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Theorem List for Metamath Proof Explorer - 38801-38900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsnopeqop 38801 Equivalence for an ordered pair equal to a singleton of an ordered pair. (Contributed by AV, 18-Sep-2020.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  D  e.  _V   =>    |-  ( { <. A ,  B >. }  =  <. C ,  D >. 
 <->  ( A  =  B  /\  C  =  D  /\  C  =  { A } ) )
 
Theorempropeqop 38802 Equivalence for an ordered pair equal to a pair of ordered pairs. (Contributed by AV, 18-Sep-2020.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  D  e.  _V   &    |-  E  e.  _V   &    |-  F  e.  _V   =>    |-  ( { <. A ,  B >. ,  <. C ,  D >. }  =  <. E ,  F >. 
 <->  ( ( A  =  C  /\  E  =  { A } )  /\  (
 ( A  =  B  /\  F  =  { A ,  D } )  \/  ( A  =  D  /\  F  =  { A ,  B } ) ) ) )
 
Theorempropssopi 38803 If a pair of ordered pairs is a subset of an ordered pair, their first components are equal. (Contributed by AV, 20-Sep-2020.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  D  e.  _V   &    |-  E  e.  _V   &    |-  F  e.  _V   =>    |-  ( { <. A ,  B >. ,  <. C ,  D >. }  C_  <. E ,  F >.  ->  A  =  C )
 
Theoremssprss 38804 A pair as subset of a pair. (Contributed by AV, 26-Oct-2020.)
 |-  (
 ( A  e.  V  /\  B  e.  W ) 
 ->  ( { A ,  B }  C_  { C ,  D }  <->  ( ( A  =  C  \/  A  =  D )  /\  ( B  =  C  \/  B  =  D )
 ) ) )
 
Theoremssprsseq 38805 A proper pair is a subset of a pair iff it is equal to the superset. (Contributed by AV, 26-Oct-2020.)
 |-  (
 ( A  e.  V  /\  B  e.  W  /\  A  =/=  B )  ->  ( { A ,  B }  C_  { C ,  D }  <->  { A ,  B }  =  { C ,  D } ) )
 
21.33.7.5  Indexed union and intersection - extension
 
Theoremiunxprg 38806* A pair index picks out two instances of an indexed union's argument. (Contributed by Alexander van der Vekens, 2-Feb-2018.)
 |-  ( x  =  A  ->  C  =  D )   &    |-  ( x  =  B  ->  C  =  E )   =>    |-  ( ( A  e.  V  /\  B  e.  W )  ->  U_ x  e.  { A ,  B } C  =  ( D  u.  E ) )
 
Theoremiunopeqop 38807* Equivalence for an ordered pair equal to an indexed union of singletons of ordered pairs. (Contributed by AV, 20-Sep-2020.)
 |-  B  e.  _V   &    |-  C  e.  _V   &    |-  D  e.  _V   =>    |-  ( A  =/=  (/)  ->  ( U_ x  e.  A  { <. x ,  B >. }  =  <. C ,  D >.  ->  E. z  A  =  { z } )
 )
 
TheoremotiunsndisjX 38808* The union of singletons consisting of ordered triples which have distinct first and third components are disjunct. (Contributed by Alexander van der Vekens, 10-Mar-2018.)
 |-  ( B  e.  X  -> Disj  a  e.  V  U_ c  e.  W  { <. a ,  B ,  c >. } )
 
21.33.7.6  Introduce the Axiom of Union - extension
 
Theoremralxfrd2 38809* Transfer universal quantification from a variable  x to another variable  y contained in expression  A. Variant of ralxfrd 4571. (Contributed by Alexander van der Vekens, 25-Apr-2018.)
 |-  (
 ( ph  /\  y  e.  C )  ->  A  e.  B )   &    |-  ( ( ph  /\  x  e.  B ) 
 ->  E. y  e.  C  x  =  A )   &    |-  (
 ( ph  /\  y  e.  C  /\  x  =  A )  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  (
 A. x  e.  B  ps 
 <-> 
 A. y  e.  C  ch ) )
 
Theoremrexxfrd2 38810* Transfer existence from a variable 
x to another variable  y contained in expression  A. Variant of rexxfrd 4572. (Contributed by Alexander van der Vekens, 25-Apr-2018.)
 |-  (
 ( ph  /\  y  e.  C )  ->  A  e.  B )   &    |-  ( ( ph  /\  x  e.  B ) 
 ->  E. y  e.  C  x  =  A )   &    |-  (
 ( ph  /\  y  e.  C  /\  x  =  A )  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( E. x  e.  B  ps 
 <-> 
 E. y  e.  C  ch ) )
 
21.33.7.7  Relations - extension
 
Theoremresresdm 38811 A restriction by an arbitrary set is a restriction by its domain. (Contributed by AV, 16-Nov-2020.)
 |-  ( F  =  ( E  |`  A )  ->  F  =  ( E  |`  dom  F ) )
 
Theoremresisresindm 38812 The restriction of a relation by a set  B is identical with the restriction by the intersection of  B with the domain of the relation. (Contributed by Alexander van der Vekens, 3-Feb-2018.)
 |-  ( Rel  F  ->  ( F  |`  B )  =  ( F  |`  ( B  i^i  dom  F ) ) )
 
21.33.7.8  Functions - extension
 
Theoremfvifeq 38813 Equality of function values with conditional arguments, see also fvif 5829. (Contributed by Alexander van der Vekens, 21-May-2018.)
 |-  ( A  =  if ( ph ,  B ,  C )  ->  ( F `  A )  =  if ( ph ,  ( F `
  B ) ,  ( F `  C ) ) )
 
