HomeHome Metamath Proof Explorer
Theorem List (p. 192 of 399)
< 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
(1-26563)
  Hilbert Space Explorer  Hilbert Space Explorer
(26564-28086)
  Users' Mathboxes  Users' Mathboxes
(28087-39884)
 

Theorem List for Metamath Proof Explorer - 19101-19200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremznbaslem 19101 Lemma for znbas 19106. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by Mario Carneiro, 14-Aug-2015.) (Revised by AV, 13-Jun-2019.)
 |-  S  =  (RSpan ` ring )   &    |-  U  =  (ring  /.s  (ring ~QG  ( S `
  { N }
 ) ) )   &    |-  Y  =  (ℤ/n `  N )   &    |-  E  = Slot  K   &    |-  K  e.  NN   &    |-  K  <  10   =>    |-  ( N  e.  NN0  ->  ( E `  U )  =  ( E `  Y ) )
 
Theoremznbas2 19102 The base set of ℤ/nℤ is the same as the quotient ring it is based on. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.)
 |-  S  =  (RSpan ` ring )   &    |-  U  =  (ring  /.s  (ring ~QG  ( S `
  { N }
 ) ) )   &    |-  Y  =  (ℤ/n `  N )   =>    |-  ( N  e.  NN0  ->  ( Base `  U )  =  ( Base `  Y )
 )
 
Theoremznadd 19103 The additive structure of ℤ/nℤ is the same as the quotient ring it is based on. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.)
 |-  S  =  (RSpan ` ring )   &    |-  U  =  (ring  /.s  (ring ~QG  ( S `
  { N }
 ) ) )   &    |-  Y  =  (ℤ/n `  N )   =>    |-  ( N  e.  NN0  ->  ( +g  `  U )  =  ( +g  `  Y ) )
 
Theoremznmul 19104 The multiplicative structure of ℤ/nℤ is the same as the quotient ring it is based on. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.)
 |-  S  =  (RSpan ` ring )   &    |-  U  =  (ring  /.s  (ring ~QG  ( S `
  { N }
 ) ) )   &    |-  Y  =  (ℤ/n `  N )   =>    |-  ( N  e.  NN0  ->  ( .r `  U )  =  ( .r `  Y ) )
 
Theoremznzrh 19105 The  ZZ ring homomorphism of ℤ/nℤ is inherited from the quotient ring it is based on. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 13-Jun-2019.)
 |-  S  =  (RSpan ` ring )   &    |-  U  =  (ring  /.s  (ring ~QG  ( S `
  { N }
 ) ) )   &    |-  Y  =  (ℤ/n `  N )   =>    |-  ( N  e.  NN0  ->  ( ZRHom `  U )  =  ( ZRHom `  Y ) )
 
Theoremznbas 19106 The base set of ℤ/nℤ structure. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.)
 |-  S  =  (RSpan ` ring )   &    |-  Y  =  (ℤ/n `  N )   &    |-  R  =  (ring ~QG  ( S `
  { N }
 ) )   =>    |-  ( N  e.  NN0  ->  ( ZZ /. R )  =  ( Base `  Y ) )
 
Theoremzncrng 19107 ℤ/nℤ is a commutative ring. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  Y  =  (ℤ/n `  N )   =>    |-  ( N  e.  NN0  ->  Y  e.  CRing )
 
Theoremznzrh2 19108* The  ZZ ring homomorphism maps elements to their equivalence classes. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.)
 |-  S  =  (RSpan ` ring )   &    |-  .~  =  (ring ~QG  ( S `  { N }
 ) )   &    |-  Y  =  (ℤ/n `  N )   &    |-  L  =  ( ZRHom `  Y )   =>    |-  ( N  e.  NN0  ->  L  =  ( x  e.  ZZ  |->  [ x ]  .~  )
 )
 
Theoremznzrhval 19109 The  ZZ ring homomorphism maps elements to their equivalence classes. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.)
 |-  S  =  (RSpan ` ring )   &    |-  .~  =  (ring ~QG  ( S `  { N }
 ) )   &    |-  Y  =  (ℤ/n `  N )   &    |-  L  =  ( ZRHom `  Y )   =>    |-  (
 ( N  e.  NN0  /\  A  e.  ZZ )  ->  ( L `  A )  =  [ A ]  .~  )
 
Theoremznzrhfo 19110 The  ZZ ring homomorphism is a surjection onto 
ZZ  /  n ZZ. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  Y  =  (ℤ/n `  N )   &    |-  B  =  ( Base `  Y )   &    |-  L  =  ( ZRHom `  Y )   =>    |-  ( N  e.  NN0  ->  L : ZZ -onto-> B )
 
Theoremzncyg 19111 The group  ZZ  /  n ZZ is cyclic for all  n (including  n  =  0). (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  Y  =  (ℤ/n `  N )   =>    |-  ( N  e.  NN0  ->  Y  e. CycGrp )
 
Theoremzndvds 19112 Express equality of equivalence classes in  ZZ 
/  n ZZ in terms of divisibility. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  Y  =  (ℤ/n `  N )   &    |-  L  =  ( ZRHom `  Y )   =>    |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( L `  A )  =  ( L `  B )  <->  N  ||  ( A  -  B ) ) )
 
Theoremzndvds0 19113 Special case of zndvds 19112 when one argument is zero. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  Y  =  (ℤ/n `  N )   &    |-  L  =  ( ZRHom `  Y )   &    |-  .0.  =  ( 0g `  Y )   =>    |-  ( ( N  e.  NN0  /\  A  e.  ZZ )  ->  ( ( L `  A )  =  .0.  <->  N  ||  A ) )
 
