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Theorem numclwwlk6 30631
Description: For a prime divisor p of k-1, the total number of closed walks of length p in an undirected simple graph with m vertices mod p is equal to the number of vertices mod p. (Contributed by Alexander van der Vekens, 7-Oct-2018.)
Hypotheses
Ref Expression
numclwwlk.c  |-  C  =  ( n  e.  NN0  |->  ( ( V ClWWalksN  E ) `
 n ) )
numclwwlk.f  |-  F  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( C `  n )  |  ( w ` 
0 )  =  v } )
Assertion
Ref Expression
numclwwlk6  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  -> 
( ( # `  ( C `  P )
)  mod  P )  =  ( ( # `  V )  mod  P
) )
Distinct variable groups:    n, E    n, V    w, C, n, v    v, V    w, E    w, V    w, F    w, P    v, E    v, K, w    P, n, v
Allowed substitution hints:    F( v, n)    K( n)

Proof of Theorem numclwwlk6
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 rusisusgra 30473 . . . . . 6  |-  ( <. V ,  E >. RegUSGrph  K  ->  V USGrph  E )
213ad2ant1 1004 . . . . 5  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  ->  V USGrph  E )
32adantr 462 . . . 4  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  V USGrph  E )
4 simp3 985 . . . . 5  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  ->  V  e.  Fin )
54adantr 462 . . . 4  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  V  e.  Fin )
6 prmnn 13762 . . . . . 6  |-  ( P  e.  Prime  ->  P  e.  NN )
76nnnn0d 10632 . . . . 5  |-  ( P  e.  Prime  ->  P  e. 
NN0 )
87ad2antrl 722 . . . 4  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  P  e.  NN0 )
9 numclwwlk.c . . . . 5  |-  C  =  ( n  e.  NN0  |->  ( ( V ClWWalksN  E ) `
 n ) )
10 numclwwlk.f . . . . 5  |-  F  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( C `  n )  |  ( w ` 
0 )  =  v } )
119, 10numclwwlk4 30628 . . . 4  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  P  e.  NN0 )  ->  ( # `  ( C `  P )
)  =  sum_ x  e.  V  ( # `  (
x F P ) ) )
123, 5, 8, 11syl3anc 1213 . . 3  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  -> 
( # `  ( C `
 P ) )  =  sum_ x  e.  V  ( # `  ( x F P ) ) )
1312oveq1d 6105 . 2  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  -> 
( ( # `  ( C `  P )
)  mod  P )  =  ( sum_ x  e.  V  ( # `  (
x F P ) )  mod  P ) )
146ad2antrl 722 . . . 4  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  P  e.  NN )
15 usgrav 23205 . . . . . . . . . . . . . . 15  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
1615simprd 460 . . . . . . . . . . . . . 14  |-  ( V USGrph  E  ->  E  e.  _V )
171, 16syl 16 . . . . . . . . . . . . 13  |-  ( <. V ,  E >. RegUSGrph  K  ->  E  e.  _V )
1817anim1i 565 . . . . . . . . . . . 12  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  V  e.  Fin )  ->  ( E  e. 
_V  /\  V  e.  Fin ) )
1918ancomd 449 . . . . . . . . . . 11  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  V  e.  Fin )  ->  ( V  e. 
Fin  /\  E  e.  _V ) )
20193adant2 1002 . . . . . . . . . 10  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  ->  ( V  e.  Fin  /\  E  e.  _V ) )
2120adantr 462 . . . . . . . . 9  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  -> 
( V  e.  Fin  /\  E  e.  _V )
)
2221adantr 462 . . . . . . . 8  |-  ( ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  /\  x  e.  V )  ->  ( V  e.  Fin  /\  E  e.  _V ) )
238anim1i 565 . . . . . . . . 9  |-  ( ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  /\  x  e.  V )  ->  ( P  e.  NN0  /\  x  e.  V ) )
2423ancomd 449 . . . . . . . 8  |-  ( ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  /\  x  e.  V )  ->  (
x  e.  V  /\  P  e.  NN0 ) )
259, 10numclwwlkffin 30600 . . . . . . . 8  |-  ( ( ( V  e.  Fin  /\  E  e.  _V )  /\  ( x  e.  V  /\  P  e.  NN0 ) )  ->  (
x F P )  e.  Fin )
2622, 24, 25syl2anc 656 . . . . . . 7  |-  ( ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  /\  x  e.  V )  ->  (
x F P )  e.  Fin )
27 hashcl 12122 . . . . . . 7  |-  ( ( x F P )  e.  Fin  ->  ( # `
 ( x F P ) )  e. 
NN0 )
2826, 27syl 16 . . . . . 6  |-  ( ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  /\  x  e.  V )  ->  ( # `
 ( x F P ) )  e. 