Theorem2f1fvneq 38814 If two one-to-one functions are applied on different arguments, also the values are different. (Contributed by Alexander van der Vekens, 25-Jan-2018.)
 |-  (
 ( ( E : D -1-1-> R  /\  F : C -1-1-> D )  /\  ( A  e.  C  /\  B  e.  C )  /\  A  =/=  B ) 
 ->  ( ( ( E `
  ( F `  A ) )  =  X  /\  ( E `
  ( F `  B ) )  =  Y )  ->  X  =/=  Y ) )
 
Theoremrnfdmpr 38815 The range of a one-to-one function 
F of an unordered pair into a set is the unordered pair of the function values. (Contributed by Alexander van der Vekens, 2-Feb-2018.)
 |-  (
 ( X  e.  V  /\  Y  e.  W ) 
 ->  ( F  Fn  { X ,  Y }  ->  ran  F  =  {
 ( F `  X ) ,  ( F `  Y ) } )
 )
 
Theoremimarnf1pr 38816 The image of the range of a function  F under a function  E if  F is a function of a pair into the domain of  E. (Contributed by Alexander van der Vekens, 2-Feb-2018.)
 |-  (
 ( X  e.  V  /\  Y  e.  W ) 
 ->  ( ( ( F : { X ,  Y } --> dom  E  /\  E : dom  E --> R ) 
 /\  ( ( E `
  ( F `  X ) )  =  A  /\  ( E `
  ( F `  Y ) )  =  B ) )  ->  ( E " ran  F )  =  { A ,  B } ) )
 
Theoremfuniun 38817* A function is a union of singletons of ordered pairs indexed by its domain. (Contributed by AV, 18-Sep-2020.)
 |-  ( Fun  F  ->  F  =  U_ x  e.  dom  F { <. x ,  ( F `  x ) >. } )
 
Theoremfunopsn 38818* If a function is an ordered pair then it is a singleton of an ordered pair. (Contributed by AV, 20-Sep-2020.)
 |-  X  e.  _V   &    |-  Y  e.  _V   =>    |-  (
 ( Fun  F  /\  F  =  <. X ,  Y >. )  ->  E. a
 ( X  =  {
 a }  /\  F  =  { <. a ,  a >. } ) )
 
Theoremfunop 38819* An ordered pair is a function iff it is a singleton of an ordered pair. (Contributed by AV, 20-Sep-2020.)
 |-  X  e.  _V   &    |-  Y  e.  _V   =>    |-  ( Fun  <. X ,  Y >.  <->  E. a ( X  =  { a }  /\  <. X ,  Y >.  =  { <. a ,  a >. } ) )
 
Theoremfunop1 38820* A function is an ordered pair iff it is a singleton of an ordered pair. (Contributed by AV, 20-Sep-2020.)
 |-  ( E. x E. y  F  =  <. x ,  y >.  ->  ( Fun  F  <->  E. x E. y  F  =  { <. x ,  y >. } ) )
 
Theoremf1ssf1 38821 A subset of an injective function is injective. (Contributed by AV, 20-Nov-2020.)
 |-  (
 ( Fun  F  /\  Fun  `' F  /\  G  C_  F )  ->  Fun  `' G )
 
Theoremfunsndifnop 38822 A singleton of an ordered pair is not an ordered pair if the components are different. (Contributed by AV, 23-Sep-2020.)
 |-  A  e.  V   &    |-  B  e.  W   &    |-  G  =  { <. A ,  B >. }   =>    |-  ( A  =/=  B  ->  -.  G  e.  ( _V  X.  _V ) )
 
Theoremfunsneqopsn 38823 A singleton of an ordered pair is an ordered pair of equal singletons if the components are equal. (Contributed by AV, 24-Sep-2020.)
 |-  A  e.  V   &    |-  B  e.  W   &    |-  G  =  { <. A ,  B >. }   =>    |-  ( A  =  B  ->  G  =  <. { A } ,  { A } >. )
 
Theoremfunsneqop 38824 A singleton of an ordered pair is an ordered pair if the components are equal. (Contributed by AV, 24-Sep-2020.)
 |-  A  e.  V   &    |-  B  e.  W   &    |-  G  =  { <. A ,  B >. }   =>    |-  ( A  =  B  ->  G  e.  ( _V 
 X.  _V ) )
 
Theoremfunsneqopb 38825 A singleton of an ordered pair is an ordered pair iff the components are equal. (Contributed by AV, 24-Sep-2020.)
 |-  A  e.  V   &    |-  B  e.  W   &    |-  G  =  { <. A ,  B >. }   =>    |-  ( A  =  B  <->  G  e.  ( _V  X.  _V ) )
 
Theoremfundmge2nop 38826 A function with a domain containing (at least) two different elements is not an ordered pair. (Contributed by AV, 12-Oct-2020.)
 |-  (
 ( Fun  G  /\  2  <_  ( # `  dom  G ) )  ->  -.  G  e.  ( _V  X.  _V ) )
 
Theoremfun2dmnop 38827 A function with a domain containing (at least) two different elements is not an ordered pair. (Contributed by AV, 21-Sep-2020.) (Proof shortened by AV, 12-Oct-2020.)
 |-  A  e.  V   &    |-  B  e.  W   =>    |-  (
 ( Fun  G  /\  A  =/=  B  /\  { A ,  B }  C_ 
 dom  G )  ->  -.  G  e.  ( _V  X.  _V ) )
 
Theoremfun2dmnopgexmpl 38828 A function with a domain containing (at least) two different elements is not an ordered pair. (Contributed by AV, 21-Sep-2020.)
 |-  ( G  =  { <. 0 ,  1 >. ,  <. 1 ,  1 >. }  ->  -.  G  e.  ( _V 
 X.  _V ) )
 