Theoremznf1o 19114 The function  F enumerates all equivalence classes in ℤ/nℤ for each  n. When  n  = 
0,  ZZ  /  0 ZZ  =  ZZ  /  {
0 }  ~~  ZZ so we let  W  =  ZZ; otherwise  W  =  { 0 , 
... ,  n  - 
1 } enumerates all the equivalence classes. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by Mario Carneiro, 2-May-2016.) (Revised by AV, 13-Jun-2019.)
 |-  Y  =  (ℤ/n `  N )   &    |-  B  =  ( Base `  Y )   &    |-  F  =  ( ( ZRHom `  Y )  |`  W )   &    |-  W  =  if ( N  =  0 ,  ZZ ,  (
 0..^ N ) )   =>    |-  ( N  e.  NN0  ->  F : W -1-1-onto-> B )
 
Theoremzzngim 19115 The  ZZ ring homomorphism is an isomorphism for 
N  =  0. (We only show group isomorphism here, but ring isomorphism follows, since it is a bijective ring homomorphism.) (Contributed by Mario Carneiro, 21-Apr-2016.) (Revised by AV, 13-Jun-2019.)
 |-  Y  =  (ℤ/n `  0
 )   &    |-  L  =  ( ZRHom `  Y )   =>    |-  L  e.  (ring GrpIso  Y )
 
Theoremznle2 19116 The ordering of the ℤ/nℤ structure. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.)
 |-  Y  =  (ℤ/n `  N )   &    |-  F  =  ( ( ZRHom `  Y )  |`  W )   &    |-  W  =  if ( N  =  0 ,  ZZ ,  ( 0..^ N ) )   &    |-  .<_  =  ( le `  Y )   =>    |-  ( N  e.  NN0  ->  .<_  =  ( ( F  o.  <_  )  o.  `' F ) )
 
Theoremznleval 19117 The ordering of the ℤ/nℤ structure. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.)
 |-  Y  =  (ℤ/n `  N )   &    |-  F  =  ( ( ZRHom `  Y )  |`  W )   &    |-  W  =  if ( N  =  0 ,  ZZ ,  ( 0..^ N ) )   &    |-  .<_  =  ( le `  Y )   &    |-  X  =  ( Base `  Y )   =>    |-  ( N  e.  NN0  ->  ( A  .<_  B  <->  ( A  e.  X  /\  B  e.  X  /\  ( `' F `  A )  <_  ( `' F `  B ) ) ) )
 
Theoremznleval2 19118 The ordering of the ℤ/nℤ structure. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.)
 |-  Y  =  (ℤ/n `  N )   &    |-  F  =  ( ( ZRHom `  Y )  |`  W )   &    |-  W  =  if ( N  =  0 ,  ZZ ,  ( 0..^ N ) )   &    |-  .<_  =  ( le `  Y )   &    |-  X  =  ( Base `  Y )   =>    |-  ( ( N  e.  NN0  /\  A  e.  X  /\  B  e.  X )  ->  ( A  .<_  B  <->  ( `' F `  A )  <_  ( `' F `  B ) ) )
 
Theoremzntoslem 19119 Lemma for zntos 19120. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.)
 |-  Y  =  (ℤ/n `  N )   &    |-  F  =  ( ( ZRHom `  Y )  |`  W )   &    |-  W  =  if ( N  =  0 ,  ZZ ,  ( 0..^ N ) )   &    |-  .<_  =  ( le `  Y )   &    |-  X  =  ( Base `  Y )   =>    |-  ( N  e.  NN0  ->  Y  e. Toset )
 
Theoremzntos 19120 The ℤ/nℤ structure is a totally ordered set. (The order is not respected by the operations, except in the case  N  =  0 when it coincides with the ordering on  ZZ.) (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  Y  =  (ℤ/n `  N )   =>    |-  ( N  e.  NN0  ->  Y  e. Toset )
 
Theoremznhash 19121 The ℤ/nℤ structure has  n elements. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  Y  =  (ℤ/n `  N )   &    |-  B  =  ( Base `  Y )   =>    |-  ( N  e.  NN  ->  ( # `  B )  =  N )
 
Theoremznfi 19122 The ℤ/nℤ structure is a finite ring. (Contributed by Mario Carneiro, 2-May-2016.)
 |-  Y  =  (ℤ/n `  N )   &    |-  B  =  ( Base `  Y )   =>    |-  ( N  e.  NN  ->  B  e.  Fin )
 
Theoremznfld 19123 The ℤ/nℤ structure is a finite field when  n is prime. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  Y  =  (ℤ/n `  N )   =>    |-  ( N  e.  Prime  ->  Y  e. Field )
 
Theoremznidomb 19124 The ℤ/nℤ structure is a domain (and hence a field) precisely when  n is prime. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  Y  =  (ℤ/n `  N )   =>    |-  ( N  e.  NN  ->  ( Y  e. IDomn  <->  N  e.  Prime ) )
 
Theoremznchr 19125 Cyclic rings are defined by their characteristic. (Contributed by Stefan O'Rear, 6-Sep-2015.)
 |-  Y  =  (ℤ/n `  N )   =>    |-  ( N  e.  NN0  ->  (chr `  Y )  =  N )
 
Theoremznunit 19126 The units of ℤ/nℤ are the integers coprime to the base. (Contributed by Mario Carneiro, 18-Apr-2016.)
 |-  Y  =  (ℤ/n `  N )   &    |-  U  =  (Unit `  Y )   &    |-  L  =  ( ZRHom `  Y )   =>    |-  (
 ( N  e.  NN0  /\  A  e.  ZZ )  ->  ( ( L `  A )  e.  U  <->  ( A  gcd  N )  =  1 ) )
 
