proof shortened by Mario Carneiro, 14-Aug-2015.)
 |-  F  =  ( g  e.  X  |->  ( a  e.  X  |->  ( g 
 .+  a ) ) )   &    |-  X  =  (
 Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  (
 ( G  e.  Grp  /\  A  e.  X ) 
 ->  ( F `  A ) : X -1-1-onto-> X )
 
Theoremgrpsubpropd 15936 Weak property deduction for the group subtraction operation. (Contributed by Mario Carneiro, 27-Mar-2015.)
 |-  ( ph  ->  ( Base `  G )  =  ( Base `  H )
 )   &    |-  ( ph  ->  ( +g  `  G )  =  ( +g  `  H ) )   =>    |-  ( ph  ->  ( -g `  G )  =  ( -g `  H ) )
 
Theoremgrpsubpropd2 15937* Strong property deduction for the group subtraction operation. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  ( ph  ->  B  =  ( Base `  G )
 )   &    |-  ( ph  ->  B  =  ( Base `  H )
 )   &    |-  ( ph  ->  G  e.  Grp )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B ) )  ->  ( x ( +g  `  G ) y )  =  ( x ( +g  `  H ) y ) )   =>    |-  ( ph  ->  ( -g `  G )  =  ( -g `  H ) )
 
Theoremmulgfval 15938* Group multiple (exponentiation) operation. (Contributed by Mario Carneiro, 11-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  I  =  ( invg `  G )   &    |-  .x. 
 =  (.g `  G )   =>    |-  .x.  =  ( n  e.  ZZ ,  x  e.  B  |->  if ( n  =  0 ,  .0.  ,  if ( 0  <  n ,  (  seq 1
 (  .+  ,  ( NN  X.  { x }
 ) ) `  n ) ,  ( I `  (  seq 1
 (  .+  ,  ( NN  X.  { x }
 ) ) `  -u n ) ) ) ) )
 
Theoremmulgval 15939 Group multiple (exponentiation) operation. (Contributed by Mario Carneiro, 11-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  I  =  ( invg `  G )   &    |-  .x. 
 =  (.g `  G )   &    |-  S  =  seq 1 (  .+  ,  ( NN  X.  { X } ) )   =>    |-  ( ( N  e.  ZZ  /\  X  e.  B )  ->  ( N  .x.  X )  =  if ( N  =  0 ,  .0.  ,  if ( 0  <  N ,  ( S `  N ) ,  ( I `  ( S `  -u N ) ) ) ) )
 
Theoremmulgfn 15940 Functionality of the group multiple function. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   =>    |- 
 .x.  Fn  ( ZZ  X.  B )
 
Theoremmulgfvi 15941 The group multiple function is compatible with identity-function protection. (Contributed by Mario Carneiro, 21-Mar-2015.)
 |- 
 .x.  =  (.g `  G )   =>    |- 
 .x.  =  (.g `  (  _I  `  G ) )
 
Theoremmulg0 15942 Group multiple (exponentiation) operation at zero. (Contributed by Mario Carneiro, 11-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  .x. 
 =  (.g `  G )   =>    |-  ( X  e.  B  ->  ( 0  .x.  X )  =  .0.  )
 
Theoremmulgnn 15943 Group multiple (exponentiation) operation at a positive integer. (Contributed by Mario Carneiro, 11-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .x.  =  (.g `  G )   &    |-  S  =  seq 1 (  .+  ,  ( NN  X.  { X }
 ) )   =>    |-  ( ( N  e.  NN  /\  X  e.  B )  ->  ( N  .x.  X )  =  ( S `
  N ) )
 
Theoremmulg1 15944 Group multiple (exponentiation) operation at one. (Contributed by Mario Carneiro, 11-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   =>    |-  ( X  e.  B  ->  ( 1  .x.  X )  =  X )
 
Theoremmulgnnp1 15945 Group multiple (exponentiation) operation at a successor. (Contributed by Mario Carneiro, 11-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( ( N  e.  NN  /\  X  e.  B )  ->  (
 ( N  +  1 )  .x.  X )  =  ( ( N  .x.  X )  .+  X ) )
 
Theoremmulg2 15946 Group multiple (exponentiation) operation at two. (Contributed by Mario Carneiro, 15-Oct-2015.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( X  e.  B  ->  ( 2  .x.  X )  =  ( X 
 .+  X ) )
 
