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Theorem clwwlkextfrlem1 30578
Description: Lemma for numclwwlk2lem1 30604. (Contributed by Alexander van der Vekens, 3-Oct-2018.)
Assertion
Ref Expression
clwwlkextfrlem1  |-  ( ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  /\  ( W  e.  ( ( V WWalksN  E
) `  N )  /\  ( W `  0
)  =  X  /\  ( lastS  `  W )  =/= 
X ) )  -> 
( ( ( W concat  <" Z "> ) `  0 )  =  X  /\  (
( W concat  <" Z "> ) `  N
)  =/=  X ) )

Proof of Theorem clwwlkextfrlem1
StepHypRef Expression
1 wwlknimpb 30247 . . . 4  |-  ( W  e.  ( ( V WWalksN  E ) `  N
)  ->  ( W  e. Word  V  /\  ( # `  W )  =  ( N  +  1 ) ) )
2 simprll 756 . . . . . . . . 9  |-  ( ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  /\  ( ( W  e. Word  V  /\  ( # `
 W )  =  ( N  +  1 ) )  /\  (
( W `  0
)  =  X  /\  ( lastS  `  W )  =/= 
X ) ) )  ->  W  e. Word  V
)
3 s1cl 12289 . . . . . . . . . . 11  |-  ( Z  e.  V  ->  <" Z ">  e. Word  V )
433ad2ant3 1006 . . . . . . . . . 10  |-  ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  ->  <" Z ">  e. Word  V )
54adantr 462 . . . . . . . . 9  |-  ( ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  /\  ( ( W  e. Word  V  /\  ( # `
 W )  =  ( N  +  1 ) )  /\  (
( W `  0
)  =  X  /\  ( lastS  `  W )  =/= 
X ) ) )  ->  <" Z ">  e. Word  V )
6 nnnn0 10582 . . . . . . . . . . . . 13  |-  ( N  e.  NN  ->  N  e.  NN0 )
7 nn0p1gt0 10605 . . . . . . . . . . . . 13  |-  ( N  e.  NN0  ->  0  < 
( N  +  1 ) )
86, 7syl 16 . . . . . . . . . . . 12  |-  ( N  e.  NN  ->  0  <  ( N  +  1 ) )
983ad2ant2 1005 . . . . . . . . . . 11  |-  ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  ->  0  <  ( N  +  1 ) )
109adantr 462 . . . . . . . . . 10  |-  ( ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  /\  ( ( W  e. Word  V  /\  ( # `
 W )  =  ( N  +  1 ) )  /\  (
( W `  0
)  =  X  /\  ( lastS  `  W )  =/= 
X ) ) )  ->  0  <  ( N  +  1 ) )
11 breq2 4293 . . . . . . . . . . . 12  |-  ( (
# `  W )  =  ( N  + 
1 )  ->  (
0  <  ( # `  W
)  <->  0  <  ( N  +  1 ) ) )
1211adantl 463 . . . . . . . . . . 11  |-  ( ( W  e. Word  V  /\  ( # `  W )  =  ( N  + 
1 ) )  -> 
( 0  <  ( # `
 W )  <->  0  <  ( N  +  1 ) ) )
1312ad2antrl 722 . . . . . . . . . 10  |-  ( ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  /\  ( ( W  e. Word  V  /\  ( # `
 W )  =  ( N  +  1 ) )  /\  (
( W `  0
)  =  X  /\  ( lastS  `  W )  =/= 
X ) ) )  ->  ( 0  < 
( # `  W )  <->  0  <  ( N  +  1 ) ) )
1410, 13mpbird 232 . . . . . . . . 9  |-  ( ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  /\  ( ( W  e. Word  V  /\  ( # `
 W )  =  ( N  +  1 ) )  /\  (
( W `  0
)  =  X  /\  ( lastS  `  W )  =/= 
X ) ) )  ->  0  <  ( # `
 W ) )
15 ccatfv0 12278 . . . . . . . . 9  |-  ( ( W  e. Word  V  /\  <" Z ">  e. Word  V  /\  0  < 
( # `  W ) )  ->  ( ( W concat  <" Z "> ) `  0 )  =  ( W ` 
0 ) )
162, 5, 14, 15syl3anc 1213 . . . . . . . 8  |-  ( ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  /\  ( ( W  e. Word  V  /\  ( # `
