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Theorem splfv1 11671
Description: Symbols to the left of a splice are unaffected. (Contributed by Stefan O'Rear, 23-Aug-2015.)
Hypotheses
Ref Expression
spllen.s  |-  ( ph  ->  S  e. Word  A )
spllen.f  |-  ( ph  ->  F  e.  ( 0 ... T ) )
spllen.t  |-  ( ph  ->  T  e.  ( 0 ... ( # `  S
) ) )
spllen.r  |-  ( ph  ->  R  e. Word  A )
splfv1.x  |-  ( ph  ->  X  e.  ( 0..^ F ) )
Assertion
Ref Expression
splfv1  |-  ( ph  ->  ( ( S splice  <. F ,  T ,  R >. ) `
 X )  =  ( S `  X
) )

Proof of Theorem splfv1
StepHypRef Expression
1 spllen.s . . . 4  |-  ( ph  ->  S  e. Word  A )
2 spllen.f . . . 4  |-  ( ph  ->  F  e.  ( 0 ... T ) )
3 spllen.t . . . 4  |-  ( ph  ->  T  e.  ( 0 ... ( # `  S
) ) )
4 spllen.r . . . 4  |-  ( ph  ->  R  e. Word  A )
5 splval 11667 . . . 4  |-  ( ( S  e. Word  A  /\  ( F  e.  (
0 ... T )  /\  T  e.  ( 0 ... ( # `  S
) )  /\  R  e. Word  A ) )  -> 
( S splice  <. F ,  T ,  R >. )  =  ( ( ( S substr  <. 0 ,  F >. ) concat  R ) concat  ( S substr  <. T ,  (
# `  S ) >. ) ) )
61, 2, 3, 4, 5syl13anc 1185 . . 3  |-  ( ph  ->  ( S splice  <. F ,  T ,  R >. )  =  ( ( ( S substr  <. 0 ,  F >. ) concat  R ) concat  ( S substr  <. T ,  (
# `  S ) >. ) ) )
76fveq1d 5634 . 2  |-  ( ph  ->  ( ( S splice  <. F ,  T ,  R >. ) `
 X )  =  ( ( ( ( S substr  <. 0 ,  F >. ) concat  R ) concat  ( S substr  <. T ,  (
# `  S ) >. ) ) `  X
) )
8 swrdcl 11653 . . . . 5  |-  ( S  e. Word  A  ->  ( S substr  <. 0 ,  F >. )  e. Word  A )
91, 8syl 15 . . . 4  |-  ( ph  ->  ( S substr  <. 0 ,  F >. )  e. Word  A
)
10 ccatcl 11630 . . . 4  |-  ( ( ( S substr  <. 0 ,  F >. )  e. Word  A  /\  R  e. Word  A )  ->  ( ( S substr  <. 0 ,  F >. ) concat  R )  e. Word  A
)
119, 4, 10syl2anc 642 . . 3  |-  ( ph  ->  ( ( S substr  <. 0 ,  F >. ) concat  R )  e. Word  A )
12 swrdcl 11653 . . . 4  |-  ( S  e. Word  A  ->  ( S substr  <. T ,  (
# `  S ) >. )  e. Word  A )
131, 12syl 15 . . 3  |-  ( ph  ->  ( S substr  <. T , 
( # `  S )
>. )  e. Word  A )
14 elfzelz 10951 . . . . . . . 8  |-  ( F  e.  ( 0 ... T )  ->  F  e.  ZZ )
15 uzid 10393 . . . . . . . 8  |-  ( F  e.  ZZ  ->  F  e.  ( ZZ>= `  F )
)
162, 14, 153syl 18 . . . . . . 7  |-  ( ph  ->  F  e.  ( ZZ>= `  F ) )
17 wrdfin 11621 . . . . . . . 8  |-  ( R  e. Word  A  ->  R  e.  Fin )
18 hashcl 11526 . . . . . . . 8  |-  ( R  e.  Fin  ->  ( # `
 R )  e. 
NN0 )
194, 17, 183syl 18 . . . . . . 7  |-  ( ph  ->  ( # `  R
)  e.  NN0 )
20 uzaddcl 10426 . . . . . . 7  |-  ( ( F  e.  ( ZZ>= `  F )  /\  ( # `
 R )  e. 
