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Theorem splcl 11773
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 2956 . . . 4  |-  ( S  e. Word  A  ->  S  e.  _V )
2 otex 4420 . . . 4  |-  <. F ,  T ,  R >.  e. 
_V
3 id 20 . . . . . . . 8  |-  ( s  =  S  ->  s  =  S )
4 fveq2 5720 . . . . . . . . . 10  |-  ( b  =  <. F ,  T ,  R >.  ->  ( 1st `  b )  =  ( 1st `  <. F ,  T ,  R >. ) )
54fveq2d 5724 . . . . . . . . 9  |-  ( b  =  <. F ,  T ,  R >.  ->  ( 1st `  ( 1st `  b
) )  =  ( 1st `  ( 1st `  <. F ,  T ,  R >. ) ) )
65opeq2d 3983 . . . . . . . 8  |-  ( b  =  <. F ,  T ,  R >.  ->  <. 0 ,  ( 1st `  ( 1st `  b ) )
>.  =  <. 0 ,  ( 1st `  ( 1st `  <. F ,  T ,  R >. ) ) >.
)
73, 6oveqan12d 6092 . . . . . . 7  |-  ( ( s  =  S  /\  b  =  <. F ,  T ,  R >. )  ->  ( s substr  <. 0 ,  ( 1st `  ( 1st `  b
) ) >. )  =  ( S substr  <. 0 ,  ( 1st `  ( 1st `  <. F ,  T ,  R >. ) ) >.
) )
8 simpr 448 . . . . . . . 8  |-  ( ( s  =  S  /\  b  =  <. F ,  T ,  R >. )  ->  b  =  <. F ,  T ,  R >. )
98fveq2d 5724 . . . . . . 7  |-  ( ( s  =  S  /\  b  =  <. F ,  T ,  R >. )  ->  ( 2nd `  b
)  =  ( 2nd `  <. F ,  T ,  R >. ) )
107, 9oveq12d 6091 . . . . . 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 444 . . . . . . 7  |-  ( ( s  =  S  /\  b  =  <. F ,  T ,  R >. )  ->  s  =  S )
128fveq2d 5724 . . . . . . . . 9  |-  ( ( s  =  S  /\  b  =  <. F ,  T ,  R >. )  ->  ( 1st `  b
)  =  ( 1st `  <. F ,  T ,  R >. ) )
1312fveq2d 5724 . . . . . . . 8  |-  ( ( s  =  S  /\  b  =  <. F ,  T ,  R >. )  ->  ( 2nd `  ( 1st `  b ) )  =  ( 2nd `  ( 1st `  <. F ,  T ,  R >. ) ) )
1411fveq2d 5724 . . . . . . . 8  |-  ( ( s  =  S  /\  b  =  <. F ,  T ,  R >. )  ->  ( # `  s
)  =  ( # `  S ) )
1513, 14opeq12d 3984 . . . . . . 7  |-  ( ( s  =  S  /\  b  =  <. F ,  T ,  R >. )  ->  <. ( 2nd `  ( 1st `  b ) ) ,  ( # `  s
) >.  =  <. ( 2nd `  ( 1st `  <. F ,  T ,  R >. ) ) ,  (
# `  S ) >. )
1611, 15oveq12d 6091 . . . . . 6  |-  ( ( s  =  S  /\  b  =  <. F ,  T ,  R >. )  ->  ( s substr  <. ( 2nd `  ( 1st `  b ) ) ,  ( # `  s
) >. )  =  ( S substr  <. ( 2nd `  ( 1st `  <. F ,  T ,  R >. ) ) ,  ( # `  S
) >. ) )
1710, 16oveq12d 6091 . . . . 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 11719 . . . . 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 6098 . . . . 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 6196 . . . 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 644 . . 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 452 . 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 11758 . . . . 5  |-  ( S  e. Word  A  ->  ( S substr  <. 0 ,  ( 1st `  ( 1st `  <. F ,  T ,  R >. ) ) >.
)  e. Word  A )
2423adantr 452 . . . 4  |-  ( ( S  e. Word  A  /\  R  e. Word  A )  ->  ( S substr  <. 0 ,  ( 1st `  ( 1st `  <. F ,  T ,  R >. ) ) >.
)  e. Word  A )
25 ot3rdg 6355 . . . . . 6  |-  ( R  e. Word  A  ->  ( 2nd `  <. F ,  T ,  R >. )  =  R )
2625adantl 453 . . . . 5  |-  ( ( S  e. Word  A  /\  R  e. Word  A )  ->  ( 2nd `  <. F ,  T ,  R >. )  =  R )
27 simpr 448 . . . . 5  |-  ( ( S  e. Word  A  /\  R  e. Word  A )  ->  R  e. Word  A )
2826, 27eqeltrd 2509 . . . 4  |-  ( ( S  e. Word  A  /\  R  e. Word  A )  ->  ( 2nd `  <. F ,  T ,  R >. )  e. Word  A )
29 ccatcl 11735 . . . 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 643 . . 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 11758 . . . 4  |-  ( S  e. Word  A  ->  ( S substr  <. ( 2nd `  ( 1st `  <. F ,  T ,  R >. ) ) ,  ( # `  S
) >. )  e. Word  A
)
3231adantr 452 . . 3  |-  ( ( S  e. Word  A  /\  R  e. Word  A )  ->  ( S substr  <. ( 2nd `  ( 1st `  <. F ,  T ,  R >. ) ) ,  (
# `  S ) >. )  e. Word  A )
33 ccatcl 11735 . . 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 643 . 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 2509 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 359    = wceq 1652    e. wcel 1725   _Vcvv 2948   <.cop 3809   <.cotp 3810   ` cfv 5446  (class class class)co 6073   1stc1st 6339   2ndc2nd 6340   0cc0 8982   #chash 11610  Word cword 11709   concat cconcat 11710   substr csubstr 11712   splice csplice 11713
This theorem is referenced by:  efglem  15340  efgtf  15346  frgpuplem  15396  psgnunilem2  27376
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-ot 3816  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-n0 10214  df-z 10275  df-uz 10481  df-fz 11036  df-fzo 11128  df-hash 11611  df-word 11715  df-concat 11716  df-substr 11718  df-splice 11719
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