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Theorem splcl 11483
Description: Closure of the substring replacement operator. (Contributed by Stefan O'Rear, 26-Aug-2015.)
Assertion
Ref Expression
splcl  |-  ( ( S  e. Word  A  /\  R  e. Word  A )  ->  ( S splice  <. F ,  T ,  R >. )  e. Word  A )

Proof of Theorem splcl
Dummy variables  s 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 2809 . . . 4  |-  ( S  e. Word  A  ->  S  e.  _V )
2 otex 4254 . . . 4  |-  <. F ,  T ,  R >.  e. 
_V
3 id 19 . . . . . . . 8  |-  ( s  =  S  ->  s  =  S )
4 fveq2 5541 . . . . . . . . . 10  |-  ( b  =  <. F ,  T ,  R >.  ->  ( 1st `  b )  =  ( 1st `  <. F ,  T ,  R >. ) )
54fveq2d 5545 . . . . . . . . 9  |-  ( b  =  <. F ,  T ,  R >.  ->  ( 1st `  ( 1st `  b
) )  =  ( 1st `  ( 1st `  <. F ,  T ,  R >. ) ) )
65opeq2d 3819 . . . . . . . 8  |-  ( b  =  <. F ,  T ,  R >.  ->  <. 0 ,  ( 1st `  ( 1st `  b ) )
>.  =  <. 0 ,  ( 1st `  ( 1st `  <. F ,  T ,  R >. ) ) >.
)
73, 6oveqan12d 5893 . . . . . . 7  |-  ( ( s  =  S  /\  b  =  <. F ,  T ,  R >. )  ->  ( s substr  <. 0 ,  ( 1st `  ( 1st `  b
) ) >. )  =  ( S substr  <. 0 ,  ( 1st `  ( 1st `  <. F ,  T ,  R >. ) ) >.
) )
8 simpr 447 . . . . . . . 8  |-  ( ( s  =  S  /\  b  =  <. F ,  T ,  R >. )  ->  b  =  <. F ,  T ,  R >. )
98fveq2d 5545 . . . . . . 7  |-  ( ( s  =  S  /\  b  =  <. F ,  T ,  R >. )  ->  ( 2nd `  b
)  =  ( 2nd `  <. F ,  T ,  R >. ) )
107, 9oveq12d 5892 . . . . . 6  |-  ( ( s  =  S  /\  b  =  <. F ,  T ,  R >. )  ->  ( ( s substr  <. 0 ,  ( 1st `  ( 1st `  b
) ) >. ) concat  ( 2nd `  b ) )  =  ( ( S substr  <. 0 ,  ( 1st `  ( 1st `  <. F ,  T ,  R >. ) ) >.
) concat  ( 2nd `  <. F ,  T ,  R >. ) ) )
11 simpl 443 . . . . . . 7  |-  ( ( s  =  S  /\  b  =  <. F ,  T ,  R >. )  ->  s  =  S )
128fveq2d 5545 . . . . . . . . 9  |-  ( ( s  =  S  /\  b  =  <. F ,  T ,  R >. )  ->  ( 1st `  b
)  =  ( 1st `  <. F ,  T ,  R >. ) )
1312fveq2d 5545 . . . . . . . 8  |-  ( ( s  =  S  /\  b  =  <. F ,  T ,  R >. )  ->  ( 2nd `  ( 1st `  b ) )  =  ( 2nd `  ( 1st `  <. F ,  T ,  R >. ) ) )
1411fveq2d 5545 . . . . . . . 8  |-  ( ( s  =  S  /\  b  =  <. F ,  T ,  R >. )  ->  ( # `  s
)  =  ( # `  S ) )
1513, 14opeq12d 3820 . . . . . . 7  |-  ( ( s  =  S  /\  b  =  <. F ,  T ,  R >. )  ->  <. ( 2nd `  ( 1st `  b ) ) ,  ( # `  s
