HomeHome Metamath Proof Explorer
Theorem List (p. 251 of 325)
< 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
(1-22374)
  Hilbert Space Explorer  Hilbert Space Explorer
(22375-23897)
  Users' Mathboxes  Users' Mathboxes
(23898-32447)
 

Theorem List for Metamath Proof Explorer - 25001-25100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
19.4.12  Models of ZF
 
Syntaxcgze 25001 The Axiom of Extensionality.
 class  AxExt
 
Syntaxcgzr 25002 The Axiom Scheme of Replacement.
 class  AxRep
 
Syntaxcgzp 25003 The Axiom of Power Sets.
 class  AxPow
 
Syntaxcgzu 25004 The Axiom of Unions.
 class  AxUn
 
Syntaxcgzg 25005 The Axiom of Regularity.
 class  AxReg
 
Syntaxcgzi 25006 The Axiom of Infinity.
 class  AxInf
 
Syntaxcgzf 25007 The set of models of ZF.
 class  ZF
 
Definitiondf-gzext 25008 The Godel-set version of the Axiom of Extensionality. (Contributed by Mario Carneiro, 14-Jul-2013.)
 |-  AxExt  =  (
 A.g 2o ( ( 2o 
 e.g  (/) )  <->g  ( 2o  e.g 
 1o ) )  ->g  ( (/)  =g  1o ) )
 
Definitiondf-gzrep 25009 The Godel-set version of the Axiom Scheme of Replacement. Since this is a scheme and not a single axiom, it manifests as a function on wffs, each giving rise to a different axiom. (Contributed by Mario Carneiro, 14-Jul-2013.)
 |-  AxRep  =  ( u  e.  ( Fmla ` 
 om )  |->  ( A.g 3o E.g 1o A.g 2o ( A.g 1o u  ->g  ( 2o  =g  1o ) )  ->g  A.g 1o A.g 2o ( ( 2o  e.g  1o )  <->g  E.g 3o ( ( 3o  e.g  (/) )  /\g  A.g
 1o u ) ) ) )
 
Definitiondf-gzpow 25010 The Godel-set version of the Axiom of Power Sets. (Contributed by Mario Carneiro, 14-Jul-2013.)
 |-  AxPow  =  E.g 1o A.g 2o ( A.g 1o (
 ( 1o  e.g  2o ) 
 <->g  ( 1o  e.g  (/) ) ) 
 ->g  ( 2o  e.g  1o ) )
 
Definitiondf-gzun 25011 The Godel-set version of the Axiom of Unions. (Contributed by Mario Carneiro, 14-Jul-2013.)
 |-  AxUn  = 
 E.g 1o A.g 2o ( E.g 1o ( ( 2o  e.g  1o )  /\g  ( 1o  e.g  (/) ) )  ->g  ( 2o 
 e.g  1o ) )
 
Definitiondf-gzreg 25012 The Godel-set version of the Axiom of Regularity. (Contributed by Mario Carneiro, 14-Jul-2013.)
 |-  AxReg  =  (
 E.g 1o ( 1o  e.g  (/) )  ->g  E.g 1o ( ( 1o  e.g  (/) )  /\g  A.g
 2o ( ( 2o 
 e.g  1o )  ->g  -.g ( 2o  e.g  (/) ) ) ) )
 
Definitiondf-gzinf 25013 The Godel-set version of the Axiom of Infinity. (Contributed by Mario Carneiro, 14-Jul-2013.)
 |-  AxInf  =  E.g 1o ( ( (/)  e.g  1o )  /\g  A.g 2o ( ( 2o  e.g  1o )  ->g 
 E.g (/) ( ( 2o 
 e.g  (/) )  /\g  ( (/) 
 e.g  1o ) ) ) )
 
Definitiondf-gzf 25014* Define the class of all (transitive) models of ZF. (Contributed by Mario Carneiro, 14-Jul-2013.)
 |-  ZF  =  { m  |  ( ( Tr  m  /\  m  |=  AxExt  /\  m  |=  AxPow
 )  /\  ( m  |= 
 AxUn  /\  m  |=  AxReg  /\  m  |= 
 AxInf )  /\  A. u  e.  ( Fmla `  om ) m 
 |=  ( AxRep `  u ) ) }
 
19.4.13  Splitting fields
 
Syntaxcitr 25015 Integral subring of a ring.
 class IntgRing
 
Syntaxccpms 25016 Completion of a metric space.
 class cplMetSp
 
Syntaxchlb 25017 Embeddings for a direct limit.
 class HomLimB
 
Syntaxchlim 25018 Direct limit structure.
 class HomLim
 
Syntaxcpfl 25019 Polynomial extension field.
 class polyFld
 
Syntaxcsf1 25020 Splitting field for a single polynomial (auxiliary).
 class splitFld1
 
Syntaxcsf 25021 Splitting field for a finite set of polynomials.
 class splitFld
 
Syntaxcpsl 25022 Splitting field for a sequence of polynomials.
 class polySplitLim
 
Definitiondf-irng 25023* Define the subring of elements of  r integral over  s in a ring. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |- IntgRing  =  ( r  e.  _V ,  s  e.  _V  |->  U_ f  e.  (Monic1p `
  ( rs  s ) ) ( `' f " { ( 0g `  r ) } )
 )
 
