HomeHome Metamath Proof Explorer
Theorem List (p. 208 of 309)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-21328)
  Hilbert Space Explorer  Hilbert Space Explorer
(21329-22851)
  Users' Mathboxes  Users' Mathboxes
(22852-30843)
 

Theorem List for Metamath Proof Explorer - 20701-20800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremgrpoidinvlem1 20701 Lemma for grpoidinv 20705. (Contributed by NM, 10-Oct-2006.) (New usage is discouraged.)
 |-  X  =  ran  G   =>    |-  (
 ( ( G  e.  GrpOp  /\  ( Y  e.  X  /\  A  e.  X ) )  /\  ( ( Y G A )  =  U  /\  ( A G A )  =  A ) )  ->  ( U G A )  =  U )
 
Theoremgrpoidinvlem2 20702 Lemma for grpoidinv 20705. (Contributed by NM, 10-Oct-2006.) (New usage is discouraged.)
 |-  X  =  ran  G   =>    |-  (
 ( ( G  e.  GrpOp  /\  ( Y  e.  X  /\  A  e.  X ) )  /\  ( ( U G Y )  =  Y  /\  ( Y G A )  =  U ) )  ->  ( ( A G Y ) G ( A G Y ) )  =  ( A G Y ) )
 
Theoremgrpoidinvlem3 20703* Lemma for grpoidinv 20705. (Contributed by NM, 11-Oct-2006.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  ( ph 
 <-> 
 A. x  e.  X  ( U G x )  =  x )   &    |-  ( ps 
 <-> 
 A. x  e.  X  E. z  e.  X  ( z G x )  =  U )   =>    |-  ( ( ( ( G  e.  GrpOp  /\  U  e.  X )  /\  ( ph  /\  ps ) ) 
 /\  A  e.  X )  ->  E. y  e.  X  ( ( y G A )  =  U  /\  ( A G y )  =  U ) )
 
Theoremgrpoidinvlem4 20704* Lemma for grpoidinv 20705. (Contributed by NM, 14-Oct-2006.) (New usage is discouraged.)
 |-  X  =  ran  G   =>    |-  (
 ( ( G  e.  GrpOp  /\  A  e.  X ) 
 /\  E. y  e.  X  ( ( y G A )  =  U  /\  ( A G y )  =  U ) )  ->  ( A G U )  =  ( U G A ) )
 
Theoremgrpoidinv 20705* A group has a left and right identity element, and every member has a left and right inverse. (Contributed by NM, 14-Oct-2006.) (New usage is discouraged.)
 |-  X  =  ran  G   =>    |-  ( G  e.  GrpOp  ->  E. u  e.  X  A. x  e.  X  ( ( ( u G x )  =  x  /\  ( x G u )  =  x )  /\  E. y  e.  X  (
 ( y G x )  =  u  /\  ( x G y )  =  u ) ) )
 
Theoremgrpoideu 20706* The left identity element of a group is unique. Lemma 2.2.1(a) of [Herstein] p. 55. (Contributed by NM, 14-Oct-2006.) (New usage is discouraged.)
 |-  X  =  ran  G   =>    |-  ( G  e.  GrpOp  ->  E! u  e.  X  A. x  e.  X  ( u G x )  =  x )
 
Theoremgrporndm 20707 A group's range in terms of its domain. (Contributed by NM, 6-Apr-2008.) (New usage is discouraged.)
 |-  ( G  e.  GrpOp  ->  ran  G  =  dom  dom  G )
 
Theorem0ngrp 20708 The empty set is not a group. (Contributed by NM, 25-Apr-2007.) (New usage is discouraged.)
 |- 
 -.  (/)  e.  GrpOp
 
Theoremgrporn 20709 The range of a group operation. Useful for satisfying group base set hypotheses of the form  X  =  ran  G. (Contributed by NM, 5-Nov-2006.) (New usage is discouraged.)
 |-  G  e.  GrpOp   &    |- 
 dom  G  =  ( X  X.  X )   =>    |-  X  =  ran  G
 
