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Theorem swrdcl 11542
Description: Closure of the subword extractor. (Contributed by Stefan O'Rear, 16-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
Assertion
Ref Expression
swrdcl  |-  ( S  e. Word  A  ->  ( S substr  <. F ,  L >. )  e. Word  A )

Proof of Theorem swrdcl
Dummy variables  s 
b  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2418 . 2  |-  ( ( S substr  <. F ,  L >. )  =  (/)  ->  (
( S substr  <. F ,  L >. )  e. Word  A  <->  (/)  e. Word  A ) )
2 n0 3540 . . . 4  |-  ( ( S substr  <. F ,  L >. )  =/=  (/)  <->  E. x  x  e.  ( S substr  <. F ,  L >. ) )
3 df-substr 11502 . . . . . . 7  |- substr  =  ( s  e.  _V , 
b  e.  ( ZZ 
X.  ZZ )  |->  if ( ( ( 1st `  b )..^ ( 2nd `  b ) )  C_  dom  s ,  ( x  e.  ( 0..^ ( ( 2nd `  b
)  -  ( 1st `  b ) ) ) 
|->  ( s `  (
x  +  ( 1st `  b ) ) ) ) ,  (/) ) )
43elmpt2cl2 6147 . . . . . 6  |-  ( x  e.  ( S substr  <. F ,  L >. )  ->  <. F ,  L >.  e.  ( ZZ 
X.  ZZ ) )
5 opelxp 4798 . . . . . 6  |-  ( <. F ,  L >.  e.  ( ZZ  X.  ZZ ) 
<->  ( F  e.  ZZ  /\  L  e.  ZZ ) )
64, 5sylib 188 . . . . 5  |-  ( x  e.  ( S substr  <. F ,  L >. )  ->  ( F  e.  ZZ  /\  L  e.  ZZ ) )
76exlimiv 1634 . . . 4  |-  ( E. x  x  e.  ( S substr  <. F ,  L >. )  ->  ( F  e.  ZZ  /\  L  e.  ZZ ) )
82, 7sylbi 187 . . 3  |-  ( ( S substr  <. F ,  L >. )  =/=  (/)  ->  ( F  e.  ZZ  /\  L  e.  ZZ ) )
9 swrdval 11540 . . . . 5  |-  ( ( S  e. Word  A  /\  F  e.  ZZ  /\  L  e.  ZZ )  ->  ( S substr  <. F ,  L >. )  =  if ( ( F..^ L ) 
C_  dom  S , 
( x  e.  ( 0..^ ( L  -  F ) )  |->  ( S `  ( x  +  F ) ) ) ,  (/) ) )
10 wrdf 11509 . . . . . . . . . . 11  |-  ( S  e. Word  A  ->  S : ( 0..^ (
# `  S )
) --> A )
11103ad2ant1 976 . . . . . . . . . 10  |-  ( ( S  e. Word  A  /\  F  e.  ZZ  /\  L  e.  ZZ )  ->  S : ( 0..^ (
# `  S )
) --> A )
1211ad2antrr 706 . . . . . . . . 9  |-  ( ( ( ( S  e. Word  A  /\  F  e.  ZZ  /\  L  e.  ZZ )  /\  ( F..^ L
)  C_  dom  S )  /\  x  e.  ( 0..^ ( L  -  F ) ) )  ->  S : ( 0..^ ( # `  S
) ) --> A )
13 simplr 731 . . . . . . . . . . 11  |-  ( ( ( ( S  e. Word  A  /\  F  e.  ZZ  /\  L  e.  ZZ )  /\  ( F..^ L
)  C_  dom  S )  /\  x  e.  ( 0..^ ( L  -  F ) ) )  ->  ( F..^ L
)  C_  dom  S )
14 simpr 447 . . . . . . . . . . . 12  |-  ( ( ( ( S  e. Word  A  /\  F  e.  ZZ  /\  L  e.  ZZ )  /\  ( F..^ L
)  C_  dom  S )  /\  x  e.  ( 0..^ ( L  -  F ) ) )  ->  x  e.  ( 0..^ ( L  -  F ) ) )