Theoremznunithash 19127 The size of the unit group of ℤ/nℤ. (Contributed by Mario Carneiro, 19-Apr-2016.)
 |-  Y  =  (ℤ/n `  N )   &    |-  U  =  (Unit `  Y )   =>    |-  ( N  e.  NN  ->  ( # `  U )  =  ( phi `  N ) )
 
Theoremznrrg 19128 The regular elements of ℤ/nℤ are exactly the units. (This theorem fails for  N  =  0, where all nonzero integers are regular, but only  pm 1 are units.) (Contributed by Mario Carneiro, 18-Apr-2016.)
 |-  Y  =  (ℤ/n `  N )   &    |-  U  =  (Unit `  Y )   &    |-  E  =  (RLReg `  Y )   =>    |-  ( N  e.  NN  ->  E  =  U )
 
Theoremcygznlem1 19129* Lemma for cygzn 19133. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  N  =  if ( B  e.  Fin ,  ( # `  B ) ,  0 )   &    |-  Y  =  (ℤ/n `  N )   &    |-  .x.  =  (.g `  G )   &    |-  L  =  ( ZRHom `  Y )   &    |-  E  =  { x  e.  B  |  ran  ( n  e. 
 ZZ  |->  ( n  .x.  x ) )  =  B }   &    |-  ( ph  ->  G  e. CycGrp )   &    |-  ( ph  ->  X  e.  E )   =>    |-  ( ( ph  /\  ( K  e.  ZZ  /\  M  e.  ZZ )
 )  ->  ( ( L `  K )  =  ( L `  M ) 
 <->  ( K  .x.  X )  =  ( M  .x.  X ) ) )
 
Theoremcygznlem2a 19130* Lemma for cygzn 19133. (Contributed by Mario Carneiro, 23-Dec-2016.)
 |-  B  =  ( Base `  G )   &    |-  N  =  if ( B  e.  Fin ,  ( # `  B ) ,  0 )   &    |-  Y  =  (ℤ/n `  N )   &    |-  .x.  =  (.g `  G )   &    |-  L  =  ( ZRHom `  Y )   &    |-  E  =  { x  e.  B  |  ran  ( n  e. 
 ZZ  |->  ( n  .x.  x ) )  =  B }   &    |-  ( ph  ->  G  e. CycGrp )   &    |-  ( ph  ->  X  e.  E )   &    |-  F  =  ran  ( m  e. 
 ZZ  |->  <. ( L `  m ) ,  ( m  .x.  X ) >. )   =>    |-  ( ph  ->  F :
 ( Base `  Y ) --> B )
 
Theoremcygznlem2 19131* Lemma for cygzn 19133. (Contributed by Mario Carneiro, 21-Apr-2016.) (Revised by Mario Carneiro, 23-Dec-2016.)
 |-  B  =  ( Base `  G )   &    |-  N  =  if ( B  e.  Fin ,  ( # `  B ) ,  0 )   &    |-  Y  =  (ℤ/n `  N )   &    |-  .x.  =  (.g `  G )   &    |-  L  =  ( ZRHom `  Y )   &    |-  E  =  { x  e.  B  |  ran  ( n  e. 
 ZZ  |->  ( n  .x.  x ) )  =  B }   &    |-  ( ph  ->  G  e. CycGrp )   &    |-  ( ph  ->  X  e.  E )   &    |-  F  =  ran  ( m  e. 
 ZZ  |->  <. ( L `  m ) ,  ( m  .x.  X ) >. )   =>    |-  ( ( ph  /\  M  e.  ZZ )  ->  ( F `  ( L `  M ) )  =  ( M  .x.  X ) )
 
Theoremcygznlem3 19132* A cyclic group with  n elements is isomorphic to  ZZ  /  n ZZ. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  N  =  if ( B  e.  Fin ,  ( # `  B ) ,  0 )   &    |-  Y  =  (ℤ/n `  N )   &    |-  .x.  =  (.g `  G )   &    |-  L  =  ( ZRHom `  Y )   &    |-  E  =  { x  e.  B  |  ran  ( n  e. 
 ZZ  |->  ( n  .x.  x ) )  =  B }   &    |-  ( ph  ->  G  e. CycGrp )   &    |-  ( ph  ->  X  e.  E )   &    |-  F  =  ran  ( m  e. 
 ZZ  |->  <. ( L `  m ) ,  ( m  .x.  X ) >. )   =>    |-  ( ph  ->  G  ~=g𝑔  Y )
 
Theoremcygzn 19133 A cyclic group with  n elements is isomorphic to  ZZ  /  n ZZ, and an infinite cyclic group is isomorphic to  ZZ 
/  0 ZZ  ~~  ZZ. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  N  =  if ( B  e.  Fin ,  ( # `  B ) ,  0 )   &    |-  Y  =  (ℤ/n `  N )   =>    |-  ( G  e. CycGrp  ->  G 
 ~=g𝑔  Y )
 
Theoremcygth 19134* The "fundamental theorem of cyclic groups". Cyclic groups are exactly the additive groups  ZZ  /  n ZZ, for 
0  <_  n (where  n  =  0 is the infinite cyclic group 
ZZ), up to isomorphism. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  ( G  e. CycGrp  <->  E. n  e.  NN0  G 
 ~=g𝑔  (ℤ/n `  n ) )
 
Theoremcyggic 19135 Cyclic groups are isomorphic precisely when they have the same order. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  C  =  (
 Base `  H )   =>    |-  ( ( G  e. CycGrp  /\  H  e. CycGrp )  ->  ( G  ~=g𝑔  H  <->  B  ~~  C ) )
 