Definitiondf-cplmet 25024* A function which completes the given metric space. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |- cplMetSp  =  ( w  e.  _V  |->  [_ ( ( w  ^s  NN )s  ( Cau `  ( dist `  w ) ) ) 
 /  r ]_ [_ ( Base `  r )  /  v ]_ [_ { <. f ,  g >.  |  ( { f ,  g }  C_  v  /\  A. x  e.  RR+  E. j  e.  ZZ  ( f  |`  ( ZZ>= `  j )
 ) : ( ZZ>= `  j ) --> ( ( g `  j ) ( ball `  ( dist `  w ) ) x ) ) }  /  e ]_ ( ( r 
 /.s 
 e ) sSet  { <. (
 dist `  ndx ) ,  { <. <. x ,  y >. ,  z >.  |  E. p  e.  v  E. q  e.  v  (
 ( x  =  [ p ] e  /\  y  =  [ q ] e
 )  /\  ( p  o F ( dist `  r
 ) q )  ~~>  z ) } >. } ) )
 
Definitiondf-homlimb 25025* The input to this function is a sequence (on  NN) of homomorphisms  F ( n ) : R ( n ) --> R ( n  +  1 ). The resulting structure is the direct limit of the direct system so defined. This function returns the pair  <. S ,  G >. where 
S is the terminal object and  G is a sequence of functions such that  G ( n ) : R ( n ) --> S and  G ( n )  =  F ( n )  o.  G
( n  +  1 ). (Contributed by Mario Carneiro, 2-Dec-2014.)
 |- HomLimB  =  ( f  e.  _V  |->  [_ U_ n  e.  NN  ( { n }  X.  dom  ( f `  n ) )  /  v ]_ [_ |^| { s  |  ( s  Er  v  /\  ( x  e.  v  |-> 
 <. ( ( 1st `  x )  +  1 ) ,  ( ( f `  ( 1st `  x )
 ) `  ( 2nd `  x ) ) >. ) 
 C_  s ) }  /  e ]_ <. ( v
 /. e ) ,  ( n  e.  NN  |->  ( x  e.  dom  ( f `  n )  |->  [ <. n ,  x >. ] e ) )
 >. )
 
Definitiondf-homlim 25026* The input to this function is a sequence (on  NN) of structures  R ( n ) and homomorphisms  F ( n ) : R ( n ) --> R ( n  +  1 ). The resulting structure is the direct limit of the direct system so defined, and maintains any structures that were present in the original objects. TODO: generalize to directed sets? (Contributed by Mario Carneiro, 2-Dec-2014.)
 |- HomLim  =  ( r  e.  _V ,  f  e.  _V  |->  [_ ( HomLimB  `  f )  /  e ]_ [_ ( 1st `  e
 )  /  v ]_ [_ ( 2nd `  e
 )  /  g ]_ ( { <. ( Base `  ndx ) ,  v >. , 
 <. ( +g  `  ndx ) ,  U_ n  e. 
 NN  ran  ( x  e.  dom  ( g `  n ) ,  y  e.  dom  ( g `  n )  |->  <. <. ( ( g `  n ) `
  x ) ,  ( ( g `  n ) `  y
 ) >. ,  ( ( g `  n ) `
  ( x (
 +g  `  ( r `  n ) ) y ) ) >. ) >. , 
 <. ( .r `  ndx ) ,  U_ n  e. 
 NN  ran  ( x  e.  dom  ( g `  n ) ,  y  e.  dom  ( g `  n )  |->  <. <. ( ( g `  n ) `
  x ) ,  ( ( g `  n ) `  y
 ) >. ,  ( ( g `  n ) `
  ( x ( .r `  ( r `
  n ) ) y ) ) >. )
 >. }  u.  { <. (
 TopOpen `  ndx ) ,  { s  e.  ~P v  |  A. n  e. 
 NN  ( `' (
 g `  n ) " s )  e.  ( TopOpen `  ( r `  n ) ) } >. , 
 <. ( dist `  ndx ) , 
 U_ n  e.  NN  ran  ( x  e.  dom  ( ( g `  n ) `  n ) ,  y  e.  dom  ( ( g `  n ) `  n )  |->  <. <. ( ( g `
  n ) `  x ) ,  (
 ( g `  n ) `  y ) >. ,  ( x ( dist `  ( r `  n ) ) y )
 >. ) >. ,  <. ( le ` 
 ndx ) ,  U_ n  e.  NN  ( `' ( g `  n )  o.  ( ( le `  ( r `  n ) )  o.  (
 g `  n )
 ) ) >. } )
 )
 
Definitiondf-plfl 25027* Define the field extension that augments a field with the root of the given irreducible polynomial, and extends the norm if one exists and the extension is unique. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |- polyFld  =  ( r  e.  _V ,  p  e.  _V  |->  [_ (Poly1 `  r )  /  s ]_ [_ ( (RSpan `  s ) `  { p } )  /  i ]_ [_ ( z  e.  ( Base `  r )  |->  [ ( z ( .s `  s ) ( 1r `  s
 ) ) ] (
 s ~QG 
 i ) )  /  f ]_ <. [_ ( s  /.s  (
 s ~QG 
 i ) )  /  t ]_ ( ( t toNrmGrp  ( iota_ n  e.  (AbsVal `  t ) ( n  o.  f )  =  ( norm `  r )
 ) ) sSet  <. ( le ` 
 ndx ) ,  [_ ( z  e.  ( Base `  t )  |->  (
 iota_ q  e.  z
 ( r deg1  q )  < 
 ( r deg1  p ) ) )  /  g ]_ ( `' g  o.  (
 ( le `  s
 )  o.  g ) ) >. ) ,  f >. )
 
Definitiondf-sfl1 25028* Temporary construction for the splitting field of a polynomial. The inputs are a field  r and a polynomial  p that we want to split, along with a tuple  j in the same format as the output. The output is a tuple  <. S ,  F >. where 
S is the splitting field and  F is an injective homomorphism from the original field  r.

