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Definition df-substr 11726
Description: Define an operation which extracts portions of words. (Contributed by Stefan O'Rear, 15-Aug-2015.)
Assertion
Ref Expression
df-substr  |- 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 ) ) ) ) ,  (/) ) )
Distinct variable group:    s, b, x

Detailed syntax breakdown of Definition df-substr
StepHypRef Expression
1 csubstr 11720 . 2  class substr
2 vs . . 3  set  s
3 vb . . 3  set  b
4 cvv 2956 . . 3  class  _V
5 cz 10282 . . . 4  class  ZZ
65, 5cxp 4876 . . 3  class  ( ZZ 
X.  ZZ )
73cv 1651 . . . . . . 7  class  b
8 c1st 6347 . . . . . . 7  class  1st
97, 8cfv 5454 . . . . . 6  class  ( 1st `  b )
10 c2nd 6348 . . . . . . 7  class  2nd
117, 10cfv 5454 . . . . . 6  class  ( 2nd `  b )
12 cfzo 11135 . . . . . 6  class ..^
139, 11, 12co 6081 . . . . 5  class  ( ( 1st `  b )..^ ( 2nd `  b
) )
142cv 1651 . . . . . 6  class  s
1514cdm 4878 . . . . 5  class  dom  s
1613, 15wss 3320 . . . 4  wff  ( ( 1st `  b )..^ ( 2nd `  b
) )  C_  dom  s
17 vx . . . . 5  set  x
18 cc0 8990 . . . . . 6  class  0
19 cmin 9291 . . . . . . 7  class  -
2011, 9, 19co 6081 . . . . . 6  class  ( ( 2nd `  b )  -  ( 1st `  b
) )
2118, 20, 12co 6081 . . . . 5  class  ( 0..^ ( ( 2nd `  b
)  -  ( 1st `  b ) ) )
2217cv 1651 . . . . . . 7  class  x
23 caddc 8993 . . . . . . 7  class  +
2422, 9, 23co 6081 . . . . . 6  class  ( x  +  ( 1st `  b
) )
2524, 14cfv 5454 . . . . 5  class  ( s `
 ( x  +  ( 1st `  b ) ) )
2617, 21, 25cmpt 4266 . . . 4  class  ( x  e.  ( 0..^ ( ( 2nd `  b
)  -  ( 1st `  b ) ) ) 
|->  ( s `  (
x  +  ( 1st `  b ) ) ) )
27 c0 3628 . . . 4  class  (/)
2816, 26, 27cif 3739 . . 3  class  if ( ( ( 1st `  b
)..^ ( 2nd `  b
) )  C_  dom  s ,  ( x  e.  ( 0..^ ( ( 2nd `  b )  -  ( 1st `  b
) ) )  |->  ( s `  ( x  +  ( 1st `  b
) ) ) ) ,  (/) )
292, 3, 4, 6, 28cmpt2 6083 . 2  class  ( 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
) ) ) ) ,  (/) ) )
301, 29wceq 1652 1  wff 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 ) ) ) ) ,  (/) ) )
Colors of variables: wff set class
This definition is referenced by:  swrdval  11764  swrd00  11765  swrdcl  11766
  Copyright terms: Public domain W3C validator