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Theorem swrdval 12652
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 12549 . . 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 756 . . 3  |-  ( ( ( S  e.  V  /\  F  e.  ZZ  /\  L  e.  ZZ )  /\  ( s  =  S  /\  b  = 
<. F ,  L >. ) )  ->  s  =  S )
4 fveq2 5872 . . . . 5  |-  ( b  =  <. F ,  L >.  ->  ( 1st `  b
)  =  ( 1st `  <. F ,  L >. ) )
54adantl 466 . . . 4  |-  ( ( s  =  S  /\  b  =  <. F ,  L >. )  ->  ( 1st `  b )  =  ( 1st `  <. F ,  L >. )
)
6 op1stg 6811 . . . . 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 2520 . . 3  |-  ( ( ( S  e.  V  /\  F  e.  ZZ  /\  L  e.  ZZ )  /\  ( s  =  S  /\  b  = 
<. F ,  L >. ) )  ->  ( 1st `  b )  =  F )
9 fveq2 5872 . . . . 5  |-  ( b  =  <. F ,  L >.  ->  ( 2nd `  b
)  =  ( 2nd `  <. F ,  L >. ) )
109adantl 466 . . . 4  |-  ( ( s  =  S  /\  b  =  <. F ,  L >. )  ->  ( 2nd `  b )  =  ( 2nd `  <. F ,  L >. )
)
11 op2ndg 6812 . . . . 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 2520 . . 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 6314 . . . . 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 5215 . . . . 5  |-  ( ( s  =  S  /\  ( 1st `  b )  =  F  /\  ( 2nd `  b )  =  L )  ->  dom  s  =  dom  S )
1916, 18sseq12d 3528 . . . 4  |-  ( ( s  =  S  /\  ( 1st `  b )  =  F  /\  ( 2nd `  b )  =  L )  ->  (
( ( 1st `  b
)..^ ( 2nd `  b
) )  C_  dom  s 
<->  ( F..^ L ) 
C_  dom  S )
)
2015, 14oveq12d 6314 . . . . . 6  |-  ( ( s  =  S  /\  ( 1st `  b )  =  F  /\  ( 2nd `  b )  =  L )  ->  (
( 2nd `  b
)  -  ( 1st `  b ) )  =  ( L  -  F
) )
2120oveq2d 6312 . . . . 5  |-  ( ( s  =  S  /\  ( 1st `  b )  =  F  /\  ( 2nd `  b )  =  L )  ->  (
0..^ ( ( 2nd `  b )  -  ( 1st `  b ) ) )  =  ( 0..^ ( L  -  F
) ) )
2214oveq2d 6312 . . . . . 6  |-  ( ( s  =  S  /\  ( 1st `  b )  =  F  /\  ( 2nd `  b )  =  L )  ->  (
x  +  ( 1st `  b ) )  =  ( x  +  F
) )
2317, 22fveq12d 5878 . . . . 5  |-  ( ( s  =  S  /\  ( 1st `  b )  =  F  /\  ( 2nd `  b )  =  L )  ->  (
s `  ( x  +  ( 1st `  b
) ) )  =  ( S `  (
x  +  F ) ) )
2421, 23mpteq12dv 4535 . . . 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 3967 . . 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 3118 . . 3  |-  ( S  e.  V  ->  S  e.  _V )
28273ad2ant1 1017 . 2  |-  ( ( S  e.  V  /\  F  e.  ZZ  /\  L  e.  ZZ )  ->  S  e.  _V )
29 opelxpi 5040 . . 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 6324 . . . . 5  |-  ( 0..^ ( L  -  F
) )  e.  _V
3231mptex 6144 . . . 4  |-  ( x  e.  ( 0..^ ( L  -  F ) )  |->  ( S `  ( x  +  F
) ) )  e. 
_V
33 0ex 4587 . . . 4  |-  (/)  e.  _V
3432, 33ifex 4013 . . 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 6429 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 1395    e. wcel 1819   _Vcvv 3109    C_ wss 3471   (/)c0 3793   ifcif 3944   <.cop 4038    |-> cmpt 4515    X. cxp 5006   dom cdm 5008   ` cfv 5594  (class class class)co 6296    |-> cmpt2 6298   1stc1st 6797   2ndc2nd 6798   0cc0 9509    + caddc 9512    - cmin 9824   ZZcz 10885  ..^cfzo 11820   substr csubstr 12541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-substr 12549
This theorem is referenced by:  swrd00  12653  swrdcl  12654  swrdval2  12655  swrdlend  12666  swrdnd  12667  swrdnd2  12668  swrd0  12669  repswswrd  12767
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