The function works by repeatedly finding the smallest monic irreducible factor, and extending the field by that factor using the polyFld construction. We keep track of a total order in each of the splitting fields so that we can pick an element definably without needing global choice. (Contributed by Mario Carneiro, 2-Dec-2014.)

 |- splitFld1  =  (
 r  e.  _V ,  j  e.  _V  |->  ( p  e.  (Poly1 `  r )  |->  ( rec ( ( s  e.  _V ,  f  e.  _V  |->  [_ ( mPoly  `  s
 )  /  m ]_ [_ { g  e.  ( (Monic1p `
  s )  i^i  (Irred `  m )
 )  |  ( g ( ||r
 `  m ) ( p  o.  f ) 
 /\  1  <  (
 s deg1  g ) ) }  /  b ]_ if (
 ( ( p  o.  f )  =  ( 0g `  m )  \/  b  =  (/) ) , 
 <. s ,  f >. , 
 [_ ( glb `  b
 )  /  h ]_ [_ (
 s polyFld  h )  /  t ]_ <. ( 1st `  t
 ) ,  ( f  o.  ( 2nd `  t
 ) ) >. ) ) ,  j ) `  ( card `  ( 1 ... ( r deg1  p ) ) ) ) ) )
 
Definitiondf-sfl 25029* Define the splitting field of a finite collection of polynomials, given a total ordered base field. The output is a tuple  <. S ,  F >. where  S is the totally ordered splitting field and  F is an injective homomorphism from the original field  r. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |- splitFld  =  ( r  e.  _V ,  p  e.  _V  |->  ( iota
 x E. f ( f  Isom  <  ,  ( lt `  r ) ( ( 1 ... ( # `
  p ) ) ,  p )  /\  x  =  (  seq  0 ( ( e  e.  _V ,  g  e.  _V  |->  ( ( r splitFld1  e ) `  g ) ) ,  ( f  u.  { <. 0 ,  <. r ,  (  _I  |`  ( Base `  r
 ) ) >. >. } )
 ) `  ( # `  p ) ) ) ) )
 
Definitiondf-psl 25030* Define the direct limit of an increasing sequence of fields produced by pasting together the splitting fields for each sequence of polynomials. That is, given a ring  r, a strict order on  r, and a sequence  p : NN --> ( ~P r  i^i  Fin ) of finite sets of polynomials to split, we construct the direct limit system of field extensions by splitting one set at a time and passing the resulting construction to HomLim. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |- polySplitLim  =  ( r  e.  _V ,  p  e.  ( ( ~P ( Base `  r )  i^i  Fin )  ^m  NN )  |->  [_ ( 1st  o.  seq  0 ( ( g  e.  _V ,  q  e.  _V  |->  [_ ( 1st `  g
 )  /  e ]_ [_ ( 1st `  e
 )  /  s ]_ [_ ( s splitFld  ran  ( x  e.  q  |->  ( x  o.  ( 2nd `  g
 ) ) ) ) 
 /  f ]_ <. f ,  ( ( 2nd `  g
 )  o.  ( 2nd `  f ) ) >. ) ,  ( p  u.  {
 <. 0 ,  <. <. r ,  (/) >. ,  (  _I  |`  ( Base `  r )
 ) >. >. } ) ) )  /  f ]_ ( ( 1st  o.  ( f  shift  1 ) ) HomLim  ( 2nd  o.  f ) ) )
 
19.4.14  p-adic number fields
 
Syntaxczr 25031 Integral elements of a ring.
 class ZRing
 
Syntaxcgf 25032 Galois finite field.
 class GF
 
Syntaxcgfo 25033 Galois limit field.
 class GF
 
Syntaxceqp 25034 Equivalence relation for df-qp 25045.
 class ~Qp
 
Syntaxcrqp 25035 Equivalence relation representatives for df-qp 25045.
 class /Qp
 
Syntaxcqp 25036 The set of  p-adic rational numbers.
 class Qp
 
Syntaxczp 25037 The set of  p-adic integers. (Not to be confused with czn 16736.)
 class Zp
 
Syntaxcqpa 25038 Algebraic completion of the  p-adic rational numbers.
 class _Qp
 
Syntaxccp 25039 Metric completion of _Qp.
 class Cp
 
Definitiondf-zrng 25040 Define the subring of integral elements in a ring. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |- ZRing  =  ( r  e.  _V  |->  ( r IntgRing  ran  ( ZRHom `  r
 ) ) )
 
Definitiondf-gf 25041* Define the Galois finite field of order  p ^ n. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |- GF  =  ( p  e.  Prime ,  n  e.  NN  |->  [_ (ℤ/n `  p )  /  r ]_ ( 1st `  (
 r splitFld  { [_ (Poly1 `  r
 )  /  s ]_ [_ (var1 `  r )  /  x ]_ ( ( ( p ^ n ) (.g `  (mulGrp `  s
 ) ) x ) ( -g `  s
 ) x ) }
 ) ) )
 
