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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 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 ClWWalksN
numclwwlk.f
numclwwlk.g
Assertion
Ref Expression
numclwwlk1 RegUSGrph
Distinct variable groups:   ,   ,   ,   ,   ,   ,,,   ,   ,,,   ,   ,   ,   ,   ,   ,
Allowed substitution hints:   ()   (,)   (,)   (,)

Proof of Theorem numclwwlk1
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovex 6324 . . . 4
2 ovex 6324 . . . . 5
3 ovex 6324 . . . . 5 Neighbors
42, 3xpex 6603 . . . 4 Neighbors
51, 4pm3.2i 455 . . 3 Neighbors
6 rusisusgra 25058 . . . . 5 RegUSGrph USGrph
76ad2antlr 726 . . . 4 RegUSGrph USGrph
8 simprl 756 . . . 4 RegUSGrph
9 simpr 461 . . . . 5
109adantl 466 . . . 4 RegUSGrph
11 numclwwlk.c . . . . 5 ClWWalksN
12 numclwwlk.f . . . . 5
13 numclwwlk.g . . . . 5
1411, 12, 13numclwlk1lem2 25224 . . . 4 USGrph Neighbors
157, 8, 10, 14syl3anc 1228 . . 3 RegUSGrph Neighbors
16 hasheqf1oi 12427 . . 3 Neighbors Neighbors Neighbors
175, 15, 16mpsyl 63 . 2 RegUSGrph Neighbors
18 usgrav 24465 . . . . . . 7 USGrph
19 simpr 461 . . . . . . 7
206, 18, 193syl 20 . . . . . 6 RegUSGrph
2120anim2i 569 . . . . 5 RegUSGrph
22 uzuzle23 11146 . . . . . . 7
23 uznn0sub 11137 . . . . . . 7
2422, 23syl 16 . . . . . 6
2524anim2i 569 . . . . 5
2611, 12numclwwlkffin 25209 . . . . 5
2721, 25, 26syl2an 477 . . . 4 RegUSGrph
286anim1i 568 . . . . . . . 8 RegUSGrph USGrph
2928ancoms 453 . . . . . . 7 RegUSGrph USGrph
30 usgrafis 24542 . . . . . . 7 USGrph
3129, 30syl 16 . . . . . 6 RegUSGrph
3231adantr 465 . . . . 5 RegUSGrph
33 nbusgrafi 24575 . . . . 5 USGrph Neighbors
347, 8, 32, 33syl3anc 1228 . . . 4 RegUSGrph Neighbors
35 hashxp 12496 . . . 4 Neighbors Neighbors Neighbors
3627, 34, 35syl2anc 661 . . 3 RegUSGrph Neighbors Neighbors
37 rusgraprop2 25069 . . . . . . . . . 10 RegUSGrph USGrph Neighbors
38 oveq2 6304 . . . . . . . . . . . . . 14 Neighbors Neighbors
3938fveq2d 5876 . . . . . . . . . . . . 13 Neighbors Neighbors
4039eqeq1d 2459 . . . . . . . . . . . 12 Neighbors Neighbors
4140rspccv 3207 . . . . . . . . . . 11 Neighbors Neighbors
42413ad2ant3 1019 . . . . . . . . . 10 USGrph Neighbors Neighbors
4337, 42syl 16 . . . . . . . . 9 RegUSGrph Neighbors
4443adantl 466 . . . . . . . 8 RegUSGrph Neighbors
4544com12 31 . . . . . . 7 RegUSGrph Neighbors
4645adantr 465 . . . . . 6 RegUSGrph Neighbors
4746impcom 430 . . . . 5 RegUSGrph Neighbors
4847oveq2d 6312 . . . 4 RegUSGrph Neighbors
49 hashcl 12431 . . . . . 6
50 nn0cn 10826 . . . . . 6
5127, 49, 503syl 20 . . . . 5 RegUSGrph
52 nn0cn 10826 . . . . . . . 8
53523ad2ant2 1018 . . . . . . 7 USGrph Neighbors
5437, 53syl 16 . . . . . 6 RegUSGrph
5554ad2antlr 726 . . . . 5 RegUSGrph
5651, 55mulcomd 9634 . . . 4 RegUSGrph
5748, 56eqtrd 2498 . . 3 RegUSGrph Neighbors
5836, 57eqtrd 2498 . 2 RegUSGrph Neighbors
5917, 58eqtrd 2498 1 RegUSGrph
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369   w3a 973   wceq 1395  wex 1613   wcel 1819  wral 2807  crab 2811  cvv 3109  cop 4038   class class class wbr 4456   cmpt 4515   cxp 5006  wf1o 5593  cfv 5594  (class class class)co 6296   cmpt2 6298  cfn 7535  cc 9507  cc0 9509   cmul 9514   cmin 9824  c2 10606  c3 10607  cn0 10816  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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