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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 ClWWalksN
numclwwlk.f
numclwwlk.g
numclwwlk.q WWalksN lastS
numclwwlk.h
Assertion
Ref Expression
numclwwlk2 RegUSGrph FriendGrph
Distinct variable groups:   ,   ,   ,   ,   ,   ,,,   ,   ,,,   ,   ,   ,   ,   ,   ,   ,   ,   ,,
Allowed substitution hints:   (,)   (,)   (,)   ()   (,)

Proof of Theorem numclwwlk2
StepHypRef Expression
1 eluzelcn 10984 . . . . . . . 8
2 2cnd 10506 . . . . . . . 8
31, 2npcand 9835 . . . . . . 7
43eqcomd 2462 . . . . . 6
543ad2ant3 1011 . . . . 5
65adantl 466 . . . 4 RegUSGrph FriendGrph
76oveq2d 6217 . . 3 RegUSGrph FriendGrph
87fveq2d 5804 . 2 RegUSGrph FriendGrph
9 simplr 754 . . 3 RegUSGrph FriendGrph FriendGrph
10 simp2 989 . . . 4
1110adantl 466 . . 3 RegUSGrph FriendGrph
12 uz3m2nn 30344 . . . . 5
13123ad2ant3 1011 . . . 4
1413adantl 466 . . 3 RegUSGrph FriendGrph
15 numclwwlk.c . . . 4 ClWWalksN
16 numclwwlk.f . . . 4
17 numclwwlk.g . . . 4
18 numclwwlk.q . . . 4 WWalksN lastS
19 numclwwlk.h . . . 4
2015, 16, 17, 18, 19numclwwlk2lem3 30848 . . 3 FriendGrph
219, 11, 14, 20syl3anc 1219 . 2 RegUSGrph FriendGrph
22 simpl 457 . . . 4 RegUSGrph FriendGrph RegUSGrph
23 simp1 988 . . . 4
2422, 23anim12i 566 . . 3 RegUSGrph FriendGrph RegUSGrph
2510, 13jca 532 . . . 4
2625adantl 466 . . 3 RegUSGrph FriendGrph
2715, 16, 17, 18numclwwlkqhash 30842 . . 3 RegUSGrph
2824, 26, 27syl2anc 661 . 2 RegUSGrph FriendGrph
298, 21, 283eqtr2d 2501 1 RegUSGrph FriendGrph
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369   w3a 965   wceq 1370   wcel 1758   wne 2648  crab 2803  cop 3992   class class class wbr 4401   cmpt 4459  cfv 5527  (class class class)co 6201   cmpt2 6203  cfn 7421  cc0 9394   caddc 9397   cmin 9707  cn 10434  c2 10483  c3 10484  cn0 10691  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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