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Theorem wwlkextsur 24858
 Description: Lemma 3 for wwlkextbij 24860. (Contributed by Alexander van der Vekens, 7-Aug-2018.)
Hypotheses
Ref Expression
wwlkextbij.d Word substr lastS lastS
wwlkextbij.r lastS
wwlkextbij.f lastS
Assertion
Ref Expression
wwlkextsur WWalksN
Distinct variable groups:   ,   ,,   ,,   ,   ,,,   ,,,
Allowed substitution hints:   (,)   (,)   ()   (,,)   ()

Proof of Theorem wwlkextsur
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wwlknprop 24813 . . 3 WWalksN Word
2 simprl 756 . . 3 Word
3 wwlkextbij.d . . . 4 Word substr lastS lastS
4 wwlkextbij.r . . . 4 lastS
5 wwlkextbij.f . . . 4 lastS
63, 4, 5wwlkextfun 24856 . . 3
71, 2, 63syl 20 . 2 WWalksN
8 preq2 4112 . . . . . 6 lastS lastS
98eleq1d 2526 . . . . 5 lastS lastS
109, 4elrab2 3259 . . . 4 lastS
11 wwlknext 24851 . . . . . . . . . . 11 WWalksN lastS ++ WWalksN
12113expb 1197 . . . . . . . . . 10 WWalksN lastS ++ WWalksN
13 wwlknimp 24814 . . . . . . . . . . . . . 14 WWalksN Word ..^
14 s1cl 12623 . . . . . . . . . . . . . . . . . . 19 Word
15 swrdccat1 12694 . . . . . . . . . . . . . . . . . . 19 Word Word ++ substr
1614, 15sylan2 474 . . . . . . . . . . . . . . . . . 18 Word ++ substr
1716ex 434 . . . . . . . . . . . . . . . . 17 Word ++ substr
1817adantr 465 . . . . . . . . . . . . . . . 16 Word ++ substr
19 opeq2 4220 . . . . . . . . . . . . . . . . . . . 20
2019eqcoms 2469 . . . . . . . . . . . . . . . . . . 19
2120oveq2d 6312 . . . . . . . . . . . . . . . . . 18 ++ substr ++ substr
2221eqeq1d 2459 . . . . . . . . . . . . . . . . 17 ++ substr ++ substr
2322adantl 466 . . . . . . . . . . . . . . . 16 Word ++ substr ++ substr
2418, 23sylibrd 234 . . . . . . . . . . . . . . 15 Word ++ substr
25243adant3 1016 . . . . . . . . . . . . . 14 Word ..^ ++ substr
2613, 25syl 16 . . . . . . . . . . . . 13 WWalksN ++ substr
2726com12 31 . . . . . . . . . . . 12 WWalksN ++ substr
2827adantr 465 . . . . . . . . . . 11 lastS WWalksN ++ substr
2928impcom 430 . . . . . . . . . 10 WWalksN lastS ++ substr
30 lswccats1 12647 . . . . . . . . . . . . . . . . . . . 20 Word lastS ++
3130eqcomd 2465 . . . . . . . . . . . . . . . . . . 19 Word lastS ++
3231ex 434 . . . . . . . . . . . . . . . . . 18 Word lastS ++
3332adantl 466 . . . . . . . . . . . . . . . . 17 Word lastS ++
3433adantl 466 . . . . . . . . . . . . . . . 16 Word lastS ++
351, 34syl 16 . . . . . . . . . . . . . . 15 WWalksN lastS ++
3635imp 429 . . . . . . . . . . . . . 14 WWalksN lastS ++
3736preq2d 4118 . . . . . . . . . . . . 13 WWalksN lastS lastS lastS ++
3837eleq1d 2526 . . . . . . . . . . . 12 WWalksN lastS lastS lastS ++
3938biimpd 207 . . . . . . . . . . 11 WWalksN lastS lastS lastS ++
4039impr 619 . . . . . . . . . 10 WWalksN lastS lastS lastS ++
4112, 29, 40jca32 535 . . . . . . . . 9 WWalksN lastS ++ WWalksN ++ substr lastS lastS ++
4235com12 31 . . . . . . . . . . 11 WWalksN lastS ++
4342adantr 465 . . . . . . . . . 10 lastS WWalksN lastS ++
4443impcom 430 . . . . . . . . 9 WWalksN lastS lastS ++
45 ovex 6324 . . . . . . . . . . 11 ++
4645a1i 11 . . . . . . . . . 10 WWalksN lastS ++
47 eleq1 2529 . . . . . . . . . . . . . . 15 ++ WWalksN ++ WWalksN
48 oveq1 6303 . . . . . . . . . . . . . . . . 17 ++ substr ++ substr
4948eqeq1d 2459 . . . . . . . . . . . . . . . 16 ++ substr ++ substr
50 fveq2 5872 . . . . . . . . . . . . . . . . . 18 ++ lastS lastS ++
5150preq2d 4118 . . . . . . . . . . . . . . . . 17 ++ lastS lastS lastS lastS ++
