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Theorem numclwwlk2 30849
Description: Huneke: "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 k^(n-2) - f(n-2)." According to rusgranumwlkg 30725, we have k^(n-2) different walks of length (n-2): v(0) ... v(n-2). From this number, the number of closed walks of length (n-2), which is f(n-2) per definition, must be subtracted, because for these walks v(n-2) =/= v(0) = v would hold. Because of the friendship condition, there is exactly one vertex v(n-1) which is a neighbor of v(n-2) as well as of v(n)=v=v(0), because v(n-2) and v(n)=v are different, so the number of walks v(0) ... v(n-2) is identical with the number of walks v(0) ... v(n), that means each (not closed) walk v(0) ... v(n-2) can be extended by two edges to a closed walk v(0) ... v(n)=v=v(0) in exactly one way. (Contributed by Alexander van der Vekens, 6-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 } )
numclwwlk.g  |-  G  =  ( v  e.  V ,  n  e.  ( ZZ>=
`  2 )  |->  { w  e.  ( C `
 n )  |  ( ( w ` 
0 )  =  v  /\  ( w `  ( n  -  2
) )  =  ( w `  0 ) ) } )
numclwwlk.q  |-  Q  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  (
( V WWalksN  E ) `  n )  |  ( ( w `  0
)  =  v  /\  ( lastS  `  w )  =/=  v ) } )
numclwwlk.h  |-  H  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( C `  n )  |  ( ( w `
 0 )  =  v  /\  ( w `
 ( n  - 
2 ) )  =/=  ( w `  0
) ) } )
Assertion
Ref Expression
numclwwlk2  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E )  /\  ( V  e.  Fin  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) ) )  ->  ( # `  ( X H N ) )  =  ( ( K ^ ( N  - 
2 ) )  -  ( # `  ( 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, Q    w, K    w, G    v, E    v, H, w
Allowed substitution hints:    Q( v, n)    F( v, n)    G( v, n)    H( n)    K( v, n)

Proof of Theorem numclwwlk2
StepHypRef Expression
1 eluzelcn 10984 . . . . . . . 8  |-  ( N  e.  ( ZZ>= `  3
)  ->  N  e.  CC )
2 2cnd 10506 . . . . . . . 8  |-  ( N  e.  ( ZZ>= `  3
)  ->  2  e.  CC )
31, 2npcand 9835 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  3
)  ->  ( ( N  -  2 )  +  2 )  =  N )
43eqcomd 2462 . . . . . 6  |-  ( N  e.  ( ZZ>= `  3
)  ->  N  =  ( ( N  - 
2 )  +  2 ) )
543ad2ant3 1011 . . . . 5  |-  ( ( V  e.  Fin  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  ->  N  =  ( ( N  -  2 )  +  2 ) )
65adantl 466 . . . 4  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E )  /\  ( V  e.  Fin  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) ) )  ->  N  =  ( ( N  -  2 )  +  2 ) )
76oveq2d 6217 . . 3  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E )  /\  ( V  e.  Fin  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) ) )  ->  ( X H N )  =  ( X H ( ( N  -  2 )  +  2 ) ) )
87fveq2d 5804 . 2  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E )  /\  ( V  e.  Fin  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) ) )  ->  ( # `  ( X H N ) )  =  ( # `  ( X H ( ( N  -  2 )  +  2 ) ) ) )
9 simplr 754 . . 3  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E )  /\  ( V  e.  Fin  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) ) )  ->  V FriendGrph  E )
10 simp2 989 . . . 4  |-  ( ( V  e.  Fin  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  ->  X  e.  V )
1110adantl 466 . . 3  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E )  /\  ( V  e.  Fin  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) ) )  ->  X  e.  V
)
12 uz3m2nn 30344 . . . . 5  |-  ( N  e.  ( ZZ>= `  3
)  ->  ( N  -  2 )  e.  NN )
13123ad2ant3 1011 . . . 4  |-  ( ( V  e.  Fin  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  -> 
( N  -  2 )  e.  NN )
1413adantl 466 . . 3  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E )  /\  ( V  e.  Fin  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) ) )  ->  ( N  - 
2 )  e.  NN )
15 numclwwlk.c . . . 4  |-  C  =  ( n  e.  NN0  |->  ( ( V ClWWalksN  E ) `
 n ) )
16 numclwwlk.f . . . 4  |-  F  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( C `  n )  |  ( w ` 
0 )  =  v } )
17 numclwwlk.g . . . 4  |-  G  =  ( v  e.  V ,  n  e.  ( ZZ>=
`  2 )  |->  { w  e.  ( C `
 n )  |  ( ( w ` 
