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Theorem swrdval 12827
Description: Value of a subword. (Contributed by Stefan O'Rear, 15-Aug-2015.)
Assertion
Ref Expression
swrdval  |-  ( ( S  e.  V  /\  F  e.  ZZ  /\  L  e.  ZZ )  ->  ( S substr  <. F ,  L >. )  =  if ( ( F..^ L ) 
C_  dom  S , 
( x  e.  ( 0..^ ( L  -  F ) )  |->  ( S `  ( x  +  F ) ) ) ,  (/) ) )
Distinct variable groups:    x, S    x, F    x, L    x, V

Proof of Theorem swrdval
Dummy variables  s 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-substr 12715 . . 3  |- 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 ) ) ) ) ,  (/) ) )
21a1i 11 . 2  |-  ( ( S  e.  V  /\  F  e.  ZZ  /\  L  e.  ZZ )  -> 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 ) ) ) ) ,  (/) ) ) )
3 simprl 772 . . 3  |-  ( ( ( S  e.  V  /\  F  e.  ZZ  /\  L  e.  ZZ )  /\  ( s  =  S  /\  b  = 
<. F ,  L >. ) )  ->  s  =  S )
4 fveq2 5879 . . . . 5  |-  ( b  =  <. F ,  L >.  ->  ( 1st `  b
)  =  ( 1st `  <. F ,  L >. ) )
54adantl 473 . . . 4  |-  ( ( s  =  S  /\  b  =  <. F ,  L >. )  ->  ( 1st `  b )  =  ( 1st `  <. F ,  L >. )
)
6 op1stg 6824 . . . . 5  |-  ( ( F  e.  ZZ  /\  L  e.  ZZ )  ->  ( 1st `  <. F ,  L >. )  =  F )
763adant1 1048 . . . 4  |-  ( ( S  e.  V  /\  F  e.  ZZ  /\  L  e.  ZZ )  ->  ( 1st `  <. F ,  L >. )  =  F )
85, 7sylan9eqr 2527 . . 3  |-  ( ( ( S  e.  V  /\  F  e.  ZZ  /\  L  e.  ZZ )  /\  ( s  =  S  /\  b  = 
<. F ,  L >. ) )  ->  ( 1st `  b )  =  F )
9 fveq2 5879 . . . . 5  |-  ( b  =  <. F ,  L >.  ->  ( 2nd `  b
)  =  ( 2nd `  <. F ,  L >. ) )
109adantl 473 . . . 4  |-  ( ( s  =  S  /\  b  =  <. F ,  L >. )  ->  ( 2nd `  b )  =  ( 2nd `  <. F ,  L >. )
)
11 op2ndg 6825 . . . . 5  |-  ( ( F  e.  ZZ  /\  L  e.  ZZ )  ->  ( 2nd `  <. F ,  L >. )  =  L )
12113adant1 1048 . . . 4  |-  ( ( S  e.  V  /\  F  e.  ZZ  /\  L  e.  ZZ )  ->  ( 2nd `  <. F ,  L >. )  =  L )
1310, 12sylan9eqr 2527 . . 3  |-  ( ( ( S  e.  V  /\  F  e.  ZZ  /\  L  e.  ZZ )  /\  ( s  =  S  /\  b  = 
<. F ,  L >. ) )  ->  ( 2nd `  b )  =  L )
14 simp2 1031 . . . . . 6  |-  ( ( s  =  S  /\  ( 1st `  b )  =  F  /\  ( 2nd `  b )  =  L )  ->  ( 1st `  b )  =  F )
15 simp3 1032 . . . . . 6  |-  ( ( s  =  S  /\  ( 1st `  b )  =  F  /\  ( 2nd `  b )  =  L )  ->  ( 2nd `  b )  =  L )
1614, 15oveq12d 6326 . . . . 5  |-  ( ( s  =  S  /\  ( 1st `  b )  =  F  /\  ( 2nd `  b )  =  L )  ->  (
( 1st `  b
)..^ ( 2nd `  b
) )  =  ( F..^ L ) )
17 simp1 1030 . . . . . 6  |-  ( ( s  =  S  /\  ( 1st `  b )  =  F  /\  ( 2nd `  b )  =  L )  ->  s  =  S )
1817dmeqd 5042 . . . . 5  |-  ( ( s  =  S  /\  ( 1st `  b )  =  F  /\  ( 2nd `  b )  =  L )  ->  dom  s  =  dom  S )
1916, 18sseq12d 3447 . . . 4  |-  ( ( s  =  S  /\  ( 1st `  b )  =  F  /\  ( 2nd `  b )  =  L )  ->  (
( ( 1st `  b
)..^ ( 2nd `  b
) )  C_  dom  s 
<->  ( F..^ L ) 
C_  dom  S )
)
2015, 14oveq12d 6326 . . . . . 6  |-  ( ( s  =  S  /\  ( 1st `  b )  =  F  /\  ( 2nd `  b )  =  L )  ->  (
( 2nd `  b
)  -  ( 1st `  b ) )  =  ( L  -  F
) )
2120oveq2d 6324 . . . . 5  |-  ( ( s  =  S  /\  ( 1st `  b )  =  F  /\  ( 2nd `  b )  =  L )  ->  (
