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Theorem extwwlkfab 30832
 Description: The set of closed walks (having a fixed length greater than 1 and starting at a fixed vertex) with the last but 2 vertex is identical with the first (and therefore last) vertex can be constructed from the set of closed walks with length smaller by 2 than the fixed length appending a neighbor of the last vertex and afterwards the last vertex (which is the first vertex) itself ("walking forth and back" from the last vertex). is required since for : , see clwwlkgt0 30583 stating that a walk of length 0 is not represented as word, at least not for an undirected simple graph.) (Contributed by Alexander van der Vekens, 18-Sep-2018.)
Hypotheses
Ref Expression
numclwwlk.c ClWWalksN
numclwwlk.f
numclwwlk.g
Assertion
Ref Expression
extwwlkfab USGrph substr Neighbors
Distinct variable groups:   ,   ,   ,   ,   ,   ,,,   ,   ,,,   ,   ,   ,   ,
Allowed substitution hints:   ()   (,)   (,,)

Proof of Theorem extwwlkfab
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 uzuzle23 30342 . . . 4
2 numclwwlk.c . . . . 5 ClWWalksN
3 numclwwlk.f . . . . 5
4 numclwwlk.g . . . . 5
52, 3, 4numclwwlkovg 30829 . . . 4
61, 5sylan2 474 . . 3
8 eluzge2nn0 30341 . . . . . . . . 9
92numclwwlkfvc 30819 . . . . . . . . 9 ClWWalksN
101, 8, 93syl 20 . . . . . . . 8 ClWWalksN
11103ad2ant3 1011 . . . . . . 7 USGrph ClWWalksN
1211eleq2d 2524 . . . . . 6 USGrph ClWWalksN
132extwwlkfablem2 30820 . . . . . . . . . . . 12 USGrph ClWWalksN substr
14 simpl 457 . . . . . . . . . . . . 13
1514adantl 466 . . . . . . . . . . . 12 USGrph ClWWalksN
1613, 15jca 532 . . . . . . . . . . 11 USGrph ClWWalksN substr
1713anim3i 1176 . . . . . . . . . . . 12 USGrph USGrph
18 extwwlkfablem1 30816 . . . . . . . . . . . 12 USGrph ClWWalksN Neighbors
1917, 18sylanl1 650 . . . . . . . . . . 11 USGrph ClWWalksN Neighbors
20 simpr 461 . . . . . . . . . . . . 13
2120, 14eqtrd 2495 . . . . . . . . . . . 12
2221adantl 466 . . . . . . . . . . 11 USGrph ClWWalksN
2316, 19, 223jca 1168 . . . . . . . . . 10 USGrph ClWWalksN substr Neighbors
2423ex 434 . . . . . . . . 9 USGrph ClWWalksN substr Neighbors
25 simpl 457 . . . . . . . . . . . . . 14
26 simpr 461 . . . . . . . . . . . . . . 15
2725eqcomd 2462 . . . . . . . . . . . . . . 15
2826, 27eqtrd 2495 . . . . . . . . . . . . . 14
2925, 28jca 532 . . . . . . . . . . . . 13
3029ex 434 . . . . . . . . . . . 12
3130a1d 25 . . . . . . . . . . 11 Neighbors
3231adantl 466 . . . . . . . . . 10 substr Neighbors
33323imp 1182 . . . . . . . . 9 substr Neighbors
3424, 33impbid1 203 . . . . . . . 8 USGrph ClWWalksN substr Neighbors
35 clwwlknimp 30588 . . . . . . . . . . . . . . . . 17 ClWWalksN Word ..^ lastS
36 ige3m2fz 30356 . . . . . . . . . . . . . . . . . . . . 21
37 oveq2 6209 . . . . . . . . . . . . . . . . . . . . . 22
3837eleq2d 2524 . . . . . . . . . . . . . . . . . . . . 21
3936, 38syl5ibr 221 . . . . . . . . . . . . . . . . . . . 20
4039adantl 466 . . . . . . . . . . . . . . . . . . 19 Word
41 simpl 457 . . . . . . . . . . . . . . . . . . 19 Word Word
4240, 41jctild 543 . . . . . . . . . . . . . . . . . 18 Word Word
43423ad2ant1 1009 . . . . . . . . . . . . . . . . 17 Word ..^ lastS Word