Theoremfrgpcyg 19136 A free group is cyclic iff it has zero or one generator. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  G  =  (freeGrp `  I
 )   =>    |-  ( I  ~<_  1o  <->  G  e. CycGrp )
 
10.11.4  Signs as subgroup of the complex numbers
 
Theoremcnmsgnsubg 19137 The signs form a multiplicative subgroup of the complex numbers. (Contributed by Stefan O'Rear, 28-Aug-2015.)
 |-  M  =  ( (mulGrp ` fld )s  ( CC  \  { 0 } ) )   =>    |-  { 1 ,  -u 1 }  e.  (SubGrp `  M )
 
Theoremcnmsgnbas 19138 The base set of the sign subgroup of the complex numbers. (Contributed by Stefan O'Rear, 28-Aug-2015.)
 |-  U  =  ( (mulGrp ` fld )s  { 1 ,  -u 1 } )   =>    |- 
 { 1 ,  -u 1 }  =  ( Base `  U )
 
Theoremcnmsgngrp 19139 The group of signs under multiplication. (Contributed by Stefan O'Rear, 28-Aug-2015.)
 |-  U  =  ( (mulGrp ` fld )s  { 1 ,  -u 1 } )   =>    |-  U  e.  Grp
 
Theorempsgnghm 19140 The sign is a homomorphism from the finitary permutation group to the numeric signs. (Contributed by Stefan O'Rear, 28-Aug-2015.)
 |-  S  =  ( SymGrp `  D )   &    |-  N  =  (pmSgn `  D )   &    |-  F  =  ( Ss 
 dom  N )   &    |-  U  =  ( (mulGrp ` fld )s  { 1 ,  -u 1 } )   =>    |-  ( D  e.  V  ->  N  e.  ( F 
 GrpHom  U ) )
 
Theorempsgnghm2 19141 The sign is a homomorphism from the finite symmetric group to the numeric signs. (Contributed by Stefan O'Rear, 28-Aug-2015.)
 |-  S  =  ( SymGrp `  D )   &    |-  N  =  (pmSgn `  D )   &    |-  U  =  ( (mulGrp ` fld )s  { 1 ,  -u 1 } )   =>    |-  ( D  e.  Fin  ->  N  e.  ( S  GrpHom  U ) )
 
Theorempsgninv 19142 The sign of a permutation equals the sign of the inverse of the permutation. (Contributed by SO, 9-Jul-2018.)
 |-  S  =  ( SymGrp `  D )   &    |-  N  =  (pmSgn `  D )   &    |-  P  =  (
 Base `  S )   =>    |-  ( ( D  e.  Fin  /\  F  e.  P )  ->  ( N `
  `' F )  =  ( N `  F ) )
 
Theorempsgnco 19143 Multiplicativity of the permutation sign function. (Contributed by SO, 9-Jul-2018.)
 |-  S  =  ( SymGrp `  D )   &    |-  N  =  (pmSgn `  D )   &    |-  P  =  (
 Base `  S )   =>    |-  ( ( D  e.  Fin  /\  F  e.  P  /\  G  e.  P )  ->  ( N `  ( F  o.  G ) )  =  (
 ( N `  F )  x.  ( N `  G ) ) )
 
10.11.5  Embedding of permutation signs into a ring
 
Theoremzrhpsgnmhm 19144 Embedding of permutation signs into an arbitrary ring is a homomorphism. (Contributed by SO, 9-Jul-2018.)
 |-  ( ( R  e.  Ring  /\  A  e.  Fin )  ->  ( ( ZRHom `  R )  o.  (pmSgn `  A ) )  e.  (
 ( SymGrp `  A ) MndHom  (mulGrp `  R ) ) )
 
Theoremzrhpsgninv 19145 The embedded sign of a permutation equals the embedded sign of the inverse of the permutation. (Contributed by SO, 9-Jul-2018.)
 |-  P  =  ( Base `  ( SymGrp `  N )
 )   &    |-  Y  =  ( ZRHom `  R )   &    |-  S  =  (pmSgn `  N )   =>    |-  ( ( R  e.  Ring  /\  N  e.  Fin  /\  F  e.  P )  ->  ( ( Y  o.  S ) `  `' F )  =  (
 ( Y  o.  S ) `  F ) )
 
Theoremevpmss 19146 Even permutations are permutations. (Contributed by SO, 9-Jul-2018.)
 |-  S  =  ( SymGrp `  D )   &    |-  P  =  (
 Base `  S )   =>    |-  (pmEven `  D )  C_  P
 
Theorempsgnevpmb 19147 A class is an even permutation if it is a permutation with sign 1. (Contributed by SO, 9-Jul-2018.)
 |-  S  =  ( SymGrp `  D )   &    |-  P  =  (
 Base `  S )   &    |-  N  =  (pmSgn `  D )   =>    |-  ( D  e.  Fin  ->  ( F  e.  (pmEven `  D ) 
 <->  ( F  e.  P  /\  ( N `  F )  =  1 )
 ) )
 
Theorempsgnodpm 19148 A permutation which is odd (i.e. not even) has sign -1. (Contributed by SO, 9-Jul-2018.)
 |-  S  =  ( SymGrp `  D )   &    |-  P  =  (
 Base `  S )   &    |-  N  =  (pmSgn `  D )   =>    |-  (
 ( D  e.  Fin  /\  F  e.  ( P 
 \  (pmEven `  D )
 ) )  ->  ( N `  F )  =  -u 1 )
 