NN0 )  ->  ( F  +  ( # `  R
) )  e.  (
ZZ>= `  F ) )
2116, 19, 20syl2anc 642 . . . . . 6  |-  ( ph  ->  ( F  +  (
# `  R )
)  e.  ( ZZ>= `  F ) )
22 fzoss2 11053 . . . . . 6  |-  ( ( F  +  ( # `  R ) )  e.  ( ZZ>= `  F )  ->  ( 0..^ F ) 
C_  ( 0..^ ( F  +  ( # `  R ) ) ) )
2321, 22syl 15 . . . . 5  |-  ( ph  ->  ( 0..^ F ) 
C_  ( 0..^ ( F  +  ( # `  R ) ) ) )
24 splfv1.x . . . . 5  |-  ( ph  ->  X  e.  ( 0..^ F ) )
2523, 24sseldd 3267 . . . 4  |-  ( ph  ->  X  e.  ( 0..^ ( F  +  (
# `  R )
) ) )
26 ccatlen 11631 . . . . . . 7  |-  ( ( ( S substr  <. 0 ,  F >. )  e. Word  A  /\  R  e. Word  A )  ->  ( # `  (
( S substr  <. 0 ,  F >. ) concat  R )
)  =  ( (
# `  ( S substr  <.
0 ,  F >. ) )  +  ( # `  R ) ) )
279, 4, 26syl2anc 642 . . . . . 6  |-  ( ph  ->  ( # `  (
( S substr  <. 0 ,  F >. ) concat  R )
)  =  ( (
# `  ( S substr  <.
0 ,  F >. ) )  +  ( # `  R ) ) )
28 elfzuz 10947 . . . . . . . . . 10  |-  ( F  e.  ( 0 ... T )  ->  F  e.  ( ZZ>= `  0 )
)
29 eluzfz1 10956 . . . . . . . . . 10  |-  ( F  e.  ( ZZ>= `  0
)  ->  0  e.  ( 0 ... F
) )
302, 28, 293syl 18 . . . . . . . . 9  |-  ( ph  ->  0  e.  ( 0 ... F ) )
31 fzass4 10982 . . . . . . . . . . . 12  |-  ( ( F  e.  ( 0 ... ( # `  S
) )  /\  T  e.  ( F ... ( # `
 S ) ) )  <->  ( F  e.  ( 0 ... T
)  /\  T  e.  ( 0 ... ( # `
 S ) ) ) )
3231bicomi 193 . . . . . . . . . . 11  |-  ( ( F  e.  ( 0 ... T )  /\  T  e.  ( 0 ... ( # `  S
) ) )  <->  ( F  e.  ( 0 ... ( # `
 S ) )  /\  T  e.  ( F ... ( # `  S ) ) ) )
3332simplbi 446 . . . . . . . . . 10  |-  ( ( F  e.  ( 0 ... T )  /\  T  e.  ( 0 ... ( # `  S
) ) )  ->  F  e.  ( 0 ... ( # `  S
) ) )
342, 3, 33syl2anc 642 . . . . . . . . 9  |-  ( ph  ->  F  e.  ( 0 ... ( # `  S
) ) )
35 swrdlen 11657 . . . . . . . . 9  |-  ( ( S  e. Word  A  /\  0  e.  ( 0 ... F )  /\  F  e.  ( 0 ... ( # `  S
) ) )  -> 
( # `  ( S substr  <. 0 ,  F >. ) )  =  ( F  -  0 ) )
361, 30, 34, 35syl3anc 1183 . . . . . . . 8  |-  ( ph  ->  ( # `  ( S substr  <. 0 ,  F >. ) )  =  ( F  -  0 ) )
372, 14syl 15 . . . . . . . . . 10  |-  ( ph  ->  F  e.  ZZ )
3837zcnd 10269 . . . . . . . . 9  |-  ( ph  ->  F  e.  CC )
3938subid1d 9293 . . . . . . . 8  |-  ( ph  ->  ( F  -  0 )  =  F )
4036, 39eqtrd 2398 . . . . . . 7  |-  ( ph  ->  ( # `  ( S substr  <. 0 ,  F >. ) )  =  F )
4140oveq1d 5996 . . . . . 6  |-  ( ph  ->  ( ( # `  ( S substr  <. 0 ,  F >. ) )  +  (
# `  R )
)  =  ( F  +  ( # `  R
) ) )
4227, 41eqtrd 2398 . . . . 5  |-  ( ph  ->  ( # `  (
( S substr  <. 0 ,  F >. ) concat  R )
)  =  ( F  +  ( # `  R
) ) )
4342oveq2d 5997 . . . 4  |-  ( ph  ->  ( 0..^ ( # `  ( ( S substr  <. 0 ,  F >. ) concat  R )
) )  =  ( 0..^ ( F  +  ( # `  R ) ) ) )
4425, 43eleqtrrd 2443 . . 3  |-  ( ph  ->  X  e.  ( 0..^ ( # `  (
( S substr  <. 0 ,  F >. ) concat  R )
) ) )
45 ccatval1 11632 . . 3  |-  ( ( ( ( S substr  <. 0 ,  F >. ) concat  R )  e. Word  A  /\  ( S substr  <. T ,  ( # `  S ) >. )  e. Word  A  /\  X  e.  ( 0..^ ( # `  ( ( S substr  <. 0 ,  F >. ) concat  R )
