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Theorem numclwwlk6 24937
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 24754 . . . . . 6  |-  ( <. V ,  E >. RegUSGrph  K  ->  V USGrph  E )
213ad2ant1 1017 . . . . 5  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  ->  V USGrph  E )
32adantr 465 . . . 4  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  V USGrph  E )
4 simp3 998 . . . . 5  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  ->  V  e.  Fin )
54adantr 465 . . . 4  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  V  e.  Fin )
6 prmnn 14096 . . . . . 6  |-  ( P  e.  Prime  ->  P  e.  NN )
76nnnn0d 10864 . . . . 5  |-  ( P  e.  Prime  ->  P  e. 
NN0 )
87ad2antrl 727 . . . 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 24934 . . . 4  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  P  e.  NN0 )  ->  ( # `  ( C `  P )
)  =  sum_ x  e.  V  ( # `  (
x F P ) ) )
123, 5, 8, 11syl3anc 1228 . . 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 6310 . 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 727 . . . 4  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  P  e.  NN )
15 usgrav 24161 . . . . . . . . . . . . . . 15  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
1615simprd 463 . . . . . . . . . . . . . 14  |-  ( V USGrph  E  ->  E  e.  _V )
171, 16syl 16 . . . . . . . . . . . . 13  |-  ( <. V ,  E >. RegUSGrph  K  ->  E  e.  _V )
1817anim1i 568 . . . . . . . . . . . 12  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  V  e.  Fin )  ->  ( E  e. 
_V  /\  V  e.  Fin ) )
1918ancomd 451 . . . . . . . . . . 11  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  V  e.  Fin )  ->  ( V  e. 
Fin  /\  E  e.  _V ) )
20193adant2 1015 . . . . . . . . . 10  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  ->  ( V  e.  Fin  /\  E  e.  _V ) )
2120adantr 465 . . . . . . . . 9  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  -> 
( V  e.  Fin  /\  E  e.  _V )
)
2221adantr 465 . . . . . . . 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 568 . . . . . . . . 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 451 . . . . . . . 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 24906 . . . . . . . 8  |-  ( ( ( V  e.  Fin  /\  E  e.  _V )  /\  ( x  e.  V  /\  P  e.  NN0 ) )  ->  (
x F P )  e.  Fin )
2622, 24, 25syl2anc 661 . . . . . . 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 12408 . . . . . . 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 10976 . . . . 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 2881 . . . 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 13588 . . 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 753 . . . . . 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 461 . . . . . 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 759 . . . . . 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 760 . . . . . 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 24936 . . . . . 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 1230 . . . . 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 13505 . . . 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 6310 . . 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 2508 . 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 9562 . . . . . . 7  |-  1  e.  CC
424, 41jctir 538 . . . . . 6  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  ->  ( V  e.  Fin  /\  1  e.  CC ) )
4342adantr 465 . . . . 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 13585 . . . . 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 12408 . . . . . . . 8  |-  ( V  e.  Fin  ->  ( # `
 V )  e. 
NN0 )
4746nn0red 10865 . . . . . . 7  |-  ( V  e.  Fin  ->  ( # `
 V )  e.  RR )
48473ad2ant3 1019 . . . . . 6  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  ->  ( # `
 V )  e.  RR )
4948adantr 465 . . . . 5  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  -> 
( # `  V )  e.  RR )
50 ax-1rid 9574 . . . . 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 2508 . . 3  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  sum_ x  e.  V  1  =  ( # `  V
) )
5352oveq1d 6310 . 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 2512 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 973    = wceq 1379    e. wcel 1767   {crab 2821   _Vcvv 3118   <.cop 4039   class class class wbr 4453    |-> cmpt 4511   ` cfv 5594  (class class class)co 6295    |-> cmpt2 6297   Fincfn 7528   CCcc 9502   RRcr 9503   0cc0 9504   1c1 9505    x. cmul 9509    - cmin 9817   NNcn 10548   NN0cn0 10807   ZZcz 10876    mod cmo 11976   #chash 12385   sum_csu 13488    || cdivides 13864   Primecprime 14093   USGrph cusg 24153   ClWWalksN cclwwlkn 24572   RegUSGrph crusgra 24746   FriendGrph cfrgra 24811
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 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
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 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-disj 4424  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-er 7323  df-map 7434  df-pm 7435  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-sup 7913  df-oi 7947  df-card 8332  df-cda 8560  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-n0 10808  df-z 10877  df-uz 11095  df-rp 11233  df-xadd 11331  df-fz 11685  df-fzo 11805  df-fl 11909  df-mod 11977  df-seq 12088  df-exp 12147  df-hash 12386  df-word 12523  df-lsw 12524  df-concat 12525  df-s1 12526  df-substr 12527  df-s2 12793  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-clim 13291  df-sum 13489  df-dvds 13865  df-gcd 14021  df-prm 14094  df-phi 14172  df-usgra 24156  df-nbgra 24243  df-wlk 24331  df-wwlk 24502  df-wwlkn 24503  df-clwwlk 24574  df-clwwlkn 24575  df-vdgr 24717  df-rgra 24747  df-rusgra 24748  df-frgra 24812
This theorem is referenced by:  numclwwlk7  24938
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