Theorempsgnevpm 19149 A permutation which is even has sign 1. (Contributed by SO, 9-Jul-2018.)
 |-  S  =  ( SymGrp `  D )   &    |-  P  =  (
 Base `  S )   &    |-  N  =  (pmSgn `  D )   =>    |-  (
 ( D  e.  Fin  /\  F  e.  (pmEven `  D ) )  ->  ( N `
  F )  =  1 )
 
Theorempsgnodpmr 19150 If a permutation has sign -1 it is odd (not even). (Contributed by SO, 9-Jul-2018.)
 |-  S  =  ( SymGrp `  D )   &    |-  P  =  (
 Base `  S )   &    |-  N  =  (pmSgn `  D )   =>    |-  (
 ( D  e.  Fin  /\  F  e.  P  /\  ( N `  F )  =  -u 1 )  ->  F  e.  ( P  \  (pmEven `  D )
 ) )
 
Theoremzrhpsgnevpm 19151 The sign of an even permutation embedded into a ring is the multiplicative neutral element of the ring. (Contributed by SO, 9-Jul-2018.)
 |-  Y  =  ( ZRHom `  R )   &    |-  S  =  (pmSgn `  N )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( ( R  e.  Ring  /\  N  e.  Fin  /\  F  e.  (pmEven `  N ) )  ->  ( ( Y  o.  S ) `
  F )  =  .1.  )
 
Theoremzrhpsgnodpm 19152 The sign of an odd permutation embedded into a ring is the additive inverse of the multiplicative neutral element of the ring. (Contributed by SO, 9-Jul-2018.)
 |-  Y  =  ( ZRHom `  R )   &    |-  S  =  (pmSgn `  N )   &    |-  .1.  =  ( 1r `  R )   &    |-  P  =  ( Base `  ( SymGrp `  N )
 )   &    |-  I  =  ( invg `  R )   =>    |-  ( ( R  e.  Ring  /\  N  e.  Fin  /\  F  e.  ( P  \  (pmEven `  N )
 ) )  ->  (
 ( Y  o.  S ) `  F )  =  ( I `  .1.  ) )
 
Theoremzrhcofipsgn 19153 Composition of a  ZRHom homomorphism and the sign function for a finite permutation. (Contributed by AV, 27-Dec-2018.)
 |-  P  =  ( Base `  ( SymGrp `  N )
 )   &    |-  Y  =  ( ZRHom `  R )   &    |-  S  =  (pmSgn `  N )   =>    |-  ( ( N  e.  Fin  /\  Q  e.  P ) 
 ->  ( ( Y  o.  S ) `  Q )  =  ( Y `  ( S `  Q ) ) )
 
Theoremzrhpsgnelbas 19154 Embedding of permutation signs into a ring results in an element of the ring. (Contributed by AV, 1-Jan-2019.)
 |-  P  =  ( Base `  ( SymGrp `  N )
 )   &    |-  S  =  (pmSgn `  N )   &    |-  Y  =  ( ZRHom `  R )   =>    |-  (
 ( R  e.  Ring  /\  N  e.  Fin  /\  Q  e.  P )  ->  ( Y `  ( S `  Q ) )  e.  ( Base `  R ) )
 
Theoremzrhcopsgnelbas 19155 Embedding of permutation signs into a ring results in an element of the ring. (Contributed by AV, 1-Jan-2019.)
 |-  P  =  ( Base `  ( SymGrp `  N )
 )   &    |-  S  =  (pmSgn `  N )   &    |-  Y  =  ( ZRHom `  R )   =>    |-  (
 ( R  e.  Ring  /\  N  e.  Fin  /\  Q  e.  P )  ->  ( ( Y  o.  S ) `  Q )  e.  ( Base `  R ) )
 
Theoremevpmodpmf1o 19156* The function for performing an even permutation after a fixed odd permutation is one to one onto all odd permutations. (Contributed by SO, 9-Jul-2018.)
 |-  S  =  ( SymGrp `  D )   &    |-  P  =  (
 Base `  S )   =>    |-  ( ( D  e.  Fin  /\  F  e.  ( P  \  (pmEven `  D ) ) )  ->  ( f  e.  (pmEven `  D )  |->  ( F ( +g  `  S ) f ) ) : (pmEven `  D )
 -1-1-onto-> ( P  \  (pmEven `  D ) ) )
 
Theorempmtrodpm 19157 A transposition is an odd permutation. (Contributed by SO, 9-Jul-2018.)
 |-  S  =  ( SymGrp `  D )   &    |-  P  =  (
 Base `  S )   &    |-  T  =  ran  (pmTrsp `  D )   =>    |-  ( ( D  e.  Fin  /\  F  e.  T ) 
 ->  F  e.  ( P 
 \  (pmEven `  D )
 ) )
 
Theorempsgnfix1 19158* A permutation of a finite set fixing one element is generated by transpositions not involving the fixed element. (Contributed by AV, 13-Jan-2019.)
 |-  P  =  ( Base `  ( SymGrp `  N )
 )   &    |-  T  =  ran  (pmTrsp `  ( N  \  { K } ) )   &    |-  S  =  ( SymGrp `  ( N  \  { K } )
 )   =>    |-  ( ( N  e.  Fin  /\  K  e.  N ) 
 ->  ( Q  e.  {
 q  e.  P  |  ( q `  K )  =  K }  ->  E. w  e. Word  T ( Q  |`  ( N 
 \  { K }
 ) )  =  ( S  gsumg  w ) ) )
 