) >.  =  <. ( 2nd `  ( 1st `  <. F ,  T ,  R >. ) ) ,  (
# `  S ) >. )
1611, 15oveq12d 5892 . . . . . 6  |-  ( ( s  =  S  /\  b  =  <. F ,  T ,  R >. )  ->  ( s substr  <. ( 2nd `  ( 1st `  b ) ) ,  ( # `  s
) >. )  =  ( S substr  <. ( 2nd `  ( 1st `  <. F ,  T ,  R >. ) ) ,  ( # `  S
) >. ) )
1710, 16oveq12d 5892 . . . . 5  |-  ( ( 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 ,  ( 1st `  ( 1st `  <. F ,  T ,  R >. ) ) >. ) concat  ( 2nd `  <. F ,  T ,  R >. ) ) concat  ( S substr  <. ( 2nd `  ( 1st `  <. F ,  T ,  R >. ) ) ,  (
# `  S ) >. ) ) )
18 df-splice 11429 . . . . 5  |- splice  =  ( s  e.  _V , 
b  e.  _V  |->  ( ( ( s substr  <. 0 ,  ( 1st `  ( 1st `  b
) ) >. ) concat  ( 2nd `  b ) ) concat  ( s substr  <. ( 2nd `  ( 1st `  b ) ) ,  ( # `  s
) >. ) ) )
19 ovex 5899 . . . . 5  |-  ( ( ( S substr  <. 0 ,  ( 1st `  ( 1st `  <. F ,  T ,  R >. ) ) >.
) concat  ( 2nd `  <. F ,  T ,  R >. ) ) concat  ( S substr  <. ( 2nd `  ( 1st `  <. F ,  T ,  R >. ) ) ,  ( # `  S
) >. ) )  e. 
_V
2017, 18, 19ovmpt2a 5994 . . . 4  |-  ( ( S  e.  _V  /\  <. F ,  T ,  R >.  e.  _V )  ->  ( S splice  <. F ,  T ,  R >. )  =  ( ( ( S substr  <. 0 ,  ( 1st `  ( 1st `  <. F ,  T ,  R >. ) ) >.
) concat  ( 2nd `  <. F ,  T ,  R >. ) ) concat  ( S substr  <. ( 2nd `  ( 1st `  <. F ,  T ,  R >. ) ) ,  ( # `  S
) >. ) ) )
211, 2, 20sylancl 643 . . 3  |-  ( S  e. Word  A  ->  ( S splice  <. F ,  T ,  R >. )  =  ( ( ( S substr  <. 0 ,  ( 1st `  ( 1st `  <. F ,  T ,  R >. ) ) >.
) concat  ( 2nd `  <. F ,  T ,  R >. ) ) concat  ( S substr  <. ( 2nd `  ( 1st `  <. F ,  T ,  R >. ) ) ,  ( # `  S
) >. ) ) )
2221adantr 451 . 2  |-  ( ( S  e. Word  A  /\  R  e. Word  A )  ->  ( S splice  <. F ,  T ,  R >. )  =  ( ( ( S substr  <. 0 ,  ( 1st `  ( 1st `  <. F ,  T ,  R >. ) ) >.
) concat  ( 2nd `  <. F ,  T ,  R >. ) ) concat  ( S substr  <. ( 2nd `  ( 1st `  <. F ,  T ,  R >. ) ) ,  ( # `  S
) >. ) ) )
23 swrdcl 11468 . . . . 5  |-  ( S  e. Word  A  ->  ( S substr  <. 0 ,  ( 1st `  ( 1st `  <. F ,  T ,  R >. ) ) >.
)  e. Word  A )
2423adantr 451 . . . 4  |-  ( ( S  e. Word  A  /\  R  e. Word  A )  ->  ( S substr  <. 0 ,  ( 1st `  ( 1st `  <. F ,  T ,  R >. ) ) >.
