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Theorem clwwlkndivn 24513
Description: The size of the set of closed walks (defined as words) of length n is divisible by n. (Contributed by Alexander van der Vekens, 17-Jun-2018.)
Assertion
Ref Expression
clwwlkndivn  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  Prime )  ->  N  ||  ( # `
 ( ( V ClWWalksN  E ) `  N
) ) )

Proof of Theorem clwwlkndivn
Dummy variables  n  t  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp2 997 . . . . . 6  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  Prime )  ->  V  e.  Fin )
2 usgrav 24014 . . . . . . . 8  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
32simprd 463 . . . . . . 7  |-  ( V USGrph  E  ->  E  e.  _V )
433ad2ant1 1017 . . . . . 6  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  Prime )  ->  E  e.  _V )
5 prmnn 14075 . . . . . . . 8  |-  ( N  e.  Prime  ->  N  e.  NN )
65nnnn0d 10848 . . . . . . 7  |-  ( N  e.  Prime  ->  N  e. 
NN0 )
763ad2ant3 1019 . . . . . 6  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  Prime )  ->  N  e.  NN0 )
8 eqid 2467 . . . . . . 7  |-  ( ( V ClWWalksN  E ) `  N
)  =  ( ( V ClWWalksN  E ) `  N
)
9 eqid 2467 . . . . . . 7  |-  { <. t ,  u >.  |  ( t  e.  ( ( V ClWWalksN  E ) `  N
)  /\  u  e.  ( ( V ClWWalksN  E ) `
 N )  /\  E. n  e.  ( 0 ... N ) t  =  ( u cyclShift  n
) ) }  =  { <. t ,  u >.  |  ( t  e.  ( ( V ClWWalksN  E ) `
 N )  /\  u  e.  ( ( V ClWWalksN  E ) `  N
)  /\  E. n  e.  ( 0 ... N
) t  =  ( u cyclShift  n ) ) }
108, 9qerclwwlknfi 24505 . . . . . 6  |-  ( ( V  e.  Fin  /\  E  e.  _V  /\  N  e.  NN0 )  ->  (
( ( V ClWWalksN  E ) `
 N ) /. { <. t ,  u >.  |  ( t  e.  ( ( V ClWWalksN  E ) `
 N )  /\  u  e.  ( ( V ClWWalksN  E ) `  N
)  /\  E. n  e.  ( 0 ... N
) t  =  ( u cyclShift  n ) ) } )  e.  Fin )
111, 4, 7, 10syl3anc 1228 . . . . 5  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  Prime )  ->  ( (
( V ClWWalksN  E ) `  N ) /. { <. t ,  u >.  |  ( t  e.  ( ( V ClWWalksN  E ) `  N )  /\  u  e.  ( ( V ClWWalksN  E ) `
 N )  /\  E. n  e.  ( 0 ... N ) t  =  ( u cyclShift  n
) ) } )  e.  Fin )
12 hashcl 12392 . . . . 5  |-  ( ( ( ( V ClWWalksN  E ) `
 N ) /. { <. t ,  u >.  |  ( t  e.  ( ( V ClWWalksN  E ) `
 N )  /\  u  e.  ( ( V ClWWalksN  E ) `  N
)  /\  E. n  e.  ( 0 ... N
) t  =  ( u cyclShift  n ) ) } )  e.  Fin  ->  (
# `  ( (
( V ClWWalksN  E ) `  N ) /. { <. t ,  u >.  |  ( t  e.  ( ( V ClWWalksN  E ) `  N )  /\  u  e.  ( ( V ClWWalksN  E ) `
 N )  /\  E. n  e.  ( 0 ... N ) t  =  ( u cyclShift  n
) ) } ) )  e.  NN0 )
1311, 12syl 16 . . . 4  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  Prime )  ->  ( # `  (
( ( V ClWWalksN  E ) `
 N ) /. { <. t ,  u >.  |  ( t  e.  ( ( V ClWWalksN  E ) `
 N )  /\  u  e.  ( ( V ClWWalksN  E ) `  N
)  /\  E. n  e.  ( 0 ... N
) t  =  ( u cyclShift  n ) ) } ) )  e.  NN0 )
1413nn0zd 10960 . . 3  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  Prime )  ->  ( # `  (
( ( V ClWWalksN  E ) `
 N ) /. { <. t ,  u >.  |  ( t  e.  ( ( V ClWWalksN  E ) `
 N )  /\  u  e.  ( ( V ClWWalksN  E ) `  N
)  /\  E. n  e.  ( 0 ... N
