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Theorem clwwlkextfrlem1 30681
Description: Lemma for numclwwlk2lem1 30707. (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 30350 . . . 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 12305 . . . . . . . . . . 11  |-  ( Z  e.  V  ->  <" Z ">  e. Word  V )
433ad2ant3 1011 . . . . . . . . . 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 10598 . . . . . . . . . . . . 13  |-  ( N  e.  NN  ->  N  e.  NN0 )
7 nn0p1gt0 10621 . . . . . . . . . . . . 13  |-  ( N  e.  NN0  ->  0  < 
( N  +  1 ) )
86, 7syl 16 . . . . . . . . . . . 12  |-  ( N  e.  NN  ->  0  <  ( N  +  1 ) )
983ad2ant2 1010 . . . . . . . . . . 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 4308 . . . . . . . . . . . 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 12294 . . . . . . . . 9  |-  ( ( W  e. Word  V  /\  <" Z ">  e. Word  V  /\  0  < 
( # `  W ) )  ->  ( ( W concat  <" Z "> ) `  0 )  =  ( W ` 
0 ) )
162, 5, 14, 15syl3anc 1218 . . . . . . . 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 2475 . . . . . . 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 6110 . . . . . . . . . . . . . . . . . . 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 10342 . . . . . . . . . . . . . . . . . . . 20  |-  ( N  e.  NN  ->  N  e.  CC )
24 pncan1 9784 . . . . . . . . . . . . . . . . . . . 20  |-  ( N  e.  CC  ->  (
( N  +  1 )  -  1 )  =  N )
2523, 24syl 16 . . . . . . . . . . . . . . . . . . 19  |-  ( N  e.  NN  ->  (
( N  +  1 )  -  1 )  =  N )
26253ad2ant2 1010 . . . . . . . . . . . . . . . . . 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 2476 . . . . . . . . . . . . . . . 16  |-  ( ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  /\  ( W  e. Word  V  /\  ( # `  W
)  =  ( N  +  1 ) ) )  ->  N  =  ( ( # `  W
)  -  1 ) )
2928fveq2d 5707 . . . . . . . . . . . . . . 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 12144 . . . . . . . . . . . . . . . . . 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 12295 . . . . . . . . . . . . . . . 16  |-  ( ( W  e. Word  V  /\  <" Z ">  e. Word  V  /\  W  =/=  (/) )  ->  ( ( W concat  <" Z "> ) `  ( (
# `  W )  -  1 ) )  =  ( lastS  `  W
) )
3930, 31, 37, 38syl3anc 1218 . . . . . . . . . . . . . . 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 2476 . . . . . . . . . . . . . 14  |-  ( ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  /\  ( W  e. Word  V  /\  ( # `  W
)  =  ( N  +  1 ) ) )  ->  ( lastS  `  W
)  =  ( ( W concat  <" Z "> ) `  N ) )
4140neeq1d 2633 . . . . . . . . . . . . 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 1181 . 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 965    = wceq 1369    e. wcel 1756    =/= wne 2618   (/)c0 3649   class class class wbr 4304   ` cfv 5430  (class class class)co 6103   CCcc 9292   0cc0 9294   1c1 9295    + caddc 9297    < clt 9430    - cmin 9607   NNcn 10334   NN0cn0 10591   #chash 12115  Word cword 12233   lastS clsw 12234   concat cconcat 12235   <"cs1 12236   WWalksN cwwlkn 30324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-int 4141  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-1st 6589  df-2nd 6590  df-recs 6844  df-rdg 6878  df-1o 6932  df-oadd 6936  df-er 7113  df-map 7228  df-pm 7229  df-en 7323  df-dom 7324  df-sdom 7325  df-fin 7326  df-card 8121  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-nn 10335  df-n0 10592  df-z 10659  df-uz 10874  df-fz 11450  df-fzo 11561  df-hash 12116  df-word 12241  df-lsw 12242  df-concat 12243  df-s1 12244  df-wwlk 30325  df-wwlkn 30326
This theorem is referenced by:  numclwwlk2lem1  30707
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