Theoremgidval 20710* The value of the identity element of a group. (Contributed by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
 |-  X  =  ran  G   =>    |-  ( G  e.  V  ->  (GId `  G )  =  (
 iota_ u  e.  X A. x  e.  X  ( ( u G x )  =  x 
 /\  ( x G u )  =  x ) ) )
 
Theoremfngid 20711 GId is a function. (Contributed by FL, 5-Feb-2010.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
 |- GId  Fn  _V
 
Theoremgrposn 20712 The group operation for the singleton group. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)
 |-  A  e.  _V   =>    |-  { <. <. A ,  A >. ,  A >. }  e.  GrpOp
 
Theoremgrpoidval 20713* Lemma for grpoidcl 20714 and others. (Contributed by NM, 5-Feb-2010.) (Proof shortened by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  U  =  (GId `  G )   =>    |-  ( G  e.  GrpOp  ->  U  =  ( iota_ u  e.  X A. x  e.  X  ( u G x )  =  x ) )
 
Theoremgrpoidcl 20714 The identity element of a group belongs to the group. (Contributed by NM, 24-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  U  =  (GId `  G )   =>    |-  ( G  e.  GrpOp  ->  U  e.  X )
 
Theoremgrpoidinv2 20715* A group's properties using the explicit identity element. (Contributed by NM, 5-Feb-2010.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  U  =  (GId `  G )   =>    |-  (
 ( G  e.  GrpOp  /\  A  e.  X ) 
 ->  ( ( ( U G A )  =  A  /\  ( A G U )  =  A )  /\  E. y  e.  X  (
 ( y G A )  =  U  /\  ( A G y )  =  U ) ) )
 
Theoremgrpolid 20716 The identity element of a group is a left identity. (Contributed by NM, 24-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  U  =  (GId `  G )   =>    |-  (
 ( G  e.  GrpOp  /\  A  e.  X ) 
 ->  ( U G A )  =  A )
 
Theoremgrporid 20717 The identity element of a group is a right identity. (Contributed by NM, 24-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  U  =  (GId `  G )   =>    |-  (
 ( G  e.  GrpOp  /\  A  e.  X ) 
 ->  ( A G U )  =  A )
 
Theoremgrporcan 20718 Right cancellation law for groups. (Contributed by NM, 26-Oct-2006.) (New usage is discouraged.)
 |-  X  =  ran  G   =>    |-  (
 ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A G C )  =  ( B G C ) 
 <->  A  =  B ) )
 
Theoremgrpoinveu 20719* The left inverse element of a group is unique. Lemma 2.2.1(b) of [Herstein] p. 55. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  U  =  (GId `  G )   =>    |-  (
 ( G  e.  GrpOp  /\  A  e.  X ) 
 ->  E! y  e.  X  ( y G A )  =  U )
 
Theoremgrpoid 20720 Two ways of saying that an element of a group is the identity element. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  U  =  (GId `  G )   =>    |-  (
 ( G  e.  GrpOp  /\  A  e.  X ) 
 ->  ( A  =  U  <->  ( A G A )  =  A ) )
 
Theoremgrpoinvfval 20721* The inverse function of a group. (Contributed by NM, 26-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  U  =  (GId `  G )   &    |-  N  =  ( inv `  G )   =>    |-  ( G  e.  GrpOp  ->  N  =  ( x  e.  X  |->  ( iota_ y  e.  X ( y G x )  =  U ) ) )
 
Theoremgrpoinvval 20722* The inverse of a group element. (Contributed by NM, 26-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  U  =  (GId `  G )   &    |-  N  =  ( inv `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X ) 
 ->  ( N `  A )  =  ( iota_ y  e.  X ( y G A )  =  U ) )
 
Theoremgrpoinvcl 20723 A group element's inverse is a group element. (Contributed by NM, 27-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  N  =  ( inv `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X ) 
 ->  ( N `  A )  e.  X )
 
Theoremgrpoinv 20724 The properties of a group element's inverse. (Contributed by NM, 27-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  U  =  (GId `  G )   &    |-  N  =  ( inv `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X ) 
 ->  ( ( ( N `
  A ) G A )  =  U  /\  ( A G ( N `  A ) )  =  U ) )
 