) t  =  ( u cyclShift  n ) ) } ) )  e.  ZZ )
15 prmz 14076 . . . 4  |-  ( N  e.  Prime  ->  N  e.  ZZ )
16153ad2ant3 1019 . . 3  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  Prime )  ->  N  e.  ZZ )
17 dvdsmul2 13863 . . 3  |-  ( ( ( # `  (
( ( V ClWWalksN  E ) `
 N ) /. { <. t ,  u >.  |  ( t  e.  ( ( V ClWWalksN  E ) `
 N )  /\  u  e.  ( ( V ClWWalksN  E ) `  N
)  /\  E. n  e.  ( 0 ... N
) t  =  ( u cyclShift  n ) ) } ) )  e.  ZZ  /\  N  e.  ZZ )  ->  N  ||  (
( # `  ( ( ( V ClWWalksN  E ) `  N ) /. { <. t ,  u >.  |  ( t  e.  ( ( V ClWWalksN  E ) `  N )  /\  u  e.  ( ( V ClWWalksN  E ) `
 N )  /\  E. n  e.  ( 0 ... N ) t  =  ( u cyclShift  n
) ) } ) )  x.  N ) )
1814, 16, 17syl2anc 661 . 2  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  Prime )  ->  N  ||  (
( # `  ( ( ( V ClWWalksN  E ) `  N ) /. { <. t ,  u >.  |  ( t  e.  ( ( V ClWWalksN  E ) `  N )  /\  u  e.  ( ( V ClWWalksN  E ) `
 N )  /\  E. n  e.  ( 0 ... N ) t  =  ( u cyclShift  n
) ) } ) )  x.  N ) )
198, 9hashclwwlkn 24512 . . 3  |-  ( ( V  e.  Fin  /\  V USGrph  E  /\  N  e. 
Prime )  ->  ( # `  ( ( V ClWWalksN  E ) `
 N ) )  =  ( ( # `  ( ( ( V ClWWalksN  E ) `  N
) /. { <. t ,  u >.  |  ( t  e.  ( ( V ClWWalksN  E ) `  N
)  /\  u  e.  ( ( V ClWWalksN  E ) `
 N )  /\  E. n  e.  ( 0 ... N ) t  =  ( u cyclShift  n
) ) } ) )  x.  N ) )
20193com12 1200 . 2  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  Prime )  ->  ( # `  (
( V ClWWalksN  E ) `  N ) )  =  ( ( # `  (
( ( V ClWWalksN  E ) `
 N ) /. { <. t ,  u >.  |  ( t  e.  ( ( V ClWWalksN  E ) `
 N )  /\  u  e.  ( ( V ClWWalksN  E ) `  N
)  /\  E. n  e.  ( 0 ... N
) t  =  ( u cyclShift  n ) ) } ) )  x.  N
) )
2118, 20breqtrrd 4473 1  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  Prime )  ->  N  ||  ( # `
 ( ( V ClWWalksN  E ) `  N
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 973    = wceq 1379    e. wcel 1767   E.wrex 2815   _Vcvv 3113   class class class wbr 4447   {copab 4504   ` cfv 5586  (class class class)co 6282   /.cqs 7307   Fincfn 7513   0cc0 9488    x. cmul 9493   NN0cn0 10791   ZZcz 10860   ...cfz 11668   #chash 12369   cyclShift ccsh 12718    || cdivides 13843   Primecprime 14072   USGrph cusg 24006   ClWWalksN cclwwlkn 24425
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-disj 4418  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-oadd 7131  df-er 7308  df-ec 7310  df-qs 7314  df-map 7419  df-pm 7420  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-sup 7897  df-oi 7931  df-card 8316  df-cda 8544  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11079  df-rp 11217  df-fz 11669  df-fzo 11789  df-fl 11893  df-mod 11961  df-seq 12072  df-exp 12131  df-hash 12370  df-word 12504  df-lsw 12505  df-concat 12506  df-substr 12508  df-reps 12511  df-csh 12719  df-cj 12891  df-re 12892  df-im 12893  df-sqrt 13027  df-abs 13028  df-clim 13270  df-sum 13468  df-dvds 13844  df-gcd 14000  df-prm 14073  df-phi 14151  df-usgra 24009  df-clwwlk 24427  df-clwwlkn 24428
This theorem is referenced by:  clwlkndivn  24529  numclwwlk8  24792
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