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Theorem clwwlkextfrlem1 24769
Description: Lemma for numclwwlk2lem1 24795. (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 24396 . . . 4  |-  ( W  e.  ( ( V WWalksN  E ) `  N
)  ->  ( W  e. Word  V  /\  ( # `  W )  =  ( N  +  1 ) ) )
2 simprll 761 . . . . . . . . 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 12576 . . . . . . . . . . 11  |-  ( Z  e.  V  ->  <" Z ">  e. Word  V )
433ad2ant3 1019 . . . . . . . . . 10  |-  ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  ->  <" Z ">  e. Word  V )
54adantr 465 . . . . . . . . 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 10801 . . . . . . . . . . . . 13  |-  ( N  e.  NN  ->  N  e.  NN0 )
7 nn0p1gt0 10824 . . . . . . . . . . . . 13  |-  ( N  e.  NN0  ->  0  < 
( N  +  1 ) )
86, 7syl 16 . . . . . . . . . . . 12  |-  ( N  e.  NN  ->  0  <  ( N  +  1 ) )
983ad2ant2 1018 . . . . . . . . . . 11  |-  ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  ->  0  <  ( N  +  1 ) )
109adantr 465 . . . . . . . . . 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 4451 . . . . . . . . . . . 12  |-  ( (
# `  W )  =  ( N  + 
1 )  ->  (
0  <  ( # `  W
)  <->  0  <  ( N  +  1 ) ) )
1211adantl 466 . . . . . . . . . . 11  |-  ( ( W  e. Word  V  /\  ( # `  W )  =  ( N  + 
1 ) )  -> 
( 0  <  ( # `
 W )  <->  0  <  ( N  +  1 ) ) )
1312ad2antrl 727 . . . . . . . . . 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 12565 . . . . . . . . 9  |-  ( ( W  e. Word  V  /\  <" Z ">  e. Word  V  /\  0  < 
( # `  W ) )  ->  ( ( W concat  <" Z "> ) `  0 )  =  ( W ` 
0 ) )
162, 5, 14, 15syl3anc 1228 . . . . . . . 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 755 . . . . . . . . 9  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  =  ( N  +  1 ) )  /\  ( ( W `
 0 )  =  X  /\  ( lastS  `  W
)  =/=  X ) )  ->  ( W `  0 )  =  X )
1817adantl 466 . . . . . . . 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 2508 . . . . . . 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 6290 . . . . . . . . . . . . . . . . . . 19  |-  ( (
# `  W )  =  ( N  + 
1 )  ->  (
( # `  W )  -  1 )  =  ( ( N  + 
1 )  -  1 ) )
2120adantl 466 . . . . . . . . . . . . . . . . . 18  |-  ( ( W  e. Word  V  /\  ( # `  W )  =  ( N  + 
1 ) )  -> 
( ( # `  W
)  -  1 )  =  ( ( N  +  1 )  - 
1 ) )
2221adantl 466 . . . . . . . . . . . . . . . . 17  |-  ( ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  /\  ( W  e. Word  V  /\  ( # `  W
)  =  ( N  +  1 ) ) )  ->  ( ( # `
 W )  - 
1 )  =  ( ( N  +  1 )  -  1 ) )
23 nncn 10543 . . . . . . . . . . . . . . . . . . . 20  |-  ( N  e.  NN  ->  N  e.  CC )
24 pncan1 9982 . . . . . . . . . . . . . . . . . . . 20  |-  ( N  e.  CC  ->  (
( N  +  1 )  -  1 )  =  N )
2523, 24syl 16 . . . . . . . . . . . . . . . . . . 19  |-  ( N  e.  NN  ->  (
( N  +  1 )  -  1 )  =  N )
26253ad2ant2 1018 . . . . . . . . . . . . . . . . . 18  |-  ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  ->  (
( N  +  1 )  -  1 )  =  N )
2726adantr 465 . . . . . . . . . . . . . . . . 17  |-  ( ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  /\  ( W  e. Word  V  /\  ( # `  W
)  =  ( N  +  1 ) ) )  ->  ( ( N  +  1 )  -  1 )  =  N )
2822, 27eqtr2d 2509 . . . . . . . . . . . . . . . 16  |-  ( ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  /\  ( W  e. Word  V  /\  ( # `  W
)  =  ( N  +  1 ) ) )  ->  N  =  ( ( # `  W
)  -  1 ) )
2928fveq2d 5869 . . . . . . . . . . . . . . 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 755 . . . . . . . . . . . . . . . 16  |-  ( ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  /\  ( W  e. Word  V  /\  ( # `  W
)  =  ( N  +  1 ) ) )  ->  W  e. Word  V )
314adantr 465 . . . . . . . . . . . . . . . 16  |-  ( ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  /\  ( W  e. Word  V  /\  ( # `  W
)  =  ( N  +  1 ) ) )  ->  <" Z ">  e. Word  V )
329adantr 465 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  /\  ( W  e. Word  V  /\  ( # `  W
)  =  ( N  +  1 ) ) )  ->  0  <  ( N  +  1 ) )
3312adantl 466 . . . . . . . . . . . . . . . . . 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 12401 . . . . . . . . . . . . . . . . . 18  |-  ( W  e. Word  V  ->  (
0  <  ( # `  W
)  <->  W  =/=  (/) ) )
3635ad2antrl 727 . . . . . . . . . . . . . . . . 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 12566 . . . . . . . . . . . . . . . 16  |-  ( ( W  e. Word  V  /\  <" Z ">  e. Word  V  /\  W  =/=  (/) )  ->  ( ( W concat  <" Z "> ) `  ( (
# `  W )  -  1 ) )  =  ( lastS  `  W
) )
3930, 31, 37, 38syl3anc 1228 . . . . . . . . . . . . . . 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 2509 . . . . . . . . . . . . . 14  |-  ( ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  /\  ( W  e. Word  V  /\  ( # `  W
)  =  ( N  +  1 ) ) )  ->  ( lastS  `  W
)  =  ( ( W concat  <" Z "> ) `  N ) )
4140neeq1d 2744 . . . . . . . . . . . . 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 466 . . . . . . . . 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 532 . . . . . 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 605 . . . 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 1190 . 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   (/)c0 3785   class class class wbr 4447   ` cfv 5587  (class class class)co 6283   CCcc 9489   0cc0 9491   1c1 9492    + caddc 9494    < clt 9627    - cmin 9804   NNcn 10535   NN0cn0 10794   #chash 12372  Word cword 12499   lastS clsw 12500   concat cconcat 12501   <"cs1 12502   WWalksN cwwlkn 24370
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-cnex 9547  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-om 6680  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-card 8319  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-nn 10536  df-n0 10795  df-z 10864  df-uz 11082  df-fz 11672  df-fzo 11792  df-hash 12373  df-word 12507  df-lsw 12508  df-concat 12509  df-s1 12510  df-wwlk 24371  df-wwlkn 24372
This theorem is referenced by:  numclwwlk2lem1  24795
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