Theoremgrpolinv 20725 The left inverse of a group element. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  U  =  (GId `  G )   &    |-  N  =  ( inv `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X ) 
 ->  ( ( N `  A ) G A )  =  U )
 
Theoremgrporinv 20726 The right inverse of a group element. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  U  =  (GId `  G )   &    |-  N  =  ( inv `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X ) 
 ->  ( A G ( N `  A ) )  =  U )
 
Theoremgrpoinvid1 20727 The inverse of a group element expressed in terms of the identity element. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  U  =  (GId `  G )   &    |-  N  =  ( inv `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( ( N `  A )  =  B  <->  ( A G B )  =  U ) )
 
Theoremgrpoinvid2 20728 The inverse of a group element expressed in terms of the identity element. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  U  =  (GId `  G )   &    |-  N  =  ( inv `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( ( N `  A )  =  B  <->  ( B G A )  =  U ) )
 
Theoremgrpoinvid 20729 The inverse of the identity element of a group. (Contributed by NM, 4-Dec-2006.) (New usage is discouraged.)
 |-  U  =  (GId `  G )   &    |-  N  =  ( inv `  G )   =>    |-  ( G  e.  GrpOp  ->  ( N `  U )  =  U )
 
Theoremgrpolcan 20730 Left cancellation law for groups. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)
 |-  X  =  ran  G   =>    |-  (
 ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( C G A )  =  ( C G B ) 
 <->  A  =  B ) )
 
Theoremgrpo2grp 20731 Convert a group operation to a group structure. (Contributed by NM, 25-Oct-2012.) (Revised by Mario Carneiro, 6-Jan-2015.) (New usage is discouraged.)
 |-  ( Base `  K )  =  ran  .+   &    |-  ( +g  `  K )  =  .+   &    |-  .+  e.  GrpOp   =>    |-  K  e.  Grp
 
Theoremisgrp2d 20732* An alternate way to show a group operation. Exercise 1 of [Herstein] p. 57. (Contributed by Mario Carneiro, 12-May-2014.) (New usage is discouraged.)
 |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  X  =/=  (/) )   &    |-  ( ph  ->  G : ( X  X.  X ) --> X )   &    |-  ( ( ph  /\  ( x  e.  X  /\  y  e.  X  /\  z  e.  X )
 )  ->  ( ( x G y ) G z )  =  ( x G ( y G z ) ) )   &    |-  ( ( ph  /\  ( x  e.  X  /\  y  e.  X ) )  ->  E. z  e.  X  ( z G x )  =  y )   &    |-  ( ( ph  /\  ( x  e.  X  /\  y  e.  X ) )  ->  E. z  e.  X  ( x G z )  =  y )   =>    |-  ( ph  ->  G  e.  GrpOp )
 
Theoremisgrp2i 20733* An alternate way to show a group operation. Exercise 1 of [Herstein] p. 57. (Contributed by NM, 5-Dec-2006.) (Revised by Mario Carneiro, 12-May-2014.) (New usage is discouraged.)
 |-  X  e.  _V   &    |-  X  =/= 
 (/)   &    |-  G : ( X  X.  X ) --> X   &    |-  (
 ( x  e.  X  /\  y  e.  X  /\  z  e.  X )  ->  ( ( x G y ) G z )  =  ( x G ( y G z ) ) )   &    |-  ( ( x  e.  X  /\  y  e.  X )  ->  E. z  e.  X  ( z G x )  =  y )   &    |-  ( ( x  e.  X  /\  y  e.  X )  ->  E. z  e.  X  ( x G z )  =  y )   =>    |-  G  e.  GrpOp
 
Theoremgrpoasscan1 20734 An associative cancellation law for groups. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  N  =  ( inv `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A G ( ( N `  A ) G B ) )  =  B )
 
Theoremgrpoasscan2 20735 An associative cancellation law for groups. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  N  =  ( inv `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( ( A G ( N `  B ) ) G B )  =  A )
 
Theoremgrpo2inv 20736 Double inverse law for groups. Lemma 2.2.1(c) of [Herstein] p. 55. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  N  =  ( inv `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X ) 
 ->  ( N `  ( N `  A ) )  =  A )
 
