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Theorem extwwlkfab 25217
 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 24898 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 11146 . . . 4
2 numclwwlk.c . . . . 5 ClWWalksN
3 numclwwlk.f . . . . 5
4 numclwwlk.g . . . . 5
52, 3, 4numclwwlkovg 25214 . . . 4
61, 5sylan2 474 . . 3
8 eluzge2nn0 11145 . . . . . . . . 9
92numclwwlkfvc 25204 . . . . . . . . 9 ClWWalksN
101, 8, 93syl 20 . . . . . . . 8 ClWWalksN
11103ad2ant3 1019 . . . . . . 7 USGrph ClWWalksN
1211eleq2d 2527 . . . . . 6 USGrph ClWWalksN
132extwwlkfablem2 25205 . . . . . . . . . . . 12 USGrph ClWWalksN substr
14 simpl 457 . . . . . . . . . . . . 13
1514adantl 466 . . . . . . . . . . . 12 USGrph ClWWalksN
1613, 15jca 532 . . . . . . . . . . 11 USGrph ClWWalksN substr
1713anim3i 1184 . . . . . . . . . . . 12 USGrph USGrph
18 extwwlkfablem1 25201 . . . . . . . . . . . 12 USGrph ClWWalksN Neighbors
1917, 18sylanl1 650 . . . . . . . . . . 11 USGrph ClWWalksN Neighbors
20 simpr 461 . . . . . . . . . . . . 13
2120, 14eqtrd 2498 . . . . . . . . . . . 12
2221adantl 466 . . . . . . . . . . 11 USGrph ClWWalksN
2316, 19, 223jca 1176 . . . . . . . . . 10 USGrph ClWWalksN substr Neighbors
2423ex 434 . . . . . . . . 9 USGrph ClWWalksN substr Neighbors
25 simpl 457 . . . . . . . . . . . . . 14
26 simpr 461 . . . . . . . . . . . . . . 15
2725eqcomd 2465 . . . . . . . . . . . . . . 15
2826, 27eqtrd 2498 . . . . . . . . . . . . . 14
2925, 28jca 532 . . . . . . . . . . . . 13
3029ex 434 . . . . . . . . . . . 12
3130a1d 25 . . . . . . . . . . 11 Neighbors
3231adantl 466 . . . . . . . . . 10 substr Neighbors
33323imp 1190 . . . . . . . . 9 substr Neighbors
3424, 33impbid1 203 . . . . . . . 8 USGrph ClWWalksN substr Neighbors
35 clwwlknimp 24903 . . . . . . . . . . . . . . . . 17 ClWWalksN Word ..^ lastS
36 ige3m2fz 11734 . . . . . . . . . . . . . . . . . . . . 21
37 oveq2 6304 . . . . . . . . . . . . . . . . . . . . . 22
3837eleq2d 2527 . . . . . . . . . . . . . . . . . . . . 21
3936, 38syl5ibr 221 . . . . . . . . . . . . . . . . . . . 20
4039adantl 466 . . . . . . . . . . . . . . . . . . 19 Word
41 simpl 457 . . . . . . . . . . . . . . . . . . 19 Word Word
4240, 41jctild 543 . . . . . . . . . . . . . . . . . 18 Word Word
43423ad2ant1 1017 . . . . . . . . . . . . . . . . 17 Word ..^ lastS Word
4435, 43syl 16 . . . . . . . . . . . . . . . 16 ClWWalksN Word
4544com12 31 . . . . . . . . . . . . . . 15 ClWWalksN Word
46453ad2ant3 1019 . . . . . . . . . . . . . 14 USGrph ClWWalksN Word
4746imp 429 . . . . . . . . . . . . 13 USGrph ClWWalksN Word
48 swrd0fv0 12676 . . . . . . . . . . . . 13 Word substr
4947, 48syl 16 . . . . . . . . . . . 12 USGrph ClWWalksN substr
5049eqcomd 2465 . . . . . . . . . . 11 USGrph ClWWalksN substr
5150eqeq1d 2459 . . . . . . . . . 10 USGrph ClWWalksN substr
5251anbi2d 703 . . . . . . . . 9 USGrph ClWWalksN substr substr substr
53523anbi1d 1303 . . . . . . . 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 11137 . . . . . . . . . . 11
591, 58syl 16 . . . . . . . . . 10
602, 3numclwwlkovf 25208 . . . . . . . . . 10
6159, 60sylan2 474 . . . . . . . . 9
62613adant1 1014 . . . . . . . 8 USGrph
6362adantr 465 . . . . . . 7 USGrph
6463eleq2d 2527 . . . . . 6 USGrph substr substr
65 fveq1 5871 . . . . . . . 8 substr substr
6665eqeq1d 2459 . . . . . . 7 substr substr
67 fveq1 5871 . . . . . . . . 9
6867eqeq1d 2459 . . . . . . . 8
6968cbvrabv 3108 . . . . . . 7
7066, 69elrab2 3259 . . . . . 6 substr substr substr
7164, 70syl6rbb 262 . . . . 5 USGrph substr substr substr
72713anbi1d 1303 . . . 4 USGrph substr substr Neighbors substr Neighbors
7357, 72bitrd 253 . . 3 USGrph substr Neighbors
7473rabbidva 3100 . 2 USGrph substr Neighbors
757, 74eqtrd 2498 1 USGrph substr Neighbors
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 369   w3a 973   wceq 1395   wcel 1819  wral 2807  crab 2811  cpr 4034  cop 4038   class class class wbr 4456   cmpt 4515   crn 5009  cfv 5594  (class class class)co 6296   cmpt2 6298  cc0 9509  c1 9510   caddc 9512   cmin 9824  c2 10606  c3 10607  cn0 10816  cuz 11106  cfz 11697  ..^cfzo 11821  chash 12408  Word cword 12538   lastS clsw 12539   substr csubstr 12542   USGrph cusg 24457   Neighbors cnbgra 24544   ClWWalksN cclwwlkn 24876 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-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-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-fz 11698  df-fzo 11822  df-hash 12409  df-word 12546  df-lsw 12547  df-substr 12550  df-usgra 24460  df-nbgra 24547  df-clwwlk 24878  df-clwwlkn 24879 This theorem is referenced by:  numclwlk1lem2foa  25218  numclwlk1lem2f  25219
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