21.33.7.9  Equinumerosity - extension
 
Theoremresfnfinfin 38829 The restriction of a function by a finite set is finite. (Contributed by Alexander van der Vekens, 3-Feb-2018.)
 |-  (
 ( F  Fn  A  /\  B  e.  Fin )  ->  ( F  |`  B )  e.  Fin )
 
Theoremresidfi 38830 A restricted identity function is finite iff the restricting class is finite. (Contributed by AV, 10-Jan-2020.)
 |-  (
 (  _I  |`  A )  e.  Fin  <->  A  e.  Fin )
 
21.33.7.10  Subtraction - extension
 
Theoremcnambpcma 38831 ((a-b)+c)-a = c-a holds for complex numbers a,b,c. (Contributed by Alexander van der Vekens, 23-Mar-2018.)
 |-  (
 ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( ( A  -  B )  +  C )  -  A )  =  ( C  -  B ) )
 
Theoremcnapbmcpd 38832 ((a+b)-c)+d = ((a+d)+b)-c holds for complex numbers a,b,c,d. (Contributed by Alexander van der Vekens, 23-Mar-2018.)
 |-  (
 ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  ( ( ( A  +  B )  -  C )  +  D )  =  (
 ( ( A  +  D )  +  B )  -  C ) )
 
21.33.7.11  Multiplication - extension
 
Theorem2txmxeqx 38833 Two times a complex number minus the number itself results in the number itself. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
 |-  ( X  e.  CC  ->  ( ( 2  x.  X )  -  X )  =  X )
 
21.33.7.12  Ordering on reals (cont.) - extension
 
Theoremleaddsuble 38834 Addition and subtraction on one side of 'less or equal'. (Contributed by Alexander van der Vekens, 18-Mar-2018.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( B  <_  C  <->  ( ( A  +  B )  -  C )  <_  A ) )
 
Theorem2leaddle2 38835 If two real numbers are less than a third real number, the sum of the real numbers is less than twice the third real number. (Contributed by Alexander van der Vekens, 21-May-2018.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( ( A  <  C 
 /\  B  <  C )  ->  ( A  +  B )  <  ( 2  x.  C ) ) )
 
Theoremltnltne 38836 Variant of trichotomy law for 'less than'. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  ( -.  B  <  A  /\  -.  B  =  A ) ) )
 
Theoremp1lep2 38837 A real number increasd by 1 is less than or equal to the number increased by 2. (Contributed by Alexander van der Vekens, 17-Sep-2018.)
 |-  ( N  e.  RR  ->  ( N  +  1 ) 
 <_  ( N  +  2 ) )
 
Theoremlelttrdi 38838 If a number is less than another number, and the other number is less than or equal to a third number, the first number is less than the third number. (Contributed by Alexander van der Vekens, 24-Mar-2018.)
 |-  ( ph  ->  ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )
 )   &    |-  ( ph  ->  B  <_  C )   =>    |-  ( ph  ->  ( A  <  B  ->  A  <  C ) )
 
Theoremltsubsubaddltsub 38839 If the result of subtracting two numbers is greater than a number, the result of adding one of these subtracted numbers to the number is less than the result of subtracting the other subtracted number only. (Contributed by Alexander van der Vekens, 9-Jun-2018.)
 |-  (
 ( J  e.  RR  /\  ( L  e.  RR  /\  M  e.  RR  /\  N  e.  RR )
 )  ->  ( J  <  ( ( L  -  M )  -  N ) 
 <->  ( J  +  M )  <  ( L  -  N ) ) )
 
Theoremzm1nn 38840 An integer minus 1 is positive under certain circumstances. (Contributed by Alexander van der Vekens, 9-Jun-2018.)
 |-  (
 ( N  e.  NN0  /\  L  e.  ZZ )  ->  ( ( J  e.  RR  /\  0  <_  J  /\  J  <  ( ( L  -  N )  -  1 ) ) 
 ->  ( L  -  1
 )  e.  NN )
 )
 
21.33.7.13  Nonnegative integers (as a subset of complex numbers) - extension
 
Theoremlesubnn0 38841 Subtracting a nonnegative integer from a nonnegative integer which is greater than or equal to the first one results in a nonnegative integer. (Contributed by Alexander van der Vekens, 6-Apr-2018.)
 |-  (
 ( A  e.  NN0  /\  B  e.  NN0 )  ->  ( B  <_  A  ->  ( A  -  B )  e.  NN0 ) )
 
Theoremltsubnn0 38842 Subtracting a nonnegative integer from a nonnegative integer which is greater than the first one results in a nonnegative integer. (Contributed by Alexander van der Vekens, 6-Apr-2018.)
 |-  (
 ( A  e.  NN0  /\  B  e.  NN0 )  ->  ( B  <  A  ->  ( A  -  B )  e.  NN0 ) )
 
Theoremnn0resubcl 38843 Closure law for subtraction of reals, restricted to nonnegative integers. (Contributed by Alexander van der Vekens, 6-Apr-2018.)
 |-  (
 ( A  e.  NN0  /\  B  e.  NN0 )  ->  ( A  -  B )  e.  RR )
 
21.33.7.14  Upper sets of integers - extension
 
Theoremeluzge0nn0 38844 If an integer is greater than or equal to a nonnegative integer, then it is a nonnegative integer. (Contributed by Alexander van der Vekens, 27-Aug-2018.)
 |-  ( N  e.  ( ZZ>= `  M )  ->  ( 0 
 <_  M  ->  N  e.  NN0 ) )
 
21.33.7.15  Finite intervals of integers - extension
 
Theoremssfz12 38845 Subset relationship for finite sets of sequential integers. (Contributed by Alexander van der Vekens, 16-Mar-2018.)
 |-  (
 ( K  e.  ZZ  /\  L  e.  ZZ  /\  K  <_  L )  ->  ( ( K ... L )  C_  ( M ... N )  ->  ( M  <_  K  /\  L  <_  N ) ) )
 