Definitiondf-gfoo 25042* Define the Galois field of order  p ^  +oo, as a direct limit of the Galois finite fields. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |- GF  =  ( p  e.  Prime  |->  [_ (ℤ/n `  p )  /  r ]_ (
 r polySplitLim  ( n  e.  NN  |->  {
 [_ (Poly1 `  r )  /  s ]_ [_ (var1 `  r
 )  /  x ]_ (
 ( ( p ^ n ) (.g `  (mulGrp `  s ) ) x ) ( -g `  s
 ) x ) }
 ) ) )
 
Definitiondf-eqp 25043* Define an equivalence relation on 
ZZ-indexed sequences of integers such that two sequences are equivalent iff the difference is equivalent to zero, and a sequence is equivalent to zero iff the sum  sum_ k  <_  n f ( k ) ( p ^
k ) is a multiple of  p ^ (
n  +  1 ) for every  n. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |- ~Qp  =  ( p  e.  Prime  |->  { <. f ,  g >.  |  ( { f ,  g }  C_  ( ZZ  ^m  ZZ )  /\  A. n  e.  ZZ  sum_ k  e.  ( ZZ>=
 `  -u n ) ( ( ( f `  -u k )  -  (
 g `  -u k ) )  /  ( p ^ ( k  +  ( n  +  1
 ) ) ) )  e.  ZZ ) }
 )
 
Definitiondf-rqp 25044* There is a unique element of  ( ZZ  ^m  (
0 ... ( p  - 
1 ) ) ) ~Qp -equivalent to any element of 
( ZZ  ^m  ZZ ), if the sequences are zero for sufficiently large negative values; this function selects that element. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |- /Qp  =  ( p  e.  Prime  |->  (~Qp  i^i  [_
 { f  e.  ( ZZ  ^m  ZZ )  | 
 E. x  e.  ran  ZZ>= ( `' f " ( ZZ  \  { 0 } )
 )  C_  x }  /  y ]_ ( y  X.  ( y  i^i  ( ZZ  ^m  (
 0 ... ( p  -  1 ) ) ) ) ) ) )
 
Definitiondf-qp 25045* Define the  p-adic completion of the rational numbers, as a normed field structure with a total order (that is not compatible with the operations). (Contributed by Mario Carneiro, 2-Dec-2014.)
 |- Qp  =  ( p  e.  Prime  |->  [_
 { h  e.  ( ZZ  ^m  ( 0 ... ( p  -  1
 ) ) )  | 
 E. x  e.  ran  ZZ>= ( `' h " ( ZZ  \  { 0 } )
 )  C_  x }  /  b ]_ ( ( { <. ( Base `  ndx ) ,  b >. , 
 <. ( +g  `  ndx ) ,  ( f  e.  b ,  g  e.  b  |->  ( (/Qp `  p ) `  (
 f  o F  +  g ) ) )
 >. ,  <. ( .r `  ndx ) ,  ( f  e.  b ,  g  e.  b  |->  ( (/Qp `  p ) `  ( n  e.  ZZ  |->  sum_ k  e.  ZZ  ( ( f `
  k )  x.  ( g `  ( n  -  k ) ) ) ) ) )
 >. }  u.  { <. ( le `  ndx ) ,  { <. f ,  g >.  |  ( { f ,  g }  C_  b  /\  sum_ k  e.  ZZ  ( ( f `  -u k )  x.  (
 ( p  +  1 ) ^ -u k
 ) )  <  sum_ k  e.  ZZ  ( ( g `
  -u k )  x.  ( ( p  +  1 ) ^ -u k
 ) ) ) } >. } ) toNrmGrp  ( f  e.  b  |->  if (
 f  =  ( ZZ 
 X.  { 0 } ) ,  0 ,  ( p ^ -u sup ( ( `' f " ( ZZ  \  { 0 } )
 ) ,  RR ,  `'  <  ) ) ) ) ) )
 
Definitiondf-zp 25046 Define the  p-adic integers, as a subset of the  p-adic rationals. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |- Zp  =  (ZRing  o. Qp )
 
Definitiondf-qpa 25047* Define the completion of the  p-adic rationals. Here we simply define it as the splitting field of a dense sequence of polynomials (using as the  n-th set the collection of polynomials with degree less than  n and with coefficients  <  ( p ^
n )). Krasner's lemma will then show that all monic polynomials have splitting fields isomorphic to a sufficiently close Eisenstein polynomial from the list, and unramified extensions are generated by the polynomial  x ^ (
p ^ n )  -  x, which is in the list. Thus, every finite extension of Qp is a subfield of this field extension, so it is algebraically closed. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |- _Qp  =  ( p  e.  Prime  |->  [_ (Qp `  p )  /  r ]_ ( r polySplitLim  ( n  e.  NN  |->  { f  e.  (Poly1 `  r )  |  (
 ( r deg1  f )  <_  n  /\  A. d  e. 
 ran  (coe1 `  f ) ( `' d " ( ZZ  \  { 0 } )
 )  C_  ( 0 ... n ) ) }
 ) ) )
 
Definitiondf-cp 25048 Define the metric completion of the algebraic completion of the  p -adic rationals. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |- Cp  =  ( cplMetSp  o. _Qp )
 