0 )  =  v  /\  ( w `  ( n  -  2
) )  =  ( w `  0 ) ) } )
18 numclwwlk.q . . . 4  |-  Q  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  (
( V WWalksN  E ) `  n )  |  ( ( w `  0
)  =  v  /\  ( lastS  `  w )  =/=  v ) } )
19 numclwwlk.h . . . 4  |-  H  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( C `  n )  |  ( ( w `
 0 )  =  v  /\  ( w `
 ( n  - 
2 ) )  =/=  ( w `  0
) ) } )
2015, 16, 17, 18, 19numclwwlk2lem3 30848 . . 3  |-  ( ( V FriendGrph  E  /\  X  e.  V  /\  ( N  -  2 )  e.  NN )  ->  ( # `
 ( X Q ( N  -  2 ) ) )  =  ( # `  ( X H ( ( N  -  2 )  +  2 ) ) ) )
219, 11, 14, 20syl3anc 1219 . 2  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E )  /\  ( V  e.  Fin  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) ) )  ->  ( # `  ( X Q ( N  - 
2 ) ) )  =  ( # `  ( X H ( ( N  -  2 )  +  2 ) ) ) )
22 simpl 457 . . . 4  |-  ( (
<. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E )  ->  <. V ,  E >. RegUSGrph  K
)
23 simp1 988 . . . 4  |-  ( ( V  e.  Fin  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  ->  V  e.  Fin )
2422, 23anim12i 566 . . 3  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E )  /\  ( V  e.  Fin  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) ) )  ->  ( <. V ,  E >. RegUSGrph  K  /\  V  e. 
Fin ) )
2510, 13jca 532 . . . 4  |-  ( ( V  e.  Fin  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  -> 
( X  e.  V  /\  ( N  -  2 )  e.  NN ) )
2625adantl 466 . . 3  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E )  /\  ( V  e.  Fin  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) ) )  ->  ( X  e.  V  /\  ( N  -  2 )  e.  NN ) )
2715, 16, 17, 18numclwwlkqhash 30842 . . 3  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V  e.  Fin )  /\  ( X  e.  V  /\  ( N  -  2 )  e.  NN ) )  -> 
( # `  ( X Q ( N  - 
2 ) ) )  =  ( ( K ^ ( N  - 
2 ) )  -  ( # `  ( X F ( N  - 
2 ) ) ) ) )
2824, 26, 27syl2anc 661 . 2  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E )  /\  ( V  e.  Fin  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) ) )  ->  ( # `  ( X Q ( N  - 
2 ) ) )  =  ( ( K ^ ( N  - 
2 ) )  -  ( # `  ( X F ( N  - 
2 ) ) ) ) )
298, 21, 283eqtr2d 2501 1  |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E )  /\  ( V  e.  Fin  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) ) )  ->  ( # `  ( X H N ) )  =  ( ( K ^ ( N  - 
2 ) )  -  ( # `  ( X F ( N  - 
2 ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2648   {crab 2803   <.cop 3992   class class class wbr 4401    |-> cmpt 4459   ` cfv 5527  (class class class)co 6201    |-> cmpt2 6203   Fincfn 7421   0cc0 9394    + caddc 9397    - cmin 9707   NNcn 10434   2c2 10483   3c3 10484   NN0cn0 10691   ZZ>=cuz 10973   ^cexp 11983   #chash 12221   lastS clsw 12341   WWalksN cwwlkn 30461   ClWWalksN cclwwlkn 30563   RegUSGrph crusgra 30689   FriendGrph cfrgra 30729
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-inf2 7959  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471  ax-pre-sup 9472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-disj 4372  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-se 4789  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-isom 5536  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-1st 6688  df-2nd 6689  df-recs 6943  df-rdg 6977  df-1o 7031  df-2o 7032  df-oadd 7035  df-er 7212  df-map 7327  df-pm 7328  df-en 7422  df-dom 7423  df-sdom 7424  df-fin 7425  df-sup 7803  df-oi 7836  df-card 8221  df-cda 8449  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-div 10106  df-nn 10435  df-2 10492  df-3 10493  df-n0 10692  df-z 10759  df-uz 10974  df-rp 11104  df-xadd 11202  df-fz 11556  df-fzo 11667  df-seq 11925  df-exp 11984  df-hash 12222  df-word 12348  df-lsw 12349  df-concat 12350  df-s1 12351  df-substr 12352  df-cj 12707  df-re 12708  df-im 12709  df-sqr 12843  df-abs 12844  df-clim 13085  df-sum 13283  df-usgra 23419  df-nbgra 23485  df-wlk 23568  df-vdgr 23717  df-wwlk 30462  df-wwlkn 30463  df-clwwlk 30565  df-clwwlkn 30566  df-rgra 30690  df-rusgra 30691  df-frgra 30730
This theorem is referenced by:  numclwwlk3  30851
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