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Theorem swrd0val 11470
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 10814 . . . . . . . 8  |-  ( L  e.  ( 0 ... ( # `  S
) )  ->  L  e.  ZZ )
21adantl 452 . . . . . . 7  |-  ( ( S  e. Word  A  /\  L  e.  ( 0 ... ( # `  S
) ) )  ->  L  e.  ZZ )
32zcnd 10134 . . . . . 6  |-  ( ( S  e. Word  A  /\  L  e.  ( 0 ... ( # `  S
) ) )  ->  L  e.  CC )
43subid1d 9162 . . . . 5  |-  ( ( S  e. Word  A  /\  L  e.  ( 0 ... ( # `  S
) ) )  -> 
( L  -  0 )  =  L )
54oveq2d 5890 . . . 4  |-  ( ( S  e. Word  A  /\  L  e.  ( 0 ... ( # `  S
) ) )  -> 
( 0..^ ( L  -  0 ) )  =  ( 0..^ L ) )
6 mpteq1 4116 . . . 4  |-  ( ( 0..^ ( L  - 
0 ) )  =  ( 0..^ L )  ->  ( x  e.  ( 0..^ ( L  -  0 ) ) 
|->  ( S `  (
x  +  0 ) ) )  =  ( x  e.  ( 0..^ L )  |->  ( S `
 ( x  + 
0 ) ) ) )
75, 6syl 15 . . 3  |-  ( ( S  e. Word  A  /\  L  e.  ( 0 ... ( # `  S
) ) )  -> 
( x  e.  ( 0..^ ( L  - 
0 ) )  |->  ( S `  ( x  +  0 ) ) )  =  ( x  e.  ( 0..^ L )  |->  ( S `  ( x  +  0
) ) ) )
8 elfzoelz 10891 . . . . . . . 8  |-  ( x  e.  ( 0..^ L )  ->  x  e.  ZZ )
98zcnd 10134 . . . . . . 7  |-  ( x  e.  ( 0..^ L )  ->  x  e.  CC )
109addid1d 9028 . . . . . 6  |-  ( x  e.  ( 0..^ L )  ->  ( x  +  0 )  =  x )
1110fveq2d 5545 . . . . 5  |-  ( x  e.  ( 0..^ L )  ->  ( S `  ( x  +  0 ) )  =  ( S `  x ) )
1211adantl 452 . . . 4  |-  ( ( ( S  e. Word  A  /\  L  e.  (
0 ... ( # `  S
) ) )  /\  x  e.  ( 0..^ L ) )  -> 
( S `  (
x  +  0 ) )  =  ( S `
 x ) )
1312mpteq2dva 4122 . . 3  |-  ( ( S  e. Word  A  /\  L  e.  ( 0 ... ( # `  S
) ) )  -> 
( x  e.  ( 0..^ L )  |->  ( S `  ( x  +  0 ) ) )  =  ( x  e.  ( 0..^ L )  |->  ( S `  x ) ) )
147, 13eqtrd 2328 . 2  |-  ( ( S  e. Word  A  /\  L  e.  ( 0 ... ( # `  S
) ) )  -> 
( x  e.  ( 0..^ ( L  - 
0 ) )  |->  ( S `  ( x  +  0 ) ) )  =  ( x  e.  ( 0..^ L )  |->  ( S `  x ) ) )
15 simpl 443 . . 3  |-  ( ( S  e. Word  A  /\  L  e.  ( 0 ... ( # `  S
) ) )  ->  S  e. Word  A )
16 elfzuz 10810 . . . . 5  |-  ( L  e.  ( 0 ... ( # `  S
) )  ->  L  e.  ( ZZ>= `  0 )
)
1716adantl 452 . . . 4  |-  ( ( S  e. Word  A  /\  L  e.  ( 0 ... ( # `  S
) ) )  ->  L  e.  ( ZZ>= ` 
0 ) )
18 eluzfz1 10819 . . . 4  |-  ( L  e.  ( ZZ>= `  0
)  ->  0  e.  ( 0 ... L
) )
1917, 18syl 15 . . 3  |-  ( ( S  e. Word  A  /\  L  e.  ( 0 ... ( # `  S
) ) )  -> 
0  e.  ( 0 ... L ) )
20 simpr 447 . . 3  |-  ( ( S  e. Word  A  /\  L  e.  ( 0 ... ( # `  S
) ) )  ->  L  e.  ( 0 ... ( # `  S
) ) )
21 swrdval2 11469 . . 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 ) ) ) )
2215, 19, 20, 21syl3anc 1182 . 2  |-  ( ( S  e. Word  A  /\  L  e.  ( 0 ... ( # `  S
) ) )  -> 
( S substr  <. 0 ,  L >. )  =  ( x  e.  ( 0..^ ( L  -  0 ) )  |->  ( S `
 ( x  + 
0 ) ) ) )
23 wrdf 11435 . . . 4  |-  ( S  e. Word  A  ->  S : ( 0..^ (
# `  S )
) --> A )
2423adantr 451 . . 3  |-  ( ( S  e. Word  A  /\  L  e.  ( 0 ... ( # `  S
) ) )  ->  S : ( 0..^ (
# `  S )
) --> A )
25 elfzuz3 10811 . . . . 5  |-  ( L  e.  ( 0 ... ( # `  S
) )  ->  ( # `
 S )  e.  ( ZZ>= `  L )
)
2625adantl 452 . . . 4  |-  ( ( S  e. Word  A  /\  L  e.  ( 0 ... ( # `  S
) ) )  -> 
( # `  S )  e.  ( ZZ>= `  L
) )
27 fzoss2 10913 . . . 4  |-  ( (
# `  S )  e.  ( ZZ>= `  L )  ->  ( 0..^ L ) 
C_  ( 0..^ (
# `  S )
) )
2826, 27syl 15 . . 3  |-  ( ( S  e. Word  A  /\  L  e.  ( 0 ... ( # `  S
) ) )  -> 
( 0..^ L ) 
C_  ( 0..^ (
# `  S )
) )
2924, 28feqresmpt 5592 . 2  |-  ( ( S  e. Word  A  /\  L  e.  ( 0 ... ( # `  S
) ) )  -> 
( S  |`  (
0..^ L ) )  =  ( x  e.  ( 0..^ L ) 
|->  ( S `  x
) ) )
3014, 22, 293eqtr4d 2338 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 358    = wceq 1632    e. wcel 1696    C_ wss 3165   <.cop 3656    e. cmpt 4093    |` cres 4707   -->wf 5267   ` cfv 5271  (class class class)co 5874   0cc0 8753    + caddc 8756    - cmin 9053   ZZcz 10040   ZZ>=cuz 10246   ...cfz 10798  ..^cfzo 10886   #chash 11353  Word cword 11419   substr csubstr 11422
This theorem is referenced by:  swrd0len  11471  swrdccat1  11476  efgsres  15063  efgredlemd  15069  efgredlem  15072  psgnunilem5  27520
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-fzo 10887  df-hash 11354  df-word 11425  df-substr 11428
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