19.5  Mathbox for Paul Chapman
 
19.5.1  Group homomorphism and isomorphism
 
Theoremghomgrpilem1 25049 Lemma for ghomgrpi 25051. (Contributed by Paul Chapman, 25-Feb-2008.)
 |-  G  e.  GrpOp   &    |-  H  e.  GrpOp   &    |-  F  e.  ( G GrpOpHom  H )   &    |-  X  =  ran  G   &    |-  U  =  (GId `  G )   &    |-  N  =  ( inv `  G )   &    |-  W  =  ran  H   &    |-  T  =  (GId `  H )   &    |-  M  =  ( inv `  H )   &    |-  Z  =  ran  F   &    |-  S  =  ( H  |`  ( Z  X.  Z ) )   =>    |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( ( F `
  A ) H ( F `  B ) )  =  ( F `  ( A G B ) ) )
 
Theoremghomgrpilem2 25050 Lemma for ghomgrpi 25051. (Contributed by Paul Chapman, 25-Feb-2008.)
 |-  G  e.  GrpOp   &    |-  H  e.  GrpOp   &    |-  F  e.  ( G GrpOpHom  H )   &    |-  X  =  ran  G   &    |-  U  =  (GId `  G )   &    |-  N  =  ( inv `  G )   &    |-  W  =  ran  H   &    |-  T  =  (GId `  H )   &    |-  M  =  ( inv `  H )   &    |-  Z  =  ran  F   &    |-  S  =  ( H  |`  ( Z  X.  Z ) )   =>    |-  S  e.  ( SubGrpOp `  H )
 
Theoremghomgrpi 25051 The image of a group homomorphism from  G to  H is a subgroup of  H (inference version). (Contributed by Paul Chapman, 25-Feb-2008.)
 |-  G  e.  GrpOp   &    |-  H  e.  GrpOp   &    |-  F  e.  ( G GrpOpHom  H )   &    |-  Y  =  ran  F   &    |-  S  =  ( H  |`  ( Y  X.  Y ) )   =>    |-  S  e.  ( SubGrpOp `  H )
 
Theoremghomsn 25052 The endomorphism of the trivial group. (Contributed by Paul Chapman, 25-Feb-2008.)
 |-  A  e.  _V   &    |-  G  =  { <.
 <. A ,  A >. ,  A >. }   =>    |-  (  _I  |`  { A } )  e.  ( G GrpOpHom  G )
 
Theoremghomgrplem 25053 Lemma for ghomgrp 25054. (Contributed by Paul Chapman, 25-Feb-2008.)
 |-  ( ph  ->  ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H )
 ) )   &    |-  S  =  { <.
 <. z ,  z >. ,  z >. }   &    |-  J  =  (  _I  |`  { z } )   =>    |-  ( ph  ->  ( H  |`  ( ran  F  X.  ran  F ) )  e.  ( SubGrpOp `  H ) )
 
Theoremghomgrp 25054 The image of a group homomorphism from  G to  H is a subgroup of  H. (Contributed by Paul Chapman, 25-Feb-2008.)
 |-  Y  =  ran  F   &    |-  S  =  ( H  |`  ( Y  X.  Y ) )   =>    |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  S  e.  ( SubGrpOp `  H ) )
 
Theoremghomfo 25055 A group homomorphism maps onto its image. (Contributed by Paul Chapman, 3-Mar-2008.)
 |-  X  =  ran  G   &    |-  Y  =  ran  F   &    |-  S  =  ( H  |`  ( Y  X.  Y ) )   &    |-  Z  =  ran  S   =>    |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H )
 )  ->  F : X -onto-> Z )
 
Theoremghomcl 25056 Closure of a group homomorphism. (Contributed by Paul Chapman, 3-Mar-2008.)
 |-  X  =  ran  G   &    |-  Y  =  ran  F   &    |-  S  =  ( H  |`  ( Y  X.  Y ) )   &    |-  Z  =  ran  S   =>    |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H )
 )  ->  ( A  e.  X  ->  ( F `  A )  e.  Z ) )
 
Theoremghomgsg 25057 A group homomorphism from  G to  H is also a group homomorphism from  G to its image in  H. (Contributed by Paul Chapman, 3-Mar-2008.)
 |-  Y  =  ran  F   &    |-  S  =  ( H  |`  ( Y  X.  Y ) )   =>    |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  F  e.  ( G GrpOpHom  S ) )
 
Theoremghomf1olem 25058* Lemma for ghomf1o 25059. (Contributed by Paul Chapman, 3-Mar-2008.)
 |-  X  =  ran  G   &    |-  Y  =  ran  F   &    |-  S  =  ( H  |`  ( Y  X.  Y ) )   &    |-  Z  =  ran  S   &    |-  U  =  (GId `  G )   &    |-  T  =  (GId `  H )   &    |-  N  =  ( inv `  G )   =>    |-  (
 ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H )
 )  ->  ( F : X -1-1-onto-> Z  <->  A. x  e.  X  ( ( F `  x )  =  T  ->  x  =  U ) ) )
 
Theoremghomf1o 25059* Two ways of saying a group homomorphism is 1-1-onto its image. (Contributed by Paul Chapman, 3-Mar-2008.)
 |-  X  =  ran  G   &    |-  Y  =  ran  F   &    |-  S  =  ( H  |`  ( Y  X.  Y ) )   &    |-  Z  =  ran  S   &    |-  U  =  (GId `  G )   &    |-  T  =  (GId `  H )   =>    |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H )
 )  ->  ( F : X -1-1-onto-> Z  <->  A. x  e.  X  ( ( F `  x )  =  T  ->  x  =  U ) ) )
 
