MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  splval Structured version   Unicode version

Theorem splval 12679
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 >. ) concat  R ) concat  ( 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 12502 . . 3  |- splice  =  ( s  e.  _V , 
b  e.  _V  |->  ( ( ( s substr  <. 0 ,  ( 1st `  ( 1st `  b
) ) >. ) concat  ( 2nd `  b ) ) concat  ( 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
) ) >. ) concat  ( 2nd `  b ) ) concat  ( s substr  <. ( 2nd `  ( 1st `  b ) ) ,  ( # `  s
) >. ) ) ) )
3 simprl 755 . . . . 5  |-  ( ( ( S  e.  V  /\  ( F  e.  W  /\  T  e.  X  /\  R  e.  Y
) )  /\  (
s  =  S  /\  b  =  <. F ,  T ,  R >. ) )  ->  s  =  S )
4 fveq2 5859 . . . . . . . . 9  |-  ( b  =  <. F ,  T ,  R >.  ->  ( 1st `  b )  =  ( 1st `  <. F ,  T ,  R >. ) )
54fveq2d 5863 . . . . . . . 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 6790 . . . . . . . 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 2525 . . . . . 6  |-  ( ( ( S  e.  V  /\  ( F  e.  W  /\  T  e.  X  /\  R  e.  Y
) )  /\  (
s  =  S  /\  b  =  <. F ,  T ,  R >. ) )  ->  ( 1st `  ( 1st `  b
) )  =  F )
109opeq2d 4215 . . . . 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 6295 . . . 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 5859 . . . . . 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 6792 . . . . . . 7  |-  ( R  e.  Y  ->  ( 2nd `  <. F ,  T ,  R >. )  =  R )
15143ad2ant3 1014 . . . . . 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 2525 . . . 4  |-  ( ( ( S  e.  V  /\  ( F  e.  W  /\  T  e.  X  /\  R  e.  Y
) )  /\  (
s  =  S  /\  b  =  <. F ,  T ,  R >. ) )  ->  ( 2nd `  b )  =  R )
1811, 17oveq12d 6295 . . 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 ) ) >.
) concat  ( 2nd `  b
) )  =  ( ( S substr  <. 0 ,  F >. ) concat  R )
)
194fveq2d 5863 . . . . . . 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 6791 . . . . . . 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 2525 . . . . 5  |-  ( ( ( S  e.  V  /\  ( F  e.  W  /\  T  e.  X  /\  R  e.  Y
) )  /\  (
s  =  S  /\  b  =  <. F ,  T ,  R >. ) )  ->  ( 2nd `  ( 1st `  b
) )  =  T )
243fveq2d 5863 . . . . 5  |-  ( ( ( S  e.  V  /\  ( F  e.  W  /\  T  e.  X  /\  R  e.  Y
) )  /\  (
s  =  S  /\  b  =  <. F ,  T ,  R >. ) )  ->  ( # `  s
)  =  ( # `  S ) )
2523, 24opeq12d 4216 . . . 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 6295 . . 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 6295 . 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 ) )
>. ) concat  ( 2nd `  b
) ) concat  ( s substr  <.
( 2nd `  ( 1st `  b ) ) ,  ( # `  s
) >. ) )  =  ( ( ( S substr  <. 0 ,  F >. ) concat  R ) concat  ( S substr  <. T , 
( # `  S )
>. ) ) )
28 elex 3117 . . 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 4707 . . 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 6302 . . 3  |-  ( ( ( S substr  <. 0 ,  F >. ) concat  R ) concat  ( 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 >. ) concat  R ) concat  ( S substr  <. T ,  (
# `  S ) >. ) )  e.  _V )
342, 27, 29, 31, 33ovmpt2d 6407 1  |-  ( ( S  e.  V  /\  ( F  e.  W  /\  T  e.  X  /\  R  e.  Y
) )  ->  ( S splice  <. F ,  T ,  R >. )  =  ( ( ( S substr  <. 0 ,  F >. ) concat  R ) concat  ( S substr  <. T ,  (
# `  S ) >. ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   _Vcvv 3108   <.cop 4028   <.cotp 4030   ` cfv 5581  (class class class)co 6277    |-> cmpt2 6279   1stc1st 6774   2ndc2nd 6775   0cc0 9483   #chash 12362   concat cconcat 12491   substr csubstr 12493   splice csplice 12494
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-ot 4031  df-uni 4241  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-iota 5544  df-fun 5583  df-fv 5589  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6776  df-2nd 6777  df-splice 12502
This theorem is referenced by:  splid  12681  spllen  12682  splfv1  12683  splfv2a  12684  splval2  12685  gsumspl  15830  efgredleme  16552  efgredlemc  16554  efgcpbllemb  16564  frgpuplem  16581
  Copyright terms: Public domain W3C validator