Theoremgrpoinvf 20737 Mapping of the inverse function of a group. (Contributed by NM, 29-Mar-2008.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  N  =  ( inv `  G )   =>    |-  ( G  e.  GrpOp  ->  N : X -1-1-onto-> X )
 
Theoremgrpoinvop 20738 The inverse of the group operation reverses the arguments. Lemma 2.2.1(d) of [Herstein] p. 55. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  N  =  ( inv `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A G B ) )  =  ( ( N `
  B ) G ( N `  A ) ) )
 
Theoremgrpodivfval 20739* Group division (or subtraction) operation. (Contributed by NM, 15-Feb-2008.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  N  =  ( inv `  G )   &    |-  D  =  (  /g  `  G )   =>    |-  ( G  e.  GrpOp  ->  D  =  ( x  e.  X ,  y  e.  X  |->  ( x G ( N `  y
 ) ) ) )
 
Theoremgrpodivval 20740 Group division (or subtraction) operation value. (Contributed by NM, 15-Feb-2008.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  N  =  ( inv `  G )   &    |-  D  =  (  /g  `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  =  ( A G ( N `  B ) ) )
 
Theoremgrpodivinv 20741 Group division by an inverse. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  N  =  ( inv `  G )   &    |-  D  =  (  /g  `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A D ( N `  B ) )  =  ( A G B ) )
 
Theoremgrpoinvdiv 20742 Inverse of a group division. (Contributed by NM, 24-Feb-2008.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  N  =  ( inv `  G )   &    |-  D  =  (  /g  `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A D B ) )  =  ( B D A ) )
 
Theoremgrpodivf 20743 Mapping for group division. (Contributed by NM, 10-Apr-2008.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  D  =  (  /g  `  G )   =>    |-  ( G  e.  GrpOp  ->  D : ( X  X.  X ) --> X )
 
Theoremgrpodivcl 20744 Closure of group division (or subtraction) operation. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  D  =  (  /g  `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  e.  X )
 
Theoremgrpodivdiv 20745 Double group division. (Contributed by NM, 24-Feb-2008.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  D  =  (  /g  `  G )   =>    |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( A D ( B D C ) )  =  ( A G ( C D B ) ) )
 
Theoremgrpomuldivass 20746 Associative-type law for multiplication and division. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  D  =  (  /g  `  G )   =>    |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A G B ) D C )  =  ( A G ( B D C ) ) )
 
Theoremgrpodivid 20747 Division of a group member by itself. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  D  =  (  /g  `  G )   &    |-  U  =  (GId `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X ) 
 ->  ( A D A )  =  U )
 
Theoremgrpopncan 20748 Cancellation law for group division. (pncan 8937 analog.) (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  D  =  (  /g  `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( ( A G B ) D B )  =  A )
 
Theoremgrponpcan 20749 Cancellation law for group division. (npcan 8940 analog.) (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  D  =  (  /g  `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( ( A D B ) G B )  =  A )
 
Theoremgrpopnpcan2 20750 Cancellation law for mixed addition and group division. (pnpcan2 8967 analog.) (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  D  =  (  /g  `  G )   =>    |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A G C ) D ( B G C ) )  =  ( A D B ) )
 
Theoremgrponnncan2 20751 Cancellation law for group division. (nnncan2 8964 analog.) (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  D  =  (  /g  `  G )   =>    |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A D C ) D ( B D C ) )  =  ( A D B ) )
 
Theoremgrponpncan 20752 Cancellation law for group division. (npncan 8949 analog.) (Contributed by NM, 24-Feb-2008.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  D  =  (  /g  `  G )   =>    |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A D B ) G ( B D C ) )  =  ( A D C ) )
 
Theoremgrpodiveq 20753 Relationship between group division and group multiplication. (Contributed by Mario Carneiro, 11-Jul-2014.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  D  =  (  /g  `  G )   =>    |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A D B )  =  C  <->  ( C G B )  =  A ) )
 