Theoremmulgnegnn 15947 Group multiple (exponentiation) operation at a negative integer. (Contributed by Mario Carneiro, 11-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  I  =  ( invg `  G )   =>    |-  ( ( N  e.  NN  /\  X  e.  B )  ->  ( -u N  .x.  X )  =  ( I `  ( N 
 .x.  X ) ) )
 
Theoremmulgnn0p1 15948 Group multiple (exponentiation) operation at a successor, extended to  NN0. (Contributed by Mario Carneiro, 11-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( ( G  e.  Mnd  /\  N  e.  NN0  /\  X  e.  B ) 
 ->  ( ( N  +  1 )  .x.  X )  =  ( ( N 
 .x.  X )  .+  X ) )
 
Theoremmulgnnsubcl 15949* Closure of the group multiple (exponentiation) operation in a subsemigroup. (Contributed by Mario Carneiro, 10-Jan-2015.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  ( ph  ->  G  e.  V )   &    |-  ( ph  ->  S  C_  B )   &    |-  ( ( ph  /\  x  e.  S  /\  y  e.  S )  ->  ( x  .+  y
 )  e.  S )   =>    |-  ( ( ph  /\  N  e.  NN  /\  X  e.  S )  ->  ( N 
 .x.  X )  e.  S )
 
Theoremmulgnn0subcl 15950* Closure of the group multiple (exponentiation) operation in a submonoid. (Contributed by Mario Carneiro, 10-Jan-2015.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  ( ph  ->  G  e.  V )   &    |-  ( ph  ->  S  C_  B )   &    |-  ( ( ph  /\  x  e.  S  /\  y  e.  S )  ->  ( x  .+  y
 )  e.  S )   &    |-  .0.  =  ( 0g `  G )   &    |-  ( ph  ->  .0. 
 e.  S )   =>    |-  ( ( ph  /\  N  e.  NN0  /\  X  e.  S )  ->  ( N  .x.  X )  e.  S )
 
Theoremmulgsubcl 15951* Closure of the group multiple (exponentiation) operation in a subgroup. (Contributed by Mario Carneiro, 10-Jan-2015.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  ( ph  ->  G  e.  V )   &    |-  ( ph  ->  S  C_  B )   &    |-  ( ( ph  /\  x  e.  S  /\  y  e.  S )  ->  ( x  .+  y
 )  e.  S )   &    |-  .0.  =  ( 0g `  G )   &    |-  ( ph  ->  .0. 
 e.  S )   &    |-  I  =  ( invg `  G )   &    |-  ( ( ph  /\  x  e.  S ) 
 ->  ( I `  x )  e.  S )   =>    |-  (
 ( ph  /\  N  e.  ZZ  /\  X  e.  S )  ->  ( N  .x.  X )  e.  S )
 
Theoremmulgnncl 15952 Closure of the group multiple (exponentiation) operation. TODO: This can be generalized to a magma if/when we introduce them. (Contributed by Mario Carneiro, 11-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   =>    |-  ( ( G  e.  Mnd  /\  N  e.  NN  /\  X  e.  B )  ->  ( N  .x.  X )  e.  B )
 
Theoremmulgnn0cl 15953 Closure of the group multiple (exponentiation) operation. (Contributed by Mario Carneiro, 11-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   =>    |-  ( ( G  e.  Mnd  /\  N  e.  NN0  /\  X  e.  B )  ->  ( N  .x.  X )  e.  B )
 
Theoremmulgcl 15954 Closure of the group multiple (exponentiation) operation. (Contributed by Mario Carneiro, 11-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   =>    |-  ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  ->  ( N  .x.  X )  e.  B )
 
Theoremmulgneg 15955 Group multiple (exponentiation) operation at a negative integer. (Contributed by Mario Carneiro, 11-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  I  =  ( invg `  G )   =>    |-  ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  ->  ( -u N  .x.  X )  =  ( I `  ( N  .x.  X ) ) )
 
Theoremmulgm1 15956 Group multiple (exponentiation) operation at negative one. (Contributed by Mario Carneiro, 20-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  I  =  ( invg `  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  B ) 
 ->  ( -u 1  .x.  X )  =  ( I `  X ) )
 
Theoremmulgnn0z 15957 A group multiple of the identity, for nonnegative multiple. (Contributed by Mario Carneiro, 13-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( ( G  e.  Mnd  /\  N  e.  NN0 )  ->  ( N  .x.  .0.  )  =  .0.  )
 
