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Theorem splval 12738
Description: Value of the substring replacement operator. (Contributed by Stefan O'Rear, 15-Aug-2015.)
Assertion
Ref Expression
splval  |-  ( ( S  e.  V  /\  ( F  e.  W  /\  T  e.  X  /\  R  e.  Y
) )  ->  ( S splice  <. F ,  T ,  R >. )  =  ( ( ( S substr  <. 0 ,  F >. ) ++  R ) ++  ( S substr  <. T , 
( # `  S )
>. ) ) )

Proof of Theorem splval
Dummy variables  s 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-splice 12550 . . 3  |- splice  =  ( s  e.  _V , 
b  e.  _V  |->  ( ( ( s substr  <. 0 ,  ( 1st `  ( 1st `  b
) ) >. ) ++  ( 2nd `  b ) ) ++  ( s substr  <. ( 2nd `  ( 1st `  b ) ) ,  ( # `  s
) >. ) ) )
21a1i 11 . 2  |-  ( ( S  e.  V  /\  ( F  e.  W  /\  T  e.  X  /\  R  e.  Y
) )  -> splice  =  ( s  e.  _V , 
b  e.  _V  |->  ( ( ( s substr  <. 0 ,  ( 1st `  ( 1st `  b
) ) >. ) ++  ( 2nd `  b ) ) ++  ( s substr  <. ( 2nd `  ( 1st `  b ) ) ,  ( # `  s
) >. ) ) ) )
3 simprl 756 . . . . 5  |-  ( ( ( S  e.  V  /\  ( F  e.  W  /\  T  e.  X  /\  R  e.  Y
) )  /\  (
s  =  S  /\  b  =  <. F ,  T ,  R >. ) )  ->  s  =  S )
4 fveq2 5872 . . . . . . . . 9  |-  ( b  =  <. F ,  T ,  R >.  ->  ( 1st `  b )  =  ( 1st `  <. F ,  T ,  R >. ) )
54fveq2d 5876 . . . . . . . 8  |-  ( b  =  <. F ,  T ,  R >.  ->  ( 1st `  ( 1st `  b
) )  =  ( 1st `  ( 1st `  <. F ,  T ,  R >. ) ) )
65adantl 466 . . . . . . 7  |-  ( ( s  =  S  /\  b  =  <. F ,  T ,  R >. )  ->  ( 1st `  ( 1st `  b ) )  =  ( 1st `  ( 1st `  <. F ,  T ,  R >. ) ) )
7 ot1stg 6813 . . . . . . . 8  |-  ( ( F  e.  W  /\  T  e.  X  /\  R  e.  Y )  ->  ( 1st `  ( 1st `  <. F ,  T ,  R >. ) )  =  F )
87adantl 466 . . . . . . 7  |-  ( ( S  e.  V  /\  ( F  e.  W  /\  T  e.  X  /\  R  e.  Y
) )  ->  ( 1st `  ( 1st `  <. F ,  T ,  R >. ) )  =  F )
96, 8sylan9eqr 2520 . . . . . 6  |-  ( ( ( S  e.  V  /\  ( F  e.  W  /\  T  e.  X  /\  R  e.  Y
) )  /\  (
s  =  S  /\  b  =  <. F ,  T ,  R >. ) )  ->  ( 1st `  ( 1st `  b
) )  =  F )
109opeq2d 4226 . . . . 5  |-  ( ( ( S  e.  V  /\  ( F  e.  W  /\  T  e.  X  /\  R  e.  Y
) )  /\  (
s  =  S  /\  b  =  <. F ,  T ,  R >. ) )  ->  <. 0 ,  ( 1st `  ( 1st `  b ) )
>.  =  <. 0 ,  F >. )
113, 10oveq12d 6314 . . . 4  |-  ( ( ( S  e.  V  /\  ( F  e.  W  /\  T  e.  X  /\  R  e.  Y
) )  /\  (
s  =  S  /\  b  =  <. F ,  T ,  R >. ) )  ->  ( s substr  <.
0 ,  ( 1st `  ( 1st `  b
) ) >. )  =  ( S substr  <. 0 ,  F >. ) )
12 fveq2 5872 . . . . . 6  |-  ( b  =  <. F ,  T ,  R >.  ->  ( 2nd `  b )  =  ( 2nd `  <. F ,  T ,  R >. ) )
1312adantl 466 . . . . 5  |-  ( ( s  =  S  /\  b  =  <. F ,  T ,  R >. )  ->  ( 2nd `  b
)  =  ( 2nd `  <. F ,  T ,  R >. ) )
14 ot3rdg 6815 . . . . . . 7  |-  ( R  e.  Y  ->  ( 2nd `  <. F ,  T ,  R >. )  =  R )
15143ad2ant3 1019 . . . . . 6  |-  ( ( F  e.  W  /\  T  e.  X  /\  R  e.  Y )  ->  ( 2nd `  <. F ,  T ,  R >. )  =  R )
1615adantl 466 . . . . 5  |-  ( ( S  e.  V  /\  ( F  e.  W  /\  T  e.  X  /\  R  e.  Y
) )  ->  ( 2nd `  <. F ,  T ,  R >. )  =  R )
1713, 16sylan9eqr 2520 . . . 4  |-  ( ( ( S  e.  V  /\  ( F  e.  W  /\  T  e.  X  /\  R  e.  Y
) )  /\  (
s  =  S  /\  b  =  <. F ,  T ,  R >. ) )  ->  ( 2nd `  b )  =  R )
1811, 17oveq12d 6314 . . 3  |-  ( ( ( S  e.  V  /\  ( F  e.  W  /\  T  e.  X  /\  R  e.  Y
) )  /\  (
s  =  S  /\  b  =  <. F ,  T ,  R >. ) )  ->  ( (
s substr  <. 0 ,  ( 1st `  ( 1st `  b ) ) >.
