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Theorem numclwwlk1 25225
Description: Statement 9 in [Huneke] p. 2: "If n > 1, then the number of closed n-walks v(0) ... v(n-2) v(n-1) v(n) from v = v(0) = v(n) with v(n-2) = v is kf(n-2)". Since  <. V ,  E >. is k-regular, the vertex v(n-2) = v has k neighbors v(n-1), so there are k walks from v(n-2) = v to v(n) = v (via each of v's neighbors) completing each of the f(n-2) walks from v=v(0) to v(n-2)=v. This theorem holds even for k=0, but only for finite graphs! (Contributed by Alexander van der Vekens, 26-Sep-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 } )
numclwwlk.g  |-  G  =  ( v  e.  V ,  n  e.  ( ZZ>=
`  2 )  |->  { w  e.  ( C `
 n )  |  ( ( w ` 
0 )  =  v  /\  ( w `  ( n  -  2
) )  =  ( w `  0 ) ) } )
Assertion
Ref Expression
numclwwlk1  |-  ( ( ( V  e.  Fin  /\ 
<. V ,  E >. RegUSGrph  K
)  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  ->  ( # `
 ( X G N ) )  =  ( K  x.  ( # `
 ( X F ( N  -  2 ) ) ) ) )
Distinct variable groups:    n, E    n, N    n, V    w, C    w, N    C, n, v, w    v, N    n, X, v, w    v, V   
w, E    w, V    w, F    w, K    w, G
Allowed substitution hints:    E( v)    F( v, n)    G( v, n)    K( v, n)

Proof of Theorem numclwwlk1
Dummy variables  x  f are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovex 6324 . . . 4  |-  ( X G N )  e. 
_V
2 ovex 6324 . . . . 5  |-  ( X F ( N  - 
2 ) )  e. 
_V
3 ovex 6324 . . . . 5  |-  ( <. V ,  E >. Neighbors  X
)  e.  _V
42, 3xpex 6603 . . . 4  |-  ( ( X F ( N  -  2 ) )  X.  ( <. V ,  E >. Neighbors  X ) )  e. 
_V
51, 4pm3.2i 455 . . 3  |-  ( ( X G N )  e.  _V  /\  (
( X F ( N  -  2 ) )  X.  ( <. V ,  E >. Neighbors  X
) )  e.  _V )
6 rusisusgra 25058 . . . . 5  |-  ( <. V ,  E >. RegUSGrph  K  ->  V USGrph  E )
76ad2antlr 726 . . . 4  |-  ( ( ( V  e.  Fin  /\ 
<. V ,  E >. RegUSGrph  K
)  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  ->  V USGrph  E )
8 simprl 756 . . . 4  |-  ( ( ( V  e.  Fin  /\ 
<. V ,  E >. RegUSGrph  K
)  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  ->  X  e.  V )
9 simpr 461 . . . . 5  |-  ( ( X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  ->  N  e.  ( ZZ>= ` 
3 ) )
109adantl 466 . . . 4  |-  ( ( ( V  e.  Fin  /\ 
<. V ,  E >. RegUSGrph  K
)  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  ->  N  e.  ( ZZ>= `  3 )
)
11 numclwwlk.c . . . . 5  |-  C  =  ( n  e.  NN0  |->  ( ( V ClWWalksN  E ) `
 n ) )
12 numclwwlk.f . . . . 5  |-  F  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( C `  n )  |  ( w ` 
0 )  =  v } )
13 numclwwlk.g . . . . 5  |-  G  =  ( v  e.  V ,  n  e.  ( ZZ>=
`  2 )  |->  { w  e.  ( C `
 n )  |  ( ( w ` 
0 )  =  v  /\  ( w `  ( n  -  2
) )  =  ( w `  0 ) ) } )
