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Theorem splfv1 11815
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 11811 . . . 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 1187 . . 3  |-  ( ph  ->  ( S splice  <. F ,  T ,  R >. )  =  ( ( ( S substr  <. 0 ,  F >. ) concat  R ) concat  ( S substr  <. T ,  (
# `  S ) >. ) ) )
76fveq1d 5759 . 2  |-  ( ph  ->  ( ( S splice  <. F ,  T ,  R >. ) `
 X )  =  ( ( ( ( S substr  <. 0 ,  F >. ) concat  R ) concat  ( S substr  <. T ,  (
# `  S ) >. ) ) `  X
) )
8 swrdcl 11797 . . . . 5  |-  ( S  e. Word  A  ->  ( S substr  <. 0 ,  F >. )  e. Word  A )
91, 8syl 16 . . . 4  |-  ( ph  ->  ( S substr  <. 0 ,  F >. )  e. Word  A
)
10 ccatcl 11774 . . . 4  |-  ( ( ( S substr  <. 0 ,  F >. )  e. Word  A  /\  R  e. Word  A )  ->  ( ( S substr  <. 0 ,  F >. ) concat  R )  e. Word  A
)
119, 4, 10syl2anc 644 . . 3  |-  ( ph  ->  ( ( S substr  <. 0 ,  F >. ) concat  R )  e. Word  A )
12 swrdcl 11797 . . . 4  |-  ( S  e. Word  A  ->  ( S substr  <. T ,  (
# `  S ) >. )  e. Word  A )
131, 12syl 16 . . 3  |-  ( ph  ->  ( S substr  <. T , 
( # `  S )
>. )  e. Word  A )
14 elfzelz 11090 . . . . . . . 8  |-  ( F  e.  ( 0 ... T )  ->  F  e.  ZZ )
15 uzid 10531 . . . . . . . 8  |-  ( F  e.  ZZ  ->  F  e.  ( ZZ>= `  F )
)
162, 14, 153syl 19 . . . . . . 7  |-  ( ph  ->  F  e.  ( ZZ>= `  F ) )
17 wrdfin 11765 . . . . . . . 8  |-  ( R  e. Word  A  ->  R  e.  Fin )
18 hashcl 11670 . . . . . . . 8  |-  ( R  e.  Fin  ->  ( # `
 R )  e. 
NN0 )
194, 17, 183syl 19 . . . . . . 7  |-  ( ph  ->  ( # `  R
)  e.  NN0 )
20 uzaddcl 10564 . . . . . . 7  |-  ( ( F  e.  ( ZZ>= `  F )  /\  ( # `
 R )  e. 
NN0 )  ->  ( F  +  ( # `  R
) )  e.  (
ZZ>= `  F ) )
2116, 19, 20syl2anc 644 . . . . . 6  |-  ( ph  ->  ( F  +  (
# `  R )
)  e.  ( ZZ>= `  F ) )
22 fzoss2 11194 . . . . . 6  |-  ( ( F  +  ( # `  R ) )  e.  ( ZZ>= `  F )  ->  ( 0..^ F ) 
C_  ( 0..^ ( F  +  ( # `  R ) ) ) )
2321, 22syl 16 . . . . 5  |-  ( ph  ->  ( 0..^ F ) 
C_  ( 0..^ ( F  +  ( # `  R ) ) ) )
24 splfv1.x . . . . 5  |-  ( ph  ->  X  e.  ( 0..^ F ) )
2523, 24sseldd 3335 . . . 4  |-  ( ph  ->  X  e.  ( 0..^ ( F  +  (
# `  R )
) ) )
26 ccatlen 11775 . . . . . . 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 644 . . . . . 6  |-  ( ph  ->  ( # `  (
( S substr  <. 0 ,  F >. ) concat  R )