) ++  ( 2nd `  b
) )  =  ( ( S substr  <. 0 ,  F >. ) ++  R ) )
194fveq2d 5876 . . . . . . 7  |-  ( b  =  <. F ,  T ,  R >.  ->  ( 2nd `  ( 1st `  b
) )  =  ( 2nd `  ( 1st `  <. F ,  T ,  R >. ) ) )
2019adantl 466 . . . . . 6  |-  ( ( s  =  S  /\  b  =  <. F ,  T ,  R >. )  ->  ( 2nd `  ( 1st `  b ) )  =  ( 2nd `  ( 1st `  <. F ,  T ,  R >. ) ) )
21 ot2ndg 6814 . . . . . . 7  |-  ( ( F  e.  W  /\  T  e.  X  /\  R  e.  Y )  ->  ( 2nd `  ( 1st `  <. F ,  T ,  R >. ) )  =  T )
2221adantl 466 . . . . . 6  |-  ( ( S  e.  V  /\  ( F  e.  W  /\  T  e.  X  /\  R  e.  Y
) )  ->  ( 2nd `  ( 1st `  <. F ,  T ,  R >. ) )  =  T )
2320, 22sylan9eqr 2520 . . . . 5  |-  ( ( ( S  e.  V  /\  ( F  e.  W  /\  T  e.  X  /\  R  e.  Y
) )  /\  (
s  =  S  /\  b  =  <. F ,  T ,  R >. ) )  ->  ( 2nd `  ( 1st `  b
) )  =  T )
243fveq2d 5876 . . . . 5  |-  ( ( ( S  e.  V  /\  ( F  e.  W  /\  T  e.  X  /\  R  e.  Y
) )  /\  (
s  =  S  /\  b  =  <. F ,  T ,  R >. ) )  ->  ( # `  s
)  =  ( # `  S ) )
2523, 24opeq12d 4227 . . . 4  |-  ( ( ( S  e.  V  /\  ( F  e.  W  /\  T  e.  X  /\  R  e.  Y
) )  /\  (
s  =  S  /\  b  =  <. F ,  T ,  R >. ) )  ->  <. ( 2nd `  ( 1st `  b
) ) ,  (
# `  s ) >.  =  <. T ,  (
# `  S ) >. )
263, 25oveq12d 6314 . . 3  |-  ( ( ( S  e.  V  /\  ( F  e.  W  /\  T  e.  X  /\  R  e.  Y
) )  /\  (
s  =  S  /\  b  =  <. F ,  T ,  R >. ) )  ->  ( s substr  <.
( 2nd `  ( 1st `  b ) ) ,  ( # `  s
) >. )  =  ( S substr  <. T ,  (
# `  S ) >. ) )
2718, 26oveq12d 6314 . 2  |-  ( ( ( S  e.  V  /\  ( F  e.  W  /\  T  e.  X  /\  R  e.  Y
) )  /\  (
s  =  S  /\  b  =  <. F ,  T ,  R >. ) )  ->  ( (
( s substr  <. 0 ,  ( 1st `  ( 1st `  b ) )
>. ) ++  ( 2nd `  b ) ) ++  ( s substr  <. ( 2nd `  ( 1st `  b ) ) ,  ( # `  s
) >. ) )  =  ( ( ( S substr  <. 0 ,  F >. ) ++  R ) ++  ( S substr  <. T ,  ( # `  S ) >. )
) )
28 elex 3118 . . 3  |-  ( S  e.  V  ->  S  e.  _V )
2928adantr 465 . 2  |-  ( ( S  e.  V  /\  ( F  e.  W  /\  T  e.  X  /\  R  e.  Y
) )  ->  S  e.  _V )
30 otex 4721 . . 3  |-  <. F ,  T ,  R >.  e. 
_V
3130a1i 11 . 2  |-  ( ( S  e.  V  /\  ( F  e.  W  /\  T  e.  X  /\  R  e.  Y
) )  ->  <. F ,  T ,  R >.  e. 
_V )
32 ovex 6324 . . 3  |-  ( ( ( S substr  <. 0 ,  F >. ) ++  R ) ++  ( S substr  <. T , 
( # `  S )
>. ) )  e.  _V
3332a1i 11 . 2  |-  ( ( S  e.  V  /\  ( F  e.  W  /\  T  e.  X  /\  R  e.  Y
) )  ->  (
( ( S substr  <. 0 ,  F >. ) ++  R ) ++  ( S substr  <. T , 
( # `  S )
>. ) )  e.  _V )
342, 27, 29, 31, 33ovmpt2d 6429 1  |-  ( ( S  e.  V  /\  ( F  e.  W  /\  T  e.  X  /\  R  e.  Y
) )  ->  ( S splice  <. F ,  T ,  R >. )  =  ( ( ( S substr  <. 0 ,  F >. ) ++  R ) ++  ( S substr  <. T , 
( # `  S )
>. ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   _Vcvv 3109   <.cop 4038   <.cotp 4040   ` cfv 5594  (class class class)co 6296    |-> cmpt2 6298   1stc1st 6797   2ndc2nd 6798   0cc0 9509   #chash 12407   ++ cconcat 12539   substr csubstr 12541   splice csplice 12542
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-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-rab 2816  df-v 3111  df-sbc 3328  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-ot 4041  df-uni 4252  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-iota 5557  df-fun 5596  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-splice 12550
This theorem is referenced by:  splid  12740  spllen  12741  splfv1  12742  splfv2a  12743  splval2  12744  gsumspl  16138  efgredleme  16887  efgredlemc  16889  efgcpbllemb  16899  frgpuplem  16916  splvalpfx  32511
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