Theoremmulgz 15958 A group multiple of the identity, for integer multiple. (Contributed by Mario Carneiro, 13-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( ( G  e.  Grp  /\  N  e.  ZZ )  ->  ( N  .x.  .0.  )  =  .0.  )
 
Theoremmulgnndir 15959 Sum of group multiples, for positive multiples. TODO: This can be generalized to a semigroup if/when we introduce them. (Contributed by Mario Carneiro, 11-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( ( G  e.  Mnd  /\  ( M  e.  NN  /\  N  e.  NN  /\  X  e.  B ) )  ->  ( ( M  +  N )  .x.  X )  =  ( ( M 
 .x.  X )  .+  ( N  .x.  X ) ) )
 
Theoremmulgnn0dir 15960 Sum of group multiples, generalized to  NN0. (Contributed by Mario Carneiro, 11-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B ) )  ->  ( ( M  +  N )  .x.  X )  =  ( ( M  .x.  X )  .+  ( N  .x.  X ) ) )
 
Theoremmulgdirlem 15961 Lemma for mulgdir 15962. (Contributed by Mario Carneiro, 13-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( ( G  e.  Grp  /\  ( M  e.  ZZ  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( M  +  N )  e. 
 NN0 )  ->  (
 ( M  +  N )  .x.  X )  =  ( ( M  .x.  X )  .+  ( N 
 .x.  X ) ) )
 
Theoremmulgdir 15962 Sum of group multiples, generalized to  ZZ. (Contributed by Mario Carneiro, 13-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( ( G  e.  Grp  /\  ( M  e.  ZZ  /\  N  e.  ZZ  /\  X  e.  B ) )  ->  ( ( M  +  N )  .x.  X )  =  ( ( M 
 .x.  X )  .+  ( N  .x.  X ) ) )
 
Theoremmulgp1 15963 Group multiple (exponentiation) operation at a successor, extended to  ZZ. (Contributed by Mario Carneiro, 11-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  ->  ( ( N  +  1 )  .x.  X )  =  ( ( N  .x.  X )  .+  X ) )
 
Theoremmulgneg2 15964 Group multiple (exponentiation) operation at a negative integer. (Contributed by Mario Carneiro, 13-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  I  =  ( invg `  G )   =>    |-  ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  ->  ( -u N  .x.  X )  =  ( N  .x.  ( I `  X ) ) )
 
Theoremmulgnnass 15965 Product of group multiples, for positive multiples. TODO: This can be generalized to a semigroup if/when we introduce them. (Contributed by Mario Carneiro, 13-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   =>    |-  ( ( G  e.  Mnd  /\  ( M  e.  NN  /\  N  e.  NN  /\  X  e.  B )
 )  ->  ( ( M  x.  N )  .x.  X )  =  ( M 
 .x.  ( N  .x.  X ) ) )
 
Theoremmulgnn0ass 15966 Product of group multiples, generalized to  NN0. (Contributed by Mario Carneiro, 13-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   =>    |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B ) )  ->  ( ( M  x.  N )  .x.  X )  =  ( M  .x.  ( N  .x.  X ) ) )
 
Theoremmulgass 15967 Product of group multiples, generalized to  ZZ. (Contributed by Mario Carneiro, 13-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   =>    |-  ( ( G  e.  Grp  /\  ( M  e.  ZZ  /\  N  e.  ZZ  /\  X  e.  B )
 )  ->  ( ( M  x.  N )  .x.  X )  =  ( M 
 .x.  ( N  .x.  X ) ) )
 
Theoremmulgsubdir 15968 Subtraction of a group element from itself. (Contributed by Mario Carneiro, 13-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  ( ( G  e.  Grp  /\  ( M  e.  ZZ  /\  N  e.  ZZ  /\  X  e.  B ) )  ->  ( ( M  -  N )  .x.  X )  =  ( ( M 
 .x.  X )  .-  ( N  .x.  X ) ) )
 
Theoremmhmmulg 15969 A homomorphism of monoids preserves group multiples. (Contributed by Mario Carneiro, 14-Jun-2015.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  .X.  =  (.g `  H )   =>    |-  ( ( F  e.  ( G MndHom  H )  /\  N  e.  NN0  /\  X  e.  B )  ->  ( F `  ( N  .x.  X ) )  =  ( N  .X.  ( F `  X ) ) )
 
