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Theorem clwwlkndivn 25241
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 998 . . . . . 6  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  Prime )  ->  V  e.  Fin )
2 usgrav 24742 . . . . . . . 8  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
32simprd 461 . . . . . . 7  |-  ( V USGrph  E  ->  E  e.  _V )
433ad2ant1 1018 . . . . . 6  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  Prime )  ->  E  e.  _V )
5 prmnn 14427 . . . . . . . 8  |-  ( N  e.  Prime  ->  N  e.  NN )
65nnnn0d 10892 . . . . . . 7  |-  ( N  e.  Prime  ->  N  e. 
NN0 )
763ad2ant3 1020 . . . . . 6  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  Prime )  ->  N  e.  NN0 )
8 eqid 2402 . . . . . . 7  |-  ( ( V ClWWalksN  E ) `  N
)  =  ( ( V ClWWalksN  E ) `  N
)
9 eqid 2402 . . . . . . 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 25233 . . . . . 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 1230 . . . . 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 12473 . . . . 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 17 . . . 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 11005 . . 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 14428 . . . 4  |-  ( N  e.  Prime  ->  N  e.  ZZ )
16153ad2ant3 1020 . . 3  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  N  e.  Prime )  ->  N  e.  ZZ )
17 dvdsmul2 14213 . . 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 659 . 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 25240 . . 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 1201 . 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 4420 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 974    = wceq 1405    e. wcel 1842   E.wrex 2754   _Vcvv 3058   class class class wbr 4394   {copab 4451   ` cfv 5568  (class class class)co 6277   /.cqs 7346   Fincfn 7553   0cc0 9521    x. cmul 9526   NN0cn0 10835   ZZcz 10904   ...cfz 11724   #chash 12450   cyclShift ccsh 12813    || cdvds 14193   Primecprime 14424   USGrph cusg 24734   ClWWalksN cclwwlkn 25153
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-inf2 8090  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598  ax-pre-sup 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-disj 4366  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-se 4782  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-isom 5577  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-1st 6783  df-2nd 6784  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-1o 7166  df-2o 7167  df-oadd 7170  df-er 7347  df-ec 7349  df-qs 7353  df-map 7458  df-pm 7459  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-sup 7934  df-oi 7968  df-card 8351  df-cda 8579  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-div 10247  df-nn 10576  df-2 10634  df-3 10635  df-n0 10836  df-z 10905  df-uz 11127  df-rp 11265  df-ico 11587  df-fz 11725  df-fzo 11853  df-fl 11964  df-mod 12033  df-seq 12150  df-exp 12209  df-hash 12451  df-word 12589  df-lsw 12590  df-concat 12591  df-substr 12593  df-reps 12596  df-csh 12814  df-cj 13079  df-re 13080  df-im 13081  df-sqrt 13215  df-abs 13216  df-clim 13458  df-sum 13656  df-dvds 14194  df-gcd 14352  df-prm 14425  df-phi 14503  df-usgra 24737  df-clwwlk 25155  df-clwwlkn 25156
This theorem is referenced by:  clwlkndivn  25257  numclwwlk8  25519
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