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Theorem numclwwlk6 30709
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 30551 . . . . . 6  |-  ( <. V ,  E >. RegUSGrph  K  ->  V USGrph  E )
213ad2ant1 1009 . . . . 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 990 . . . . 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 13769 . . . . . 6  |-  ( P  e.  Prime  ->  P  e.  NN )
76nnnn0d 10639 . . . . 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 30706 . . . 4  |-  ( ( V USGrph  E  /\  V  e. 
Fin  /\  P  e.  NN0 )  ->  ( # `  ( C `  P )
)  =  sum_ x  e.  V  ( # `  (
x F P ) ) )
123, 5, 8, 11syl3anc 1218 . . 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 6109 . 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 23273 . . . . . . . . . . . . . . 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 1007 . . . . . . . . . 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 30678 . . . . . . . 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 12129 . . . . . . 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 10748 . . . . 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 2802 . . . 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 30248 . . 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 30708 . . . . . 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 1220 . . . . 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 13183 . . . 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 6109 . . 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 2475 . 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 9343 . . . . . . 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 13260 . . . . 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 12129 . . . . . . . 8  |-  ( V  e.  Fin  ->  ( # `
 V )  e. 
NN0 )
4746nn0red 10640 . . . . . . 7  |-  ( V  e.  Fin  ->  ( # `
 V )  e.  RR )
48473ad2ant3 1011 . . . . . 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 9355 . . . . 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 2475 . . 3  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  sum_ x  e.  V  1  =  ( # `  V
) )
5352oveq1d 6109 . 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 2479 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 965    = wceq 1369    e. wcel 1756   {crab 2722   _Vcvv 2975   <.cop 3886   class class class wbr 4295    e. cmpt 4353   ` cfv 5421  (class class class)co 6094    e. cmpt2 6096   Fincfn 7313   CCcc 9283   RRcr 9284   0cc0 9285   1c1 9286    x. cmul 9290    - cmin 9598   NNcn 10325   NN0cn0 10582   ZZcz 10649    mod cmo 11711   #chash 12106   sum_csu 13166    || cdivides 13538   Primecprime 13766   USGrph cusg 23267   ClWWalksN cclwwlkn 30417   RegUSGrph crusgra 30543   FriendGrph cfrgra 30583
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4406  ax-sep 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375  ax-inf2 7850  ax-cnex 9341  ax-resscn 9342  ax-1cn 9343  ax-icn 9344  ax-addcl 9345  ax-addrcl 9346  ax-mulcl 9347  ax-mulrcl 9348  ax-mulcom 9349  ax-addass 9350  ax-mulass 9351  ax-distr 9352  ax-i2m1 9353  ax-1ne0 9354  ax-1rid 9355  ax-rnegex 9356  ax-rrecex 9357  ax-cnre 9358  ax-pre-lttri 9359  ax-pre-lttrn 9360  ax-pre-ltadd 9361  ax-pre-mulgt0 9362  ax-pre-sup 9363
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-nel 2612  df-ral 2723  df-rex 2724  df-reu 2725  df-rmo 2726  df-rab 2727  df-v 2977  df-sbc 3190  df-csb 3292  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-pss 3347  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-tp 3885  df-op 3887  df-uni 4095  df-int 4132  df-iun 4176  df-disj 4266  df-br 4296  df-opab 4354  df-mpt 4355  df-tr 4389  df-eprel 4635  df-id 4639  df-po 4644  df-so 4645  df-fr 4682  df-se 4683  df-we 4684  df-ord 4725  df-on 4726  df-lim 4727  df-suc 4728  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-riota 6055  df-ov 6097  df-oprab 6098  df-mpt2 6099  df-om 6480  df-1st 6580  df-2nd 6581  df-recs 6835  df-rdg 6869  df-1o 6923  df-2o 6924  df-oadd 6927  df-er 7104  df-map 7219  df-pm 7220  df-en 7314  df-dom 7315  df-sdom 7316  df-fin 7317  df-sup 7694  df-oi 7727  df-card 8112  df-cda 8340  df-pnf 9423  df-mnf 9424  df-xr 9425  df-ltxr 9426  df-le 9427  df-sub 9600  df-neg 9601  df-div 9997  df-nn 10326  df-2 10383  df-3 10384  df-n0 10583  df-z 10650  df-uz 10865  df-rp 10995  df-xadd 11093  df-fz 11441  df-fzo 11552  df-fl 11645  df-mod 11712  df-seq 11810  df-exp 11869  df-hash 12107  df-word 12232  df-lsw 12233  df-concat 12234  df-s1 12235  df-substr 12236  df-s2 12478  df-cj 12591  df-re 12592  df-im 12593  df-sqr 12727  df-abs 12728  df-clim 12969  df-sum 13167  df-dvds 13539  df-gcd 13694  df-prm 13767  df-phi 13844  df-usgra 23269  df-nbgra 23335  df-wlk 23418  df-vdgr 23567  df-wwlk 30316  df-wwlkn 30317  df-clwwlk 30419  df-clwwlkn 30420  df-rgra 30544  df-rusgra 30545  df-frgra 30584
This theorem is referenced by:  numclwwlk7  30710
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