Theorempsgnfix2 19159* A permutation of a finite set fixing one element is generated by transpositions not involving the fixed element. (Contributed by AV, 17-Jan-2019.)
 |-  P  =  ( Base `  ( SymGrp `  N )
 )   &    |-  T  =  ran  (pmTrsp `  ( N  \  { K } ) )   &    |-  S  =  ( SymGrp `  ( N  \  { K } )
 )   &    |-  Z  =  ( SymGrp `  N )   &    |-  R  =  ran  (pmTrsp `  N )   =>    |-  ( ( N  e.  Fin  /\  K  e.  N )  ->  ( Q  e.  { q  e.  P  |  ( q `
  K )  =  K }  ->  E. w  e. Word  R Q  =  ( Z  gsumg  w ) ) )
 
TheorempsgndiflemB 19160* Lemma 1 for psgndif 19162. (Contributed by AV, 27-Jan-2019.)
 |-  P  =  ( Base `  ( SymGrp `  N )
 )   &    |-  T  =  ran  (pmTrsp `  ( N  \  { K } ) )   &    |-  S  =  ( SymGrp `  ( N  \  { K } )
 )   &    |-  Z  =  ( SymGrp `  N )   &    |-  R  =  ran  (pmTrsp `  N )   =>    |-  ( ( ( N  e.  Fin  /\  K  e.  N )  /\  Q  e.  { q  e.  P  |  ( q `
  K )  =  K } )  ->  ( ( W  e. Word  T 
 /\  ( Q  |`  ( N 
 \  { K }
 ) )  =  ( S  gsumg 
 W ) )  ->  ( ( U  e. Word  R 
 /\  ( # `  W )  =  ( # `  U )  /\  A. i  e.  ( 0..^ ( # `  W ) ) ( ( ( U `  i ) `  K )  =  K  /\  A. n  e.  ( N 
 \  { K }
 ) ( ( W `
  i ) `  n )  =  (
 ( U `  i
 ) `  n )
 ) )  ->  Q  =  ( Z  gsumg 
 U ) ) ) )
 
TheorempsgndiflemA 19161* Lemma 2 for psgndif 19162. (Contributed by AV, 31-Jan-2019.)
 |-  P  =  ( Base `  ( SymGrp `  N )
 )   &    |-  T  =  ran  (pmTrsp `  ( N  \  { K } ) )   &    |-  S  =  ( SymGrp `  ( N  \  { K } )
 )   &    |-  Z  =  ( SymGrp `  N )   &    |-  R  =  ran  (pmTrsp `  N )   =>    |-  ( ( ( N  e.  Fin  /\  K  e.  N )  /\  Q  e.  { q  e.  P  |  ( q `
  K )  =  K } )  ->  ( ( W  e. Word  T 
 /\  ( Q  |`  ( N 
 \  { K }
 ) )  =  ( S  gsumg 
 W )  /\  U  e. Word  R )  ->  ( Q  =  ( ( SymGrp `
  N )  gsumg  U ) 
 ->  ( -u 1 ^ ( # `
  W ) )  =  ( -u 1 ^ ( # `  U ) ) ) ) )
 
Theorempsgndif 19162* Embedding of permutation signs restricted to a set without a single element into a ring. (Contributed by AV, 31-Jan-2019.)
 |-  P  =  ( Base `  ( SymGrp `  N )
 )   &    |-  S  =  (pmSgn `  N )   &    |-  Z  =  (pmSgn `  ( N  \  { K } ) )   =>    |-  ( ( N  e.  Fin  /\  K  e.  N )  ->  ( Q  e.  { q  e.  P  |  ( q `
  K )  =  K }  ->  ( Z `  ( Q  |`  ( N 
 \  { K }
 ) ) )  =  ( S `  Q ) ) )
 
Theoremzrhcopsgndif 19163* Embedding of permutation signs restricted to a set without a single element into a ring. (Contributed by AV, 31-Jan-2019.)
 |-  P  =  ( Base `  ( SymGrp `  N )
 )   &    |-  S  =  (pmSgn `  N )   &    |-  Z  =  (pmSgn `  ( N  \  { K } ) )   &    |-  Y  =  ( ZRHom `  R )   =>    |-  ( ( N  e.  Fin  /\  K  e.  N ) 
 ->  ( Q  e.  {
 q  e.  P  |  ( q `  K )  =  K }  ->  ( ( Y  o.  Z ) `  ( Q  |`  ( N  \  { K } ) ) )  =  ( ( Y  o.  S ) `
  Q ) ) )
 
10.11.6  The ordered field of real numbers
 
Syntaxcrefld 19164 Extend class notation with the field of real numbers.
 class RRfld
 
Definitiondf-refld 19165 The field of real numbers. (Contributed by Thierry Arnoux, 30-Jun-2019.)
 |- RRfld  =  (flds  RR )
 
Theoremrebase 19166 The base of the field of reals. (Contributed by Thierry Arnoux, 1-Nov-2017.)
 |- 
 RR  =  ( Base ` RRfld
 )
 
Theoremremulg 19167 The multiplication (group power) operation of the group of reals. (Contributed by Thierry Arnoux, 1-Nov-2017.)
 |-  ( ( N  e.  ZZ  /\  A  e.  RR )  ->  ( N (.g ` RRfld
 ) A )  =  ( N  x.  A ) )
 
Theoremresubdrg 19168 The real numbers form a division subring of the complex numbers. (Contributed by Mario Carneiro, 4-Dec-2014.) (Revised by Thierry Arnoux, 30-Jun-2019.)
 |-  ( RR  e.  (SubRing ` fld ) 
 /\ RRfld  e.  DivRing )
 
