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Theorem swrd0val 12313
Description: Value of the subword extractor for left-anchored subwords. (Contributed by Stefan O'Rear, 24-Aug-2015.)
Assertion
Ref Expression
swrd0val  |-  ( ( S  e. Word  A  /\  L  e.  ( 0 ... ( # `  S
) ) )  -> 
( S substr  <. 0 ,  L >. )  =  ( S  |`  ( 0..^ L ) ) )

Proof of Theorem swrd0val
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 elfzelz 11449 . . . . . . . 8  |-  ( L  e.  ( 0 ... ( # `  S
) )  ->  L  e.  ZZ )
21adantl 463 . . . . . . 7  |-  ( ( S  e. Word  A  /\  L  e.  ( 0 ... ( # `  S
) ) )  ->  L  e.  ZZ )
32zcnd 10744 . . . . . 6  |-  ( ( S  e. Word  A  /\  L  e.  ( 0 ... ( # `  S
) ) )  ->  L  e.  CC )
43subid1d 9704 . . . . 5  |-  ( ( S  e. Word  A  /\  L  e.  ( 0 ... ( # `  S
) ) )  -> 
( L  -  0 )  =  L )
54oveq2d 6106 . . . 4  |-  ( ( S  e. Word  A  /\  L  e.  ( 0 ... ( # `  S
) ) )  -> 
( 0..^ ( L  -  0 ) )  =  ( 0..^ L ) )
65mpteq1d 4370 . . 3  |-  ( ( S  e. Word  A  /\  L  e.  ( 0 ... ( # `  S
) ) )  -> 
( x  e.  ( 0..^ ( L  - 
0 ) )  |->  ( S `  ( x  +  0 ) ) )  =  ( x  e.  ( 0..^ L )  |->  ( S `  ( x  +  0
) ) ) )
7 elfzoelz 11549 . . . . . . . 8  |-  ( x  e.  ( 0..^ L )  ->  x  e.  ZZ )
87zcnd 10744 . . . . . . 7  |-  ( x  e.  ( 0..^ L )  ->  x  e.  CC )
98addid1d 9565 . . . . . 6  |-  ( x  e.  ( 0..^ L )  ->  ( x  +  0 )  =  x )
109fveq2d 5692 . . . . 5  |-  ( x  e.  ( 0..^ L )  ->  ( S `  ( x  +  0 ) )  =  ( S `  x ) )
1110adantl 463 . . . 4  |-  ( ( ( S  e. Word  A  /\  L  e.  (
0 ... ( # `  S
) ) )  /\  x  e.  ( 0..^ L ) )  -> 
( S `  (
x  +  0 ) )  =  ( S `
 x ) )
1211mpteq2dva 4375 . . 3  |-  ( ( S  e. Word  A  /\  L  e.  ( 0 ... ( # `  S
) ) )  -> 
( x  e.  ( 0..^ L )  |->  ( S `  ( x  +  0 ) ) )  =  ( x  e.  ( 0..^ L )  |->  ( S `  x ) ) )
136, 12eqtrd 2473 . 2  |-  ( ( S  e. Word  A  /\  L  e.  ( 0 ... ( # `  S
) ) )  -> 
( x  e.  ( 0..^ ( L  - 
0 ) )  |->  ( S `  ( x  +  0 ) ) )  =  ( x  e.  ( 0..^ L )  |->  ( S `  x ) ) )
14 simpl 454 . . 3  |-  ( ( S  e. Word  A  /\  L  e.  ( 0 ... ( # `  S
) ) )  ->  S  e. Word  A )
15 elfzuz 11445 . . . . 5  |-  ( L  e.  ( 0 ... ( # `  S
) )  ->  L  e.  ( ZZ>= `  0 )
)
1615adantl 463 . . . 4  |-  ( ( S  e. Word  A  /\  L  e.  ( 0 ... ( # `  S
) ) )  ->  L  e.  ( ZZ>= ` 
0 ) )
17 eluzfz1 11454 . . . 4  |-  ( L  e.  ( ZZ>= `  0
)  ->  0  e.  ( 0 ... L
) )
1816, 17syl 16 . . 3  |-  ( ( S  e. Word  A  /\  L  e.  ( 0 ... ( # `  S
) ) )  -> 
0  e.  ( 0 ... L ) )
19 simpr 458 . . 3  |-  ( ( S  e. Word  A  /\  L  e.  ( 0 ... ( # `  S
) ) )  ->  L  e.  ( 0 ... ( # `  S
) ) )
20 swrdval2 12312 . . 3  |-  ( ( S  e. Word  A  /\  0  e.  ( 0 ... L )  /\  L  e.  ( 0 ... ( # `  S
) ) )  -> 
( S substr  <. 0 ,  L >. )  =  ( x  e.  ( 0..^ ( L  -  0 ) )  |->  ( S `
 ( x  + 
0 ) ) ) )
2114, 18, 19, 20syl3anc 1213 . 2  |-  ( ( S  e. Word  A  /\  L  e.  ( 0 ... ( # `  S
) ) )  -> 
( S substr  <. 0 ,  L >. )  =  ( x  e.  ( 0..^ ( L  -  0 ) )  |->  ( S `
 ( x  + 
0 ) ) ) )
22 wrdf 12236 . . . 4  |-  ( S  e. Word  A  ->  S : ( 0..^ (
# `  S )
) --> A )
2322adantr 462 . . 3  |-  ( ( S  e. Word  A  /\  L  e.  ( 0 ... ( # `  S
) ) )  ->  S : ( 0..^ (
# `  S )
) --> A )
24 elfzuz3 11446 . . . . 5  |-  ( L  e.  ( 0 ... ( # `  S
) )  ->  ( # `
 S )  e.  ( ZZ>= `  L )
)
2524adantl 463 . . . 4  |-  ( ( S  e. Word  A  /\  L  e.  ( 0 ... ( # `  S
) ) )  -> 
( # `  S )  e.  ( ZZ>= `  L
) )
26 fzoss2 11573 . . . 4  |-  ( (
# `  S )  e.  ( ZZ>= `  L )  ->  ( 0..^ L ) 
C_  ( 0..^ (
# `  S )
) )
2725, 26syl 16 . . 3  |-  ( ( S  e. Word  A  /\  L  e.  ( 0 ... ( # `  S
) ) )  -> 
( 0..^ L ) 
C_  ( 0..^ (
# `  S )
) )
2823, 27feqresmpt 5742 . 2  |-  ( ( S  e. Word  A  /\  L  e.  ( 0 ... ( # `  S
) ) )  -> 
( S  |`  (
0..^ L ) )  =  ( x  e.  ( 0..^ L ) 
|->  ( S `  x
) ) )
2913, 21, 283eqtr4d 2483 1  |-  ( ( S  e. Word  A  /\  L  e.  ( 0 ... ( # `  S
) ) )  -> 
( S substr  <. 0 ,  L >. )  =  ( S  |`  ( 0..^ L ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761    C_ wss 3325   <.cop 3880    e. cmpt 4347    |` cres 4838   -->wf 5411   ` cfv 5415  (class class class)co 6090   0cc0 9278    + caddc 9281    - cmin 9591   ZZcz 10642   ZZ>=cuz 10857   ...cfz 11433  ..^cfzo 11544   #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 2261  df-mo 2262  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:  swrd0len  12314  swrdn0  12320  swrdccat1  12347  psgnunilem5  15993  efgsres  16228  efgredlemd  16234  efgredlem  16237  iwrdsplit  26700  wrdsplex  26869  signsvtn0  26901  wwlkm1edg  30292
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