NN0 )
2928nn0zd 10741 . . . . 5  |-  ( ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  /\  x  e.  V )  ->  ( # `
 ( x F P ) )  e.  ZZ )
3029ralrimiva 2797 . . . 4  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  A. x  e.  V  ( # `  ( x F P ) )  e.  ZZ )
3114, 5, 30modfsummod 30170 . . 3  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  -> 
( sum_ x  e.  V  ( # `  ( x F P ) )  mod  P )  =  ( sum_ x  e.  V  ( ( # `  (
x F P ) )  mod  P )  mod  P ) )
32 simpll 748 . . . . . 6  |-  ( ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  /\  x  e.  V )  ->  ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin ) )
33 simpr 458 . . . . . 6  |-  ( ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  /\  x  e.  V )  ->  x  e.  V )
34 simplrl 754 . . . . . 6  |-  ( ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  /\  x  e.  V )  ->  P  e.  Prime )
35 simplrr 755 . . . . . 6  |-  ( ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  /\  x  e.  V )  ->  P  ||  ( K  -  1 ) )
369, 10numclwwlk5 30630 . . . . . 6  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  (
x  e.  V  /\  P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  -> 
( ( # `  (
x F P ) )  mod  P )  =  1 )
3732, 33, 34, 35, 36syl13anc 1215 . . . . 5  |-  ( ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  /\  x  e.  V )  ->  (
( # `  ( x F P ) )  mod  P )  =  1 )
3837sumeq2dv 13176 . . . 4  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  sum_ x  e.  V  ( ( # `  (
x F P ) )  mod  P )  =  sum_ x  e.  V 
1 )
3938oveq1d 6105 . . 3  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  -> 
( sum_ x  e.  V  ( ( # `  (
x F P ) )  mod  P )  mod  P )  =  ( sum_ x  e.  V 
1  mod  P )
)
4031, 39eqtrd 2473 . 2  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  -> 
( sum_ x  e.  V  ( # `  ( x F P ) )  mod  P )  =  ( sum_ x  e.  V 
1  mod  P )
)
41 ax-1cn 9336 . . . . . . 7  |-  1  e.  CC
424, 41jctir 535 . . . . . 6  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  ->  ( V  e.  Fin  /\  1  e.  CC ) )
4342adantr 462 . . . . 5  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  -> 
( V  e.  Fin  /\  1  e.  CC ) )
44 fsumconst 13253 . . . . 5  |-  ( ( V  e.  Fin  /\  1  e.  CC )  -> 
sum_ x  e.  V 
1  =  ( (
# `  V )  x.  1 ) )
4543, 44syl 16 . . . 4  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  sum_ x  e.  V  1  =  ( ( # `  V )  x.  1 ) )
46 hashcl 12122 . . . . . . . 8  |-  ( V  e.  Fin  ->  ( # `
 V )  e. 
NN0 )
4746nn0red 10633 . . . . . . 7  |-  ( V  e.  Fin  ->  ( # `
 V )  e.  RR )
48473ad2ant3 1006 . . . . . 6  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  ->  ( # `
 V )  e.  RR )
4948adantr 462 . . . . 5  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  -> 
( # `  V )  e.  RR )
50 ax-1rid 9348 . . . . 5  |-  ( (
# `  V )  e.  RR  ->  ( ( # `
 V )  x.  1 )  =  (
# `  V )
)
5149, 50syl 16 . . . 4  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  -> 
( ( # `  V
)  x.  1 )  =  ( # `  V
) )
5245, 51eqtrd 2473 . . 3  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  sum_ x  e.  V  1  =  ( # `  V
) )
5352oveq1d 6105 . 2  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  -> 
( sum_ x  e.  V 
1  mod  P )  =  ( ( # `  V )  mod  P
) )
5413, 40, 533eqtrd 2477 1  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  -> 
( ( # `  ( C `  P )
)  mod  P )  =  ( ( # `  V )  mod  P
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761   {crab 2717   _Vcvv 2970   <.cop 3880   class class class wbr 4289    e. cmpt 4347   ` cfv 5415  (class class class)co 6090    e. cmpt2 6092   Fincfn 7306   CCcc 9276   RRcr 9277   0cc0 9278   1c1 9279    x. cmul 9283    - cmin 9591   NNcn 10318   NN0cn0 10575   ZZcz 10642    mod cmo 11704   #chash 12099   sum_csu 13159    || cdivides 13531   Primecprime 13759   USGrph cusg 23199   ClWWalksN cclwwlkn 30339   RegUSGrph crusgra 30465   FriendGrph cfrgra 30505
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-disj 4260  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-map 7212  df-pm 7213  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-sup 7687  df-oi 7720  df-card 8105  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-n0 10576  df-z 10643  df-uz 10858  df-rp 10988  df-xadd 11086  df-fz 11434  df-fzo 11545  df-fl 11638  df-mod 11705  df-seq 11803  df-exp 11862  df-hash 12100  df-word 12225  df-lsw 12226  df-concat 12227  df-s1 12228  df-substr 12229  df-s2 12471  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-clim 12962  df-sum 13160  df-dvds 13532  df-gcd 13687  df-prm 13760  df-phi 13837  df-usgra 23201  df-nbgra 23267  df-wlk 23350  df-vdgr 23499  df-wwlk 30238  df-wwlkn 30239  df-clwwlk 30341  df-clwwlkn 30342  df-rgra 30466  df-rusgra 30467  df-frgra 30506
This theorem is referenced by:  numclwwlk7  30632
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