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Theorem swrdval 12601
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 12506 . . 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 755 . . 3  |-  ( ( ( S  e.  V  /\  F  e.  ZZ  /\  L  e.  ZZ )  /\  ( s  =  S  /\  b  = 
<. F ,  L >. ) )  ->  s  =  S )
4 fveq2 5864 . . . . 5  |-  ( b  =  <. F ,  L >.  ->  ( 1st `  b
)  =  ( 1st `  <. F ,  L >. ) )
54adantl 466 . . . 4  |-  ( ( s  =  S  /\  b  =  <. F ,  L >. )  ->  ( 1st `  b )  =  ( 1st `  <. F ,  L >. )
)
6 op1stg 6793 . . . . 5  |-  ( ( F  e.  ZZ  /\  L  e.  ZZ )  ->  ( 1st `  <. F ,  L >. )  =  F )
763adant1 1014 . . . 4  |-  ( ( S  e.  V  /\  F  e.  ZZ  /\  L  e.  ZZ )  ->  ( 1st `  <. F ,  L >. )  =  F )
85, 7sylan9eqr 2530 . . 3  |-  ( ( ( S  e.  V  /\  F  e.  ZZ  /\  L  e.  ZZ )  /\  ( s  =  S  /\  b  = 
<. F ,  L >. ) )  ->  ( 1st `  b )  =  F )
9 fveq2 5864 . . . . 5  |-  ( b  =  <. F ,  L >.  ->  ( 2nd `  b
)  =  ( 2nd `  <. F ,  L >. ) )
109adantl 466 . . . 4  |-  ( ( s  =  S  /\  b  =  <. F ,  L >. )  ->  ( 2nd `  b )  =  ( 2nd `  <. F ,  L >. )
)
11 op2ndg 6794 . . . . 5  |-  ( ( F  e.  ZZ  /\  L  e.  ZZ )  ->  ( 2nd `  <. F ,  L >. )  =  L )
12113adant1 1014 . . . 4  |-  ( ( S  e.  V  /\  F  e.  ZZ  /\  L  e.  ZZ )  ->  ( 2nd `  <. F ,  L >. )  =  L )
1310, 12sylan9eqr 2530 . . 3  |-  ( ( ( S  e.  V  /\  F  e.  ZZ  /\  L  e.  ZZ )  /\  ( s  =  S  /\  b  = 
<. F ,  L >. ) )  ->  ( 2nd `  b )  =  L )
14 simp2 997 . . . . . 6  |-  ( ( s  =  S  /\  ( 1st `  b )  =  F  /\  ( 2nd `  b )  =  L )  ->  ( 1st `  b )  =  F )
15 simp3 998 . . . . . 6  |-  ( ( s  =  S  /\  ( 1st `  b )  =  F  /\  ( 2nd `  b )  =  L )  ->  ( 2nd `  b )  =  L )
1614, 15oveq12d 6300 . . . . 5  |-  ( ( s  =  S  /\  ( 1st `  b )  =  F  /\  ( 2nd `  b )  =  L )  ->  (
( 1st `  b
)..^ ( 2nd `  b
) )  =  ( F..^ L ) )
17 simp1 996 . . . . . 6  |-  ( ( s  =  S  /\  ( 1st `  b )  =  F  /\  ( 2nd `  b )  =  L )  ->  s  =  S )
1817dmeqd 5203 . . . . 5  |-  ( ( s  =  S  /\  ( 1st `  b )  =  F  /\  ( 2nd `  b )  =  L )  ->  dom  s  =  dom  S )
1916, 18sseq12d 3533 . . . 4  |-  ( ( s  =  S  /\  ( 1st `  b )  =  F  /\  ( 2nd `  b )  =  L )  ->  (
( ( 1st `  b
)..^ ( 2nd `  b
) )  C_  dom  s 
<->  ( F..^ L ) 
C_  dom  S )
)
2015, 14oveq12d 6300 . . . . . 6  |-  ( ( s  =  S  /\  ( 1st `  b )  =  F  /\  ( 2nd `  b )  =  L )  ->  (
( 2nd `  b
)  -  ( 1st `  b ) )  =  ( L  -  F
) )
2120oveq2d 6298 . . . . 5  |-  ( ( s  =  S  /\  ( 1st `  b )  =  F  /\  ( 2nd `  b )  =  L )  ->  (
0..^ ( ( 2nd `  b )  -  ( 1st `  b ) ) )  =  ( 0..^ ( L  -  F
) ) )
2214oveq2d 6298 . . . . . 6  |-  ( ( s  =  S  /\  ( 1st `  b )  =  F  /\  ( 2nd `  b )  =  L )  ->  (
x  +  ( 1st `  b ) )  =  ( x  +  F
) )
2317, 22fveq12d 5870 . . . . 5  |-  ( ( s  =  S  /\  ( 1st `  b )  =  F  /\  ( 2nd `  b )  =  L )  ->  (
s `  ( x  +  ( 1st `  b
) ) )  =  ( S `  (
x  +  F ) ) )
2421, 23mpteq12dv 4525 . . . 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 3962 . . 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 1228 . 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 3122 . . 3  |-  ( S  e.  V  ->  S  e.  _V )
28273ad2ant1 1017 . 2  |-  ( ( S  e.  V  /\  F  e.  ZZ  /\  L  e.  ZZ )  ->  S  e.  _V )
29 opelxpi 5030 . . 3  |-  ( ( F  e.  ZZ  /\  L  e.  ZZ )  -> 
<. F ,  L >.  e.  ( ZZ  X.  ZZ ) )
30293adant1 1014 . 2  |-  ( ( S  e.  V  /\  F  e.  ZZ  /\  L  e.  ZZ )  ->  <. F ,  L >.  e.  ( ZZ 
X.  ZZ ) )
31 ovex 6307 . . . . 5  |-  ( 0..^ ( L  -  F
) )  e.  _V
3231mptex 6129 . . . 4  |-  ( x  e.  ( 0..^ ( L  -  F ) )  |->  ( S `  ( x  +  F
) ) )  e. 
_V
33 0ex 4577 . . . 4  |-  (/)  e.  _V
3432, 33ifex 4008 . . 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 6412 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 369    /\ w3a 973    = wceq 1379    e. wcel 1767   _Vcvv 3113    C_ wss 3476   (/)c0 3785   ifcif 3939   <.cop 4033    |-> cmpt 4505    X. cxp 4997   dom cdm 4999   ` cfv 5586  (class class class)co 6282    |-> cmpt2 6284   1stc1st 6779   2ndc2nd 6780   0cc0 9488    + caddc 9491    - cmin 9801   ZZcz 10860  ..^cfzo 11788   substr csubstr 12498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782  df-substr 12506
This theorem is referenced by:  swrd00  12602  swrdcl  12603  swrdval2  12604  swrdlend  12613  swrdnd  12614  swrd0  12615  swrdvalodm2  12621  swrdvalodm  12622  repswswrd  12713
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