0..^ ( ( 2nd `  b )  -  ( 1st `  b ) ) )  =  ( 0..^ ( L  -  F
) ) )
2214oveq2d 6324 . . . . . 6  |-  ( ( s  =  S  /\  ( 1st `  b )  =  F  /\  ( 2nd `  b )  =  L )  ->  (
x  +  ( 1st `  b ) )  =  ( x  +  F
) )
2317, 22fveq12d 5885 . . . . 5  |-  ( ( s  =  S  /\  ( 1st `  b )  =  F  /\  ( 2nd `  b )  =  L )  ->  (
s `  ( x  +  ( 1st `  b
) ) )  =  ( S `  (
x  +  F ) ) )
2421, 23mpteq12dv 4474 . . . 4  |-  ( ( s  =  S  /\  ( 1st `  b )  =  F  /\  ( 2nd `  b )  =  L )  ->  (
x  e.  ( 0..^ ( ( 2nd `  b
)  -  ( 1st `  b ) ) ) 
|->  ( s `  (
x  +  ( 1st `  b ) ) ) )  =  ( x  e.  ( 0..^ ( L  -  F ) )  |->  ( S `  ( x  +  F
) ) ) )
2519, 24ifbieq1d 3895 . . 3  |-  ( ( s  =  S  /\  ( 1st `  b )  =  F  /\  ( 2nd `  b )  =  L )  ->  if ( ( ( 1st `  b )..^ ( 2nd `  b ) )  C_  dom  s ,  ( x  e.  ( 0..^ ( ( 2nd `  b
)  -  ( 1st `  b ) ) ) 
|->  ( s `  (
x  +  ( 1st `  b ) ) ) ) ,  (/) )  =  if ( ( F..^ L )  C_  dom  S ,  ( x  e.  ( 0..^ ( L  -  F ) ) 
|->  ( S `  (
x  +  F ) ) ) ,  (/) ) )
263, 8, 13, 25syl3anc 1292 . 2  |-  ( ( ( S  e.  V  /\  F  e.  ZZ  /\  L  e.  ZZ )  /\  ( s  =  S  /\  b  = 
<. F ,  L >. ) )  ->  if (
( ( 1st `  b
)..^ ( 2nd `  b
) )  C_  dom  s ,  ( x  e.  ( 0..^ ( ( 2nd `  b )  -  ( 1st `  b
) ) )  |->  ( s `  ( x  +  ( 1st `  b
) ) ) ) ,  (/) )  =  if ( ( F..^ L
)  C_  dom  S , 
( x  e.  ( 0..^ ( L  -  F ) )  |->  ( S `  ( x  +  F ) ) ) ,  (/) ) )
27 elex 3040 . . 3  |-  ( S  e.  V  ->  S  e.  _V )
28273ad2ant1 1051 . 2  |-  ( ( S  e.  V  /\  F  e.  ZZ  /\  L  e.  ZZ )  ->  S  e.  _V )
29 opelxpi 4871 . . 3  |-  ( ( F  e.  ZZ  /\  L  e.  ZZ )  -> 
<. F ,  L >.  e.  ( ZZ  X.  ZZ ) )
30293adant1 1048 . 2  |-  ( ( S  e.  V  /\  F  e.  ZZ  /\  L  e.  ZZ )  ->  <. F ,  L >.  e.  ( ZZ 
X.  ZZ ) )
31 ovex 6336 . . . . 5  |-  ( 0..^ ( L  -  F
) )  e.  _V
3231mptex 6152 . . . 4  |-  ( x  e.  ( 0..^ ( L  -  F ) )  |->  ( S `  ( x  +  F
) ) )  e. 
_V
33 0ex 4528 . . . 4  |-  (/)  e.  _V
3432, 33ifex 3940 . . 3  |-  if ( ( F..^ L ) 
C_  dom  S , 
( x  e.  ( 0..^ ( L  -  F ) )  |->  ( S `  ( x  +  F ) ) ) ,  (/) )  e. 
_V
3534a1i 11 . 2  |-  ( ( S  e.  V  /\  F  e.  ZZ  /\  L  e.  ZZ )  ->  if ( ( F..^ L
)  C_  dom  S , 
( x  e.  ( 0..^ ( L  -  F ) )  |->  ( S `  ( x  +  F ) ) ) ,  (/) )  e. 
_V )
362, 26, 28, 30, 35ovmpt2d 6443 1  |-  ( ( S  e.  V  /\  F  e.  ZZ  /\  L  e.  ZZ )  ->  ( S substr  <. F ,  L >. )  =  if ( ( F..^ L ) 
C_  dom  S , 
( x  e.  ( 0..^ ( L  -  F ) )  |->  ( S `  ( x  +  F ) ) ) ,  (/) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   _Vcvv 3031    C_ wss 3390   (/)c0 3722   ifcif 3872   <.cop 3965    |-> cmpt 4454    X. cxp 4837   dom cdm 4839   ` cfv 5589  (class class class)co 6308    |-> cmpt2 6310   1stc1st 6810   2ndc2nd 6811   0cc0 9557    + caddc 9560    - cmin 9880   ZZcz 10961  ..^cfzo 11942   substr csubstr 12707
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-1st 6812  df-2nd 6813  df-substr 12715
This theorem is referenced by:  swrd00  12828  swrdcl  12829  swrdval2  12830  swrdlend  12841  swrdnd  12842  swrdnd2  12843  swrd0  12844  repswswrd  12941
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