)  =  ( (
# `  ( S substr  <.
0 ,  F >. ) )  +  ( # `  R ) ) )
28 elfzuz 11086 . . . . . . . . . 10  |-  ( F  e.  ( 0 ... T )  ->  F  e.  ( ZZ>= `  0 )
)
29 eluzfz1 11095 . . . . . . . . . 10  |-  ( F  e.  ( ZZ>= `  0
)  ->  0  e.  ( 0 ... F
) )
302, 28, 293syl 19 . . . . . . . . 9  |-  ( ph  ->  0  e.  ( 0 ... F ) )
31 fzass4 11121 . . . . . . . . . . . 12  |-  ( ( F  e.  ( 0 ... ( # `  S
) )  /\  T  e.  ( F ... ( # `
 S ) ) )  <->  ( F  e.  ( 0 ... T
)  /\  T  e.  ( 0 ... ( # `
 S ) ) ) )
3231bicomi 195 . . . . . . . . . . 11  |-  ( ( F  e.  ( 0 ... T )  /\  T  e.  ( 0 ... ( # `  S
) ) )  <->  ( F  e.  ( 0 ... ( # `
 S ) )  /\  T  e.  ( F ... ( # `  S ) ) ) )
3332simplbi 448 . . . . . . . . . 10  |-  ( ( F  e.  ( 0 ... T )  /\  T  e.  ( 0 ... ( # `  S
) ) )  ->  F  e.  ( 0 ... ( # `  S
) ) )
342, 3, 33syl2anc 644 . . . . . . . . 9  |-  ( ph  ->  F  e.  ( 0 ... ( # `  S
) ) )
35 swrdlen 11801 . . . . . . . . 9  |-  ( ( S  e. Word  A  /\  0  e.  ( 0 ... F )  /\  F  e.  ( 0 ... ( # `  S
) ) )  -> 
( # `  ( S substr  <. 0 ,  F >. ) )  =  ( F  -  0 ) )
361, 30, 34, 35syl3anc 1185 . . . . . . . 8  |-  ( ph  ->  ( # `  ( S substr  <. 0 ,  F >. ) )  =  ( F  -  0 ) )
372, 14syl 16 . . . . . . . . . 10  |-  ( ph  ->  F  e.  ZZ )
3837zcnd 10407 . . . . . . . . 9  |-  ( ph  ->  F  e.  CC )
3938subid1d 9431 . . . . . . . 8  |-  ( ph  ->  ( F  -  0 )  =  F )
4036, 39eqtrd 2474 . . . . . . 7  |-  ( ph  ->  ( # `  ( S substr  <. 0 ,  F >. ) )  =  F )
4140oveq1d 6125 . . . . . 6  |-  ( ph  ->  ( ( # `  ( S substr  <. 0 ,  F >. ) )  +  (
# `  R )
)  =  ( F  +  ( # `  R
) ) )
4227, 41eqtrd 2474 . . . . 5  |-  ( ph  ->  ( # `  (
( S substr  <. 0 ,  F >. ) concat  R )
)  =  ( F  +  ( # `  R
) ) )
4342oveq2d 6126 . . . 4  |-  ( ph  ->  ( 0..^ ( # `  ( ( S substr  <. 0 ,  F >. ) concat  R )
) )  =  ( 0..^ ( F  +  ( # `  R ) ) ) )
4425, 43eleqtrrd 2519 . . 3  |-  ( ph  ->  X  e.  ( 0..^ ( # `  (
( S substr  <. 0 ,  F >. ) concat  R )
) ) )
45 ccatval1 11776 . . 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 1185 . 2  |-  ( ph  ->  ( ( ( ( S substr  <. 0 ,  F >. ) concat  R ) concat  ( S substr  <. T ,  (
# `  S ) >. ) ) `  X
)  =  ( ( ( S substr  <. 0 ,  F >. ) concat  R ) `  X ) )
4740oveq2d 6126 . . . . 5  |-  ( ph  ->  ( 0..^ ( # `  ( S substr  <. 0 ,  F >. ) ) )  =  ( 0..^ F ) )
4824, 47eleqtrrd 2519 . . . 4  |-  ( ph  ->  X  e.  ( 0..^ ( # `  ( S substr  <. 0 ,  F >. ) ) ) )