Theoremresubgval 19169 Subtraction in the field of real numbers. (Contributed by Thierry Arnoux, 30-Jun-2019.)
 |-  .-  =  ( -g ` RRfld
 )   =>    |-  ( ( X  e.  RR  /\  Y  e.  RR )  ->  ( X  -  Y )  =  ( X  .-  Y ) )
 
Theoremreplusg 19170 The addition operation of the field of reals. (Contributed by Thierry Arnoux, 21-Jan-2018.)
 |- 
 +  =  ( +g  ` RRfld
 )
 
Theoremremulr 19171 The multiplication operation of the field of reals. (Contributed by Thierry Arnoux, 1-Nov-2017.)
 |- 
 x.  =  ( .r
 ` RRfld )
 
Theoremre0g 19172 The neutral element of the field of reals. (Contributed by Thierry Arnoux, 1-Nov-2017.)
 |-  0  =  ( 0g
 ` RRfld )
 
Theoremre1r 19173 The multiplicative neutral element of the field of reals. (Contributed by Thierry Arnoux, 1-Nov-2017.)
 |-  1  =  ( 1r
 ` RRfld )
 
Theoremrele2 19174 The ordering relation of the field of reals. (Contributed by Thierry Arnoux, 21-Jan-2018.)
 |- 
 <_  =  ( le ` RRfld
 )
 
Theoremrelt 19175 The ordering relation of the field of reals. (Contributed by Thierry Arnoux, 21-Jan-2018.)
 |- 
 <  =  ( lt ` RRfld
 )
 
Theoremreds 19176 The distance of the field of reals. (Contributed by Thierry Arnoux, 20-Jun-2019.)
 |-  ( abs  o.  -  )  =  ( dist ` RRfld
 )
 
Theoremredvr 19177 The division operation of the field of reals. (Contributed by Thierry Arnoux, 1-Nov-2017.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  B  =/=  0 ) 
 ->  ( A (/r ` RRfld ) B )  =  ( A 
 /  B ) )
 
Theoremretos 19178 The real numbers are a totally ordered set. (Contributed by Thierry Arnoux, 21-Jan-2018.)
 |- RRfld  e. Toset
 
Theoremrefld 19179 The real numbers form a field. (Contributed by Thierry Arnoux, 1-Nov-2017.)
 |- RRfld  e. Field
 
Theoremrefldcj 19180 The conjugation operation of the field of real numbers. (Contributed by Thierry Arnoux, 30-Jun-2019.)
 |-  *  =  ( *r ` RRfld )
 
Theoremrecrng 19181 The real numbers form a star ring. (Contributed by Thierry Arnoux, 19-Apr-2019.)
 |- RRfld  e.  *Ring
 
Theoremregsumsupp 19182* The group sum over the real numbers, expressed as a finite sum. (Contributed by Thierry Arnoux, 22-Jun-2019.) (Proof shortened by AV, 19-Jul-2019.)
 |-  ( ( F : I
 --> RR  /\  F finSupp  0  /\  I  e.  V )  ->  (RRfld  gsumg 
 F )  =  sum_ x  e.  ( F supp  0
 ) ( F `  x ) )
 
10.12  Generalized pre-Hilbert and Hilbert spaces
 
10.12.1  Definition and basic properties
 
Syntaxcphl 19183 Extend class notation with class all pre-Hilbert spaces.
 class  PreHil
 
Syntaxcipf 19184 Extend class notation with inner product function.
 class  .if
 
Definitiondf-phl 19185* Define class all generalized pre-Hilbert (inner product) spaces. (Contributed by NM, 22-Sep-2011.)
 |-  PreHil  =  { g  e. 
 LVec  |  [. ( Base `  g )  /  v ]. [. ( .i `  g )  /  h ].
 [. (Scalar `  g )  /  f ]. ( f  e.  *Ring  /\  A. x  e.  v  ( ( y  e.  v  |->  ( y h x ) )  e.  ( g LMHom  (ringLMod `  f ) )  /\  ( ( x h x )  =  ( 0g `  f ) 
 ->  x  =  ( 0g `  g ) ) 
 /\  A. y  e.  v  ( ( *r `
  f ) `  ( x h y ) )  =  ( y h x ) ) ) }
 
Definitiondf-ipf 19186* Define group addition function. Usually we will use  +g directly instead of  +f, and they have the same behavior in most cases. The main advantage of  +f is that it is a guaranteed function (mndplusf 16548), while  +g only has closure (mndcl 16538). (Contributed by Mario Carneiro, 12-Aug-2015.)
 |- 
 .if  =  ( g  e.  _V  |->  ( x  e.  ( Base `  g ) ,  y  e.  ( Base `  g )  |->  ( x ( .i
 `  g ) y ) ) )
 
Theoremisphl 19187* The predicate "is a generalized pre-Hilbert (inner product) space". (Contributed by NM, 22-Sep-2011.) (Revised by Mario Carneiro, 7-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  F  =  (Scalar `  W )   &    |-  .,  =  ( .i `  W )   &    |-  .0.  =  ( 0g `  W )   &    |- 
 .*  =  ( *r `  F )   &    |-  Z  =  ( 0g `  F )   =>    |-  ( W  e.  PreHil  <->  ( W  e.  LVec  /\  F  e.  *Ring  /\  A. x  e.  V  ( ( y  e.  V  |->  ( y 
 .,  x ) )  e.  ( W LMHom  (ringLMod `  F ) )  /\  ( ( x  .,  x )  =  Z  ->  x  =  .0.  )  /\  A. y  e.  V  (  .*  `  ( x  .,  y ) )  =  ( y  .,  x ) ) ) )
 
