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Theorem List for Metamath Proof Explorer - 14501-14600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremisgrpd 14501* Deduce a group from its properties. Unlike isgrpd2 14499, this one goes straight from the base properties rather than going through  Mnd.  N (negative) is normally dependent on  x i.e. read it as  N ( x ). (Contributed by NM, 6-Jun-2013.) (Revised by Mario Carneiro, 6-Jan-2015.)
 |-  ( ph  ->  B  =  ( Base `  G )
 )   &    |-  ( ph  ->  .+  =  ( +g  `  G )
 )   &    |-  ( ( ph  /\  x  e.  B  /\  y  e.  B )  ->  ( x  .+  y )  e.  B )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  ->  ( ( x  .+  y ) 
 .+  z )  =  ( x  .+  (
 y  .+  z )
 ) )   &    |-  ( ph  ->  .0. 
 e.  B )   &    |-  (
 ( ph  /\  x  e.  B )  ->  (  .0.  .+  x )  =  x )   &    |-  ( ( ph  /\  x  e.  B ) 
 ->  N  e.  B )   &    |-  ( ( ph  /\  x  e.  B )  ->  ( N  .+  x )  =  .0.  )   =>    |-  ( ph  ->  G  e.  Grp )
 
Theoremisgrpi 14502* Properties that determine a group. 
N (negative) is normally dependent on  x i.e. read it as  N ( x ). (Contributed by NM, 3-Sep-2011.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  (
 ( x  e.  B  /\  y  e.  B )  ->  ( x  .+  y )  e.  B )   &    |-  ( ( x  e.  B  /\  y  e.  B  /\  z  e.  B )  ->  (
 ( x  .+  y
 )  .+  z )  =  ( x  .+  (
 y  .+  z )
 ) )   &    |-  .0.  e.  B   &    |-  ( x  e.  B  ->  (  .0.  .+  x )  =  x )   &    |-  ( x  e.  B  ->  N  e.  B )   &    |-  ( x  e.  B  ->  ( N  .+  x )  =  .0.  )   =>    |-  G  e.  Grp
 
Theoremisgrpix 14503* Properties that determine a group. Read  N as  N ( x ). Note: This theorem has hard-coded structure indices for demonstration purposes. It is not intended for general use. (New usage is discouraged.) (Contributed by NM, 4-Sep-2011.)
 |-  B  e.  _V   &    |-  .+  e.  _V   &    |-  G  =  { <. 1 ,  B >. ,  <. 2 ,  .+  >. }   &    |-  ( ( x  e.  B  /\  y  e.  B )  ->  ( x  .+  y )  e.  B )   &    |-  ( ( x  e.  B  /\  y  e.  B  /\  z  e.  B )  ->  (
 ( x  .+  y
 )  .+  z )  =  ( x  .+  (
 y  .+  z )
 ) )   &    |-  .0.  e.  B   &    |-  ( x  e.  B  ->  (  .0.  .+  x )  =  x )   &    |-  ( x  e.  B  ->  N  e.  B )   &    |-  ( x  e.  B  ->  ( N  .+  x )  =  .0.  )   =>    |-  G  e.  Grp
 
Theoremgrpidcl 14504 The identity element of a group belongs to the group. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( G  e.  Grp  ->  .0.  e.  B )
 
Theoremgrpbn0 14505 The base set of a group is not empty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.)
 |-  B  =  ( Base `  G )   =>    |-  ( G  e.  Grp  ->  B  =/=  (/) )
 
Theoremgrplid 14506 The identity element of a group is a left identity. (Contributed by NM, 18-Aug-2011.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  B ) 
 ->  (  .0.  .+  X )  =  X )
 
Theoremgrprid 14507 The identity element of a group is a right identity. (Contributed by NM, 18-Aug-2011.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  B ) 
 ->  ( X  .+  .0.  )  =  X )
 
Theoremgrpn0 14508 A group is not empty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.) (Revised by Mario Carneiro, 2-Dec-2014.)
 |-  ( G  e.  Grp  ->  G  =/=  (/) )
 
Theoremgrprcan 14509 Right cancellation law for groups. (Contributed by NM, 24-Aug-2011.) (Proof shortened by Mario Carneiro, 6-Jan-2015.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B ) )  ->  ( ( X  .+  Z )  =  ( Y  .+  Z )  <->  X  =  Y ) )
 