5251eleq1d 2526 . . . . . . . . . . . . . . . 16 ++ lastS lastS lastS lastS ++
5349, 52anbi12d 710 . . . . . . . . . . . . . . 15 ++ substr lastS lastS ++ substr lastS lastS ++
5447, 53anbi12d 710 . . . . . . . . . . . . . 14 ++ WWalksN substr lastS lastS ++ WWalksN ++ substr lastS lastS ++
5550eqeq2d 2471 . . . . . . . . . . . . . 14 ++ lastS lastS ++
5654, 55anbi12d 710 . . . . . . . . . . . . 13 ++ WWalksN substr lastS lastS lastS ++ WWalksN ++ substr lastS lastS ++ lastS ++
5756bicomd 201 . . . . . . . . . . . 12 ++ ++ WWalksN ++ substr lastS lastS ++ lastS ++ WWalksN substr lastS lastS lastS
5857adantl 466 . . . . . . . . . . 11 WWalksN lastS ++ ++ WWalksN ++ substr lastS lastS ++ lastS ++ WWalksN substr lastS lastS lastS
5958biimpd 207 . . . . . . . . . 10 WWalksN lastS ++ ++ WWalksN ++ substr lastS lastS ++ lastS ++ WWalksN substr lastS lastS lastS
6046, 59spcimedv 3193 . . . . . . . . 9 WWalksN lastS ++ WWalksN ++ substr lastS lastS ++ lastS ++ WWalksN substr lastS lastS lastS
6141, 44, 60mp2and 679 . . . . . . . 8 WWalksN lastS WWalksN substr lastS lastS lastS
62 oveq1 6303 . . . . . . . . . . . . 13 substr substr
6362eqeq1d 2459 . . . . . . . . . . . 12 substr substr
64 fveq2 5872 . . . . . . . . . . . . . 14 lastS lastS
6564preq2d 4118 . . . . . . . . . . . . 13 lastS lastS lastS lastS
6665eleq1d 2526 . . . . . . . . . . . 12 lastS lastS lastS lastS
6763, 66anbi12d 710 . . . . . . . . . . 11 substr lastS lastS substr lastS lastS
6867elrab 3257 . . . . . . . . . 10 WWalksN substr lastS lastS WWalksN substr lastS lastS
6968anbi1i 695 . . . . . . . . 9 WWalksN substr lastS lastS lastS WWalksN substr lastS lastS lastS
7069exbii 1668 . . . . . . . 8 WWalksN substr lastS lastS lastS WWalksN substr lastS lastS lastS
7161, 70sylibr 212 . . . . . . 7 WWalksN lastS WWalksN substr lastS lastS lastS
72 df-rex 2813 . . . . . . 7 WWalksN substr lastS lastS lastS WWalksN substr lastS lastS lastS
7371, 72sylibr 212 . . . . . 6 WWalksN lastS WWalksN substr lastS lastS lastS
743wwlkextwrd 24855 . . . . . . . 8 WWalksN WWalksN substr lastS lastS
7574adantr 465 . . . . . . 7 WWalksN lastS WWalksN substr lastS lastS
7675rexeqdv 3061 . . . . . 6 WWalksN lastS lastS WWalksN substr lastS lastS lastS
7773, 76mpbird 232 . . . . 5 WWalksN lastS lastS
78 fveq2 5872 . . . . . . . 8 lastS lastS
79 fvex 5882 . . . . . . . 8 lastS
8078, 5, 79fvmpt 5956 . . . . . . 7 lastS
8180eqeq2d 2471 . . . . . 6 lastS
8281rexbiia 2958 . . . . 5 lastS
8377, 82sylibr 212 . . . 4 WWalksN lastS
8410, 83sylan2b 475 . . 3 WWalksN
8584ralrimiva 2871 . 2 WWalksN
86 dffo3 6047 . 2
877, 85, 86sylanbrc 664 1 WWalksN
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 369   w3a 973   wceq 1395  wex 1613   wcel 1819  wral 2807  wrex 2808  crab 2811  cvv 3109  cpr 4034  cop 4038   cmpt 4515   crn 5009  wf 5590  wfo 5592  cfv 5594  (class class class)co 6296  cc0 9509  c1 9510   caddc 9512  c2 10606  cn0 10816  ..^cfzo 11821  chash 12408  Word cword 12538   lastS clsw 12539   ++ cconcat 12540  cs1 12541   substr csubstr 12542   WWalksN cwwlkn 24805 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-n0 10817  df-z 10886  df-uz 11107  df-rp 11246  df-fz 11698  df-fzo 11822  df-hash 12409  df-word 12546  df-lsw 12547  df-concat 12548  df-s1 12549  df-substr 12550  df-wwlk 24806  df-wwlkn 24807 This theorem is referenced by:  wwlkextbij0  24859
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