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Theorem clwwlkextfrlem1 25278
Description: Lemma for numclwwlk2lem1 25304. (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 ++ 
<" Z "> ) `  0 )  =  X  /\  (
( W ++  <" Z "> ) `  N
)  =/=  X ) )

Proof of Theorem clwwlkextfrlem1
StepHypRef Expression
1 wwlknimpb 24906 . . . 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 12603 . . . . . . . . . . 11  |-  ( Z  e.  V  ->  <" Z ">  e. Word  V )
433ad2ant3 1017 . . . . . . . . . 10  |-  ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  ->  <" Z ">  e. Word  V )
54adantr 463 . . . . . . . . 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 10798 . . . . . . . . . . . . 13  |-  ( N  e.  NN  ->  N  e.  NN0 )
7 nn0p1gt0 10821 . . . . . . . . . . . . 13  |-  ( N  e.  NN0  ->  0  < 
( N  +  1 ) )
86, 7syl 16 . . . . . . . . . . . 12  |-  ( N  e.  NN  ->  0  <  ( N  +  1 ) )
983ad2ant2 1016 . . . . . . . . . . 11  |-  ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  ->  0  <  ( N  +  1 ) )
109adantr 463 . . . . . . . . . 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 4443 . . . . . . . . . . . 12  |-  ( (
# `  W )  =  ( N  + 
1 )  ->  (
0  <  ( # `  W
)  <->  0  <  ( N  +  1 ) ) )
1211adantl 464 . . . . . . . . . . 11  |-  ( ( W  e. Word  V  /\  ( # `  W )  =  ( N  + 
1 ) )  -> 
( 0  <  ( # `
 W )  <->  0  <  ( N  +  1 ) ) )
1312ad2antrl 725 . . . . . . . . . 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 12590 . . . . . . . . 9  |-  ( ( W  e. Word  V  /\  <" Z ">  e. Word  V  /\  0  < 
( # `  W ) )  ->  ( ( W ++  <" Z "> ) `  0 )  =  ( W ` 
0 ) )
162, 5, 14, 15syl3anc 1226 . . . . . . . 8  |-  ( ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  /\  ( ( W  e. Word  V  /\  ( # `
 W )  =  ( N  +  1 ) )  /\  (
( W `  0
)  =  X  /\  ( lastS  `  W )  =/= 
X ) ) )  ->  ( ( W ++ 
<" Z "> ) `  0 )  =  ( W ` 
0 ) )
17 simprl 754 . . . . . . . . 9  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  =  ( N  +  1 ) )  /\  ( ( W `
 0 )  =  X  /\  ( lastS  `  W
)  =/=  X ) )  ->  ( W `  0 )  =  X )
1817adantl 464 . . . . . . . 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 2495 . . . . . . 7  |-  ( ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  /\  ( ( W  e. Word  V  /\  ( # `
 W )  =  ( N  +  1 ) )  /\  (
( W `  0
)  =  X  /\  ( lastS  `  W )  =/= 
X ) ) )  ->  ( ( W ++ 
<" Z "> ) `  0 )  =  X )
20 oveq1 6277 . . . . . . . . . . . . . . . . . . 19  |-  ( (
# `  W )  =  ( N  + 
1 )  ->  (
( # `  W )  -  1 )  =  ( ( N  + 
1 )  -  1 ) )
2120adantl 464 . . . . . . . . . . . . . . . . . 18  |-  ( ( W  e. Word  V  /\  ( # `  W )  =  ( N  + 
1 ) )  -> 
( ( # `  W
)  -  1 )  =  ( ( N  +  1 )  - 
1 ) )
2221adantl 464 . . . . . . . . . . . . . . . . 17  |-  ( ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  /\  ( W  e. Word  V  /\  ( # `  W
)  =  ( N  +  1 ) ) )  ->  ( ( # `
 W )  - 
1 )  =  ( ( N  +  1 )  -  1 ) )
23 nncn 10539 . . . . . . . . . . . . . . . . . . . 20  |-  ( N  e.  NN  ->  N  e.  CC )
24 pncan1 9979 . . . . . . . . . . . . . . . . . . . 20  |-  ( N  e.  CC  ->  (
( N  +  1 )  -  1 )  =  N )
2523, 24syl 16 . . . . . . . . . . . . . . . . . . 19  |-  ( N  e.  NN  ->  (
( N  +  1 )  -  1 )  =  N )
26253ad2ant2 1016 . . . . . . . . . . . . . . . . . 18  |-  ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  ->  (
( N  +  1 )  -  1 )  =  N )
2726adantr 463 . . . . . . . . . . . . . . . . 17  |-  ( ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  /\  ( W  e. Word  V  /\  ( # `  W
)  =  ( N  +  1 ) ) )  ->  ( ( N  +  1 )  -  1 )  =  N )
2822, 27eqtr2d 2496 . . . . . . . . . . . . . . . 16  |-  ( ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  /\  ( W  e. Word  V  /\  ( # `  W
)  =  ( N  +  1 ) ) )  ->  N  =  ( ( # `  W
)  -  1 ) )
2928fveq2d 5852 . . . . . . . . . . . . . . 15  |-  ( ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  /\  ( W  e. Word  V  /\  ( # `  W
)  =  ( N  +  1 ) ) )  ->  ( ( W ++  <" Z "> ) `  N )  =  ( ( W ++ 
<" Z "> ) `  ( ( # `
 W )  - 
1 ) ) )
30 simprl 754 . . . . . . . . . . . . . . . 16  |-  ( ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  /\  ( W  e. Word  V  /\  ( # `  W
)  =  ( N  +  1 ) ) )  ->  W  e. Word  V )
314adantr 463 . . . . . . . . . . . . . . . 16  |-  ( ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  /\  ( W  e. Word  V  /\  ( # `  W
