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Theorem wwlkextproplem1 24564
Description: Lemma 1 for wwlkextprop 24567. (Contributed by Alexander van der Vekens, 31-Jul-2018.)
Hypothesis
Ref Expression
wwlkextprop.x  |-  X  =  ( ( V WWalksN  E
) `  ( N  +  1 ) )
Assertion
Ref Expression
wwlkextproplem1  |-  ( ( W  e.  X  /\  N  e.  NN0 )  -> 
( ( W substr  <. 0 ,  ( N  + 
1 ) >. ) `  0 )  =  ( W `  0
) )

Proof of Theorem wwlkextproplem1
Dummy variable  i is distinct from all other variables.
StepHypRef Expression
1 wwlknimp 24510 . . . . 5  |-  ( W  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  ->  ( W  e. Word  V  /\  ( # `  W )  =  ( ( N  +  1 )  +  1 )  /\  A. i  e.  ( 0..^ ( N  +  1 ) ) { ( W `  i ) ,  ( W `  ( i  +  1 ) ) }  e.  ran  E
) )
2 1z 10906 . . . . . . . . . . . 12  |-  1  e.  ZZ
32a1i 11 . . . . . . . . . . 11  |-  ( N  e.  NN0  ->  1  e.  ZZ )
4 nn0z 10899 . . . . . . . . . . . . 13  |-  ( N  e.  NN0  ->  N  e.  ZZ )
54peano2zd 10981 . . . . . . . . . . . 12  |-  ( N  e.  NN0  ->  ( N  +  1 )  e.  ZZ )
65peano2zd 10981 . . . . . . . . . . 11  |-  ( N  e.  NN0  ->  ( ( N  +  1 )  +  1 )  e.  ZZ )
73, 6, 53jca 1176 . . . . . . . . . 10  |-  ( N  e.  NN0  ->  ( 1  e.  ZZ  /\  (
( N  +  1 )  +  1 )  e.  ZZ  /\  ( N  +  1 )  e.  ZZ ) )
8 nn0ge0 10833 . . . . . . . . . . . 12  |-  ( N  e.  NN0  ->  0  <_  N )
9 1re 9607 . . . . . . . . . . . . . 14  |-  1  e.  RR
109a1i 11 . . . . . . . . . . . . 13  |-  ( N  e.  NN0  ->  1  e.  RR )
11 nn0re 10816 . . . . . . . . . . . . 13  |-  ( N  e.  NN0  ->  N  e.  RR )
1210, 11addge02d 10153 . . . . . . . . . . . 12  |-  ( N  e.  NN0  ->  ( 0  <_  N  <->  1  <_  ( N  +  1 ) ) )
138, 12mpbid 210 . . . . . . . . . . 11  |-  ( N  e.  NN0  ->  1  <_ 
( N  +  1 ) )
14 peano2re 9764 . . . . . . . . . . . . 13  |-  ( N  e.  RR  ->  ( N  +  1 )  e.  RR )
1511, 14syl 16 . . . . . . . . . . . 12  |-  ( N  e.  NN0  ->  ( N  +  1 )  e.  RR )
1615lep1d 10489 . . . . . . . . . . 11  |-  ( N  e.  NN0  ->  ( N  +  1 )  <_ 
( ( N  + 
1 )  +  1 ) )
1713, 16jca 532 . . . . . . . . . 10  |-  ( N  e.  NN0  ->  ( 1  <_  ( N  + 
1 )  /\  ( N  +  1 )  <_  ( ( N  +  1 )  +  1 ) ) )
18 elfz2 11691 . . . . . . . . . 10  |-  ( ( N  +  1 )  e.  ( 1 ... ( ( N  + 
1 )  +  1 ) )  <->  ( (
1  e.  ZZ  /\  ( ( N  + 
1 )  +  1 )  e.  ZZ  /\  ( N  +  1
)  e.  ZZ )  /\  ( 1  <_ 
( N  +  1 )  /\  ( N  +  1 )  <_ 
( ( N  + 
1 )  +  1 ) ) ) )
197, 17, 18sylanbrc 664 . . . . . . . . 9  |-  ( N  e.  NN0  ->  ( N  +  1 )  e.  ( 1 ... (
( N  +  1 )  +  1 ) ) )
20 oveq2 6303 . . . . . . . . . 10  |-  ( (
# `  W )  =  ( ( N  +  1 )  +  1 )  ->  (
1 ... ( # `  W
) )  =  ( 1 ... ( ( N  +  1 )  +  1 ) ) )
2120eleq2d 2537 . . . . . . . . 9  |-  ( (