Theoremmulgpropd 15970* Two structures with the same group-nature have the same group multiple function.  K is expected to either be  _V (when strong equality is available) or  B (when closure is available). (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |- 
 .x.  =  (.g `  G )   &    |- 
 .X.  =  (.g `  H )   &    |-  ( ph  ->  B  =  ( Base `  G )
 )   &    |-  ( ph  ->  B  =  ( Base `  H )
 )   &    |-  ( ph  ->  B  C_  K )   &    |-  ( ( ph  /\  ( x  e.  K  /\  y  e.  K ) )  ->  ( x ( +g  `  G ) y )  e.  K )   &    |-  ( ( ph  /\  ( x  e.  K  /\  y  e.  K ) )  ->  ( x ( +g  `  G ) y )  =  ( x ( +g  `  H ) y ) )   =>    |-  ( ph  ->  .x.  =  .X.  )
 
Theoremsubmmulgcl 15971 Closure of the group multiple (exponentiation) operation in a submonoid. (Contributed by Mario Carneiro, 13-Jan-2015.)
 |-  .xb  =  (.g `  G )   =>    |-  ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S ) 
 ->  ( N  .xb  X )  e.  S )
 
Theoremsubmmulg 15972 A group multiple is the same if evaluated in a submonoid. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  .xb  =  (.g `  G )   &    |-  H  =  ( Gs  S )   &    |-  .x.  =  (.g `  H )   =>    |-  ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  ->  ( N  .xb  X )  =  ( N  .x.  X ) )
 
Theoremprdsinvlem 15973* Characterization of inverses in a structure product. (Contributed by Mario Carneiro, 10-Jan-2015.)
 |-  Y  =  ( S
 X_s
 R )   &    |-  B  =  (
 Base `  Y )   &    |-  .+  =  ( +g  `  Y )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R : I --> Grp )   &    |-  ( ph  ->  F  e.  B )   &    |- 
 .0.  =  ( 0g  o.  R )   &    |-  N  =  ( y  e.  I  |->  ( ( invg `  ( R `  y ) ) `  ( F `
  y ) ) )   =>    |-  ( ph  ->  ( N  e.  B  /\  ( N  .+  F )  =  .0.  ) )
 
Theoremprdsgrpd 15974 The product of a family of groups is a group. (Contributed by Stefan O'Rear, 10-Jan-2015.)
 |-  Y  =  ( S
 X_s
 R )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  R : I --> Grp )   =>    |-  ( ph  ->  Y  e.  Grp )
 
Theoremprdsinvgd 15975* Negation in a product of groups. (Contributed by Stefan O'Rear, 10-Jan-2015.)
 |-  Y  =  ( S
 X_s
 R )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  R : I --> Grp )   &    |-  B  =  (
 Base `  Y )   &    |-  N  =  ( invg `  Y )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  ( N `  X )  =  ( x  e.  I  |->  ( ( invg `  ( R `
  x ) ) `
  ( X `  x ) ) ) )
 
Theorempwsgrp 15976 The product of a family of groups is a group. (Contributed by Mario Carneiro, 11-Jan-2015.)
 |-  Y  =  ( R 
 ^s  I )   =>    |-  ( ( R  e.  Grp  /\  I  e.  V )  ->  Y  e.  Grp )
 
Theorempwsinvg 15977 Negation in a group power. (Contributed by Mario Carneiro, 11-Jan-2015.)
 |-  Y  =  ( R 
 ^s  I )   &    |-  B  =  (
 Base `  Y )   &    |-  M  =  ( invg `  R )   &    |-  N  =  ( invg `  Y )   =>    |-  ( ( R  e.  Grp  /\  I  e.  V  /\  X  e.  B ) 
 ->  ( N `  X )  =  ( M  o.  X ) )
 
Theorempwssub 15978 Subtraction in a group power. (Contributed by Mario Carneiro, 12-Jan-2015.)
 |-  Y  =  ( R 
 ^s  I )   &    |-  B  =  (
 Base `  Y )   &    |-  M  =  ( -g `  R )   &    |-  .-  =  ( -g `  Y )   =>    |-  ( ( ( R  e.  Grp  /\  I  e.  V )  /\  ( F  e.  B  /\  G  e.  B )
 )  ->  ( F  .-  G )  =  ( F  oF M G ) )
 
Theorempwsmulg 15979 Value of a group multiple in a structure power. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  Y  =  ( R 
 ^s  I )   &    |-  B  =  (
 Base `  Y )   &    |-  .xb  =  (.g `  Y )   &    |-  .x.  =  (.g `  R )   =>    |-  ( ( ( R  e.  Mnd  /\  I  e.  V )  /\  ( N  e.  NN0  /\  X  e.  B  /\  A  e.  I ) )  ->  ( ( N  .xb  X ) `  A )  =  ( N  .x.  ( X `  A ) ) )
 
