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Theorem swrd0val 11723
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 11015 . . . . . . . 8  |-  ( L  e.  ( 0 ... ( # `  S
) )  ->  L  e.  ZZ )
21adantl 453 . . . . . . 7  |-  ( ( S  e. Word  A  /\  L  e.  ( 0 ... ( # `  S
) ) )  ->  L  e.  ZZ )
32zcnd 10332 . . . . . 6  |-  ( ( S  e. Word  A  /\  L  e.  ( 0 ... ( # `  S
) ) )  ->  L  e.  CC )
43subid1d 9356 . . . . 5  |-  ( ( S  e. Word  A  /\  L  e.  ( 0 ... ( # `  S
) ) )  -> 
( L  -  0 )  =  L )
54oveq2d 6056 . . . 4  |-  ( ( S  e. Word  A  /\  L  e.  ( 0 ... ( # `  S
) ) )  -> 
( 0..^ ( L  -  0 ) )  =  ( 0..^ L ) )
65mpteq1d 4250 . . 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 11095 . . . . . . . 8  |-  ( x  e.  ( 0..^ L )  ->  x  e.  ZZ )
87zcnd 10332 . . . . . . 7  |-  ( x  e.  ( 0..^ L )  ->  x  e.  CC )
98addid1d 9222 . . . . . 6  |-  ( x  e.  ( 0..^ L )  ->  ( x  +  0 )  =  x )
109fveq2d 5691 . . . . 5  |-  ( x  e.  ( 0..^ L )  ->  ( S `  ( x  +  0 ) )  =  ( S `  x ) )
1110adantl 453 . . . 4  |-  ( ( ( S  e. Word  A  /\  L  e.  (
0 ... ( # `  S
) ) )  /\  x  e.  ( 0..^ L ) )  -> 
( S `  (
x  +  0 ) )  =  ( S `
 x ) )
1211mpteq2dva 4255 . . 3  |-  ( ( S  e. Word  A  /\  L  e.  ( 0 ... ( # `  S
) ) )  -> 
( x  e.  ( 0..^ L )  |->  ( S `  ( x  +  0 ) ) )  =  ( x  e.  ( 0..^ L )  |->  ( S `  x ) ) )
136, 12eqtrd 2436 . 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 444 . . 3  |-  ( ( S  e. Word  A  /\  L  e.  ( 0 ... ( # `  S
) ) )  ->  S  e. Word  A )
15 elfzuz 11011 . . . . 5  |-  ( L  e.  ( 0 ... ( # `  S
) )  ->  L  e.  ( ZZ>= `  0 )
)
1615adantl 453 . . . 4  |-  ( ( S  e. Word  A  /\  L  e.  ( 0 ... ( # `  S
) ) )  ->  L  e.  ( ZZ>= ` 
0 ) )
17 eluzfz1 11020 . . . 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 448 . . 3  |-  ( ( S  e. Word  A  /\  L  e.  ( 0 ... ( # `  S
) ) )  ->  L  e.  ( 0 ... ( # `  S
) ) )
20 swrdval2 11722 . . 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 1184 . 2  |-  ( ( S  e. Word  A  /\  L  e.  ( 0 ... ( # `  S
) ) )  -> 
( S substr  <. 0 ,  L >. )  =  ( x  e.  ( 0..^ ( L  -  0 ) )  |->  ( S `
 ( x  + 
0 ) ) ) )
22 wrdf 11688 . . . 4  |-  ( S  e. Word  A  ->  S : ( 0..^ (
# `  S )
) --> A )
2322adantr 452 . . 3  |-  ( ( S  e. Word  A  /\  L  e.  ( 0 ... ( # `  S
) ) )  ->  S : ( 0..^ (
# `  S )
) --> A )
24 elfzuz3 11012 . . . . 5  |-  ( L  e.  ( 0 ... ( # `  S
) )  ->  ( # `
 S )  e.  ( ZZ>= `  L )
)
2524adantl 453 . . . 4  |-  ( ( S  e. Word  A  /\  L  e.  ( 0 ... ( # `  S
) ) )  -> 
( # `  S )  e.  ( ZZ>= `  L
) )
26 fzoss2 11118 . . . 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 5739 . 2  |-  ( ( S  e. Word  A  /\  L  e.  ( 0 ... ( # `  S
) ) )  -> 
( S  |`  (
0..^ L ) )  =  ( x  e.  ( 0..^ L ) 
|->  ( S `  x
) ) )
2913, 21, 283eqtr4d 2446 1  |-  ( ( S  e. Word  A  /\  L  e.  ( 0 ... ( # `  S
) ) )  -> 
( S substr  <. 0 ,  L >. )  =  ( S  |`  ( 0..^ L ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721    C_ wss 3280   <.cop 3777    e. cmpt 4226    |` cres 4839   -->wf 5409   ` cfv 5413  (class class class)co 6040   0cc0 8946    + caddc 8949    - cmin 9247   ZZcz 10238   ZZ>=cuz 10444   ...cfz 10999  ..^cfzo 11090   #chash 11573  Word cword 11672   substr csubstr 11675
This theorem is referenced by:  swrd0len  11724  swrdccat1  11729  efgsres  15325  efgredlemd  15331  efgredlem  15334  psgnunilem5  27285  swrd0swrd  28009
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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