Theoremelfz2z 38846 Membership of an integer in a finite set of sequential integers starting at 0. (Contributed by Alexander van der Vekens, 25-May-2018.)
 |-  (
 ( K  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  e.  (
 0 ... N )  <->  ( 0  <_  K  /\  K  <_  N ) ) )
 
Theorem2elfz3nn0 38847 If there are two elements in a finite set of sequential integers starting at 0, these two elements as well as the upper bound are nonnegative integers. (Contributed by Alexander van der Vekens, 7-Apr-2018.)
 |-  (
 ( A  e.  (
 0 ... N )  /\  B  e.  ( 0 ... N ) )  ->  ( A  e.  NN0  /\  B  e.  NN0  /\  N  e.  NN0 ) )
 
Theoremfz0addcom 38848 The addition of two members of a finite set of sequential integers starting at 0 is commutative. (Contributed by Alexander van der Vekens, 22-May-2018.) (Revised by Alexander van der Vekens, 9-Jun-2018.)
 |-  (
 ( A  e.  (
 0 ... N )  /\  B  e.  ( 0 ... N ) )  ->  ( A  +  B )  =  ( B  +  A ) )
 
Theorem2elfz2melfz 38849 If the sum of two integers of a 0 based finite set of sequential integers is greater than the upper bound, the difference between one of the integers and the difference between the upper bound and the other integer is in the 0 based finite set of sequential integers with the first integer as upper bound. (Contributed by Alexander van der Vekens, 7-Apr-2018.) (Revised by Alexander van der Vekens, 31-May-2018.)
 |-  (
 ( A  e.  (
 0 ... N )  /\  B  e.  ( 0 ... N ) )  ->  ( N  <  ( A  +  B )  ->  ( B  -  ( N  -  A ) )  e.  ( 0 ...
 A ) ) )
 
Theoremfz0addge0 38850 The sum of two integers in 0 based finite sets of sequential integers is greater than or equal to zero. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
 |-  (
 ( A  e.  (
 0 ... M )  /\  B  e.  ( 0 ... N ) )  -> 
 0  <_  ( A  +  B ) )
 
Theoremelfzlble 38851 Membership of an integer in a finite set of sequential integers with the integer as upper bound and a lower bound less than or equal to the integer. (Contributed by AV, 21-Oct-2018.)
 |-  (
 ( N  e.  ZZ  /\  M  e.  NN0 )  ->  N  e.  ( ( N  -  M )
 ... N ) )
 
Theoremelfzelfzlble 38852 Membership of an element of a finite set of sequential integers in a finite set of sequential integers with the same upper bound and a lower bound less than the upper bound. (Contributed by AV, 21-Oct-2018.)
 |-  (
 ( M  e.  ZZ  /\  K  e.  ( 0
 ... N )  /\  N  <  ( M  +  K ) )  ->  K  e.  ( ( N  -  M ) ... N ) )
 
21.33.7.16  Half-open integer ranges - extension
 
Theoremsubsubelfzo0 38853 Subtracting a difference from a number which is not less than the difference results in a bounded nonnegative integer. (Contributed by Alexander van der Vekens, 21-May-2018.)
 |-  (
 ( A  e.  (
 0..^ N )  /\  I  e.  ( 0..^ N )  /\  -.  I  <  ( N  -  A ) )  ->  ( I  -  ( N  -  A ) )  e.  ( 0..^ A ) )
 
Theoremfzoopth 38854 A half-open integer range can represent an ordered pair, analogous to fzopth 11779. (Contributed by Alexander van der Vekens, 1-Jul-2018.)
 |-  (
 ( M  e.  ZZ  /\  N  e.  ZZ  /\  M  <  N )  ->  ( ( M..^ N )  =  ( J..^ K )  <->  ( M  =  J  /\  N  =  K ) ) )
 
Theorem2ffzoeq 38855* Two functions over a half-open range of nonnegative integers are equal if and only if their domains have the same length and the function values are the same at each position. (Contributed by Alexander van der Vekens, 1-Jul-2018.)
 |-  (
 ( ( M  e.  NN0  /\  N  e.  NN0 )  /\  ( F : ( 0..^ M ) --> X  /\  P : ( 0..^ N )
 --> Y ) )  ->  ( F  =  P  <->  ( M  =  N  /\  A. i  e.  ( 0..^ M ) ( F `
  i )  =  ( P `  i
 ) ) ) )
 
Theoremfzosplitpr 38856 Extending a half-open integer range by an unordered pair at the end. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
 |-  ( B  e.  ( ZZ>= `  A )  ->  ( A..^ ( B  +  2 ) )  =  ( ( A..^ B )  u.  { B ,  ( B  +  1
 ) } ) )
 
21.33.7.17  The ` # ` (set size) function - extension
 
Theoremnfile 38857 The size of any infinite set is always greater than or equal to the the size of any set. (Contributed by AV, 13-Nov-2020.)
 |-  (
 ( A  e.  V  /\  B  e.  W  /\  -.  B  e.  Fin )  ->  ( # `  A )  <_  ( # `  B ) )
 
Theoremprprrab 38858 The set of proper pairs of elements of a given set expressed in two ways. (Contributed by AV, 24-Nov-2020.)
 |-  { x  e.  ( ~P A  \  { (/) } )  |  ( # `  x )  =  2 }  =  { x  e.  ~P A  |  ( # `  x )  =  2 }
 