 W )  =  ( N  +  1 ) )  /\  (
( W `  0
)  =  X  /\  ( lastS  `  W )  =/= 
X ) ) )  ->  ( ( W concat  <" Z "> ) `  0 )  =  ( W ` 
0 ) )
17 simprl 750 . . . . . . . . 9  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  =  ( N  +  1 ) )  /\  ( ( W `
 0 )  =  X  /\  ( lastS  `  W
)  =/=  X ) )  ->  ( W `  0 )  =  X )
1817adantl 463 . . . . . . . 8  |-  ( ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  /\  ( ( W  e. Word  V  /\  ( # `
 W )  =  ( N  +  1 ) )  /\  (
( W `  0
)  =  X  /\  ( lastS  `  W )  =/= 
X ) ) )  ->  ( W ` 
0 )  =  X )
1916, 18eqtrd 2473 . . . . . . 7  |-  ( ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  /\  ( ( W  e. Word  V  /\  ( # `
 W )  =  ( N  +  1 ) )  /\  (
( W `  0
)  =  X  /\  ( lastS  `  W )  =/= 
X ) ) )  ->  ( ( W concat  <" Z "> ) `  0 )  =  X )
20 oveq1 6097 . . . . . . . . . . . . . . . . . . 19  |-  ( (
# `  W )  =  ( N  + 
1 )  ->  (
( # `  W )  -  1 )  =  ( ( N  + 
1 )  -  1 ) )
2120adantl 463 . . . . . . . . . . . . . . . . . 18  |-  ( ( W  e. Word  V  /\  ( # `  W )  =  ( N  + 
1 ) )  -> 
( ( # `  W
)  -  1 )  =  ( ( N  +  1 )  - 
1 ) )
2221adantl 463 . . . . . . . . . . . . . . . . 17  |-  ( ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  /\  ( W  e. Word  V  /\  ( # `  W
)  =  ( N  +  1 ) ) )  ->  ( ( # `
 W )  - 
1 )  =  ( ( N  +  1 )  -  1 ) )
23 nncn 10326 . . . . . . . . . . . . . . . . . . . 20  |-  ( N  e.  NN  ->  N  e.  CC )
24 pncan1 9768 . . . . . . . . . . . . . . . . . . . 20  |-  ( N  e.  CC  ->  (
( N  +  1 )  -  1 )  =  N )
2523, 24syl 16 . . . . . . . . . . . . . . . . . . 19  |-  ( N  e.  NN  ->  (
( N  +  1 )  -  1 )  =  N )
26253ad2ant2 1005 . . . . . . . . . . . . . . . . . 18  |-  ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  ->  (
( N  +  1 )  -  1 )  =  N )
2726adantr 462 . . . . . . . . . . . . . . . . 17  |-  ( ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  /\  ( W  e. Word  V  /\  ( # `  W
)  =  ( N  +  1 ) ) )  ->  ( ( N  +  1 )  -  1 )  =  N )
2822, 27eqtr2d 2474 . . . . . . . . . . . . . . . 16  |-  ( ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  /\  ( W  e. Word  V  /\  ( # `  W
)  =  ( N  +  1 ) ) )  ->  N  =  ( ( # `  W
)  -  1 ) )
2928fveq2d 5692 . . . . . . . . . . . . . . 15  |-  ( ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  /\  ( W  e. Word  V  /\  ( # `  W
)  =  ( N  +  1 ) ) )  ->  ( ( W concat  <" Z "> ) `  N )  =  ( ( W concat  <" Z "> ) `  ( ( # `
 W )  - 
1 ) ) )
30 simprl 750 . . . . . . . . . . . . . . . 16  |-  ( ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  /\  ( W  e. Word  V  /\  ( # `  W
)  =  ( N  +  1 ) ) )  ->  W  e. Word  V )
314adantr 462 . . . . . . . . . . . . . . . 16  |-  ( ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  /\  ( W  e. Word  V  /\  ( # `  W
)  =  ( N  +  1 ) ) )  ->  <" Z ">  e. Word  V )
329adantr 462 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  /\  ( W  e. Word  V  /\  ( # `  W