Theoremgrpinveu 14510* The left inverse element of a group is unique. Lemma 2.2.1(b) of [Herstein] p. 55. (Contributed by NM, 24-Aug-2011.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  B ) 
 ->  E! y  e.  B  ( y  .+  X )  =  .0.  )
 
Theoremgrpid 14511 Two ways of saying that an element of a group is the identity element. Provides a convenient way to compute the value of the identity element. (Contributed by NM, 24-Aug-2011.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  B ) 
 ->  ( ( X  .+  X )  =  X  <->  .0. 
 =  X ) )
 
Theoremisgrpid2 14512 Properties showing that an element 
Z is the identity element of a group. (Contributed by NM, 7-Aug-2013.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( G  e.  Grp  ->  ( ( Z  e.  B  /\  ( Z  .+  Z )  =  Z ) 
 <->  .0.  =  Z ) )
 
Theoremgrpidd2 14513* Deduce the identity element of a group from its properties. Useful in conjunction with isgrpd 14501. (Contributed by Mario Carneiro, 14-Jun-2015.)
 |-  ( ph  ->  B  =  ( Base `  G )
 )   &    |-  ( ph  ->  .+  =  ( +g  `  G )
 )   &    |-  ( ph  ->  .0.  e.  B )   &    |-  ( ( ph  /\  x  e.  B ) 
 ->  (  .0.  .+  x )  =  x )   &    |-  ( ph  ->  G  e.  Grp )   =>    |-  ( ph  ->  .0.  =  ( 0g `  G ) )
 
Theoremgrpinvfval 14514* The inverse function of a group. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 7-Aug-2013.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  N  =  ( inv
 g `  G )   =>    |-  N  =  ( x  e.  B  |->  ( iota_ y  e.  B ( y  .+  x )  =  .0.  ) )
 
Theoremgrpinvval 14515* The inverse of a group element. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 7-Aug-2013.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  N  =  ( inv
 g `  G )   =>    |-  ( X  e.  B  ->  ( N `  X )  =  ( iota_ y  e.  B ( y  .+  X )  =  .0.  ) )
 
Theoremgrpinvfn 14516 Functionality of the group inverse function. (Contributed by Stefan O'Rear, 21-Mar-2015.)
 |-  B  =  ( Base `  G )   &    |-  N  =  ( inv g `  G )   =>    |-  N  Fn  B
 
Theoremgrpinvfvi 14517 The group inverse function is compatible with identity-function protection. (Contributed by Stefan O'Rear, 21-Mar-2015.)
 |-  N  =  ( inv
 g `  G )   =>    |-  N  =  ( inv g `  (  _I  `  G )
 )
 
Theoremgrpsubfval 14518* Group subtraction (division) operation. (Contributed by NM, 31-Mar-2014.) (Revised by Stefan O'Rear, 27-Mar-2015.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  I  =  ( inv g `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  .-  =  ( x  e.  B ,  y  e.  B  |->  ( x 
 .+  ( I `  y ) ) )
 
Theoremgrpsubval 14519 Group subtraction (division) operation. (Contributed by NM, 31-Mar-2014.) (Revised by Mario Carneiro, 13-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  I  =  ( inv g `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( X  .-  Y )  =  ( X  .+  ( I `  Y ) ) )
 
Theoremgrpinvf 14520 The group inversion operation is a function on the base set. (Contributed by Mario Carneiro, 4-May-2015.)
 |-  B  =  ( Base `  G )   &    |-  N  =  ( inv g `  G )   =>    |-  ( G  e.  Grp  ->  N : B --> B )
 
Theoremgrpinvcl 14521 A group element's inverse is a group element. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 4-May-2015.)
 |-  B  =  ( Base `  G )   &    |-  N  =  ( inv g `  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  B ) 
 ->  ( N `  X )  e.  B )
 
Theoremgrplinv 14522 The left inverse of a group element. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  N  =  ( inv
 g `  G )   =>    |-  (
 ( G  e.  Grp  /\  X  e.  B ) 
 ->  ( ( N `  X )  .+  X )  =  .0.  )
 
Theoremgrprinv 14523 The right inverse of a group element. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  N  =  ( inv
 g `  G )   =>    |-  (
 ( G  e.  Grp  /\  X  e.  B ) 
 ->  ( X  .+  ( N `  X ) )  =  .0.  )
 