21.33.7.18  Finite and infinite sums - extension
 
Theoremfsummsndifre 38859* A finite sum with one of its integer summands removed is a real number. (Contributed by Alexander van der Vekens, 31-Aug-2018.)
 |-  (
 ( A  e.  Fin  /\ 
 A. k  e.  A  B  e.  ZZ )  -> 
 sum_ k  e.  ( A  \  { X }
 ) B  e.  RR )
 
Theoremfsumsplitsndif 38860* Separate out a term in a finite sum by splitting the sum into two parts. (Contributed by Alexander van der Vekens, 31-Aug-2018.)
 |-  (
 ( A  e.  Fin  /\  X  e.  A  /\  A. k  e.  A  B  e.  ZZ )  ->  sum_ k  e.  A  B  =  (
 sum_ k  e.  ( A  \  { X }
 ) B  +  [_ X  /  k ]_ B ) )
 
Theoremfsummmodsndifre 38861* A finite sum of summands modulo a positive number with one of its summands removed is a real number. (Contributed by Alexander van der Vekens, 31-Aug-2018.)
 |-  (
 ( A  e.  Fin  /\  N  e.  NN  /\  A. k  e.  A  B  e.  ZZ )  ->  sum_ k  e.  ( A  \  { X } ) ( B 
 mod  N )  e.  RR )
 
Theoremfsummmodsnunz 38862* A finite sum of summands modulo a positive number with an additional summand is an integer. (Contributed by Alexander van der Vekens, 1-Sep-2018.)
 |-  (
 ( A  e.  Fin  /\  N  e.  NN  /\  A. k  e.  ( A  u.  { z }
 ) B  e.  ZZ )  ->  sum_ k  e.  ( A  u.  { z }
 ) ( B  mod  N )  e.  ZZ )
 
21.33.8  Graph theory (revised)
 
21.33.8.1  The edge function extractor for extensible structures
 
Syntaxcedgf 38863 Extend class notation with an edge function.
 class .ef
 
Definitiondf-edgf 38864 Define the edge function (indexed edges) of a graph. (Contributed by AV, 18-Jan-2020.)
 |- .ef  = Slot ; 1 8
 
Theoremedgfndxnn 38865 The index value of the edge function extractor is a positive integer. This property should be ensured for every concrete coding because otherwise it could not be used in an extensible structure (slots must be positive integers). (Contributed by AV, 21-Sep-2020.)
 |-  (.ef ` 
 ndx )  e.  NN
 
Theoremedgfndxid 38866 The value of the edge function extractor is the value of the corresponding slot of the structure. (Contributed by AV, 21-Sep-2020.)
 |-  ( G  e.  V  ->  (.ef `  G )  =  ( G `  (.ef `  ndx ) ) )
 
Theorembaseltedgf 38867 The index value of the  Base slot is less than the index value of the .ef slot. (Contributed by AV, 21-Sep-2020.)
 |-  ( Base `  ndx )  < 
 (.ef `  ndx )
 
Theoremslotsbaseefdif 38868 The slots  Base and .ef are different. (Contributed by AV, 21-Sep-2020.)
 |-  ( Base `  ndx )  =/=  (.ef `  ndx )
 
21.33.8.2  Vertices and edges

The key concepts in graph theory are vertices and edges. In general, a graph "consists" (at least) of two sets: the set of vertices and the set of edges. The edges "connect" vertices. The meaning of "connect" is different for different kinds of graphs (directed/undirected graphs, hyper-/multi-/simple graphs, etc.). The simplest way to represent a graph (of any kind) is to define a graph as "an ordered pair of disjoint sets (V, E)" (see section I.1 in [Bollobas] p. 1), or in the notation of Metamath:  <. V ,  E >.. Another way is to regard a graph as a mathematical structure, which can be enhanced by additional features (see Wikipedia "Mathematical structure", 24-Sep-2020, https://en.wikipedia.org/wiki/Mathematical_structure): "In mathematics, a structure is a set endowed with some additional features on the set (e.g., operation, relation, metric, topology). Often, the additional features are attached or related to the set, so as to provide it with some additional meaning or significance.". Such structures are provided as "extensible structures" in Metamath, see df-struct 15059.

To allow for expressing and proving most of the theorems for graphs independently from their representation, the functions Vtx and iEdg are defined (see df-vtx 38871 and df-iedg 38872), which provide the vertices resp. (indexed) edges of an arbitrary class  G which represents a graph:  (Vtx `  G
) resp.  (iEdg `  G ). Instead of providing edges themselves, iEdg is intended to provide a function as mapping of "indices" (the domain of the function) to the edges (therefore called "set of indexed edges"), which allows for hyper-/pseudo-/multigraphs with more than one edge between two (or more) vertices. In literature, these functions are often denoted also by "V" and "E", see section I.1 in [Bollobas] p. 1 ("If G is a graph, then V = V(G) is the vertex set of G, and E = E(G) is the edge set.") or section 1.1 in [Diestel] p. 2 ("The vertex set of graph G is referred to as V(G), its edge set as E(G)."). For example, e1 = e(1) = { a, b } and e2 = e(2) = { a, b } are two different edges connecting the same two vertices a and b (in a pseudograph). In section 1.10 of [Diestel] p. 28, the edge function is defined differently: as "map E -> V u. [V]^2 assigning to every edge either one or two vertices, its end.". Here, the domain is the set of abstract edges: for two different edges e1 and e2 connecting the same two vertices a and b, we would have e(e1) = e(e2) = { a, b }. Since the set of abstract edges can be chosen as index set, these definitions are equivalent.