Theoremgxfval 20754* The value of the group power operator function. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  U  =  (GId `  G )   &    |-  N  =  ( inv `  G )   &    |-  P  =  ( ^g `  G )   =>    |-  ( G  e.  GrpOp  ->  P  =  ( x  e.  X ,  y  e. 
 ZZ  |->  if ( y  =  0 ,  U ,  if ( 0  <  y ,  (  seq  1 ( G ,  ( NN 
 X.  { x } )
 ) `  y ) ,  ( N `  (  seq  1 ( G ,  ( NN  X.  { x } ) ) `  -u y ) ) ) ) ) )
 
Theoremgxval 20755 The result of the group power operator. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  U  =  (GId `  G )   &    |-  N  =  ( inv `  G )   &    |-  P  =  ( ^g `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  ZZ )  ->  ( A P K )  =  if ( K  =  0 ,  U ,  if (
 0  <  K ,  (  seq  1 ( G ,  ( NN  X.  { A } ) ) `
  K ) ,  ( N `  (  seq  1 ( G ,  ( NN  X.  { A } ) ) `  -u K ) ) ) ) )
 
Theoremgxpval 20756 The result of the group power operator when the exponent is positive. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  P  =  ( ^g `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  NN )  ->  ( A P K )  =  (  seq  1 ( G ,  ( NN  X.  { A } ) ) `  K ) )
 
Theoremgxnval 20757 The result of the group power operator when the exponent is negative. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  P  =  ( ^g `  G )   &    |-  N  =  ( inv `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( K  e.  ZZ  /\  K  <  0 ) )  ->  ( A P K )  =  ( N `  (  seq  1 ( G ,  ( NN  X.  { A } ) ) `  -u K ) ) )
 
Theoremgx0 20758 The result of the group power operator when the exponent is zero. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  U  =  (GId `  G )   &    |-  P  =  ( ^g `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X ) 
 ->  ( A P 0 )  =  U )
 
Theoremgx1 20759 The result of the group power operator when the exponent is one. (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  P  =  ( ^g `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X ) 
 ->  ( A P 1 )  =  A )
 
Theoremgxnn0neg 20760 A negative group power is the inverse of the positive power (lemma with nonnegative exponent - use gxneg 20763 instead). (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  N  =  ( inv `  G )   &    |-  P  =  ( ^g `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  NN0 )  ->  ( A P -u K )  =  ( N `  ( A P K ) ) )
 
Theoremgxnn0suc 20761 Induction on group power (lemma with nonnegative exponent - use gxsuc 20769 instead). (Contributed by Paul Chapman, 17-Apr-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  P  =  ( ^g `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  NN0 )  ->  ( A P ( K  +  1 ) )  =  ( ( A P K ) G A ) )
 
Theoremgxcl 20762 Closure of the group power operator. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  P  =  ( ^g `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  ZZ )  ->  ( A P K )  e.  X )
 
Theoremgxneg 20763 A negative group power is the inverse of the positive power. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  N  =  ( inv `  G )   &    |-  P  =  ( ^g `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  ZZ )  ->  ( A P -u K )  =  ( N `  ( A P K ) ) )
 
Theoremgxneg2 20764 The inverse of a negative group power is the positive power. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  N  =  ( inv `  G )   &    |-  P  =  ( ^g `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  ZZ )  ->  ( N `  ( A P -u K ) )  =  ( A P K ) )
 
Theoremgxm1 20765 The result of the group power operator when the exponent is minus one. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  N  =  ( inv `  G )   &    |-  P  =  ( ^g `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X ) 
 ->  ( A P -u 1 )  =  ( N `  A ) )
 
Theoremgxcom 20766 The group power operator commutes with the group operation. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  P  =  ( ^g `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  ZZ )  ->  ( ( A P K ) G A )  =  ( A G ( A P K ) ) )
 
Theoremgxinv 20767 The group power operator commutes with the group inverse function. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  N  =  ( inv `  G )   &    |-  P  =  ( ^g `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  ZZ )  ->  ( ( N `  A ) P K )  =  ( N `  ( A P K ) ) )
 
Theoremgxinv2 20768 The group power operator commutes with the group inverse function. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  N  =  ( inv `  G )   &    |-  P  =  ( ^g `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  ZZ )  ->  ( N `  (
 ( N `  A ) P K ) )  =  ( A P K ) )
 
