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Theorem List for Metamath Proof Explorer - 14501-14600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremgrpprop 14501 If two structures have the same group components (properties), one is a group iff the other one is. (Contributed by NM, 11-Oct-2013.)
 |-  ( Base `  K )  =  ( Base `  L )   &    |-  ( +g  `  K )  =  ( +g  `  L )   =>    |-  ( K  e.  Grp  <->  L  e.  Grp )
 
Theoremgrppropstr 14502 Generalize a specific 2-element group  L to show that any set  K with the same (relevant) properties is also a group. (Contributed by NM, 28-Oct-2012.) (Revised by Mario Carneiro, 6-Jan-2015.)
 |-  ( Base `  K )  =  B   &    |-  ( +g  `  K )  =  .+   &    |-  L  =  { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. }   =>    |-  ( K  e.  Grp  <->  L  e.  Grp )
 
Theoremgrpss 14503 Show that a structure extending a constructed group (e.g. a ring) is also a group. This allows us to prove that a constructed potential ring  R is a group before we know that it is also a ring. (Theorem rnggrp 15346, on the other hand, requires that we know in advance that  R is a ring.) (Contributed by NM, 11-Oct-2013.)
 |-  G  =  { <. (
 Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. }   &    |-  R  e.  _V   &    |-  G  C_  R   &    |-  Fun  R   =>    |-  ( G  e.  Grp  <->  R  e.  Grp )
 
Theoremisgrpd2e 14504* Deduce a group from its properties. In this version of isgrpd2 14505, we don't assume there is an expression for the inverse of  x. (Contributed by NM, 10-Aug-2013.)
 |-  ( ph  ->  B  =  ( Base `  G )
 )   &    |-  ( ph  ->  .+  =  ( +g  `  G )
 )   &    |-  ( ph  ->  .0.  =  ( 0g `  G ) )   &    |-  ( ph  ->  G  e.  Mnd )   &    |-  (
 ( ph  /\  x  e.  B )  ->  E. y  e.  B  ( y  .+  x )  =  .0.  )   =>    |-  ( ph  ->  G  e.  Grp )
 
Theoremisgrpd2 14505* Deduce a group from its properties. 
N (negative) is normally dependent on  x i.e. read it as  N ( x ). Note: normally we don't use a  ph antecedent on hypotheses that name structure components, since they can be eliminated with eqid 2283, but we make an exception for theorems such as isgrpd2 14505, ismndd 14396, and islmodd 15633 since theorems using them often rewrite the structure components. (Contributed by NM, 10-Aug-2013.)
 |-  ( ph  ->  B  =  ( Base `  G )
 )   &    |-  ( ph  ->  .+  =  ( +g  `  G )
 )   &    |-  ( ph  ->  .0.  =  ( 0g `  G ) )   &    |-  ( ph  ->  G  e.  Mnd )   &    |-  (
 ( ph  /\  x  e.  B )  ->  N  e.  B )   &    |-  ( ( ph  /\  x  e.  B ) 
 ->  ( N  .+  x )  =  .0.  )   =>    |-  ( ph  ->  G  e.  Grp )
 
Theoremisgrpde 14506* Deduce a group from its properties. In this version of isgrpd 14507, we don't assume there is an expression for the inverse of  x. (Contributed by NM, 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 ) 
 ->  E. y  e.  B  ( y  .+  x )  =  .0.  )   =>    |-  ( ph  ->  G  e.  Grp )
 
Theoremisgrpd 14507* Deduce a group from its properties. Unlike isgrpd2 14505, 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 14508* 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 14509* 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 14510 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 14511 The base set of a group is not empty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.)
 |-  B  =  ( Base `  G )   =>    |-  ( G  e.  Grp  ->  B  =/=  (/) )
 
Theoremgrplid 14512 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 14513 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 14514 A group is not empty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.) (Revised by Mario Carneiro, 2-Dec-2014.)
 |-  ( G  e.  Grp  ->  G  =/=  (/) )
 
Theoremgrprcan 14515 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 14516* 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 14517 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 14518 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 14519* Deduce the identity element of a group from its properties. Useful in conjunction with isgrpd 14507. (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 14520* 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 14521* 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 14522 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 14523 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 14524* 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 14525 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 14526 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 14527 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 14528 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 14529 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 14530 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 14531 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 14532* 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 14533 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 14534 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 14535 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 14536 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 14537 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 14538 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 14539 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 14540 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 14541 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 14542* 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 14543* 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 14544 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 14545 Functionality of group subtraction. (Contributed by Mario Carneiro, 9-Sep-2014.)
 |-  B  =  ( Base `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  ( G  e.  Grp 
 ->  .-  : ( B  X.  B ) --> B )
 
Theoremgrpsubcl 14546 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 14547 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 14548 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 14549 A df-neg 9040-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 14550 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 14551 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 14552 If the difference between two group elements is zero, they are equal. (subeq0 9073 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 14553 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 14554 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 14555 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 14556 Cancellation law for subtraction (pncan 9057 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 14557 Cancellation law for subtraction (npcan 9060 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 14558 Double group subtraction (subsub4 9080 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 14559 Cancellation law for mixed addition and subtraction. (pnpcan2 9087 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 14560 Cancellation law for group subtraction. (npncan 9069 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 14561 Cancellation law for group subtraction. (nnncan2 9084 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 14562* 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 14563* 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 14564* 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 14565* 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 14566 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 14567* 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 14568* 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 14569 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 14570 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 14571 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 14572 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 14573 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 14574 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 14575 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 14576 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 14577 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 14578 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 14579* 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 14580* 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 14581* 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 14582 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 14583 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 14584 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 14585 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 14586 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 14587 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 14588 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 14589 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 14590 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 14591 Lemma for mulgdir 14592. (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 14592 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 14593 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 14594 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 14595 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 14596 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 14597 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 14598 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 14599 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 14600* 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.  )
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