15 simpll3 996 . . . . . . . . . . . 12  |-  ( ( ( ( S  e. Word  A  /\  F  e.  ZZ  /\  L  e.  ZZ )  /\  ( F..^ L
)  C_  dom  S )  /\  x  e.  ( 0..^ ( L  -  F ) ) )  ->  L  e.  ZZ )
16 simpll2 995 . . . . . . . . . . . 12  |-  ( ( ( ( S  e. Word  A  /\  F  e.  ZZ  /\  L  e.  ZZ )  /\  ( F..^ L
)  C_  dom  S )  /\  x  e.  ( 0..^ ( L  -  F ) ) )  ->  F  e.  ZZ )
17 fzoaddel2 10996 . . . . . . . . . . . 12  |-  ( ( x  e.  ( 0..^ ( L  -  F
) )  /\  L  e.  ZZ  /\  F  e.  ZZ )  ->  (
x  +  F )  e.  ( F..^ L
) )
1814, 15, 16, 17syl3anc 1182 . . . . . . . . . . 11  |-  ( ( ( ( S  e. Word  A  /\  F  e.  ZZ  /\  L  e.  ZZ )  /\  ( F..^ L
)  C_  dom  S )  /\  x  e.  ( 0..^ ( L  -  F ) ) )  ->  ( x  +  F )  e.  ( F..^ L ) )
1913, 18sseldd 3257 . . . . . . . . . 10  |-  ( ( ( ( S  e. Word  A  /\  F  e.  ZZ  /\  L  e.  ZZ )  /\  ( F..^ L
)  C_  dom  S )  /\  x  e.  ( 0..^ ( L  -  F ) ) )  ->  ( x  +  F )  e.  dom  S )
20 fdm 5473 . . . . . . . . . . 11  |-  ( S : ( 0..^ (
# `  S )
) --> A  ->  dom  S  =  ( 0..^ (
# `  S )
) )
2112, 20syl 15 . . . . . . . . . 10  |-  ( ( ( ( S  e. Word  A  /\  F  e.  ZZ  /\  L  e.  ZZ )  /\  ( F..^ L
)  C_  dom  S )  /\  x  e.  ( 0..^ ( L  -  F ) ) )  ->  dom  S  =  ( 0..^ ( # `  S
) ) )
2219, 21eleqtrd 2434 . . . . . . . . 9  |-  ( ( ( ( S  e. Word  A  /\  F  e.  ZZ  /\  L  e.  ZZ )  /\  ( F..^ L
)  C_  dom  S )  /\  x  e.  ( 0..^ ( L  -  F ) ) )  ->  ( x  +  F )  e.  ( 0..^ ( # `  S
) ) )
23 ffvelrn 5743 . . . . . . . . 9  |-  ( ( S : ( 0..^ ( # `  S
) ) --> A  /\  ( x  +  F
)  e.  ( 0..^ ( # `  S
) ) )  -> 
( S `  (
x  +  F ) )  e.  A )
2412, 22, 23syl2anc 642 . . . . . . . 8  |-  ( ( ( ( S  e. Word  A  /\  F  e.  ZZ  /\  L  e.  ZZ )  /\  ( F..^ L
)  C_  dom  S )  /\  x  e.  ( 0..^ ( L  -  F ) ) )  ->  ( S `  ( x  +  F
) )  e.  A
)
25 eqid 2358 . . . . . . . 8  |-  ( x  e.  ( 0..^ ( L  -  F ) )  |->  ( S `  ( x  +  F
) ) )  =  ( x  e.  ( 0..^ ( L  -  F ) )  |->  ( S `  ( x  +  F ) ) )
2624, 25fmptd 5764 . . . . . . 7  |-  ( ( ( S  e. Word  A  /\  F  e.  ZZ  /\  L  e.  ZZ )  /\  ( F..^ L
)  C_  dom  S )  ->  ( x  e.  ( 0..^ ( L  -  F ) ) 
|->  ( S `  (
x  +  F ) ) ) : ( 0..^ ( L  -  F ) ) --> A )
27 iswrdi 11507 . . . . . . 7  |-  ( ( x  e.  ( 0..^ ( L  -  F
) )  |->  ( S `
 ( x  +  F ) ) ) : ( 0..^ ( L  -  F ) ) --> A  ->  (
x  e.  ( 0..^ ( L  -  F
) )  |->  ( S `
 ( x  +  F ) ) )  e. Word  A )
2826, 27syl 15 . . . . . 6  |-  ( ( ( S  e. Word  A  /\  F  e.  ZZ  /\  L  e.  ZZ )  /\  ( F..^ L