The result of these functions are as expected: for a graph represented as ordered pair ( G  e.  ( _V  X.  _V )), the set of vertices is  (Vtx `  G
)  =  ( 1st `  G ) (see opvtxval 38875) and the set of (indexed) edges is  (iEdg `  G
)  =  ( 2nd `  G ) (see opiedgval 38878), or if  G is given as ordered pair  G  =  <. V ,  E >., the set of vertices is  (Vtx `  G
)  =  V (see opvtxfv 38876) and the set of (indexed) edges is  (iEdg `  G
)  =  E (see opiedgfv 38879).

And for a graph represented as extensible structure ( G Struct  <. ( Base `  ndx ) ,  (.ef `  ndx ) >.), the set of vertices is  (Vtx `  G
)  =  ( Base `  G ) (see funvtxval 38888) and the set of (indexed) edges is  (iEdg `  G
)  =  (.ef `  G ) (see funiedgval 38889), or if  G is given in its simplest form as extensible structure with two slots ( G  =  { <. ( Base `  ndx ) ,  V >. , 
<. (.ef `  ndx ) ,  E >. }), the set of vertices is  (Vtx `  G )  =  V (see struct2grvtx 38897) and the set of (indexed) edges is  (iEdg `  G )  =  E (see struct2griedg 38898).

These two representations are convertible, see graop 38899 and grastruct 38900: If  G is a graph (for example  G  =  <. V ,  E >.), then  H  =  { <. ( Base `  ndx ) ,  (Vtx `  G
) >. ,  <. (.ef `  ndx ) ,  (iEdg `  G ) >. } represents essentially the same graph, and if  G is a graph (for example  G  =  { <. ( Base `  ndx ) ,  V >. , 
<. (.ef `  ndx ) ,  E >. }), then  H  =  <. (Vtx
`  G ) ,  (iEdg `  G ) >. represents essentially the same graph. In both cases,  (Vtx `  G )  =  (Vtx `  H ) and  (iEdg `  G
)  =  (iEdg `  H ) hold. Theorems gropd 38901 and gropeld 38903 show that if any representation of a graph with vertices  V and edges  E has a certain property, then the ordered pair  <. V ,  E >. of the set of vertices and the set of edges (which is such a representation of a graph with vertices  V and edges  E) has this property. Analogously, theorems grstructd 38902 and grstructeld 38904 show that if any representation of a graph with vertices  V and edges  E has a certain property, then any extensible structure with base set  V and value  E in the slot for edge functions (which is also such a representation of a graph with vertices  V and edges  E) has this property.

Besides the usual way to represent graphs without edges (consisting of unconnected vertices only), which would be  G  =  <. V ,  (/) >. or  G  =  { <. ( Base `  ndx ) ,  V >. , 
<. (.ef `  ndx ) ,  (/) >. }, a structure without a slot for edges can be used:  G  =  { <. ( Base `  ndx ) ,  V >. }, see snstrvtxval 38905 and snstriedgval 38906. Analogously, the empty set 
(/) can be used to represent the null graph, see vtxval0 38907 and iedgval0 38908, which can also be represented by  G  =  <.
(/) ,  (/) >. or  G  =  { <. ( Base `  ndx ) ,  (/) >. ,  <. (.ef
`  ndx ) ,  (/) >. }. Even proper classes can be used to represent the null graph, see vtxvalprc 38913 and iedgvalprc 38914.

Other classes should not be used to represent graphs, because there could be a degenerated behavior of the vertex set and (indexed) edge functions, see vtxvalsnop 38909 resp. iedgvalsnop 38910, and vtxval3sn 38911 resp. iedgval3sn 38912.

 
Syntaxcvtx 38869 Extend class notation with the vertices of "graphs".
 class Vtx
 
Syntaxciedg 38870 Extend class notation with the indexed edges of "graphs".
 class iEdg
 
Definitiondf-vtx 38871 Define the function mapping a graph to the set of its vertices. This definition is very general: It defines the set of vertices for any ordered pair as its first component, and for any other class as its "base set". It is meaningful, however, only if the ordered pair represents a graph resp. the class is an extensible structure representing a graph. (Contributed by AV, 9-Jan-2020.) (Revised by AV, 20-Sep-2020.)
 |- Vtx  =  ( g  e.  _V  |->  if ( g  e.  ( _V  X.  _V ) ,  ( 1st `  g
 ) ,  ( Base `  g ) ) )
 
Definitiondf-iedg 38872 Define the function mapping a graph to its indexed edges. This definition is very general: It defines the indexed edges for any ordered pair as its second component, and for any other class as its "edge function". It is meaningful, however, only if the ordered pair represents a graph resp. the class is an extensible structure (containing a slot for "edge functions") representing a graph. (Contributed by AV, 20-Sep-2020.)
 |- iEdg  =  ( g  e.  _V  |->  if ( g  e.  ( _V  X.  _V ) ,  ( 2nd `  g
 ) ,  (.ef `  g ) ) )
 
Theoremvtxval 38873 The set of vertices of a graph. (Contributed by AV, 9-Jan-2020.) (Revised by AV, 21-Sep-2020.)
 |-  ( G  e.  V  ->  (Vtx `  G )  =  if ( G  e.  ( _V  X.  _V ) ,  ( 1st `  G ) ,  ( Base `  G ) ) )
 
Theoremiedgval 38874 The set of indexed edges of a graph. (Contributed by AV, 21-Sep-2020.)
 |-  ( G  e.  V  ->  (iEdg `  G )  =  if ( G  e.  ( _V  X.  _V ) ,  ( 2nd `  G ) ,  (.ef `  G ) ) )
 
Theoremopvtxval 38875 The set of vertices of a graph represented as an ordered pair of vertices and indexed edges. (Contributed by AV, 9-Jan-2020.) (Revised by AV, 21-Sep-2020.)
 |-  ( G  e.  ( _V  X. 
 _V )  ->  (Vtx `  G )  =  ( 1st `  G )
 )
 