49 ccatval1 11776 . . . 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 1185 . . 3  |-  ( ph  ->  ( ( ( S substr  <. 0 ,  F >. ) concat  R ) `  X
)  =  ( ( S substr  <. 0 ,  F >. ) `  X ) )
5139oveq2d 6126 . . . . 5  |-  ( ph  ->  ( 0..^ ( F  -  0 ) )  =  ( 0..^ F ) )
5224, 51eleqtrrd 2519 . . . 4  |-  ( ph  ->  X  e.  ( 0..^ ( F  -  0 ) ) )
53 swrdfv 11802 . . . 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 1188 . . 3  |-  ( ph  ->  ( ( S substr  <. 0 ,  F >. ) `  X
)  =  ( S `
 ( X  + 
0 ) ) )
55 elfzoelz 11171 . . . . . . 7  |-  ( X  e.  ( 0..^ F )  ->  X  e.  ZZ )
5655zcnd 10407 . . . . . 6  |-  ( X  e.  ( 0..^ F )  ->  X  e.  CC )
5724, 56syl 16 . . . . 5  |-  ( ph  ->  X  e.  CC )
5857addid1d 9297 . . . 4  |-  ( ph  ->  ( X  +  0 )  =  X )
5958fveq2d 5761 . . 3  |-  ( ph  ->  ( S `  ( X  +  0 ) )  =  ( S `
 X ) )
6050, 54, 593eqtrd 2478 . 2  |-  ( ph  ->  ( ( ( S substr  <. 0 ,  F >. ) concat  R ) `  X
)  =  ( S `
 X ) )
617, 46, 603eqtrd 2478 1  |-  ( ph  ->  ( ( S splice  <. F ,  T ,  R >. ) `
 X )  =  ( S `  X
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1727    C_ wss 3306   <.cop 3841   <.cotp 3842   ` cfv 5483  (class class class)co 6110   Fincfn 7138   CCcc 9019   0cc0 9021    + caddc 9024    - cmin 9322   NN0cn0 10252   ZZcz 10313   ZZ>=cuz 10519   ...cfz 11074  ..^cfzo 11166   #chash 11649  Word cword 11748   concat cconcat 11749   substr csubstr 11751   splice csplice 11752
This theorem is referenced by:  psgnunilem2  27433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-cnex 9077  ax-resscn 9078  ax-1cn 9079  ax-icn 9080  ax-addcl 9081  ax-addrcl 9082  ax-mulcl 9083  ax-mulrcl 9084  ax-mulcom 9085  ax-addass 9086  ax-mulass 9087  ax-distr 9088  ax-i2m1 9089  ax-1ne0 9090  ax-1rid 9091  ax-rnegex 9092  ax-rrecex 9093  ax-cnre 9094  ax-pre-lttri 9095  ax-pre-lttrn 9096  ax-pre-ltadd 9097  ax-pre-mulgt0 9098
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-ot 3848  df-uni 4040  df-int 4075  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-om 4875  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6378  df-2nd 6379  df-riota 6578  df-recs 6662  df-rdg 6697  df-1o 6753  df-oadd 6757  df-er 6934  df-en 7139  df-dom 7140  df-sdom 7141  df-fin 7142  df-card 7857  df-pnf 9153  df-mnf 9154  df-xr 9155  df-ltxr 9156  df-le 9157  df-sub 9324  df-neg 9325  df-nn 10032  df-n0 10253  df-z 10314  df-uz 10520  df-fz 11075  df-fzo 11167  df-hash 11650  df-word 11754  df-concat 11755  df-substr 11757  df-splice 11758
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