)  C_  dom  S )  ->  ( x  e.  ( 0..^ ( L  -  F ) ) 
|->  ( S `  (
x  +  F ) ) )  e. Word  A
)
29 wrd0 11508 . . . . . . 7  |-  (/)  e. Word  A
3029a1i 10 . . . . . 6  |-  ( ( ( S  e. Word  A  /\  F  e.  ZZ  /\  L  e.  ZZ )  /\  -.  ( F..^ L )  C_  dom  S )  ->  (/)  e. Word  A
)
3128, 30ifclda 3668 . . . . 5  |-  ( ( S  e. Word  A  /\  F  e.  ZZ  /\  L  e.  ZZ )  ->  if ( ( F..^ L
)  C_  dom  S , 
( x  e.  ( 0..^ ( L  -  F ) )  |->  ( S `  ( x  +  F ) ) ) ,  (/) )  e. Word  A )
329, 31eqeltrd 2432 . . . 4  |-  ( ( S  e. Word  A  /\  F  e.  ZZ  /\  L  e.  ZZ )  ->  ( S substr  <. F ,  L >. )  e. Word  A )
33323expb 1152 . . 3  |-  ( ( S  e. Word  A  /\  ( F  e.  ZZ  /\  L  e.  ZZ ) )  ->  ( S substr  <. F ,  L >. )  e. Word  A )
348, 33sylan2 460 . 2  |-  ( ( S  e. Word  A  /\  ( S substr  <. F ,  L >. )  =/=  (/) )  -> 
( S substr  <. F ,  L >. )  e. Word  A
)
3529a1i 10 . 2  |-  ( S  e. Word  A  ->  (/)  e. Word  A
)
361, 34, 35pm2.61ne 2596 1  |-  ( S  e. Word  A  ->  ( S substr  <. F ,  L >. )  e. Word  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934   E.wex 1541    = wceq 1642    e. wcel 1710    =/= wne 2521   _Vcvv 2864    C_ wss 3228   (/)c0 3531   ifcif 3641   <.cop 3719    e. cmpt 4156    X. cxp 4766   dom cdm 4768   -->wf 5330   ` cfv 5334  (class class class)co 5942   1stc1st 6204   2ndc2nd 6205   0cc0 8824    + caddc 8827    - cmin 9124   ZZcz 10113  ..^cfzo 10959   #chash 11427  Word cword 11493   substr csubstr 11496
This theorem is referenced by:  swrdid  11548  ccatswrd  11549  swrdccat2  11551  splcl  11557  spllen  11559  splfv1  11560  splfv2a  11561  splval2  11562  swrds1  11563  wrdind  11567  gsumspl  14559  efgsres  15140  efgredleme  15145  efgredlemc  15147  efgcpbllemb  15157  frgpuplem  15174  psgnunilem5  26740  psgnunilem2  26741
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-cnex 8880  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900  ax-pre-mulgt0 8901
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-int 3942  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-1st 6206  df-2nd 6207  df-riota 6388  df-recs 6472  df-rdg 6507  df-1o 6563  df-oadd 6567  df-er 6744  df-en 6949  df-dom 6950  df-sdom 6951  df-fin 6952  df-card 7659  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-sub 9126  df-neg 9127  df-nn 9834  df-n0 10055  df-z 10114  df-uz 10320  df-fz 10872  df-fzo 10960  df-hash 11428  df-word 11499  df-substr 11502
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