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Theorem swrdswrd0 12352
Description: A subword of a prefix. (Contributed by Alexander van der Vekens, 6-Apr-2018.)
Assertion
Ref Expression
swrdswrd0  |-  ( ( W  e. Word  V  /\  N  e.  ( 0 ... ( # `  W
) ) )  -> 
( ( K  e.  ( 0 ... N
)  /\  L  e.  ( K ... N ) )  ->  ( ( W substr  <. 0 ,  N >. ) substr  <. K ,  L >. )  =  ( W substr  <. K ,  L >. ) ) )

Proof of Theorem swrdswrd0
StepHypRef Expression
1 simpl 454 . . . . . 6  |-  ( ( W  e. Word  V  /\  N  e.  ( 0 ... ( # `  W
) ) )  ->  W  e. Word  V )
2 simpr 458 . . . . . 6  |-  ( ( W  e. Word  V  /\  N  e.  ( 0 ... ( # `  W
) ) )  ->  N  e.  ( 0 ... ( # `  W
) ) )
3 elfznn0 11477 . . . . . . . 8  |-  ( N  e.  ( 0 ... ( # `  W
) )  ->  N  e.  NN0 )
4 0elfz 11479 . . . . . . . 8  |-  ( N  e.  NN0  ->  0  e.  ( 0 ... N
) )
53, 4syl 16 . . . . . . 7  |-  ( N  e.  ( 0 ... ( # `  W
) )  ->  0  e.  ( 0 ... N
) )
65adantl 463 . . . . . 6  |-  ( ( W  e. Word  V  /\  N  e.  ( 0 ... ( # `  W
) ) )  -> 
0  e.  ( 0 ... N ) )
71, 2, 63jca 1163 . . . . 5  |-  ( ( W  e. Word  V  /\  N  e.  ( 0 ... ( # `  W
) ) )  -> 
( W  e. Word  V  /\  N  e.  (
0 ... ( # `  W
) )  /\  0  e.  ( 0 ... N
) ) )
87adantr 462 . . . 4  |-  ( ( ( W  e. Word  V  /\  N  e.  (
0 ... ( # `  W
) ) )  /\  ( K  e.  (
0 ... N )  /\  L  e.  ( K ... N ) ) )  ->  ( W  e. Word  V  /\  N  e.  ( 0 ... ( # `  W ) )  /\  0  e.  ( 0 ... N ) ) )
9 elfzelz 11449 . . . . . . . . . 10  |-  ( N  e.  ( 0 ... ( # `  W
) )  ->  N  e.  ZZ )
10 zcn 10647 . . . . . . . . . . . 12  |-  ( N  e.  ZZ  ->  N  e.  CC )
1110subid1d 9704 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  ( N  -  0 )  =  N )
1211eqcomd 2446 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  N  =  ( N  - 
0 ) )
139, 12syl 16 . . . . . . . . 9  |-  ( N  e.  ( 0 ... ( # `  W
) )  ->  N  =  ( N  - 
0 ) )
1413adantl 463 . . . . . . . 8  |-  ( ( W  e. Word  V  /\  N  e.  ( 0 ... ( # `  W
) ) )  ->  N  =  ( N  -  0 ) )
1514oveq2d 6106 . . . . . . 7  |-  ( ( W  e. Word  V  /\  N  e.  ( 0 ... ( # `  W
) ) )  -> 
( 0 ... N
)  =  ( 0 ... ( N  - 
0 ) ) )
1615eleq2d 2508 . . . . . 6  |-  ( ( W  e. Word  V  /\  N  e.  ( 0 ... ( # `  W
) ) )  -> 
( K  e.  ( 0 ... N )  <-> 
K  e.  ( 0 ... ( N  - 
0 ) ) ) )
1714oveq2d 6106 . . . . . . 7  |-  ( ( W  e. Word  V  /\  N  e.  ( 0 ... ( # `  W
) ) )  -> 
( K ... N
)  =  ( K ... ( N  - 
0 ) ) )
1817eleq2d 2508 . . . . . 6  |-  ( ( W  e. Word  V  /\  N  e.  ( 0 ... ( # `  W
) ) )  -> 
( L  e.  ( K ... N )  <-> 
L  e.  ( K ... ( N  - 
0 ) ) ) )
1916, 18anbi12d 705 . . . . 5  |-  ( ( W  e. Word  V  /\  N  e.  ( 0 ... ( # `  W
) ) )  -> 
( ( K  e.  ( 0 ... N
)  /\  L  e.  ( K ... N ) )  <->  ( K  e.  ( 0 ... ( N  -  0 ) )  /\  L  e.  ( K ... ( N  -  0 ) ) ) ) )
2019biimpa 481 . . . 4  |-  ( ( ( W  e. Word  V  /\  N  e.  (
0 ... ( # `  W
) ) )  /\  ( K  e.  (
0 ... N )  /\  L  e.  ( K ... N ) ) )  ->  ( K  e.  ( 0 ... ( N  -  0 ) )  /\  L  e.  ( K ... ( N  -  0 ) ) ) )
21 swrdswrd 12350 . . . 4  |-  ( ( W  e. Word  V  /\  N  e.  ( 0 ... ( # `  W
) )  /\  0  e.  ( 0 ... N
) )  ->  (
( K  e.  ( 0 ... ( N  -  0 ) )  /\  L  e.  ( K ... ( N  -  0 ) ) )  ->  ( ( W substr  <. 0 ,  N >. ) substr  <. K ,  L >. )  =  ( W substr  <. ( 0  +  K
) ,  ( 0  +  L ) >.