Theoremgxsuc 20769 Induction on group power. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  P  =  ( ^g `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  K  e.  ZZ )  ->  ( A P ( K  +  1 ) )  =  ( ( A P K ) G A ) )
 
Theoremgxid 20770 The identity element of a group to any power remains unchanged. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)
 |-  U  =  (GId `  G )   &    |-  P  =  ( ^g `  G )   =>    |-  ( ( G  e.  GrpOp  /\  K  e.  ZZ )  ->  ( U P K )  =  U )
 
Theoremgxnn0add 20771 The group power of a sum is the group product of the powers (lemma with nonnegative exponent - use gxadd 20772 instead). (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  P  =  ( ^g `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( J  e.  ZZ  /\  K  e.  NN0 )
 )  ->  ( A P ( J  +  K ) )  =  ( ( A P J ) G ( A P K ) ) )
 
Theoremgxadd 20772 The group power of a sum is the group product of the powers. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  P  =  ( ^g `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( J  e.  ZZ  /\  K  e.  ZZ )
 )  ->  ( A P ( J  +  K ) )  =  ( ( A P J ) G ( A P K ) ) )
 
Theoremgxsub 20773 The group power of a difference is the group quotient of the powers. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  N  =  ( inv `  G )   &    |-  P  =  ( ^g `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( J  e.  ZZ  /\  K  e.  ZZ )
 )  ->  ( A P ( J  -  K ) )  =  ( ( A P J ) G ( N `  ( A P K ) ) ) )
 
Theoremgxnn0mul 20774 The group power of a product is the composition of the powers (lemma with nonnegative exponent - use gxmul 20775 instead). (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  P  =  ( ^g `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( J  e.  ZZ  /\  K  e.  NN0 )
 )  ->  ( A P ( J  x.  K ) )  =  ( ( A P J ) P K ) )
 
Theoremgxmul 20775 The group power of a product is the composition of the powers. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  P  =  ( ^g `  G )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  ( J  e.  ZZ  /\  K  e.  ZZ )
 )  ->  ( A P ( J  x.  K ) )  =  ( ( A P J ) P K ) )
 
Theoremgxmodid 20776 Casting out powers of the identity element leaves the group power unchanged. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  U  =  (GId `  G )   &    |-  P  =  ( ^g `  G )   =>    |-  ( ( G  e.  GrpOp  /\  ( K  e.  ZZ  /\  M  e.  NN )  /\  ( A  e.  X  /\  ( A P M )  =  U )
 )  ->  ( A P ( K  mod  M ) )  =  ( A P K ) )
 
Theoremresgrprn 20777 The underlying set of a group operation which is a restriction of a mapping. (Contributed by Paul Chapman, 25-Mar-2008.) (New usage is discouraged.)
 |-  H  =  ( G  |`  ( Y  X.  Y ) )   =>    |-  ( ( dom  G  =  ( X  X.  X )  /\  H  e.  GrpOp  /\  Y  C_  X )  ->  Y  =  ran  H )
 
15.1.2  Definition and basic properties of Abelian groups
 
Syntaxcablo 20778 Extend class notation with the class of all Abelian group operations.
 class  AbelOp
 
Definitiondf-ablo 20779* Define the class of all Abelian group operations. (Contributed by NM, 2-Nov-2006.) (New usage is discouraged.)
 |-  AbelOp  =  { g  e. 
 GrpOp  |  A. x  e. 
 ran  g A. y  e.  ran  g ( x g y )  =  ( y g x ) }
 
Theoremisablo 20780* The predicate "is an Abelian (commutative) group operation." (Contributed by NM, 2-Nov-2006.) (New usage is discouraged.)
 |-  X  =  ran  G   =>    |-  ( G  e.  AbelOp  <->  ( G  e.  GrpOp  /\  A. x  e.  X  A. y  e.  X  ( x G y )  =  ( y G x ) ) )
 