Theoremimasgrp2 15980* The image structure of a group is a group. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 5-Sep-2015.)
 |-  ( ph  ->  U  =  ( F  "s  R )
 )   &    |-  ( ph  ->  V  =  ( Base `  R )
 )   &    |-  ( ph  ->  .+  =  ( +g  `  R )
 )   &    |-  ( ph  ->  F : V -onto-> B )   &    |-  ( ( ph  /\  ( a  e.  V  /\  b  e.  V )  /\  ( p  e.  V  /\  q  e.  V ) )  ->  ( ( ( F `
  a )  =  ( F `  p )  /\  ( F `  b )  =  ( F `  q ) ) 
 ->  ( F `  (
 a  .+  b )
 )  =  ( F `
  ( p  .+  q ) ) ) )   &    |-  ( ph  ->  R  e.  W )   &    |-  (
 ( ph  /\  x  e.  V  /\  y  e.  V )  ->  ( x  .+  y )  e.  V )   &    |-  ( ( ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  ->  ( F `
  ( ( x 
 .+  y )  .+  z ) )  =  ( F `  ( x  .+  ( y  .+  z ) ) ) )   &    |-  ( ph  ->  .0. 
 e.  V )   &    |-  (
 ( ph  /\  x  e.  V )  ->  ( F `  (  .0.  .+  x ) )  =  ( F `  x ) )   &    |-  ( ( ph  /\  x  e.  V ) 
 ->  N  e.  V )   &    |-  ( ( ph  /\  x  e.  V )  ->  ( F `  ( N  .+  x ) )  =  ( F `  .0.  ) )   =>    |-  ( ph  ->  ( U  e.  Grp  /\  ( F `  .0.  )  =  ( 0g `  U ) ) )
 
Theoremimasgrp 15981* The image structure of a group is a group. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 5-Sep-2015.)
 |-  ( ph  ->  U  =  ( F  "s  R )
 )   &    |-  ( ph  ->  V  =  ( Base `  R )
 )   &    |-  ( ph  ->  .+  =  ( +g  `  R )
 )   &    |-  ( ph  ->  F : V -onto-> B )   &    |-  ( ( ph  /\  ( a  e.  V  /\  b  e.  V )  /\  ( p  e.  V  /\  q  e.  V ) )  ->  ( ( ( F `
  a )  =  ( F `  p )  /\  ( F `  b )  =  ( F `  q ) ) 
 ->  ( F `  (
 a  .+  b )
 )  =  ( F `
  ( p  .+  q ) ) ) )   &    |-  ( ph  ->  R  e.  Grp )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ph  ->  ( U  e.  Grp  /\  ( F `  .0.  )  =  ( 0g `  U ) ) )
 
Theoremimasgrpf1 15982 The image of a group under an injection is a group. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  U  =  ( F 
 "s 
 R )   &    |-  V  =  (
 Base `  R )   =>    |-  ( ( F : V -1-1-> B  /\  R  e.  Grp )  ->  U  e.  Grp )
 
Theoremdivsgrp2 15983* Prove that a quotient structure is a group. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  ( ph  ->  U  =  ( R  /.s  .~  ) )   &    |-  ( ph  ->  V  =  ( Base `  R )
 )   &    |-  ( ph  ->  .+  =  ( +g  `  R )
 )   &    |-  ( ph  ->  .~  Er  V )   &    |-  ( ph  ->  R  e.  X )   &    |-  ( ph  ->  ( ( a 
 .~  p  /\  b  .~  q )  ->  (
 a  .+  b )  .~  ( p  .+  q
 ) ) )   &    |-  (
 ( ph  /\  x  e.  V  /\  y  e.  V )  ->  ( x  .+  y )  e.  V )   &    |-  ( ( ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  ->  ( ( x  .+  y ) 
 .+  z )  .~  ( x  .+  ( y 
 .+  z ) ) )   &    |-  ( ph  ->  .0. 
 e.  V )   &    |-  (
 ( ph  /\  x  e.  V )  ->  (  .0.  .+  x )  .~  x )   &    |-  ( ( ph  /\  x  e.  V ) 
 ->  N  e.  V )   &    |-  ( ( ph  /\  x  e.  V )  ->  ( N  .+  x )  .~  .0.  )   =>    |-  ( ph  ->  ( U  e.  Grp  /\  [  .0.  ]  .~  =  ( 0g `  U ) ) )
 