Theoremelgiso 25060 Membership in the set of group isomorphisms from  G to  H. (Contributed by Paul Chapman, 25-Feb-2008.)
 |-  (
 ( G  e.  GrpOp  /\  H  e.  GrpOp )  ->  ( F  e.  ( G  GrpOpIso  H )  <->  ( F  e.  ( G GrpOpHom  H )  /\  F : ran  G -1-1-onto-> ran  H ) ) )
 
19.5.2  Real and complex numbers (cont.)
 
Theoremclimuzcnv 25061* Utility lemma to convert between  m  <_  k and  k  e.  ( ZZ>= `  m ) in limit theorems. (Contributed by Paul Chapman, 10-Nov-2012.)
 |-  ( m  e.  NN  ->  ( ( k  e.  ( ZZ>=
 `  m )  ->  ph )  <->  ( k  e. 
 NN  ->  ( m  <_  k  ->  ph ) ) ) )
 
Theoremsinccvglem 25062*  ( ( sin `  x )  /  x )  ~~>  1 as (real)  x  ~~>  0. (Contributed by Paul Chapman, 10-Nov-2012.) (Revised by Mario Carneiro, 21-May-2014.)
 |-  ( ph  ->  F : NN --> ( RR  \  { 0 } ) )   &    |-  ( ph  ->  F  ~~>  0 )   &    |-  G  =  ( x  e.  ( RR  \  { 0 } )  |->  ( ( sin `  x )  /  x ) )   &    |-  H  =  ( x  e.  CC  |->  ( 1  -  ( ( x ^ 2 ) 
 /  3 ) ) )   &    |-  ( ph  ->  M  e.  NN )   &    |-  (
 ( ph  /\  k  e.  ( ZZ>= `  M )
 )  ->  ( abs `  ( F `  k
 ) )  <  1
 )   =>    |-  ( ph  ->  ( G  o.  F )  ~~>  1 )
 
Theoremsinccvg 25063*  ( ( sin `  x )  /  x )  ~~>  1 as (real)  x  ~~>  0. (Contributed by Paul Chapman, 10-Nov-2012.) (Proof shortened by Mario Carneiro, 21-May-2014.)
 |-  (
 ( F : NN --> ( RR  \  { 0 } )  /\  F  ~~>  0 )  ->  ( ( x  e.  ( RR  \  { 0 } )  |->  ( ( sin `  x )  /  x ) )  o.  F )  ~~>  1 )
 
Theoremcircum 25064* The circumference of a circle of radius  R, defined as the limit as  n  ~~>  +oo of the perimeter of an inscribed n-sided isogons, is  ( (
2  x.  pi )  x.  R ). (Contributed by Paul Chapman, 10-Nov-2012.) (Proof shortened by Mario Carneiro, 21-May-2014.)
 |-  A  =  ( ( 2  x.  pi )  /  n )   &    |-  P  =  ( n  e.  NN  |->  ( ( 2  x.  n )  x.  ( R  x.  ( sin `  ( A  /  2 ) ) ) ) )   &    |-  R  e.  RR   =>    |-  P  ~~>  ( ( 2  x.  pi )  x.  R )
 
19.5.3  Miscellaneous theorems
 
Theoremelfzm12 25065 Membership in a curtailed finite sequence of integers. (Contributed by Paul Chapman, 17-Nov-2012.)
 |-  ( N  e.  NN  ->  ( M  e.  ( 1
 ... ( N  -  1 ) )  ->  M  e.  ( 1 ... N ) ) )
 
Theoremnn0seqcvg 25066* A strictly-decreasing nonnegative integer sequence with initial term  N reaches zero by the  N th term. Inference version. (Contributed by Paul Chapman, 31-Mar-2011.)
 |-  F : NN0 --> NN0   &    |-  N  =  ( F `
  0 )   &    |-  (
 k  e.  NN0  ->  ( ( F `  (
 k  +  1 ) )  =/=  0  ->  ( F `  ( k  +  1 ) )  <  ( F `  k ) ) )   =>    |-  ( F `  N )  =  0
 
Theoremzmodid2 25067 Identity law for modulo restricted to integers. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  (
 ( M  e.  ZZ  /\  N  e.  NN )  ->  ( ( M  mod  N )  =  M  <->  M  e.  (
 0 ... ( N  -  1 ) ) ) )
 
Theoremmodaddabs 25068 Absorbtion law for modulo. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR+ )  ->  ( ( ( A 
 mod  C )  +  ( B  mod  C ) ) 
 mod  C )  =  ( ( A  +  B )  mod  C ) )
 
Theoremelfzp1b 25069 An integer is a member of a 0-based finite set of sequential integers iff its successor is a member of the corresponding 1-based set. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  (
 ( K  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  e.  (
 0 ... ( N  -  1 ) )  <->  ( K  +  1 )  e.  (
 1 ... N ) ) )
 
Theoremlediv2aALT 25070 Division of both sides of 'less than or equal to' by a nonnegative number. (Contributed by Paul Chapman, 7-Sep-2007.)
 |-  (
 ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  <  B )  /\  ( C  e.  RR  /\  0  <_  C ) )  ->  ( A 
 <_  B  ->  ( C  /  B )  <_  ( C  /  A ) ) )
 
Theoremabs2sqlei 25071 The absolute values of two numbers compare as their squares. (Contributed by Paul Chapman, 7-Sep-2007.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  (
 ( abs `  A )  <_  ( abs `  B ) 
 <->  ( ( abs `  A ) ^ 2 )  <_  ( ( abs `  B ) ^ 2 ) )
 