)  =  ( N  +  1 ) ) )  ->  <" Z ">  e. Word  V )
329adantr 463 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  /\  ( W  e. Word  V  /\  ( # `  W
)  =  ( N  +  1 ) ) )  ->  0  <  ( N  +  1 ) )
3312adantl 464 . . . . . . . . . . . . . . . . . 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 12417 . . . . . . . . . . . . . . . . . 18  |-  ( W  e. Word  V  ->  (
0  <  ( # `  W
)  <->  W  =/=  (/) ) )
3635ad2antrl 725 . . . . . . . . . . . . . . . . 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 12591 . . . . . . . . . . . . . . . 16  |-  ( ( W  e. Word  V  /\  <" Z ">  e. Word  V  /\  W  =/=  (/) )  ->  ( ( W ++  <" Z "> ) `  ( (
# `  W )  -  1 ) )  =  ( lastS  `  W
) )
3930, 31, 37, 38syl3anc 1226 . . . . . . . . . . . . . . 15  |-  ( ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  /\  ( W  e. Word  V  /\  ( # `  W
)  =  ( N  +  1 ) ) )  ->  ( ( W ++  <" Z "> ) `  ( (
# `  W )  -  1 ) )  =  ( lastS  `  W
) )
4029, 39eqtr2d 2496 . . . . . . . . . . . . . 14  |-  ( ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  /\  ( W  e. Word  V  /\  ( # `  W
)  =  ( N  +  1 ) ) )  ->  ( lastS  `  W
)  =  ( ( W ++  <" Z "> ) `  N ) )
4140neeq1d 2731 . . . . . . . . . . . . 13  |-  ( ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  /\  ( W  e. Word  V  /\  ( # `  W
)  =  ( N  +  1 ) ) )  ->  ( ( lastS  `  W )  =/=  X  <->  ( ( W ++  <" 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 ++  <" Z "> ) `  N )  =/=  X
) )
4342ex 432 . . . . . . . . . . 11  |-  ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  ->  (
( W  e. Word  V  /\  ( # `  W
)  =  ( N  +  1 ) )  ->  ( ( lastS  `  W
)  =/=  X  -> 
( ( W ++  <" 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 ++  <" Z "> ) `  N
)  =/=  X ) ) )
4544adantl 464 . . . . . . . . 9  |-  ( ( ( W `  0
)  =  X  /\  ( lastS  `  W )  =/= 
X )  ->  (
( W  e. Word  V  /\  ( # `  W
)  =  ( N  +  1 ) )  ->  ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  ->  (
( W ++  <" 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 ++  <" Z "> ) `  N )  =/=  X ) ) )
4746imp32 431 . . . . . . 7  |-  ( ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  /\  ( ( W  e. Word  V  /\  ( # `
 W )  =  ( N  +  1 ) )  /\  (
( W `  0
)  =  X  /\  ( lastS  `  W )  =/= 
X ) ) )  ->  ( ( W ++ 
<" Z "> ) `  N )  =/=  X )
4819, 47jca 530 . . . . . 6  |-  ( ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  /\  ( ( W  e. Word  V  /\  ( # `
 W )  =  ( N  +  1 ) )  /\  (
( W `  0
)  =  X  /\  ( lastS  `  W )  =/= 
X ) ) )  ->  ( ( ( W ++  <" Z "> ) `  0 )  =  X  /\  (
( W ++  <" Z "> ) `  N
)  =/=  X ) )
4948expcom 433 . . . . 5  |-  ( ( ( W  e. Word  V  /\  ( # `  W
)  =  ( N  +  1 ) )  /\  ( ( W `
 0 )  =  X  /\  ( lastS  `  W
)  =/=  X ) )  ->  ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  ->  (
( ( W ++  <" Z "> ) `  0 )  =  X  /\  ( ( W ++  <" Z "> ) `  N )  =/=  X ) ) )
5049exp32 603 . . . 4  |-  ( ( W  e. Word  V  /\  ( # `  W )  =  ( N  + 
1 ) )  -> 
( ( W ` 
0 )  =  X  ->  ( ( lastS  `  W
)  =/=  X  -> 
( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  ->  (
( ( W ++  <" Z "> ) `  0 )  =  X  /\  ( ( W ++  <" 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 ++  <" Z "> ) `  0 )  =  X  /\  ( ( W ++  <" Z "> ) `  N )  =/=  X ) ) ) ) )
52513imp 1188 . 2  |-  ( ( W  e.  ( ( V WWalksN  E ) `  N
)  /\  ( W `  0 )  =  X  /\  ( lastS  `  W
)  =/=  X )  ->  ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  ->  (
( ( W ++  <" Z "> ) `  0 )  =  X  /\  ( ( W ++  <" Z "> ) `  N )  =/=  X ) ) )
5352impcom 428 1  |-  ( ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  /\  ( W  e.  ( ( V WWalksN  E
) `  N )  /\  ( W `  0
)  =  X  /\  ( lastS  `  W )  =/= 
X ) )  -> 
( ( ( W ++ 
<" Z "> ) `  0 )  =  X  /\  (
( W ++  <" Z "> ) `  N
)  =/=  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   (/)c0 3783   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   CCcc 9479   0cc0 9481   1c1 9482    + caddc 9484    < clt 9617    - cmin 9796   NNcn 10531   NN0cn0 10791   #chash 12387  Word cword 12518   lastS clsw 12519   ++ cconcat 12520   <"cs1 12521   WWalksN cwwlkn 24880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-fzo 11800  df-hash 12388  df-word 12526  df-lsw 12527  df-concat 12528  df-s1 12529  df-wwlk 24881  df-wwlkn 24882
This theorem is referenced by:  numclwwlk2lem1  25304
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