Theoremphllvec 19188 A pre-Hilbert space is a left vector space. (Contributed by Mario Carneiro, 7-Oct-2015.)
 |-  ( W  e.  PreHil  ->  W  e.  LVec )
 
Theoremphllmod 19189 A pre-Hilbert space is a left module. (Contributed by Mario Carneiro, 7-Oct-2015.)
 |-  ( W  e.  PreHil  ->  W  e.  LMod )
 
Theoremphlsrng 19190 The scalar ring of a pre-Hilbert space is a star ring. (Contributed by Mario Carneiro, 7-Oct-2015.)
 |-  F  =  (Scalar `  W )   =>    |-  ( W  e.  PreHil  ->  F  e.  *Ring )
 
Theoremphllmhm 19191* The inner product of a pre-Hilbert space is linear in its left argument. (Contributed by Mario Carneiro, 7-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |- 
 .,  =  ( .i
 `  W )   &    |-  V  =  ( Base `  W )   &    |-  G  =  ( x  e.  V  |->  ( x  .,  A ) )   =>    |-  ( ( W  e.  PreHil  /\  A  e.  V ) 
 ->  G  e.  ( W LMHom 
 (ringLMod `  F ) ) )
 
Theoremipcl 19192 Closure of the inner product operation in a pre-Hilbert space. (Contributed by Mario Carneiro, 7-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |- 
 .,  =  ( .i
 `  W )   &    |-  V  =  ( Base `  W )   &    |-  K  =  ( Base `  F )   =>    |-  (
 ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  ( A  .,  B )  e.  K )
 
Theoremipcj 19193 Conjugate of an inner product in a pre-Hilbert space. (Contributed by Mario Carneiro, 7-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |- 
 .,  =  ( .i
 `  W )   &    |-  V  =  ( Base `  W )   &    |-  .*  =  ( *r `  F )   =>    |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  (  .*  `  ( A  .,  B ) )  =  ( B  .,  A ) )
 
Theoremiporthcom 19194 Orthogonality (meaning inner product is 0) is commutative. (Contributed by NM, 17-Apr-2008.) (Revised by Mario Carneiro, 7-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |- 
 .,  =  ( .i
 `  W )   &    |-  V  =  ( Base `  W )   &    |-  Z  =  ( 0g `  F )   =>    |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  ( ( A  .,  B )  =  Z  <->  ( B  .,  A )  =  Z ) )
 
Theoremip0l 19195 Inner product with a zero first argument. Part of proof of Theorem 6.44 of [Ponnusamy] p. 361. (Contributed by NM, 5-Feb-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |- 
 .,  =  ( .i
 `  W )   &    |-  V  =  ( Base `  W )   &    |-  Z  =  ( 0g `  F )   &    |- 
 .0.  =  ( 0g `  W )   =>    |-  ( ( W  e.  PreHil  /\  A  e.  V ) 
 ->  (  .0.  .,  A )  =  Z )
 
Theoremip0r 19196 Inner product with a zero second argument. (Contributed by NM, 5-Feb-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |- 
 .,  =  ( .i
 `  W )   &    |-  V  =  ( Base `  W )   &    |-  Z  =  ( 0g `  F )   &    |- 
 .0.  =  ( 0g `  W )   =>    |-  ( ( W  e.  PreHil  /\  A  e.  V ) 
 ->  ( A  .,  .0.  )  =  Z )
 
Theoremipeq0 19197 The inner product of a vector with itself is zero iff the vector is zero. Part of Definition 3.1-1 of [Kreyszig] p. 129. (Contributed by NM, 24-Jan-2008.) (Revised by Mario Carneiro, 7-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |- 
 .,  =  ( .i
 `  W )   &    |-  V  =  ( Base `  W )   &    |-  Z  =  ( 0g `  F )   &    |- 
 .0.  =  ( 0g `  W )   =>    |-  ( ( W  e.  PreHil  /\  A  e.  V ) 
 ->  ( ( A  .,  A )  =  Z  <->  A  =  .0.  ) )
 
Theoremipdir 19198 Distributive law for inner product (right-distributivity). Equation I3 of [Ponnusamy] p. 362. (Contributed by NM, 25-Aug-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |- 
 .,  =  ( .i
 `  W )   &    |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  .+^  =  (
 +g  `  F )   =>    |-  (
 ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
 )  ->  ( ( A  .+  B )  .,  C )  =  (
 ( A  .,  C )  .+^  ( B  .,  C ) ) )
 
Theoremipdi 19199 Distributive law for inner product (left-distributivity). (Contributed by NM, 20-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |- 
 .,  =  ( .i
 `  W )   &    |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  .+^  =  (
 +g  `  F )   =>    |-  (
 ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
 )  ->  ( A  .,  ( B  .+  C ) )  =  (
 ( A  .,  B )  .+^  ( A  .,  C ) ) )
 
Theoremip2di 19200 Distributive law for inner product. (Contributed by NM, 17-Apr-2008.) (Revised by Mario Carneiro, 7-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |- 
 .,  =  ( .i
 `  W )   &    |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  .+^  =  (
 +g  `  F )   &    |-  ( ph  ->  W  e.  PreHil )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  V )   &    |-  ( ph  ->  C  e.  V )   &    |-  ( ph  ->  D  e.  V )   =>    |-  ( ph  ->  (
 ( A  .+  B )  .,  ( C  .+  D ) )  =  ( ( ( A 
 .,  C )  .+^  ( B  .,  D ) )  .+^  ( ( A  .,  D )  .+^  ( B  .,  C ) ) ) )
    < 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 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39884
  Copyright terms: Public domain < Previous  Next >