Theoremabs2sqlti 25072 The absolute values of two numbers compare as their squares. (Contributed by Paul Chapman, 7-Sep-2007.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  (
 ( abs `  A )  <  ( abs `  B ) 
 <->  ( ( abs `  A ) ^ 2 )  < 
 ( ( abs `  B ) ^ 2 ) )
 
Theoremabs2sqle 25073 The absolute values of two numbers compare as their squares. (Contributed by Paul Chapman, 7-Sep-2007.)
 |-  (
 ( A  e.  CC  /\  B  e.  CC )  ->  ( ( abs `  A )  <_  ( abs `  B ) 
 <->  ( ( abs `  A ) ^ 2 )  <_  ( ( abs `  B ) ^ 2 ) ) )
 
Theoremabs2sqlt 25074 The absolute values of two numbers compare as their squares. (Contributed by Paul Chapman, 7-Sep-2007.)
 |-  (
 ( A  e.  CC  /\  B  e.  CC )  ->  ( ( abs `  A )  <  ( abs `  B ) 
 <->  ( ( abs `  A ) ^ 2 )  < 
 ( ( abs `  B ) ^ 2 ) ) )
 
Theoremabs2difi 25075 Difference of absolute values. (Contributed by Paul Chapman, 7-Sep-2007.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  (
 ( abs `  A )  -  ( abs `  B ) )  <_  ( abs `  ( A  -  B ) )
 
Theoremabs2difabsi 25076 Absolute value of difference of absolute values. (Contributed by Paul Chapman, 7-Sep-2007.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( abs `  ( ( abs `  A )  -  ( abs `  B ) ) )  <_  ( abs `  ( A  -  B ) )
 
19.6  Mathbox for Drahflow

This is the mathbox of Jens-Wolfhard Schicke-Uffmann, reachable at drahflow@gmx.de / drahflow.name

 
Theoremsbcung 25077* Distribution of class substitution over union of two classes. (Contributed by Drahflow, 23-Sep-2015.) (Revised by Mario Carneiro, 11-Dec-2016.)
 |-  ( A  e.  _V  ->  [_ A  /  x ]_ ( C  u.  D )  =  ( [_ A  /  x ]_ C  u.  [_ A  /  x ]_ D ) )
 
Theoremsbcuni 25078* Distribution of class substitution over union of two classes, inference version. (Contributed by Drahflow, 23-Sep-2015.)
 |-  A  e.  _V   =>    |-  [_ A  /  x ]_ ( C  u.  D )  =  ( [_ A  /  x ]_ C  u.  [_ A  /  x ]_ D )
 
Theoremsbcopg 25079* Distribution of class substitution over ordered pairs. (Contributed by Drahflow, 25-Sep-2015.) (Revised by Mario Carneiro, 29-Oct-2015.)
 |-  ( A  e.  _V  ->  [_ A  /  x ]_ <. C ,  D >.  = 
 <. [_ A  /  x ]_ C ,  [_ A  /  x ]_ D >. )
 
Syntaxcrelexp 25080 Extend class notation to include relation exponentiation.
 class  ^ r
 
Definitiondf-relexp 25081* Definition of repeated composition of a relation with itself, aka relation exponentiation. (Contributed by Drahflow, 12-Nov-2015.)
 |-  ^ r  =  ( r  e.  _V ,  n  e.  NN0  |->  (  seq  0 ( ( x  e.  _V ,  y  e.  _V  |->  ( x  o.  r ) ) ,  ( z  e.  _V  |->  (  _I  |`  U. U. r
 ) ) ) `  n ) )
 
Theoremrelexp0 25082 A relation composed zero times is the (restricted) identity. (Contributed by Drahflow, 12-Nov-2015.)
 |-  ( ph  ->  Rel  R )   &    |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  ( R ^ r 0 )  =  (  _I  |`  U. U. R ) )
 
Theoremrelexpsucr 25083 A reduction for relation exponentiation to the right. (Contributed by Drahflow, 12-Nov-2015.)
 |-  ( ph  ->  Rel  R )   &    |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  ( N  e.  NN0  ->  ( R ^ r ( N  +  1 ) )  =  ( ( R ^ r N )  o.  R ) ) )
 
Theoremrelexp1 25084 A relation composed once is itself. (Contributed by Drahflow, 12-Nov-2015.)
 |-  ( ph  ->  Rel  R )   &    |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  ( R ^ r 1 )  =  R )
 
Theoremrelexpsucl 25085 A reduction for relation exponentiation to the left. (Contributed by Drahflow, 12-Nov-2015.)
 |-  ( ph  ->  Rel  R )   &    |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  ( N  e.  NN0  ->  ( R ^ r ( N  +  1 ) )  =  ( R  o.  ( R ^ r N ) ) ) )
 
Theoremrelexpcnv 25086 Distributivity of converse and relation exponentiation. (Contributed by Drahflow, 12-Nov-2015.)
 |-  ( ph  ->  Rel  R )   &    |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  ( N  e.  NN0  ->  `' ( R ^ r N )  =  ( `' R ^ r N ) ) )
 
Theoremrelexprel 25087 The exponentiation of a relation is a relation. (Contributed by Drahflow, 12-Nov-2015.)
 |-  ( ph  ->  Rel  R )   &    |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  ( N  e.  NN0  ->  Rel  ( R ^ r N ) ) )
 
Theoremrelexpdm 25088 The domain of an exponentiation of a relation a subset of the relation's field. (Contributed by Drahflow, 12-Nov-2015.)
 |-  ( ph  ->  Rel  R )   &    |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  ( N  e.  NN0  ->  dom  ( R ^ r N ) 
 C_  U. U. R ) )
 
