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Theorem wwlkextproplem1 30586
Description: Lemma 1 for wwlkextprop 30589. (Contributed by Alexander van der Vekens, 31-Jul-2018.)
Hypothesis
Ref Expression
hashrabrex.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 30347 . . . . 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 10697 . . . . . . . . . . . 12  |-  1  e.  ZZ
32a1i 11 . . . . . . . . . . 11  |-  ( N  e.  NN0  ->  1  e.  ZZ )
4 nn0z 10690 . . . . . . . . . . . . 13  |-  ( N  e.  NN0  ->  N  e.  ZZ )
54peano2zd 10771 . . . . . . . . . . . 12  |-  ( N  e.  NN0  ->  ( N  +  1 )  e.  ZZ )
65peano2zd 10771 . . . . . . . . . . 11  |-  ( N  e.  NN0  ->  ( ( N  +  1 )  +  1 )  e.  ZZ )
73, 6, 53jca 1168 . . . . . . . . . 10  |-  ( N  e.  NN0  ->  ( 1  e.  ZZ  /\  (
( N  +  1 )  +  1 )  e.  ZZ  /\  ( N  +  1 )  e.  ZZ ) )
8 nn0ge0 10626 . . . . . . . . . . . 12  |-  ( N  e.  NN0  ->  0  <_  N )
9 1re 9406 . . . . . . . . . . . . . 14  |-  1  e.  RR
109a1i 11 . . . . . . . . . . . . 13  |-  ( N  e.  NN0  ->  1  e.  RR )
11 nn0re 10609 . . . . . . . . . . . . 13  |-  ( N  e.  NN0  ->  N  e.  RR )
1210, 11addge02d 9949 . . . . . . . . . . . 12  |-  ( N  e.  NN0  ->  ( 0  <_  N  <->  1  <_  ( N  +  1 ) ) )
138, 12mpbid 210 . . . . . . . . . . 11  |-  ( N  e.  NN0  ->  1  <_ 
( N  +  1 ) )
14 peano2re 9563 . . . . . . . . . . . . 13  |-  ( N  e.  RR  ->  ( N  +  1 )  e.  RR )
1511, 14syl 16 . . . . . . . . . . . 12  |-  ( N  e.  NN0  ->  ( N  +  1 )  e.  RR )
1615lep1d 10285 . . . . . . . . . . 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 11465 . . . . . . . . . 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 6120 . . . . . . . . . 10  |-  ( (
# `  W )  =  ( ( N  +  1 )  +  1 )  ->  (
1 ... ( # `  W
) )  =  ( 1 ... ( ( N  +  1 )  +  1 ) ) )
2120eleq2d 2510 . . . . . . . . 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 1008 . . . . 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 hashrabrex.x . . . 4  |-  X  =  ( ( V WWalksN  E
) `  ( N  +  1 ) )
2927, 28eleq2s 2535 . . 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 12357 . 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 965    = wceq 1369    e. wcel 1756   A.wral 2736   {cpr 3900   <.cop 3904   class class class wbr 4313   ran crn 4862   ` cfv 5439  (class class class)co 6112   RRcr 9302   0cc0 9303   1c1 9304    + caddc 9306    <_ cle 9440   NN0cn0 10600   ZZcz 10667   ...cfz 11458  ..^cfzo 11569   #chash 12124  Word cword 12242   substr csubstr 12246   WWalksN cwwlkn 30338
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 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380
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 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-recs 6853  df-rdg 6887  df-1o 6941  df-oadd 6945  df-er 7122  df-map 7237  df-pm 7238  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-card 8130  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-nn 10344  df-n0 10601  df-z 10668  df-uz 10883  df-fz 11459  df-fzo 11570  df-hash 12125  df-word 12250  df-substr 12254  df-wwlk 30339  df-wwlkn 30340
This theorem is referenced by:  wwlkextproplem3  30588  wwlkextprop  30589
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