Theoremgrpinvid1 14524 The inverse of a group element expressed in terms of the identity element. (Contributed by NM, 24-Aug-2011.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  N  =  ( inv
 g `  G )   =>    |-  (
 ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( N `  X )  =  Y  <->  ( X  .+  Y )  =  .0.  ) )
 
Theoremgrpinvid2 14525 The inverse of a group element expressed in terms of the identity element. (Contributed by NM, 24-Aug-2011.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  N  =  ( inv
 g `  G )   =>    |-  (
 ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( N `  X )  =  Y  <->  ( Y  .+  X )  =  .0.  ) )
 
Theoremisgrpinv 14526* Properties showing that a function 
M is the inverse function of a group. (Contributed by NM, 7-Aug-2013.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  N  =  ( inv
 g `  G )   =>    |-  ( G  e.  Grp  ->  (
 ( M : B --> B  /\  A. x  e.  B  ( ( M `
  x )  .+  x )  =  .0.  ) 
 <->  N  =  M ) )
 
Theoremgrpinvid 14527 The inverse of the identity element of a group. (Contributed by NM, 24-Aug-2011.)
 |- 
 .0.  =  ( 0g `  G )   &    |-  N  =  ( inv g `  G )   =>    |-  ( G  e.  Grp  ->  ( N `  .0.  )  =  .0.  )
 
Theoremgrplcan 14528 Left cancellation law for groups. (Contributed by NM, 25-Aug-2011.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B ) )  ->  ( ( Z  .+  X )  =  ( Z  .+  Y )  <->  X  =  Y ) )
 
Theoremgrpinvinv 14529 Double inverse law for groups. Lemma 2.2.1(c) of [Herstein] p. 55. (Contributed by NM, 31-Mar-2014.)
 |-  B  =  ( Base `  G )   &    |-  N  =  ( inv g `  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  B ) 
 ->  ( N `  ( N `  X ) )  =  X )
 
Theoremgrpinvcnv 14530 The group inverse is its own inverse function. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  B  =  ( Base `  G )   &    |-  N  =  ( inv g `  G )   =>    |-  ( G  e.  Grp  ->  `' N  =  N )
 
Theoremgrpinv11 14531 The group inverse is one-to-one. (Contributed by NM, 22-Mar-2015.)
 |-  B  =  ( Base `  G )   &    |-  N  =  ( inv g `  G )   &    |-  ( ph  ->  G  e.  Grp )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  (
 ( N `  X )  =  ( N `  Y )  <->  X  =  Y ) )
 
Theoremgrpinvf1o 14532 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  =  ( inv g `  G )   &    |-  ( ph  ->  G  e.  Grp )   =>    |-  ( ph  ->  N : B -1-1-onto-> B )
 
Theoremgrpinvnz 14533 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  =  ( inv g `
  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  B  /\  X  =/=  .0.  )  ->  ( N `  X )  =/=  .0.  )
 
Theoremgrpinvnzcl 14534 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  =  ( inv g `
  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  ( B  \  {  .0.  } ) )  ->  ( N `  X )  e.  ( B  \  {  .0.  } ) )
 
Theoremgrpsubinv 14535 Subtraction of an inverse. (Contributed by NM, 7-Apr-2015.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .-  =  ( -g `  G )   &    |-  N  =  ( inv g `
  G )   &    |-  ( ph  ->  G  e.  Grp )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( X  .-  ( N `  Y ) )  =  ( X  .+  Y ) )
 
Theoremgrplmulf1o 14536* Left multiplication by a 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 14537* 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  ->  ( inv g `
  K )  =  ( inv g `  L ) )
 
Theoremgrpinvadd 14538 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  =  ( inv g `  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( N `  ( X  .+  Y ) )  =  ( ( N `
  Y )  .+  ( N `  X ) ) )
 
Theoremgrpsubf 14539 Functionality of group subtraction. (Contributed by Mario Carneiro, 9-Sep-2014.)
 |-  B  =  ( Base `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  ( G  e.  Grp 
 ->  .-  : ( B  X.  B ) --> B )
 