)  e. Word  A )
25 ot3rdg 6152 . . . . . 6  |-  ( R  e. Word  A  ->  ( 2nd `  <. F ,  T ,  R >. )  =  R )
2625adantl 452 . . . . 5  |-  ( ( S  e. Word  A  /\  R  e. Word  A )  ->  ( 2nd `  <. F ,  T ,  R >. )  =  R )
27 simpr 447 . . . . 5  |-  ( ( S  e. Word  A  /\  R  e. Word  A )  ->  R  e. Word  A )
2826, 27eqeltrd 2370 . . . 4  |-  ( ( S  e. Word  A  /\  R  e. Word  A )  ->  ( 2nd `  <. F ,  T ,  R >. )  e. Word  A )
29 ccatcl 11445 . . . 4  |-  ( ( ( S substr  <. 0 ,  ( 1st `  ( 1st `  <. F ,  T ,  R >. ) ) >.
)  e. Word  A  /\  ( 2nd `  <. F ,  T ,  R >. )  e. Word  A )  -> 
( ( S substr  <. 0 ,  ( 1st `  ( 1st `  <. F ,  T ,  R >. ) ) >.
) concat  ( 2nd `  <. F ,  T ,  R >. ) )  e. Word  A
)
3024, 28, 29syl2anc 642 . . 3  |-  ( ( S  e. Word  A  /\  R  e. Word  A )  ->  ( ( S substr  <. 0 ,  ( 1st `  ( 1st `  <. F ,  T ,  R >. ) ) >.
) concat  ( 2nd `  <. F ,  T ,  R >. ) )  e. Word  A
)
31 swrdcl 11468 . . . 4  |-  ( S  e. Word  A  ->  ( S substr  <. ( 2nd `  ( 1st `  <. F ,  T ,  R >. ) ) ,  ( # `  S
) >. )  e. Word  A
)
3231adantr 451 . . 3  |-  ( ( S  e. Word  A  /\  R  e. Word  A )  ->  ( S substr  <. ( 2nd `  ( 1st `  <. F ,  T ,  R >. ) ) ,  (
# `  S ) >. )  e. Word  A )
33 ccatcl 11445 . . 3  |-  ( ( ( ( S substr  <. 0 ,  ( 1st `  ( 1st `  <. F ,  T ,  R >. ) ) >.
) concat  ( 2nd `  <. F ,  T ,  R >. ) )  e. Word  A  /\  ( S substr  <. ( 2nd `  ( 1st `  <. F ,  T ,  R >. ) ) ,  (
# `  S ) >. )  e. Word  A )  ->  ( ( ( S substr  <. 0 ,  ( 1st `  ( 1st `  <. F ,  T ,  R >. ) ) >.
) concat  ( 2nd `  <. F ,  T ,  R >. ) ) concat  ( S substr  <. ( 2nd `  ( 1st `  <. F ,  T ,  R >. ) ) ,  ( # `  S
) >. ) )  e. Word  A )
3430, 32, 33syl2anc 642 . 2  |-  ( ( S  e. Word  A  /\  R  e. Word  A )  ->  ( ( ( S substr  <. 0 ,  ( 1st `  ( 1st `  <. F ,  T ,  R >. ) ) >. ) concat  ( 2nd `  <. F ,  T ,  R >. ) ) concat  ( S substr  <. ( 2nd `  ( 1st `  <. F ,  T ,  R >. ) ) ,  (
# `  S ) >. ) )  e. Word  A
)
3522, 34eqeltrd 2370 1  |-  ( ( S  e. Word  A  /\  R  e. Word  A )  ->  ( S splice  <. F ,  T ,  R >. )  e. Word  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801   <.cop 3656   <.cotp 3657   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137   0cc0 8753   #chash 11353  Word cword 11419   concat cconcat 11420   substr csubstr 11422   splice csplice 11423
This theorem is referenced by:  efglem  15041  efgtf  15047  frgpuplem  15097  psgnunilem2  27521
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-ot 3663  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-fzo 10887  df-hash 11354  df-word 11425  df-concat 11426  df-substr 11428  df-splice 11429
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