) ) )  -> 
( ( ( ( S substr  <. 0 ,  F >. ) concat  R ) concat  ( S substr  <. T ,  (
# `  S ) >. ) ) `  X
)  =  ( ( ( S substr  <. 0 ,  F >. ) concat  R ) `  X ) )
4611, 13, 44, 45syl3anc 1183 . 2  |-  ( ph  ->  ( ( ( ( S substr  <. 0 ,  F >. ) concat  R ) concat  ( S substr  <. T ,  (
# `  S ) >. ) ) `  X
)  =  ( ( ( S substr  <. 0 ,  F >. ) concat  R ) `  X ) )
4740oveq2d 5997 . . . . 5  |-  ( ph  ->  ( 0..^ ( # `  ( S substr  <. 0 ,  F >. ) ) )  =  ( 0..^ F ) )
4824, 47eleqtrrd 2443 . . . 4  |-  ( ph  ->  X  e.  ( 0..^ ( # `  ( S substr  <. 0 ,  F >. ) ) ) )
49 ccatval1 11632 . . . 4  |-  ( ( ( S substr  <. 0 ,  F >. )  e. Word  A  /\  R  e. Word  A  /\  X  e.  ( 0..^ ( # `  ( S substr  <. 0 ,  F >. ) ) ) )  ->  ( ( ( S substr  <. 0 ,  F >. ) concat  R ) `  X )  =  ( ( S substr  <. 0 ,  F >. ) `  X
) )
509, 4, 48, 49syl3anc 1183 . . 3  |-  ( ph  ->  ( ( ( S substr  <. 0 ,  F >. ) concat  R ) `  X
)  =  ( ( S substr  <. 0 ,  F >. ) `  X ) )
5139oveq2d 5997 . . . . 5  |-  ( ph  ->  ( 0..^ ( F  -  0 ) )  =  ( 0..^ F ) )
5224, 51eleqtrrd 2443 . . . 4  |-  ( ph  ->  X  e.  ( 0..^ ( F  -  0 ) ) )
53 swrdfv 11658 . . . 4  |-  ( ( ( S  e. Word  A  /\  0  e.  (
0 ... F )  /\  F  e.  ( 0 ... ( # `  S
) ) )  /\  X  e.  ( 0..^ ( F  -  0 ) ) )  -> 
( ( S substr  <. 0 ,  F >. ) `  X
)  =  ( S `
 ( X  + 
0 ) ) )
541, 30, 34, 52, 53syl31anc 1186 . . 3  |-  ( ph  ->  ( ( S substr  <. 0 ,  F >. ) `  X
)  =  ( S `
 ( X  + 
0 ) ) )
55 elfzoelz 11030 . . . . . . 7  |-  ( X  e.  ( 0..^ F )  ->  X  e.  ZZ )
5655zcnd 10269 . . . . . 6  |-  ( X  e.  ( 0..^ F )  ->  X  e.  CC )
5724, 56syl 15 . . . . 5  |-  ( ph  ->  X  e.  CC )
5857addid1d 9159 . . . 4  |-  ( ph  ->  ( X  +  0 )  =  X )
5958fveq2d 5636 . . 3  |-  ( ph  ->  ( S `  ( X  +  0 ) )  =  ( S `
 X ) )
6050, 54, 593eqtrd 2402 . 2  |-  ( ph  ->  ( ( ( S substr  <. 0 ,  F >. ) concat  R ) `  X
)  =  ( S `
 X ) )
617, 46, 603eqtrd 2402 1  |-  ( ph  ->  ( ( S splice  <. F ,  T ,  R >. ) `
 X )  =  ( S `  X
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1647    e. wcel 1715    C_ wss 3238   <.cop 3732   <.cotp 3733   ` cfv 5358  (class class class)co 5981   Fincfn 7006   CCcc 8882   0cc0 8884    + caddc 8887    - cmin 9184   NN0cn0 10114   ZZcz 10175   ZZ>=cuz 10381   ...cfz 10935  ..^cfzo 11025   #chash 11505  Word cword 11604   concat cconcat 11605   substr csubstr 11607   splice csplice 11608
This theorem is referenced by:  psgnunilem2  27009
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-ot 3739  df-uni 3930  df-int 3965  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-riota 6446  df-recs 6530  df-rdg 6565  df-1o 6621  df-oadd 6625  df-er 6802  df-en 7007  df-dom 7008  df-sdom 7009  df-fin 7010  df-card 7719  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-nn 9894  df-n0 10115  df-z 10176  df-uz 10382  df-fz 10936  df-fzo 11026  df-hash 11506  df-word 11610  df-concat 11611  df-substr 11613  df-splice 11614
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