Theoremgrpsubcl 14540 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 14541 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 14542 Inverse of a group subtraction. (Contributed by NM, 9-Sep-2014.)
 |-  B  =  ( Base `  G )   &    |-  .-  =  ( -g `  G )   &    |-  N  =  ( inv g `  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( N `  ( X  .-  Y ) )  =  ( Y  .-  X ) )
 
Theoremgrpinvval2 14543 A df-neg 9035-like equation for inverse in terms of group subtraction. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  B  =  ( Base `  G )   &    |-  .-  =  ( -g `  G )   &    |-  N  =  ( inv g `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  B ) 
 ->  ( N `  X )  =  (  .0.  .-  X ) )
 
Theoremgrpsubid 14544 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 14545 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 14546 If the difference between two group elements is zero, they are equal. (subeq0 9068 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 ) )
 
Theoremgrpsubadd 14547 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 14548 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 14549 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 14550 Cancellation law for subtraction (pncan 9052 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 14551 Cancellation law for subtraction (npcan 9055 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 14552 Double group subtraction (subsub4 9075 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 14553 Cancellation law for mixed addition and subtraction. (pnpcan2 9082 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 14554 Cancellation law for group subtraction. (npncan 9064 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 ) )
 
Theoremgrpnnncan2 14555 Cancellation law for group subtraction. (nnncan2 9079 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 14556* 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 14557* 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 14558* 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  =  ( inv g `  G )   =>    |-  ( ( G  e.  Grp  /\  A  e.  X ) 
 ->  ( ( F `  A ) : X -1-1-onto-> X  /\  `' ( F `  A )  =  ( F `  ( I `  A ) ) ) )
 
Theoremgrplactf1o 14559* 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 14560 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 14561* 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 14562* Group multiple (exponentiation) operation. (Contributed by Mario Carneiro, 11-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  I  =  ( inv
 g `  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 14563 Group multiple (exponentiation) operation. (Contributed by Mario Carneiro, 11-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  I  =  ( inv
 g `  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 14564 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 14565 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 14566 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 14567 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 14568 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 14569 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 14570 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 14571 Group multiple (exponentiation) operation at a negative integer. (Contributed by Mario Carneiro, 11-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  I  =  ( inv g `  G )   =>    |-  ( ( N  e.  NN  /\  X  e.  B )  ->  ( -u N  .x.  X )  =  ( I `  ( N 
 .x.  X ) ) )
 
Theoremmulgnn0p1 14572 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 14573* 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 14574* 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 14575* 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  =  ( inv g `  G )   &    |-  ( ( ph  /\  x  e.  S ) 
 ->  ( I `  x )  e.  S )   =>    |-  (
 ( ph  /\  N  e.  ZZ  /\  X  e.  S )  ->  ( N  .x.  X )  e.  S )
 
Theoremmulgnncl 14576 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 14577 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 14578 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 14579 Group multiple (exponentiation) operation at a negative integer. (Contributed by Mario Carneiro, 11-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  I  =  ( inv g `  G )   =>    |-  ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  ->  ( -u N  .x.  X )  =  ( I `  ( N  .x.  X ) ) )
 
Theoremmulgm1 14580 Group multiple (exponentiation) operation at negative one. (Contributed by Mario Carneiro, 20-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  I  =  ( inv g `  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  B ) 
 ->  ( -u 1  .x.  X )  =  ( I `  X ) )
 
Theoremmulgnn0z 14581 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 14582 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 14583 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 14584 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 14585 Lemma for mulgdir 14586. (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 14586 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 14587 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 14588 Group multiple (exponentiation) operation at a negative integer. (Contributed by Mario Carneiro, 13-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  I  =  ( inv g `  G )   =>    |-  ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  ->  ( -u N  .x.  X )  =  ( N  .x.  ( I `  X ) ) )
 
Theoremmulgnnass 14589 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 14590 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 14591 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 14592 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 14593 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 14594* 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 14595 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 14596 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 14597* 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  |->  ( ( inv g `  ( R `  y ) ) `  ( F `
  y ) ) )   =>    |-  ( ph  ->  ( N  e.  B  /\  ( N  .+  F )  =  .0.  ) )
 
Theoremprdsgrpd 14598 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 14599* 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  =  ( inv g `  Y )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  ( N `  X )  =  ( x  e.  I  |->  ( ( inv
 g `  ( R `  x ) ) `  ( X `  x ) ) ) )
 
Theorempwsgrp 14600 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 )
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