4435, 43syl 16 . . . . . . . . . . . . . . . 16 ClWWalksN Word
4544com12 31 . . . . . . . . . . . . . . 15 ClWWalksN Word
46453ad2ant3 1011 . . . . . . . . . . . . . 14 USGrph ClWWalksN Word
4746imp 429 . . . . . . . . . . . . 13 USGrph ClWWalksN Word
48 swrd0fv0 12455 . . . . . . . . . . . . 13 Word substr
4947, 48syl 16 . . . . . . . . . . . 12 USGrph ClWWalksN substr
5049eqcomd 2462 . . . . . . . . . . 11 USGrph ClWWalksN substr
5150eqeq1d 2456 . . . . . . . . . 10 USGrph ClWWalksN substr
5251anbi2d 703 . . . . . . . . 9 USGrph ClWWalksN substr substr substr
53523anbi1d 1294 . . . . . . . 8 USGrph ClWWalksN substr Neighbors substr substr Neighbors
5434, 53bitrd 253 . . . . . . 7 USGrph ClWWalksN substr substr Neighbors
5554ex 434 . . . . . 6 USGrph ClWWalksN substr substr Neighbors
5612, 55sylbid 215 . . . . 5 USGrph substr substr Neighbors
5756imp 429 . . . 4 USGrph substr substr Neighbors
58 uznn0sub 11004 . . . . . . . . . . 11
591, 58syl 16 . . . . . . . . . 10
602, 3numclwwlkovf 30823 . . . . . . . . . 10
6159, 60sylan2 474 . . . . . . . . 9
62613adant1 1006 . . . . . . . 8 USGrph
6362adantr 465 . . . . . . 7 USGrph
6463eleq2d 2524 . . . . . 6 USGrph substr substr
65 fveq1 5799 . . . . . . . 8 substr substr
6665eqeq1d 2456 . . . . . . 7 substr substr
67 fveq1 5799 . . . . . . . . 9
6867eqeq1d 2456 . . . . . . . 8
6968cbvrabv 3077 . . . . . . 7
7066, 69elrab2 3226 . . . . . 6 substr substr substr
7164, 70syl6rbb 262 . . . . 5 USGrph substr substr substr
72713anbi1d 1294 . . . 4 USGrph substr substr Neighbors substr Neighbors
7357, 72bitrd 253 . . 3 USGrph substr Neighbors
7473rabbidva 3069 . 2 USGrph substr Neighbors
757, 74eqtrd 2495 1 USGrph substr Neighbors
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 369   w3a 965   wceq 1370   wcel 1758  wral 2799  crab 2803  cpr 3988  cop 3992   class class class wbr 4401   cmpt 4459   crn 4950  cfv 5527  (class class class)co 6201   cmpt2 6203  cc0 9394  c1 9395   caddc 9397   cmin 9707  c2 10483  c3 10484  cn0 10691  cuz 10973  cfz 11555  ..^cfzo 11666  chash 12221  Word cword 12340   lastS clsw 12341   substr csubstr 12344   USGrph cusg 23417   Neighbors cnbgra 23482   ClWWalksN cclwwlkn 30563 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-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 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  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-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-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-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-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-oadd 7035  df-er 7212  df-map 7327  df-pm 7328  df-en 7422  df-dom 7423  df-sdom 7424  df-fin 7425  df-card 8221  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-nn 10435  df-2 10492  df-3 10493  df-n0 10692  df-z 10759  df-uz 10974  df-fz 11556  df-fzo 11667  df-hash 12222  df-word 12348  df-lsw 12349  df-substr 12352  df-usgra 23419  df-nbgra 23485  df-clwwlk 30565  df-clwwlkn 30566 This theorem is referenced by:  numclwlk1lem2foa  30833  numclwlk1lem2f  30834
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