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Theorem swrdswrd0 28013
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 444 . . . . . 6  |-  ( ( W  e. Word  V  /\  N  e.  ( 0 ... ( # `  W
) ) )  ->  W  e. Word  V )
2 simpr 448 . . . . . 6  |-  ( ( W  e. Word  V  /\  N  e.  ( 0 ... ( # `  W
) ) )  ->  N  e.  ( 0 ... ( # `  W
) ) )
3 elfznn0 11039 . . . . . . . 8  |-  ( N  e.  ( 0 ... ( # `  W
) )  ->  N  e.  NN0 )
4 0elfz 27983 . . . . . . . 8  |-  ( N  e.  NN0  ->  0  e.  ( 0 ... N
) )
53, 4syl 16 . . . . . . 7  |-  ( N  e.  ( 0 ... ( # `  W
) )  ->  0  e.  ( 0 ... N
) )
65adantl 453 . . . . . 6  |-  ( ( W  e. Word  V  /\  N  e.  ( 0 ... ( # `  W
) ) )  -> 
0  e.  ( 0 ... N ) )
71, 2, 63jca 1134 . . . . 5  |-  ( ( W  e. Word  V  /\  N  e.  ( 0 ... ( # `  W
) ) )  -> 
( W  e. Word  V  /\  N  e.  (
0 ... ( # `  W
) )  /\  0  e.  ( 0 ... N
) ) )
87adantr 452 . . . 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 11015 . . . . . . . . . 10  |-  ( N  e.  ( 0 ... ( # `  W
) )  ->  N  e.  ZZ )
10 zcn 10243 . . . . . . . . . . . 12  |-  ( N  e.  ZZ  ->  N  e.  CC )
1110subid1d 9356 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  ( N  -  0 )  =  N )
1211eqcomd 2409 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  N  =  ( N  - 
0 ) )
139, 12syl 16 . . . . . . . . 9  |-  ( N  e.  ( 0 ... ( # `  W
) )  ->  N  =  ( N  - 
0 ) )
1413adantl 453 . . . . . . . 8  |-  ( ( W  e. Word  V  /\  N  e.  ( 0 ... ( # `  W
) ) )  ->  N  =  ( N  -  0 ) )
1514oveq2d 6056 . . . . . . 7  |-  ( ( W  e. Word  V  /\  N  e.  ( 0 ... ( # `  W
) ) )  -> 
( 0 ... N
)  =  ( 0 ... ( N  - 
0 ) ) )
1615eleq2d 2471 . . . . . 6  |-  ( ( W  e. Word  V  /\  N  e.  ( 0 ... ( # `  W
) ) )  -> 
( K  e.  ( 0 ... N )  <-> 
K  e.  ( 0 ... ( N  - 
0 ) ) ) )
1714oveq2d 6056 . . . . . . 7  |-  ( ( W  e. Word  V  /\  N  e.  ( 0 ... ( # `  W
) ) )  -> 
( K ... N
)  =  ( K ... ( N  - 
0 ) ) )
1817eleq2d 2471 . . . . . 6  |-  ( ( W  e. Word  V  /\  N  e.  ( 0 ... ( # `  W
) ) )  -> 
( L  e.  ( K ... N )  <-> 
L  e.  ( K ... ( N  - 
0 ) ) ) )
1916, 18anbi12d 692 . . . . 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 471 . . . 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 28011 . . . 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 58 . . 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 11015 . . . . . . . . 9  |-  ( K  e.  ( 0 ... N )  ->  K  e.  ZZ )
2423zcnd 10332 . . . . . . . 8  |-  ( K  e.  ( 0 ... N )  ->  K  e.  CC )
2524adantr 452 . . . . . . 7  |-  ( ( K  e.  ( 0 ... N )  /\  L  e.  ( K ... N ) )  ->  K  e.  CC )
2625adantl 453 . . . . . 6  |-  ( ( ( W  e. Word  V  /\  N  e.  (
0 ... ( # `  W
) ) )  /\  ( K  e.  (
0 ... N )  /\  L  e.  ( K ... N ) ) )  ->  K  e.  CC )
2726addid2d 9223 . . . . 5  |-  ( ( ( W  e. Word  V  /\  N  e.  (
0 ... ( # `  W
) ) )  /\  ( K  e.  (
0 ... N )  /\  L  e.  ( K ... N ) ) )  ->  ( 0  +  K )  =  K )
28 elfzelz 11015 . . . . . . . . 9  |-  ( L  e.  ( K ... N )  ->  L  e.  ZZ )
2928zcnd 10332 . . . . . . . 8  |-  ( L  e.  ( K ... N )  ->  L  e.  CC )
3029adantl 453 . . . . . . 7  |-  ( ( K  e.  ( 0 ... N )  /\  L  e.  ( K ... N ) )  ->  L  e.  CC )
3130adantl 453 . . . . . 6  |-  ( ( ( W  e. Word  V  /\  N  e.  (
0 ... ( # `  W
) ) )  /\  ( K  e.  (
0 ... N )  /\  L  e.  ( K ... N ) ) )  ->  L  e.  CC )
3231addid2d 9223 . . . . 5  |-  ( ( ( W  e. Word  V  /\  N  e.  (
0 ... ( # `  W
) ) )  /\  ( K  e.  (
0 ... N )  /\  L  e.  ( K ... N ) ) )  ->  ( 0  +  L )  =  L )
3327, 32opeq12d 3952 . . . 4  |-  ( ( ( W  e. Word  V  /\  N  e.  (
0 ... ( # `  W
) ) )  /\  ( K  e.  (
0 ... N )  /\  L  e.  ( K ... N ) ) )  ->  <. ( 0  +  K ) ,  ( 0  +  L )
>.  =  <. K ,  L >. )
3433oveq2d 6056 . . 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 2436 . 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 424 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 set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   <.cop 3777   ` cfv 5413  (class class class)co 6040   CCcc 8944   0cc0 8946    + caddc 8949    - cmin 9247   NN0cn0 10177   ZZcz 10238   ...cfz 10999   #chash 11573  Word cword 11672   substr csubstr 11675
This theorem is referenced by:  swrd0swrd0  28014
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-fzo 11091  df-hash 11574  df-word 11678  df-substr 11681
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