Theoremablogrpo 20781 An Abelian group operation is a group operation. (Contributed by NM, 2-Nov-2006.) (New usage is discouraged.)
 |-  ( G  e.  AbelOp  ->  G  e.  GrpOp )
 
Theoremablocom 20782 An Abelian group operation is commutative. (Contributed by NM, 2-Nov-2006.) (New usage is discouraged.)
 |-  X  =  ran  G   =>    |-  (
 ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  =  ( B G A ) )
 
Theoremablo32 20783 Commutative/associative law for Abelian groups. (Contributed by NM, 26-Apr-2007.) (New usage is discouraged.)
 |-  X  =  ran  G   =>    |-  (
 ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A G B ) G C )  =  ( ( A G C ) G B ) )
 
Theoremablo4 20784 Commutative/associative law for Abelian groups. (Contributed by NM, 26-Apr-2007.) (New usage is discouraged.)
 |-  X  =  ran  G   =>    |-  (
 ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X ) 
 /\  ( C  e.  X  /\  D  e.  X ) )  ->  ( ( A G B ) G ( C G D ) )  =  ( ( A G C ) G ( B G D ) ) )
 
Theoremisabloi 20785* Properties that determine an Abelian group operation. (Contributed by NM, 5-Nov-2006.) (New usage is discouraged.)
 |-  G  e.  GrpOp   &    |- 
 dom  G  =  ( X  X.  X )   &    |-  (
 ( x  e.  X  /\  y  e.  X )  ->  ( x G y )  =  ( y G x ) )   =>    |-  G  e.  AbelOp
 
Theoremablomuldiv 20786 Law for group multiplication and division. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  D  =  (  /g  `  G )   =>    |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A G B ) D C )  =  ( ( A D C ) G B ) )
 
Theoremablodivdiv 20787 Law for double group division. (Contributed by NM, 29-Feb-2008.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  D  =  (  /g  `  G )   =>    |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( A D ( B D C ) )  =  ( ( A D B ) G C ) )
 
Theoremablodivdiv4 20788 Law for double group division. (Contributed by NM, 29-Feb-2008.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  D  =  (  /g  `  G )   =>    |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A D B ) D C )  =  ( A D ( B G C ) ) )
 
Theoremablodiv32 20789 Swap the second and third terms in a double division. (Contributed by NM, 29-Feb-2008.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  D  =  (  /g  `  G )   =>    |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A D B ) D C )  =  ( ( A D C ) D B ) )
 
Theoremablonnncan 20790 Cancellation law for group division. (nnncan 8962 analog.) (Contributed by NM, 29-Feb-2008.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  D  =  (  /g  `  G )   =>    |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A D ( B D C ) ) D C )  =  ( A D B ) )
 
Theoremablonncan 20791 Cancellation law for group division. (nncan 8956 analog.) (Contributed by NM, 7-Mar-2008.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  D  =  (  /g  `  G )   =>    |-  ( ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A D ( A D B ) )  =  B )
 
Theoremablonnncan1 20792 Cancellation law for group division. (nnncan1 8963 analog.) (Contributed by NM, 7-Mar-2008.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  D  =  (  /g  `  G )   =>    |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A D B ) D ( A D C ) )  =  ( C D B ) )
 
Theoremgxdi 20793 Distribution of group power over group operation for abelian groups. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  P  =  ( ^g `  G )   =>    |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X ) 
 /\  K  e.  ZZ )  ->  ( ( A G B ) P K )  =  ( ( A P K ) G ( B P K ) ) )
 
Theoremisgrpda 20794* Properties that determine a group operation. (Contributed by Jeff Madsen, 1-Dec-2009.) (New usage is discouraged.)
 |-  ( ph  ->  X  e.  _V )   &    |-  ( ph  ->  G : ( X  X.  X ) --> X )   &    |-  ( ( ph  /\  ( x  e.  X  /\  y  e.  X  /\  z  e.  X )
 )  ->  ( ( x G y ) G z )  =  ( x G ( y G z ) ) )   &    |-  ( ph  ->  U  e.  X )   &    |-  (
 ( ph  /\  x  e.  X )  ->  ( U G x )  =  x )   &    |-  ( ( ph  /\  x  e.  X ) 
 ->  E. n  e.  X  ( n G x )  =  U )   =>    |-  ( ph  ->  G  e.  GrpOp )
 