Theoremopvtxfv 38876 The set of vertices of a graph represented as an ordered pair of vertices and indexed edges as function value. (Contributed by AV, 21-Sep-2020.)
 |-  (
 ( V  e.  X  /\  E  e.  Y ) 
 ->  (Vtx `  <. V ,  E >. )  =  V )
 
Theoremopvtxov 38877 The set of vertices of a graph represented as an ordered pair of vertices and indexed edges as operation value. (Contributed by AV, 21-Sep-2020.)
 |-  (
 ( V  e.  X  /\  E  e.  Y ) 
 ->  ( VVtx E )  =  V )
 
Theoremopiedgval 38878 The set of indexed edges of a graph represented as an ordered pair of vertices and indexed edges. (Contributed by AV, 21-Sep-2020.)
 |-  ( G  e.  ( _V  X. 
 _V )  ->  (iEdg `  G )  =  ( 2nd `  G )
 )
 
Theoremopiedgfv 38879 The set of indexed edges of a graph represented as an ordered pair of vertices and indexed edges as function value. (Contributed by AV, 21-Sep-2020.)
 |-  (
 ( V  e.  X  /\  E  e.  Y ) 
 ->  (iEdg `  <. V ,  E >. )  =  E )
 
Theoremopiedgov 38880 The set of indexed edges of a graph represented as an ordered pair of vertices and indexed edges as operation value. (Contributed by AV, 21-Sep-2020.)
 |-  (
 ( V  e.  X  /\  E  e.  Y ) 
 ->  ( ViEdg E )  =  E )
 
Theoremfunvtxdm2val 38881 The set of vertices of an extensible structure with (at least) two slots. (Contributed by AV, 22-Sep-2020.)
 |-  A  e.  W   &    |-  B  e.  Z   =>    |-  (
 ( ( G  e.  V  /\  Fun  G )  /\  A  =/=  B  /\  { A ,  B }  C_ 
 dom  G )  ->  (Vtx `  G )  =  (
 Base `  G ) )
 
Theoremfuniedgdm2val 38882 The set of indexed edges of an extensible structure with (at least) two slots. (Contributed by AV, 22-Sep-2020.)
 |-  A  e.  W   &    |-  B  e.  Z   =>    |-  (
 ( ( G  e.  V  /\  Fun  G )  /\  A  =/=  B  /\  { A ,  B }  C_ 
 dom  G )  ->  (iEdg `  G )  =  (.ef `  G ) )
 
Theoremfunvtxval0 38883 The set of vertices of an extensible structure with a base set and (at least) another slot. (Contributed by AV, 22-Sep-2020.)
 |-  S  e.  Z   =>    |-  ( ( ( G  e.  V  /\  Fun  G )  /\  S  =/=  ( Base `  ndx )  /\  { ( Base `  ndx ) ,  S }  C_  dom  G )  ->  (Vtx `  G )  =  ( Base `  G ) )
 
Theoremfunvtxdmge2val 38884 The set of vertices of an extensible structure with (at least) two slots. (Contributed by AV, 12-Oct-2020.)
 |-  (
 ( G  e.  V  /\  Fun  G  /\  2  <_  ( # `  dom  G ) )  ->  (Vtx `  G )  =  (
 Base `  G ) )
 
Theoremfuniedgdmge2val 38885 The set of indexed edges of an extensible structure with (at least) two slots. (Contributed by AV, 12-Oct-2020.)
 |-  (
 ( G  e.  V  /\  Fun  G  /\  2  <_  ( # `  dom  G ) )  ->  (iEdg `  G )  =  (.ef `  G ) )
 
Theorembasvtxval 38886 The set of vertices of a graph represented as an extensible structure with the set of vertices as base set. (Contributed by AV, 14-Oct-2020.)
 |-  ( ph  ->  G  e.  X )   &    |-  ( ph  ->  Fun  G )   &    |-  ( ph  ->  2  <_  ( # `  dom  G ) )   &    |-  ( ph  ->  V  e.  Y )   &    |-  ( ph  ->  <. ( Base `  ndx ) ,  V >.  e.  G )   =>    |-  ( ph  ->  (Vtx `  G )  =  V )
 
Theoremedgfiedgval 38887 The set of indexed edges of a graph represented as an extensible structure with the indexed edges in the slot for edge functions. (Contributed by AV, 14-Oct-2020.)
 |-  ( ph  ->  G  e.  X )   &    |-  ( ph  ->  Fun  G )   &    |-  ( ph  ->  2  <_  ( # `  dom  G ) )   &    |-  ( ph  ->  E  e.  Y )   &    |-  ( ph  ->  <. (.ef `  ndx ) ,  E >.  e.  G )   =>    |-  ( ph  ->  (iEdg `  G )  =  E )
 
Theoremfunvtxval 38888 The set of vertices of a graph represented as an extensible structure with vertices as base set and indexed edges. (Contributed by AV, 22-Sep-2020.)
 |-  (
 ( G  e.  V  /\  Fun  G  /\  {
 ( Base `  ndx ) ,  (.ef `  ndx ) }  C_ 
 dom  G )  ->  (Vtx `  G )  =  (
 Base `  G ) )
 
Theoremfuniedgval 38889 The set of indexed edges of a graph represented as an extensible structure with vertices as base set and indexed edges. (Contributed by AV, 21-Sep-2020.)
 |-  (
 ( G  e.  V  /\  Fun  G  /\  {
 ( Base `  ndx ) ,  (.ef `  ndx ) }  C_ 
 dom  G )  ->  (iEdg `  G )  =  (.ef `  G ) )
 