1411, 12, 13numclwlk1lem2 25224 . . . 4  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  E. f 
f : ( X G N ) -1-1-onto-> ( ( X F ( N  -  2 ) )  X.  ( <. V ,  E >. Neighbors  X ) ) )
157, 8, 10, 14syl3anc 1228 . . 3  |-  ( ( ( V  e.  Fin  /\ 
<. V ,  E >. RegUSGrph  K
)  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  ->  E. f 
f : ( X G N ) -1-1-onto-> ( ( X F ( N  -  2 ) )  X.  ( <. V ,  E >. Neighbors  X ) ) )
16 hasheqf1oi 12427 . . 3  |-  ( ( ( X G N )  e.  _V  /\  ( ( X F ( N  -  2 ) )  X.  ( <. V ,  E >. Neighbors  X
) )  e.  _V )  ->  ( E. f 
f : ( X G N ) -1-1-onto-> ( ( X F ( N  -  2 ) )  X.  ( <. V ,  E >. Neighbors  X ) )  -> 
( # `  ( X G N ) )  =  ( # `  (
( X F ( N  -  2 ) )  X.  ( <. V ,  E >. Neighbors  X
) ) ) ) )
175, 15, 16mpsyl 63 . 2  |-  ( ( ( V  e.  Fin  /\ 
<. V ,  E >. RegUSGrph  K
)  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  ->  ( # `
 ( X G N ) )  =  ( # `  (
( X F ( N  -  2 ) )  X.  ( <. V ,  E >. Neighbors  X
) ) ) )
18 usgrav 24465 . . . . . . 7  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
19 simpr 461 . . . . . . 7  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  E  e.  _V )
206, 18, 193syl 20 . . . . . 6  |-  ( <. V ,  E >. RegUSGrph  K  ->  E  e.  _V )
2120anim2i 569 . . . . 5  |-  ( ( V  e.  Fin  /\  <. V ,  E >. RegUSGrph  K
)  ->  ( V  e.  Fin  /\  E  e. 
_V ) )
22 uzuzle23 11146 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  3
)  ->  N  e.  ( ZZ>= `  2 )
)
23 uznn0sub 11137 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  2
)  ->  ( N  -  2 )  e. 
NN0 )
2422, 23syl 16 . . . . . 6  |-  ( N  e.  ( ZZ>= `  3
)  ->  ( N  -  2 )  e. 
NN0 )
2524anim2i 569 . . . . 5  |-  ( ( X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  -> 
( X  e.  V  /\  ( N  -  2 )  e.  NN0 )
)
2611, 12numclwwlkffin 25209 . . . . 5  |-  ( ( ( V  e.  Fin  /\  E  e.  _V )  /\  ( X  e.  V  /\  ( N  -  2 )  e.  NN0 )
)  ->  ( X F ( N  - 
2 ) )  e. 
Fin )
2721, 25, 26syl2an 477 . . . 4  |-  ( ( ( V  e.  Fin  /\ 
<. V ,  E >. RegUSGrph  K
)  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  ->  ( X F ( N  - 
2 ) )  e. 
Fin )
286anim1i 568 . . . . . . . 8  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  V  e.  Fin )  ->  ( V USGrph  E  /\  V  e.  Fin ) )
2928ancoms 453 . . . . . . 7  |-  ( ( V  e.  Fin  /\  <. V ,  E >. RegUSGrph  K
)  ->  ( V USGrph  E  /\  V  e.  Fin ) )
30 usgrafis 24542 . . . . . . 7  |-  ( ( V USGrph  E  /\  V  e. 
Fin )  ->  E  e.  Fin )
3129, 30syl 16 . . . . . 6  |-  ( ( V  e.  Fin  /\  <. V ,  E >. RegUSGrph  K
)  ->  E  e.  Fin )
3231adantr 465 . . . . 5  |-  ( ( ( V  e.  Fin  /\ 
<. V ,  E >. RegUSGrph  K
)  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  ->  E  e.  Fin )
33 nbusgrafi 24575 . . . . 5  |-  ( ( V USGrph  E  /\  X  e.  V  /\  E  e. 
