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Theorem numclwwlk6 25920
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 25738 . . . . . 6  |-  ( <. V ,  E >. RegUSGrph  K  ->  V USGrph  E )
213ad2ant1 1051 . . . . 5  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  ->  V USGrph  E )
32adantr 472 . . . 4  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  V USGrph  E )
4 simp3 1032 . . . . 5  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  ->  V  e.  Fin )
54adantr 472 . . . 4  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  V  e.  Fin )
6 prmnn 14704 . . . . . 6  |-  ( P  e.  Prime  ->  P  e.  NN )
76nnnn0d 10949 . . . . 5  |-  ( P  e.  Prime  ->  P  e. 
NN0 )
87ad2antrl 742 . . . 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 25917 . . . 4  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  P  e.  NN0 )  ->  ( # `  ( C `  P )
)  =  sum_ x  e.  V  ( # `  (
x F P ) ) )
123, 5, 8, 11syl3anc 1292 . . 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 6323 . 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 742 . . . 4  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  P  e.  NN )
15 usgrav 25144 . . . . . . . . . . . . . . 15  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
1615simprd 470 . . . . . . . . . . . . . 14  |-  ( V USGrph  E  ->  E  e.  _V )
171, 16syl 17 . . . . . . . . . . . . 13  |-  ( <. V ,  E >. RegUSGrph  K  ->  E  e.  _V )
1817anim1i 578 . . . . . . . . . . . 12  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  V  e.  Fin )  ->  ( E  e. 
_V  /\  V  e.  Fin ) )
1918ancomd 458 . . . . . . . . . . 11  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  V  e.  Fin )  ->  ( V  e. 
Fin  /\  E  e.  _V ) )
20193adant2 1049 . . . . . . . . . 10  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  ->  ( V  e.  Fin  /\  E  e.  _V ) )
2120adantr 472 . . . . . . . . 9  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  -> 
( V  e.  Fin  /\  E  e.  _V )
)
2221adantr 472 . . . . . . . 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 578 . . . . . . . . 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 458 . . . . . . . 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 25889 . . . . . . . 8  |-  ( ( ( V  e.  Fin  /\  E  e.  _V )  /\  ( x  e.  V  /\  P  e.  NN0 ) )  ->  (
x F P )  e.  Fin )
2622, 24, 25syl2anc 673 . . . . . . 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 12576 . . . . . . 7  |-  ( ( x F P )  e.  Fin  ->  ( # `
 ( x F P ) )  e. 
NN0 )
2826, 27syl 17 . . . . . 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 11061 . . . . 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 2809 . . . 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 13931 . . 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 768 . . . . . 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 468 . . . . . 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 778 . . . . . 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 779 . . . . . 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 25919 . . . . . 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 1294 . . . . 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 13846 . . . 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 6323 . . 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 2505 . 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 9615 . . . . . . 7  |-  1  e.  CC
424, 41jctir 547 . . . . . 6  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  ->  ( V  e.  Fin  /\  1  e.  CC ) )
4342adantr 472 . . . . 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 13928 . . . . 5  |-  ( ( V  e.  Fin  /\  1  e.  CC )  -> 
sum_ x  e.  V 
1  =  ( (
# `  V )  x.  1 ) )
4543, 44syl 17 . . . 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 12576 . . . . . . . 8  |-  ( V  e.  Fin  ->  ( # `
 V )  e. 
NN0 )
4746nn0red 10950 . . . . . . 7  |-  ( V  e.  Fin  ->  ( # `
 V )  e.  RR )
48473ad2ant3 1053 . . . . . 6  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  ->  ( # `
 V )  e.  RR )
4948adantr 472 . . . . 5  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  -> 
( # `  V )  e.  RR )
50 ax-1rid 9627 . . . . 5  |-  ( (
# `  V )  e.  RR  ->  ( ( # `
 V )  x.  1 )  =  (
# `  V )
)
5149, 50syl 17 . . . 4  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  -> 
( ( # `  V
)  x.  1 )  =  ( # `  V
) )
5245, 51eqtrd 2505 . . 3  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  sum_ x  e.  V  1  =  ( # `  V
) )
5352oveq1d 6323 . 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 2509 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 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   {crab 2760   _Vcvv 3031   <.cop 3965   class class class wbr 4395    |-> cmpt 4454   ` cfv 5589  (class class class)co 6308    |-> cmpt2 6310   Fincfn 7587   CCcc 9555   RRcr 9556   0cc0 9557   1c1 9558    x. cmul 9562    - cmin 9880   NNcn 10631   NN0cn0 10893   ZZcz 10961    mod cmo 12129   #chash 12553   sum_csu 13829    || cdvds 14382   Primecprime 14701   USGrph cusg 25136   ClWWalksN cclwwlkn 25556   RegUSGrph crusgra 25730   FriendGrph cfrgra 25795
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-disj 4367  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-rp 11326  df-xadd 11433  df-fz 11811  df-fzo 11943  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-hash 12554  df-word 12711  df-lsw 12712  df-concat 12713  df-s1 12714  df-substr 12715  df-s2 13003  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-clim 13629  df-sum 13830  df-dvds 14383  df-gcd 14548  df-prm 14702  df-phi 14793  df-usgra 25139  df-nbgra 25227  df-wlk 25315  df-wwlk 25486  df-wwlkn 25487  df-clwwlk 25558  df-clwwlkn 25559  df-vdgr 25701  df-rgra 25731  df-rusgra 25732  df-frgra 25796
This theorem is referenced by:  numclwwlk7  25921
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