Theoremstructvtxvallem 38890 Lemma for structvtxval 38891 and structiedg0val 38892. (Contributed by AV, 23-Sep-2020.)
 |-  S  e.  NN   &    |-  ( Base `  ndx )  <  S   &    |-  G  =  { <. ( Base `  ndx ) ,  V >. ,  <. S ,  E >. }   =>    |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( G  e.  _V 
 /\  Fun  G  /\  { ( Base `  ndx ) ,  S }  C_  dom  G ) )
 
Theoremstructvtxval 38891 The set of vertices of an extensible structure with a base set and another slot. (Contributed by AV, 23-Sep-2020.)
 |-  S  e.  NN   &    |-  ( Base `  ndx )  <  S   &    |-  G  =  { <. ( Base `  ndx ) ,  V >. ,  <. S ,  E >. }   =>    |-  ( ( V  e.  X  /\  E  e.  Y )  ->  (Vtx `  G )  =  V )
 
Theoremstructiedg0val 38892 The set of indexed edges of an extensible structure with a base set and another slot not being the slot for edge functions is empty. (Contributed by AV, 23-Sep-2020.)
 |-  S  e.  NN   &    |-  ( Base `  ndx )  <  S   &    |-  G  =  { <. ( Base `  ndx ) ,  V >. ,  <. S ,  E >. }   =>    |-  ( ( V  e.  X  /\  E  e.  Y  /\  S  =/=  (.ef `  ndx ) )  ->  (iEdg `  G )  =  (/) )
 
Theoremstructgrssvtxlem 38893 Lemma for structgrssvtx 38894 and structgrssiedg 38895. (Contributed by AV, 14-Oct-2020.)
 |-  ( ph  ->  G  e.  X )   &    |-  ( ph  ->  Fun  G )   &    |-  ( ph  ->  V  e.  Y )   &    |-  ( ph  ->  E  e.  Z )   &    |-  ( ph  ->  { <. ( Base ` 
 ndx ) ,  V >. ,  <. (.ef `  ndx ) ,  E >. } 
 C_  G )   =>    |-  ( ph  ->  2 
 <_  ( # `  dom  G ) )
 
Theoremstructgrssvtx 38894 The set of vertices of a graph represented as an extensible structure with vertices as base set and indexed edges. (Contributed by AV, 14-Oct-2020.)
 |-  ( ph  ->  G  e.  X )   &    |-  ( ph  ->  Fun  G )   &    |-  ( ph  ->  V  e.  Y )   &    |-  ( ph  ->  E  e.  Z )   &    |-  ( ph  ->  { <. ( Base ` 
 ndx ) ,  V >. ,  <. (.ef `  ndx ) ,  E >. } 
 C_  G )   =>    |-  ( ph  ->  (Vtx `  G )  =  V )
 
Theoremstructgrssiedg 38895 The set of indexed edges of a graph represented as an ordered pair of vertices and indexed edges. (Contributed by AV, 14-Oct-2020.)
 |-  ( ph  ->  G  e.  X )   &    |-  ( ph  ->  Fun  G )   &    |-  ( ph  ->  V  e.  Y )   &    |-  ( ph  ->  E  e.  Z )   &    |-  ( ph  ->  { <. ( Base ` 
 ndx ) ,  V >. ,  <. (.ef `  ndx ) ,  E >. } 
 C_  G )   =>    |-  ( ph  ->  (iEdg `  G )  =  E )
 
Theoremstruct2grstr 38896 A graph represented as an extensible structure with vertices as base set and indexed edges is actually an extensible structure. (Contributed by AV, 23-Nov-2020.)
 |-  G  =  { <. ( Base `  ndx ) ,  V >. , 
 <. (.ef `  ndx ) ,  E >. }   =>    |-  G Struct  <. ( Base `  ndx ) ,  (.ef `  ndx ) >.
 
Theoremstruct2grvtx 38897 The set of vertices of a graph represented as an extensible structure with vertices as base set and indexed edges. (Contributed by AV, 23-Sep-2020.)
 |-  G  =  { <. ( Base `  ndx ) ,  V >. , 
 <. (.ef `  ndx ) ,  E >. }   =>    |-  ( ( V  e.  X  /\  E  e.  Y )  ->  (Vtx `  G )  =  V )
 
Theoremstruct2griedg 38898 The set of indexed edges of a graph represented as an ordered pair of vertices and indexed edges. (Contributed by AV, 23-Sep-2020.)
 |-  G  =  { <. ( Base `  ndx ) ,  V >. , 
 <. (.ef `  ndx ) ,  E >. }   =>    |-  ( ( V  e.  X  /\  E  e.  Y )  ->  (iEdg `  G )  =  E )
 
Theoremgraop 38899 Any representation of a graph  G (especially as extensible structure  G  =  { <. ( Base `  ndx ) ,  V >. , 
<. (.ef `  ndx ) ,  E >. }) is convertible in a representation of the graph as ordered pair. (Contributed by AV, 7-Oct-2020.)
 |-  H  =  <. (Vtx `  G ) ,  (iEdg `  G ) >.   =>    |-  ( (Vtx `  G )  =  (Vtx `  H )  /\  (iEdg `  G )  =  (iEdg `  H ) )
 
Theoremgrastruct 38900 Any representation of a graph  G (especially as ordered pair  G  =  <. V ,  E >.) is convertible in a representation of the graph as extensible structure. (Contributed by AV, 8-Oct-2020.)
 |-  H  =  { <. ( Base `  ndx ) ,  (Vtx `  G ) >. ,  <. (.ef `  ndx ) ,  (iEdg `  G ) >. }   =>    |-  ( (Vtx `  G )  =  (Vtx `  H )  /\  (iEdg `  G )  =  (iEdg `  H ) )
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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 39801-39900 400 39901-40000 401 40001-40100 402 40101-40127
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