Theoremisgrpod 20795* Properties that determine a group operation. (Renamed from isgrpd 14342 to isgrpod 20795 to prevent naming conflict. -NM 5-Jun-2013) (Contributed by Jeff Madsen, 1-Dec-2009.) (New usage is discouraged.)
 |-  ( ph  ->  X  e.  _V )   &    |-  ( ph  ->  G : ( X  X.  X ) --> X )   &    |-  ( ( ph  /\  ( x  e.  X  /\  y  e.  X  /\  z  e.  X )
 )  ->  ( ( x G y ) G z )  =  ( x G ( y G z ) ) )   &    |-  ( ph  ->  U  e.  X )   &    |-  (
 ( ph  /\  x  e.  X )  ->  ( U G x )  =  x )   &    |-  ( ( ph  /\  x  e.  X ) 
 ->  N  e.  X )   &    |-  ( ( ph  /\  x  e.  X )  ->  ( N G x )  =  U )   =>    |-  ( ph  ->  G  e.  GrpOp )
 
Theoremisabloda 20796* Properties that determine an Abelian group operation. (Contributed by Jeff Madsen, 11-Jun-2010.) (New usage is discouraged.)
 |-  ( ph  ->  X  e.  _V )   &    |-  ( ph  ->  G : ( X  X.  X ) --> X )   &    |-  ( ( ph  /\  ( x  e.  X  /\  y  e.  X  /\  z  e.  X )
 )  ->  ( ( x G y ) G z )  =  ( x G ( y G z ) ) )   &    |-  ( ph  ->  U  e.  X )   &    |-  (
 ( ph  /\  x  e.  X )  ->  ( U G x )  =  x )   &    |-  ( ( ph  /\  x  e.  X ) 
 ->  E. n  e.  X  ( n G x )  =  U )   &    |-  (
 ( ph  /\  ( x  e.  X  /\  y  e.  X ) )  ->  ( x G y )  =  ( y G x ) )   =>    |-  ( ph  ->  G  e.  AbelOp )
 
Theoremisablod 20797* Properties that determine an Abelian group operation. (Changed label from isabld 14937 to isablod 20797-NM 6-Aug-2013.) (Contributed by Jeff Madsen, 5-Dec-2009.) (New usage is discouraged.)
 |-  ( ph  ->  X  e.  _V )   &    |-  ( ph  ->  G : ( X  X.  X ) --> X )   &    |-  ( ( ph  /\  ( x  e.  X  /\  y  e.  X  /\  z  e.  X )
 )  ->  ( ( x G y ) G z )  =  ( x G ( y G z ) ) )   &    |-  ( ph  ->  U  e.  X )   &    |-  (
 ( ph  /\  x  e.  X )  ->  ( U G x )  =  x )   &    |-  ( ( ph  /\  x  e.  X ) 
 ->  N  e.  X )   &    |-  ( ( ph  /\  x  e.  X )  ->  ( N G x )  =  U )   &    |-  ( ( ph  /\  ( x  e.  X  /\  y  e.  X ) )  ->  ( x G y )  =  ( y G x ) )   =>    |-  ( ph  ->  G  e.  AbelOp )
 
15.1.3  Subgroups
 
Syntaxcsubgo 20798 Extend class notation to include the class of subgroups.
 class  SubGrpOp
 
Definitiondf-subgo 20799 Define the set of subgroups of  g. (Contributed by Paul Chapman, 3-Mar-2008.) (New usage is discouraged.)
 |-  SubGrpOp 
 =  ( g  e. 
 GrpOp  |->  ( GrpOp  i^i  ~P g ) )
 
Theoremissubgo 20800 The predicate "is a subgroup of  G." (Contributed by Paul Chapman, 3-Mar-2008.) (Revised by Mario Carneiro, 12-Jul-2014.) (New usage is discouraged.)
 |-  ( H  e.  ( SubGrpOp `  G )  <->  ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  H  C_  G ) )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30843
  Copyright terms: Public domain < Previous  Next >