Fin )  ->  ( <. V ,  E >. Neighbors  X
)  e.  Fin )
347, 8, 32, 33syl3anc 1228 . . . 4  |-  ( ( ( V  e.  Fin  /\ 
<. V ,  E >. RegUSGrph  K
)  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  ->  ( <. V ,  E >. Neighbors  X
)  e.  Fin )
35 hashxp 12496 . . . 4  |-  ( ( ( X F ( N  -  2 ) )  e.  Fin  /\  ( <. V ,  E >. Neighbors  X )  e.  Fin )  ->  ( # `  (
( X F ( N  -  2 ) )  X.  ( <. V ,  E >. Neighbors  X
) ) )  =  ( ( # `  ( X F ( N  - 
2 ) ) )  x.  ( # `  ( <. V ,  E >. Neighbors  X
) ) ) )
3627, 34, 35syl2anc 661 . . 3  |-  ( ( ( V  e.  Fin  /\ 
<. V ,  E >. RegUSGrph  K
)  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  ->  ( # `
 ( ( X F ( N  - 
2 ) )  X.  ( <. V ,  E >. Neighbors  X ) ) )  =  ( ( # `  ( X F ( N  -  2 ) ) )  x.  ( # `
 ( <. V ,  E >. Neighbors  X ) ) ) )
37 rusgraprop2 25069 . . . . . . . . . 10  |-  ( <. V ,  E >. RegUSGrph  K  ->  ( V USGrph  E  /\  K  e.  NN0  /\  A. x  e.  V  ( # `
 ( <. V ,  E >. Neighbors  x ) )  =  K ) )
38 oveq2 6304 . . . . . . . . . . . . . 14  |-  ( x  =  X  ->  ( <. V ,  E >. Neighbors  x
)  =  ( <. V ,  E >. Neighbors  X
) )
3938fveq2d 5876 . . . . . . . . . . . . 13  |-  ( x  =  X  ->  ( # `
 ( <. V ,  E >. Neighbors  x ) )  =  ( # `  ( <. V ,  E >. Neighbors  X
) ) )
4039eqeq1d 2459 . . . . . . . . . . . 12  |-  ( x  =  X  ->  (
( # `  ( <. V ,  E >. Neighbors  x
) )  =  K  <-> 
( # `  ( <. V ,  E >. Neighbors  X
) )  =  K ) )
4140rspccv 3207 . . . . . . . . . . 11  |-  ( A. x  e.  V  ( # `
 ( <. V ,  E >. Neighbors  x ) )  =  K  ->  ( X  e.  V  ->  ( # `  ( <. V ,  E >. Neighbors  X ) )  =  K ) )
42413ad2ant3 1019 . . . . . . . . . 10  |-  ( ( V USGrph  E  /\  K  e. 
NN0  /\  A. x  e.  V  ( # `  ( <. V ,  E >. Neighbors  x
) )  =  K )  ->  ( X  e.  V  ->  ( # `  ( <. V ,  E >. Neighbors  X ) )  =  K ) )
4337, 42syl 16 . . . . . . . . 9  |-  ( <. V ,  E >. RegUSGrph  K  ->  ( X  e.  V  ->  ( # `  ( <. V ,  E >. Neighbors  X
) )  =  K ) )
4443adantl 466 . . . . . . . 8  |-  ( ( V  e.  Fin  /\  <. V ,  E >. RegUSGrph  K
)  ->  ( X  e.  V  ->  ( # `  ( <. V ,  E >. Neighbors  X ) )  =  K ) )
4544com12 31 . . . . . . 7  |-  ( X  e.  V  ->  (
( V  e.  Fin  /\ 
<. V ,  E >. RegUSGrph  K
)  ->  ( # `  ( <. V ,  E >. Neighbors  X
) )  =  K ) )
4645adantr 465 . . . . . 6  |-  ( ( X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  -> 
( ( V  e. 
Fin  /\  <. V ,  E >. RegUSGrph  K )  ->  ( # `
 ( <. V ,  E >. Neighbors  X ) )  =  K ) )
4746impcom 430 . . . . 5  |-  ( ( ( V  e.  Fin  /\ 
<. V ,  E >. RegUSGrph  K
)  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  ->  ( # `
 ( <. V ,  E >. Neighbors  X ) )  =  K )
4847oveq2d 6312 . . . 4  |-  ( ( ( V  e.  Fin  /\ 
<. V ,  E >. RegUSGrph  K
)  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  ->  (
( # `  ( X F ( N  - 
2 ) ) )  x.  ( # `  ( <. V ,  E >. Neighbors  X
) ) )  =  ( ( # `  ( X F ( N  - 
2 ) ) )  x.  K ) )
49 hashcl 12431 . . . . . 6  |-  ( ( X F ( N  -  2 ) )  e.  Fin  ->  ( # `
 ( X F ( N  -  2 ) ) )  e. 