)  =  ( N  +  1 ) ) )  ->  0  <  ( N  +  1 ) )
3312adantl 463 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  /\  ( W  e. Word  V  /\  ( # `  W
)  =  ( N  +  1 ) ) )  ->  ( 0  <  ( # `  W
)  <->  0  <  ( N  +  1 ) ) )
3432, 33mpbird 232 . . . . . . . . . . . . . . . . 17  |-  ( ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  /\  ( W  e. Word  V  /\  ( # `  W
)  =  ( N  +  1 ) ) )  ->  0  <  (
# `  W )
)
35 hashneq0 12128 . . . . . . . . . . . . . . . . . 18  |-  ( W  e. Word  V  ->  (
0  <  ( # `  W
)  <->  W  =/=  (/) ) )
3635ad2antrl 722 . . . . . . . . . . . . . . . . 17  |-  ( ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  /\  ( W  e. Word  V  /\  ( # `  W
)  =  ( N  +  1 ) ) )  ->  ( 0  <  ( # `  W
)  <->  W  =/=  (/) ) )
3734, 36mpbid 210 . . . . . . . . . . . . . . . 16  |-  ( ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  /\  ( W  e. Word  V  /\  ( # `  W
)  =  ( N  +  1 ) ) )  ->  W  =/=  (/) )
38 ccatval1lsw 12279 . . . . . . . . . . . . . . . 16  |-  ( ( W  e. Word  V  /\  <" Z ">  e. Word  V  /\  W  =/=  (/) )  ->  ( ( W concat  <" Z "> ) `  ( (
# `  W )  -  1 ) )  =  ( lastS  `  W
) )
3930, 31, 37, 38syl3anc 1213 . . . . . . . . . . . . . . 15  |-  ( ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  /\  ( W  e. Word  V  /\  ( # `  W
)  =  ( N  +  1 ) ) )  ->  ( ( W concat  <" Z "> ) `  ( (
# `  W )  -  1 ) )  =  ( lastS  `  W
) )
4029, 39eqtr2d 2474 . . . . . . . . . . . . . 14  |-  ( ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  /\  ( W  e. Word  V  /\  ( # `  W
)  =  ( N  +  1 ) ) )  ->  ( lastS  `  W
)  =  ( ( W concat  <" Z "> ) `  N ) )
4140neeq1d 2619 . . . . . . . . . . . . 13  |-  ( ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  /\  ( W  e. Word  V  /\  ( # `  W
)  =  ( N  +  1 ) ) )  ->  ( ( lastS  `  W )  =/=  X  <->  ( ( W concat  <" Z "> ) `  N
)  =/=  X ) )
4241biimpd 207 . . . . . . . . . . . 12  |-  ( ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  /\  ( W  e. Word  V  /\  ( # `  W
)  =  ( N  +  1 ) ) )  ->  ( ( lastS  `  W )  =/=  X  ->  ( ( W concat  <" Z "> ) `  N
)  =/=  X ) )
4342ex 434 . . . . . . . . . . 11  |-  ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  ->  (
( W  e. Word  V  /\  ( # `  W
)  =  ( N  +  1 ) )  ->  ( ( lastS  `  W
)  =/=  X  -> 
( ( W concat  <" Z "> ) `  N
)  =/=  X ) ) )
4443com13 80 . . . . . . . . . 10  |-  ( ( lastS  `  W )  =/=  X  ->  ( ( W  e. Word  V  /\  ( # `  W
)  =  ( N  +  1 ) )  ->  ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  ->  (
( W concat  <" Z "> ) `  N
)  =/=  X ) ) )
4544adantl 463 . . . . . . . . 9  |-  ( ( ( W `  0
)  =  X  /\  ( lastS  `  W )  =/= 
X )  ->  (
( W  e. Word  V  /\  ( # `  W
)  =  ( N  +  1 ) )  ->  ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  ->  (
( W concat  <" Z "> ) `  N
)  =/=  X ) ) )
4645com13 80 . . . . . . . 8  |-  ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  ->  (
