Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  hashclwwlkn Structured version   Unicode version

Theorem hashclwwlkn 30510
Description: The size of the set of closed walks (defined as words) with a fixed length which is a prime number is the product of the number of equivalence classes for 
.~ over the set of closed walks and the fixed length. (Contributed by Alexander van der Vekens, 17-Jun-2018.)
Hypotheses
Ref Expression
erclwwlkn.w  |-  W  =  ( ( V ClWWalksN  E ) `
 N )
erclwwlkn.r  |-  .~  =  { <. t ,  u >.  |  ( t  e.  W  /\  u  e.  W  /\  E. n  e.  ( 0 ... N
) t  =  ( u cyclShift  n ) ) }
Assertion
Ref Expression
hashclwwlkn  |-  ( ( V  e.  Fin  /\  V USGrph  E  /\  N  e. 
Prime )  ->  ( # `  W )  =  ( ( # `  ( W /.  .~  ) )  x.  N ) )
Distinct variable groups:    t, E, u    t, N, u    n, V, t, u    t, W, u    n, N    n, W    n, E
Allowed substitution hints:    .~ ( u, t, n)

Proof of Theorem hashclwwlkn
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 id 22 . . 3  |-  ( V  e.  Fin  ->  V  e.  Fin )
2 usgrav 23270 . . . 4  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
32simprd 463 . . 3  |-  ( V USGrph  E  ->  E  e.  _V )
4 prmnn 13766 . . . 4  |-  ( N  e.  Prime  ->  N  e.  NN )
54nnnn0d 10636 . . 3  |-  ( N  e.  Prime  ->  N  e. 
NN0 )
6 erclwwlkn.w . . . 4  |-  W  =  ( ( V ClWWalksN  E ) `
 N )
7 erclwwlkn.r . . . 4  |-  .~  =  { <. t ,  u >.  |  ( t  e.  W  /\  u  e.  W  /\  E. n  e.  ( 0 ... N
) t  =  ( u cyclShift  n ) ) }
86, 7hashclwwlkn0 30504 . . 3  |-  ( ( V  e.  Fin  /\  E  e.  _V  /\  N  e.  NN0 )  ->  ( # `
 W )  = 
sum_ x  e.  ( W /.  .~  ) (
# `  x )
)
91, 3, 5, 8syl3an 1260 . 2  |-  ( ( V  e.  Fin  /\  V USGrph  E  /\  N  e. 
Prime )  ->  ( # `  W )  =  sum_ x  e.  ( W /.  .~  ) ( # `  x
) )
106, 7usghashecclwwlk 30509 . . . . 5  |-  ( ( V USGrph  E  /\  N  e. 
Prime )  ->  ( x  e.  ( W /.  .~  )  ->  ( # `  x
)  =  N ) )
11103adant1 1006 . . . 4  |-  ( ( V  e.  Fin  /\  V USGrph  E  /\  N  e. 
Prime )  ->  ( x  e.  ( W /.  .~  )  ->  ( # `  x
)  =  N ) )
1211imp 429 . . 3  |-  ( ( ( V  e.  Fin  /\  V USGrph  E  /\  N  e. 
Prime )  /\  x  e.  ( W /.  .~  ) )  ->  ( # `
 x )  =  N )
1312sumeq2dv 13180 . 2  |-  ( ( V  e.  Fin  /\  V USGrph  E  /\  N  e. 
Prime )  ->  sum_ x  e.  ( W /.  .~  ) ( # `  x
)  =  sum_ x  e.  ( W /.  .~  ) N )
146, 7qerclwwlknfi 30503 . . . 4  |-  ( ( V  e.  Fin  /\  E  e.  _V  /\  N  e.  NN0 )  ->  ( W /.  .~  )  e. 
Fin )
151, 3, 5, 14syl3an 1260 . . 3  |-  ( ( V  e.  Fin  /\  V USGrph  E  /\  N  e. 
Prime )  ->  ( W /.  .~  )  e. 
Fin )
164nncnd 10338 . . . 4  |-  ( N  e.  Prime  ->  N  e.  CC )
17163ad2ant3 1011 . . 3  |-  ( ( V  e.  Fin  /\  V USGrph  E  /\  N  e. 
Prime )  ->  N  e.  CC )
18 fsumconst 13257 . . 3  |-  ( ( ( W /.  .~  )  e.  Fin  /\  N  e.  CC )  ->  sum_ x  e.  ( W /.  .~  ) N  =  (
( # `  ( W /.  .~  ) )  x.  N ) )
1915, 17, 18syl2anc 661 . 2  |-  ( ( V  e.  Fin  /\  V USGrph  E  /\  N  e. 
Prime )  ->  sum_ x  e.  ( W /.  .~  ) N  =  (
( # `  ( W /.  .~  ) )  x.  N ) )
209, 13, 193eqtrd 2479 1  |-  ( ( V  e.  Fin  /\  V USGrph  E  /\  N  e. 
Prime )  ->  ( # `  W )  =  ( ( # `  ( W /.  .~  ) )  x.  N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 965    = wceq 1369    e. wcel 1756   E.wrex 2716   _Vcvv 2972   class class class wbr 4292   {copab 4349   ` cfv 5418  (class class class)co 6091   /.cqs 7100   Fincfn 7310   CCcc 9280   0cc0 9282    x. cmul 9287   NN0cn0 10579   ...cfz 11437   #chash 12103   cyclShift ccsh 12425   sum_csu 13163   Primecprime 13763   USGrph cusg 23264   ClWWalksN cclwwlkn 30414
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-disj 4263  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-2o 6921  df-oadd 6924  df-er 7101  df-ec 7103  df-qs 7107  df-map 7216  df-pm 7217  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-sup 7691  df-oi 7724  df-card 8109  df-cda 8337  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-n0 10580  df-z 10647  df-uz 10862  df-rp 10992  df-fz 11438  df-fzo 11549  df-fl 11642  df-mod 11709  df-seq 11807  df-exp 11866  df-hash 12104  df-word 12229  df-lsw 12230  df-concat 12231  df-substr 12233  df-reps 12236  df-csh 12426  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-clim 12966  df-sum 13164  df-dvds 13536  df-gcd 13691  df-prm 13764  df-phi 13841  df-usgra 23266  df-clwwlk 30416  df-clwwlkn 30417
This theorem is referenced by:  clwwlkndivn  30511
  Copyright terms: Public domain W3C validator