NN0 )
50 nn0cn 10826 . . . . . 6  |-  ( (
# `  ( X F ( N  - 
2 ) ) )  e.  NN0  ->  ( # `  ( X F ( N  -  2 ) ) )  e.  CC )
5127, 49, 503syl 20 . . . . 5  |-  ( ( ( V  e.  Fin  /\ 
<. V ,  E >. RegUSGrph  K
)  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  ->  ( # `
 ( X F ( N  -  2 ) ) )  e.  CC )
52 nn0cn 10826 . . . . . . . 8  |-  ( K  e.  NN0  ->  K  e.  CC )
53523ad2ant2 1018 . . . . . . 7  |-  ( ( V USGrph  E  /\  K  e. 
NN0  /\  A. x  e.  V  ( # `  ( <. V ,  E >. Neighbors  x
) )  =  K )  ->  K  e.  CC )
5437, 53syl 16 . . . . . 6  |-  ( <. V ,  E >. RegUSGrph  K  ->  K  e.  CC )
5554ad2antlr 726 . . . . 5  |-  ( ( ( V  e.  Fin  /\ 
<. V ,  E >. RegUSGrph  K
)  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  ->  K  e.  CC )
5651, 55mulcomd 9634 . . . 4  |-  ( ( ( V  e.  Fin  /\ 
<. V ,  E >. RegUSGrph  K
)  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  ->  (
( # `  ( X F ( N  - 
2 ) ) )  x.  K )  =  ( K  x.  ( # `
 ( X F ( N  -  2 ) ) ) ) )
5748, 56eqtrd 2498 . . 3  |-  ( ( ( V  e.  Fin  /\ 
<. V ,  E >. RegUSGrph  K
)  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  ->  (
( # `  ( X F ( N  - 
2 ) ) )  x.  ( # `  ( <. V ,  E >. Neighbors  X
) ) )  =  ( K  x.  ( # `
 ( X F ( N  -  2 ) ) ) ) )
5836, 57eqtrd 2498 . 2  |-  ( ( ( V  e.  Fin  /\ 
<. V ,  E >. RegUSGrph  K
)  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  ->  ( # `
 ( ( X F ( N  - 
2 ) )  X.  ( <. V ,  E >. Neighbors  X ) ) )  =  ( K  x.  ( # `  ( X F ( N  - 
2 ) ) ) ) )
5917, 58eqtrd 2498 1  |-  ( ( ( V  e.  Fin  /\ 
<. V ,  E >. RegUSGrph  K
)  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  ->  ( # `
 ( X G N ) )  =  ( K  x.  ( # `
 ( X F ( N  -  2 ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1395   E.wex 1613    e. wcel 1819   A.wral 2807   {crab 2811   _Vcvv 3109   <.cop 4038   class class class wbr 4456    |-> cmpt 4515    X. cxp 5006   -1-1-onto->wf1o 5593   ` cfv 5594  (class class class)co 6296    |-> cmpt2 6298   Fincfn 7535   CCcc 9507   0cc0 9509    x. cmul 9514    - cmin 9824   2c2 10606   3c3 10607   NN0cn0 10816   ZZ>=cuz 11106   #chash 12408   USGrph cusg 24457   Neighbors cnbgra 24544   ClWWalksN cclwwlkn 24876   RegUSGrph crusgra 25050
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  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-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-rp 11246  df-xadd 11344  df-fz 11698  df-fzo 11822  df-seq 12111  df-exp 12170  df-hash 12409  df-word 12546  df-lsw 12547  df-concat 12548  df-s1 12549  df-substr 12550  df-s2 12825  df-usgra 24460  df-nbgra 24547  df-clwwlk 24878  df-clwwlkn 24879  df-vdgr 25021  df-rgra 25051  df-rusgra 25052
This theorem is referenced by:  numclwwlk3  25236
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