Theoremrelexprn 25089 The range of an exponentiation of a relation a subset of the relation's field. (Contributed by Drahflow, 12-Nov-2015.)
 |-  ( ph  ->  Rel  R )   &    |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  ( N  e.  NN0  ->  ran  ( R ^ r N ) 
 C_  U. U. R ) )
 
Theoremrelexpfld 25090 The field of an exponentiation of a relation a subset of the relation's field. (Contributed by Drahflow, 12-Nov-2015.)
 |-  ( ph  ->  Rel  R )   &    |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  ( N  e.  NN0  ->  U. U. ( R ^ r N )  C_  U. U. R ) )
 
Theoremrelexpadd 25091 Relation composition becomes addition under exponentiation. (Contributed by Drahflow, 12-Nov-2015.)
 |-  ( ph  ->  Rel  R )   &    |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  (
 ( N  e.  NN0  /\  M  e.  NN0 )  ->  ( ( R ^
 r N )  o.  ( R ^ r M ) )  =  ( R ^ r
 ( N  +  M ) ) ) )
 
Theoremrelexpindlem 25092* Principle of transitive induction, finite and non-class version. The first three hypotheses give various existences, the next three give necessary substitutions and the last two are the basis and the induction hypothesis. (Contributed by Drahflow, 12-Nov-2015.)
 |-  ( et  ->  Rel  R )   &    |-  ( et  ->  R  e.  _V )   &    |-  ( et  ->  S  e.  _V )   &    |-  ( i  =  S  ->  ( ph  <->  ch ) )   &    |-  ( i  =  x  ->  ( ph  <->  ps ) )   &    |-  ( i  =  j  ->  ( ph  <->  th ) )   &    |-  ( et  ->  ch )   &    |-  ( et  ->  ( j R x  ->  ( th  ->  ps )
 ) )   =>    |-  ( et  ->  ( n  e.  NN0  ->  ( S ( R ^
 r n ) x 
 ->  ps ) ) )
 
Theoremrelexpind 25093* Principle of transitive induction, finite version. The first three hypotheses give various existences, the next three give necessary substitutions and the last two are the basis and the induction hypothesis. (Contributed by Drahflow, 12-Nov-2015.)
 |-  ( et  ->  Rel  R )   &    |-  ( et  ->  R  e.  _V )   &    |-  ( et  ->  S  e.  _V )   &    |-  ( et  ->  X  e.  _V )   &    |-  (
 i  =  S  ->  (
 ph 
 <->  ch ) )   &    |-  (
 i  =  x  ->  ( ph  <->  ps ) )   &    |-  (
 i  =  j  ->  ( ph  <->  th ) )   &    |-  ( x  =  X  ->  ( ps  <->  ta ) )   &    |-  ( et  ->  ch )   &    |-  ( et  ->  ( j R x  ->  ( th  ->  ps )
 ) )   =>    |-  ( et  ->  ( n  e.  NN0  ->  ( S ( R ^
 r n ) X 
 ->  ta ) ) )
 
Syntaxcrtrcl 25094 Extend class notation with recursively defined reflexive, transitive closure.
 class  t *rec
 
Definitiondf-rtrclrec 25095* The reflexive, transitive closure of a relation constructed as the union of all finite exponentiations. (Contributed by Drahflow, 12-Nov-2015.)
 |-  t *rec  =  ( r  e. 
 _V  |->  U_ n  e.  NN0  ( r ^ r n ) )
 
Theoremdfrtrclrec2 25096* If two elements are connected by a reflexive, transitive closure, then they are connected via  n instances the relation, for some  n. (Contributed by Drahflow, 12-Nov-2015.)
 |-  ( ph  ->  Rel  R )   &    |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  ( A ( t *rec `  R ) B  <->  E. n  e.  NN0  A ( R ^ r n ) B ) )
 
Theoremrtrclreclem.refl 25097 The reflexive, transitive closure is indeed reflexive. (Contributed by Drahflow, 12-Nov-2015.)
 |-  ( ph  ->  Rel  R )   &    |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  (  _I  |`  U. U. R )  C_  ( t *rec `  R ) )
 
Theoremrtrclreclem.subset 25098 The reflexive, transitive closure is indeed a closure. (Contributed by Drahflow, 12-Nov-2015.)
 |-  ( ph  ->  Rel  R )   &    |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  R  C_  ( t *rec `  R ) )
 
Theoremrtrclreclem.trans 25099 The reflexive, transitive closure is indeed transitive. (Contributed by Drahflow, 12-Nov-2015.)
 |-  ( ph  ->  Rel  R )   &    |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  (
 ( t *rec `  R )  o.  (
 t *rec `  R ) )  C_  ( t *rec `  R )
 )
 
Theoremrtrclreclem.min 25100* The reflexive, transitive closure of  R is the smallest reflexive, transitive relation which contains  R and the identity. (Contributed by Drahflow, 12-Nov-2015.)
 |-  ( ph  ->  Rel  R )   &    |-  ( ph  ->  R  e.  _V )   =>    |-  ( ph  ->  A. s
 ( ( (  _I  |`  ( dom  R  u.  ran 
 R ) )  C_  s  /\  R  C_  s  /\  ( s  o.  s
 )  C_  s )  ->  ( t *rec `  R )  C_  s ) )
    < 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 32401-32447
  Copyright terms: Public domain < Previous  Next >