) ) )
228, 20, 21sylc 60 . . 3  |-  ( ( ( W  e. Word  V  /\  N  e.  (
0 ... ( # `  W
) ) )  /\  ( K  e.  (
0 ... N )  /\  L  e.  ( K ... N ) ) )  ->  ( ( W substr  <. 0 ,  N >. ) substr  <. K ,  L >. )  =  ( W substr  <. (
0  +  K ) ,  ( 0  +  L ) >. )
)
23 elfzelz 11449 . . . . . . . . 9  |-  ( K  e.  ( 0 ... N )  ->  K  e.  ZZ )
2423zcnd 10744 . . . . . . . 8  |-  ( K  e.  ( 0 ... N )  ->  K  e.  CC )
2524adantr 462 . . . . . . 7  |-  ( ( K  e.  ( 0 ... N )  /\  L  e.  ( K ... N ) )  ->  K  e.  CC )
2625adantl 463 . . . . . 6  |-  ( ( ( W  e. Word  V  /\  N  e.  (
0 ... ( # `  W
) ) )  /\  ( K  e.  (
0 ... N )  /\  L  e.  ( K ... N ) ) )  ->  K  e.  CC )
2726addid2d 9566 . . . . 5  |-  ( ( ( W  e. Word  V  /\  N  e.  (
0 ... ( # `  W
) ) )  /\  ( K  e.  (
0 ... N )  /\  L  e.  ( K ... N ) ) )  ->  ( 0  +  K )  =  K )
28 elfzelz 11449 . . . . . . . . 9  |-  ( L  e.  ( K ... N )  ->  L  e.  ZZ )
2928zcnd 10744 . . . . . . . 8  |-  ( L  e.  ( K ... N )  ->  L  e.  CC )
3029adantl 463 . . . . . . 7  |-  ( ( K  e.  ( 0 ... N )  /\  L  e.  ( K ... N ) )  ->  L  e.  CC )
3130adantl 463 . . . . . 6  |-  ( ( ( W  e. Word  V  /\  N  e.  (
0 ... ( # `  W
) ) )  /\  ( K  e.  (
0 ... N )  /\  L  e.  ( K ... N ) ) )  ->  L  e.  CC )
3231addid2d 9566 . . . . 5  |-  ( ( ( W  e. Word  V  /\  N  e.  (
0 ... ( # `  W
) ) )  /\  ( K  e.  (
0 ... N )  /\  L  e.  ( K ... N ) ) )  ->  ( 0  +  L )  =  L )
3327, 32opeq12d 4064 . . . 4  |-  ( ( ( W  e. Word  V  /\  N  e.  (
0 ... ( # `  W
) ) )  /\  ( K  e.  (
0 ... N )  /\  L  e.  ( K ... N ) ) )  ->  <. ( 0  +  K ) ,  ( 0  +  L )
>.  =  <. K ,  L >. )
3433oveq2d 6106 . . 3  |-  ( ( ( W  e. Word  V  /\  N  e.  (
0 ... ( # `  W
) ) )  /\  ( K  e.  (
0 ... N )  /\  L  e.  ( K ... N ) ) )  ->  ( W substr  <. (
0  +  K ) ,  ( 0  +  L ) >. )  =  ( W substr  <. K ,  L >. ) )
3522, 34eqtrd 2473 . 2  |-  ( ( ( W  e. Word  V  /\  N  e.  (
0 ... ( # `  W
) ) )  /\  ( K  e.  (
0 ... N )  /\  L  e.  ( K ... N ) ) )  ->  ( ( W substr  <. 0 ,  N >. ) substr  <. K ,  L >. )  =  ( W substr  <. K ,  L >. ) )
3635ex 434 1  |-  ( ( W  e. Word  V  /\  N  e.  ( 0 ... ( # `  W
) ) )  -> 
( ( K  e.  ( 0 ... N
)  /\  L  e.  ( K ... N ) )  ->  ( ( W substr  <. 0 ,  N >. ) substr  <. K ,  L >. )  =  ( W substr  <. K ,  L >. ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761   <.cop 3880   ` cfv 5415  (class class class)co 6090   CCcc 9276   0cc0 9278    + caddc 9281    - cmin 9591   NN0cn0 10575   ZZcz 10642   ...cfz 11433   #chash 12099  Word cword 12217   substr csubstr 12221
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-card 8105  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-n0 10576  df-z 10643  df-uz 10858  df-fz 11434  df-fzo 11545  df-hash 12100  df-word 12225  df-substr 12229
This theorem is referenced by:  swrd0swrd0  12353
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