( W  e. Word  V  /\  ( # `  W
)  =  ( N  +  1 ) )  ->  ( ( ( W `  0 )  =  X  /\  ( lastS  `  W )  =/=  X
)  ->  ( ( W concat  <" Z "> ) `  N )  =/=  X ) ) )
4746imp32 433 . . . . . . 7  |-  ( ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  /\  ( ( W  e. Word  V  /\  ( # `
 W )  =  ( N  +  1 ) )  /\  (
( W `  0
)  =  X  /\  ( lastS  `  W )  =/= 
X ) ) )  ->  ( ( W concat  <" Z "> ) `  N )  =/=  X )
4819, 47jca 529 . . . . . 6  |-  ( ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  /\  ( ( W  e. Word  V  /\  ( # `
 W )  =  ( N  +  1 ) )  /\  (
( W `  0
)  =  X  /\  ( lastS  `  W )  =/= 
X ) ) )  ->  ( ( ( W concat  <" Z "> ) `  0 )  =  X  /\  (
( W concat  <" Z "> ) `  N
)  =/=  X ) )
4948expcom 435 . . . . 5  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  =  ( N  +  1 ) )  /\  ( ( W `
 0 )  =  X  /\  ( lastS  `  W
)  =/=  X ) )  ->  ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  ->  (
( ( W concat  <" Z "> ) `  0
)  =  X  /\  ( ( W concat  <" Z "> ) `  N
)  =/=  X ) ) )
5049exp32 602 . . . 4  |-  ( ( W  e. Word  V  /\  ( # `  W )  =  ( N  + 
1 ) )  -> 
( ( W ` 
0 )  =  X  ->  ( ( lastS  `  W
)  =/=  X  -> 
( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  ->  (
( ( W concat  <" Z "> ) `  0
)  =  X  /\  ( ( W concat  <" Z "> ) `  N
)  =/=  X ) ) ) ) )
511, 50syl 16 . . 3  |-  ( W  e.  ( ( V WWalksN  E ) `  N
)  ->  ( ( W `  0 )  =  X  ->  ( ( lastS  `  W )  =/=  X  ->  ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  ->  (
( ( W concat  <" Z "> ) `  0
)  =  X  /\  ( ( W concat  <" Z "> ) `  N
)  =/=  X ) ) ) ) )
52513imp 1176 . 2  |-  ( ( W  e.  ( ( V WWalksN  E ) `  N
)  /\  ( W `  0 )  =  X  /\  ( lastS  `  W
)  =/=  X )  ->  ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  ->  (
( ( W concat  <" Z "> ) `  0
)  =  X  /\  ( ( W concat  <" Z "> ) `  N
)  =/=  X ) ) )
5352impcom 430 1  |-  ( ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  /\  ( W  e.  ( ( V WWalksN  E
) `  N )  /\  ( W `  0
)  =  X  /\  ( lastS  `  W )  =/= 
X ) )  -> 
( ( ( W concat  <" Z "> ) `  0 )  =  X  /\  (
( W concat  <" Z "> ) `  N
)  =/=  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761    =/= wne 2604   (/)c0 3634   class class class wbr 4289   ` cfv 5415  (class class class)co 6090   CCcc 9276   0cc0 9278   1c1 9279    + caddc 9281    < clt 9414    - cmin 9591   NNcn 10318   NN0cn0 10575   #chash 12099  Word cword 12217   lastS clsw 12218   concat cconcat 12219   <"cs1 12220   WWalksN cwwlkn 30221
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-pm 7213  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-card 8105  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-n0 10576  df-z 10643  df-uz 10858  df-fz 11434  df-fzo 11545  df-hash 12100  df-word 12225  df-lsw 12226  df-concat 12227  df-s1 12228  df-wwlk 30222  df-wwlkn 30223
This theorem is referenced by:  numclwwlk2lem1  30604
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