Theoremxpsgrp 15984 The binary product of groups is a group. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  T  =  ( R  X.s  S )   =>    |-  ( ( R  e.  Grp  /\  S  e.  Grp )  ->  T  e.  Grp )
 
10.2.2  Subgroups and Quotient groups
 
Syntaxcsubg 15985 Extend class notation with all subgroups of a group.
 class SubGrp
 
Syntaxcnsg 15986 Extend class notation with all normal subgroups of a group.
 class NrmSGrp
 
Syntaxcqg 15987 Quotient group equivalence class.
 class ~QG
 
Definitiondf-subg 15988* Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 16006), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 16001), contains the neutral element of the group (see subg0 15997) and contains the inverses for all of its elements (see subginvcl 16000). (Contributed by Mario Carneiro, 2-Dec-2014.)
 |- SubGrp  =  ( w  e.  Grp  |->  { s  e.  ~P ( Base `  w )  |  ( ws  s )  e.  Grp } )
 
Definitiondf-nsg 15989* Define the equivalence relation in a quotient ring or quotient group (where  i is a two-sided ideal or a normal subgroup). For non-normal subgroups this generates the left cosets. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |- NrmSGrp  =  ( w  e.  Grp  |->  { s  e.  (SubGrp `  w )  |  [. ( Base `  w )  /  b ]. [. ( +g  `  w )  /  p ]. A. x  e.  b  A. y  e.  b  ( ( x p y )  e.  s  <->  ( y p x )  e.  s
 ) } )
 
Definitiondf-eqg 15990* Define the equivalence relation in a quotient ring or quotient group (where  i is a two-sided ideal or a normal subgroup). For non-normal subgroups this generates the left cosets. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |- ~QG  =  ( r  e.  _V ,  i  e.  _V  |->  {
 <. x ,  y >.  |  ( { x ,  y }  C_  ( Base `  r )  /\  (
 ( ( invg `  r ) `  x ) ( +g  `  r
 ) y )  e.  i ) } )
 
Theoremissubg 15991 The subgroup predicate. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |-  B  =  ( Base `  G )   =>    |-  ( S  e.  (SubGrp `  G )  <->  ( G  e.  Grp  /\  S  C_  B  /\  ( Gs  S )  e.  Grp ) )
 
Theoremsubgss 15992 A subgroup is a subset. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |-  B  =  ( Base `  G )   =>    |-  ( S  e.  (SubGrp `  G )  ->  S  C_  B )
 
Theoremsubgid 15993 A group is a subgroup of itself. (Contributed by Mario Carneiro, 7-Dec-2014.)
 |-  B  =  ( Base `  G )   =>    |-  ( G  e.  Grp  ->  B  e.  (SubGrp `  G ) )
 
Theoremsubggrp 15994 A subgroup is a group. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |-  H  =  ( Gs  S )   =>    |-  ( S  e.  (SubGrp `  G )  ->  H  e.  Grp )
 
Theoremsubgbas 15995 The base of the restricted group in a subgroup. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |-  H  =  ( Gs  S )   =>    |-  ( S  e.  (SubGrp `  G )  ->  S  =  ( Base `  H )
 )
 
Theoremsubgrcl 15996 Reverse closure for the subgroup predicate. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |-  ( S  e.  (SubGrp `  G )  ->  G  e.  Grp )
 
Theoremsubg0 15997 A subgroup of a group must have the same identity as the group. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  H  =  ( Gs  S )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( S  e.  (SubGrp `  G )  ->  .0.  =  ( 0g `  H ) )
 
Theoremsubginv 15998 The inverse of an element in a subgroup is the same as the inverse in the larger group. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |-  H  =  ( Gs  S )   &    |-  I  =  ( invg `  G )   &    |-  J  =  ( invg `  H )   =>    |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S )  ->  ( I `  X )  =  ( J `  X ) )
 
Theoremsubg0cl 15999 The group identity is an element of any subgroup. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |- 
 .0.  =  ( 0g `  G )   =>    |-  ( S  e.  (SubGrp `  G )  ->  .0.  e.  S )
 
Theoremsubginvcl 16000 The inverse of an element is closed in a subgroup. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |-  I  =  ( invg `  G )   =>    |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S )  ->  ( I `  X )  e.  S )
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