# `  W )  =  ( ( N  +  1 )  +  1 )  ->  (
( N  +  1 )  e.  ( 1 ... ( # `  W
) )  <->  ( N  +  1 )  e.  ( 1 ... (
( N  +  1 )  +  1 ) ) ) )
2219, 21syl5ibr 221 . . . . . . . 8  |-  ( (
# `  W )  =  ( ( N  +  1 )  +  1 )  ->  ( N  e.  NN0  ->  ( N  +  1 )  e.  ( 1 ... ( # `  W
) ) ) )
2322adantl 466 . . . . . . 7  |-  ( ( W  e. Word  V  /\  ( # `  W )  =  ( ( N  +  1 )  +  1 ) )  -> 
( N  e.  NN0  ->  ( N  +  1 )  e.  ( 1 ... ( # `  W
) ) ) )
24 simpl 457 . . . . . . 7  |-  ( ( W  e. Word  V  /\  ( # `  W )  =  ( ( N  +  1 )  +  1 ) )  ->  W  e. Word  V )
2523, 24jctild 543 . . . . . 6  |-  ( ( W  e. Word  V  /\  ( # `  W )  =  ( ( N  +  1 )  +  1 ) )  -> 
( N  e.  NN0  ->  ( W  e. Word  V  /\  ( N  +  1 )  e.  ( 1 ... ( # `  W
) ) ) ) )
26253adant3 1016 . . . . 5  |-  ( ( W  e. Word  V  /\  ( # `  W )  =  ( ( N  +  1 )  +  1 )  /\  A. i  e.  ( 0..^ ( N  +  1 ) ) { ( W `  i ) ,  ( W `  ( i  +  1 ) ) }  e.  ran  E )  ->  ( N  e.  NN0  ->  ( W  e. Word  V  /\  ( N  +  1 )  e.  ( 1 ... ( # `  W
) ) ) ) )
271, 26syl 16 . . . 4  |-  ( W  e.  ( ( V WWalksN  E ) `  ( N  +  1 ) )  ->  ( N  e.  NN0  ->  ( W  e. Word  V  /\  ( N  +  1 )  e.  ( 1 ... ( # `
 W ) ) ) ) )
28 wwlkextprop.x . . . 4  |-  X  =  ( ( V WWalksN  E
) `  ( N  +  1 ) )
2927, 28eleq2s 2575 . . 3  |-  ( W  e.  X  ->  ( N  e.  NN0  ->  ( W  e. Word  V  /\  ( N  +  1 )  e.  ( 1 ... ( # `  W
) ) ) ) )
3029imp 429 . 2  |-  ( ( W  e.  X  /\  N  e.  NN0 )  -> 
( W  e. Word  V  /\  ( N  +  1 )  e.  ( 1 ... ( # `  W
) ) ) )
31 swrd0fv0 12647 . 2  |-  ( ( W  e. Word  V  /\  ( N  +  1
)  e.  ( 1 ... ( # `  W
) ) )  -> 
( ( W substr  <. 0 ,  ( N  + 
1 ) >. ) `  0 )  =  ( W `  0
) )
3230, 31syl 16 1  |-  ( ( W  e.  X  /\  N  e.  NN0 )  -> 
( ( W substr  <. 0 ,  ( N  + 
1 ) >. ) `  0 )  =  ( W `  0
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2817   {cpr 4035   <.cop 4039   class class class wbr 4453   ran crn 5006   ` cfv 5594  (class class class)co 6295   RRcr 9503   0cc0 9504   1c1 9505    + caddc 9507    <_ cle 9641   NN0cn0 10807   ZZcz 10876   ...cfz 11684  ..^cfzo 11804   #chash 12385  Word cword 12515   substr csubstr 12519   WWalksN cwwlkn 24501
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 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
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 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-pm 7435  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-card 8332  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-n0 10808  df-z 10877  df-uz 11095  df-fz 11685  df-fzo 11805  df-hash 12386  df-word 12523  df-substr 12527  df-wwlk 24502  df-wwlkn 24503
This theorem is referenced by:  wwlkextproplem3  24566  wwlkextprop  24567
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