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Theorem splfv1 11747
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 11743 . . . 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 1186 . . 3  |-  ( ph  ->  ( S splice  <. F ,  T ,  R >. )  =  ( ( ( S substr  <. 0 ,  F >. ) concat  R ) concat  ( S substr  <. T ,  (
# `  S ) >. ) ) )
76fveq1d 5697 . 2  |-  ( ph  ->  ( ( S splice  <. F ,  T ,  R >. ) `
 X )  =  ( ( ( ( S substr  <. 0 ,  F >. ) concat  R ) concat  ( S substr  <. T ,  (
# `  S ) >. ) ) `  X
) )
8 swrdcl 11729 . . . . 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 11706 . . . 4  |-  ( ( ( S substr  <. 0 ,  F >. )  e. Word  A  /\  R  e. Word  A )  ->  ( ( S substr  <. 0 ,  F >. ) concat  R )  e. Word  A
)
119, 4, 10syl2anc 643 . . 3  |-  ( ph  ->  ( ( S substr  <. 0 ,  F >. ) concat  R )  e. Word  A )
12 swrdcl 11729 . . . 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 11023 . . . . . . . 8  |-  ( F  e.  ( 0 ... T )  ->  F  e.  ZZ )
15 uzid 10464 . . . . . . . 8  |-  ( F  e.  ZZ  ->  F  e.  ( ZZ>= `  F )
)
162, 14, 153syl 19 . . . . . . 7  |-  ( ph  ->  F  e.  ( ZZ>= `  F ) )
17 wrdfin 11697 . . . . . . . 8  |-  ( R  e. Word  A  ->  R  e.  Fin )
18 hashcl 11602 . . . . . . . 8  |-  ( R  e.  Fin  ->  ( # `
 R )  e. 
NN0 )
194, 17, 183syl 19 . . . . . . 7  |-  ( ph  ->  ( # `  R
)  e.  NN0 )
20 uzaddcl 10497 . . . . . . 7  |-  ( ( F  e.  ( ZZ>= `  F )  /\  ( # `
 R )  e. 
NN0 )  ->  ( F  +  ( # `  R
) )  e.  (
ZZ>= `  F ) )
2116, 19, 20syl2anc 643 . . . . . 6  |-  ( ph  ->  ( F  +  (
# `  R )
)  e.  ( ZZ>= `  F ) )
22 fzoss2 11126 . . . . . 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 3317 . . . 4  |-  ( ph  ->  X  e.  ( 0..^ ( F  +  (
# `  R )
) ) )
26 ccatlen 11707 . . . . . . 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 643 . . . . . 6  |-  ( ph  ->  ( # `  (
( S substr  <. 0 ,  F >. ) concat  R )
)  =  ( (
# `  ( S substr  <.
0 ,  F >. ) )  +  ( # `  R ) ) )
28 elfzuz 11019 . . . . . . . . . 10  |-  ( F  e.  ( 0 ... T )  ->  F  e.  ( ZZ>= `  0 )
)
29 eluzfz1 11028 . . . . . . . . . 10  |-  ( F  e.  ( ZZ>= `  0
)  ->  0  e.  ( 0 ... F
) )
302, 28, 293syl 19 . . . . . . . . 9  |-  ( ph  ->  0  e.  ( 0 ... F ) )
31 fzass4 11054 . . . . . . . . . . . 12  |-  ( ( F  e.  ( 0 ... ( # `  S
) )  /\  T  e.  ( F ... ( # `
 S ) ) )  <->  ( F  e.  ( 0 ... T
)  /\  T  e.  ( 0 ... ( # `
 S ) ) ) )
3231bicomi 194 . . . . . . . . . . 11  |-  ( ( F  e.  ( 0 ... T )  /\  T  e.  ( 0 ... ( # `  S
) ) )  <->  ( F  e.  ( 0 ... ( # `
 S ) )  /\  T  e.  ( F ... ( # `  S ) ) ) )
3332simplbi 447 . . . . . . . . . 10  |-  ( ( F  e.  ( 0 ... T )  /\  T  e.  ( 0 ... ( # `  S
) ) )  ->  F  e.  ( 0 ... ( # `  S
) ) )
342, 3, 33syl2anc 643 . . . . . . . . 9  |-  ( ph  ->  F  e.  ( 0 ... ( # `  S
) ) )
35 swrdlen 11733 . . . . . . . . 9  |-  ( ( S  e. Word  A  /\  0  e.  ( 0 ... F )  /\  F  e.  ( 0 ... ( # `  S
) ) )  -> 
( # `  ( S substr  <. 0 ,  F >. ) )  =  ( F  -  0 ) )
361, 30, 34, 35syl3anc 1184 . . . . . . . 8  |-  ( ph  ->  ( # `  ( S substr  <. 0 ,  F >. ) )  =  ( F  -  0 ) )
372, 14syl 16 . . . . . . . . . 10  |-  ( ph  ->  F  e.  ZZ )
3837zcnd 10340 . . . . . . . . 9  |-  ( ph  ->  F  e.  CC )
3938subid1d 9364 . . . . . . . 8  |-  ( ph  ->  ( F  -  0 )  =  F )
4036, 39eqtrd 2444 . . . . . . 7  |-  ( ph  ->  ( # `  ( S substr  <. 0 ,  F >. ) )  =  F )
4140oveq1d 6063 . . . . . 6  |-  ( ph  ->  ( ( # `  ( S substr  <. 0 ,  F >. ) )  +  (
# `  R )
)  =  ( F  +  ( # `  R
) ) )
4227, 41eqtrd 2444 . . . . 5  |-  ( ph  ->  ( # `  (
( S substr  <. 0 ,  F >. ) concat  R )
)  =  ( F  +  ( # `  R
) ) )
4342oveq2d 6064 . . . 4  |-  ( ph  ->  ( 0..^ ( # `  ( ( S substr  <. 0 ,  F >. ) concat  R )
) )  =  ( 0..^ ( F  +  ( # `  R ) ) ) )
4425, 43eleqtrrd 2489 . . 3  |-  ( ph  ->  X  e.  ( 0..^ ( # `  (
( S substr  <. 0 ,  F >. ) concat  R )
) ) )
45 ccatval1 11708 . . 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 1184 . 2  |-  ( ph  ->  ( ( ( ( S substr  <. 0 ,  F >. ) concat  R ) concat  ( S substr  <. T ,  (
# `  S ) >. ) ) `  X
)  =  ( ( ( S substr  <. 0 ,  F >. ) concat  R ) `  X ) )
4740oveq2d 6064 . . . . 5  |-  ( ph  ->  ( 0..^ ( # `  ( S substr  <. 0 ,  F >. ) ) )  =  ( 0..^ F ) )
4824, 47eleqtrrd 2489 . . . 4  |-  ( ph  ->  X  e.  ( 0..^ ( # `  ( S substr  <. 0 ,  F >. ) ) ) )
49 ccatval1 11708 . . . 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 1184 . . 3  |-  ( ph  ->  ( ( ( S substr  <. 0 ,  F >. ) concat  R ) `  X
)  =  ( ( S substr  <. 0 ,  F >. ) `  X ) )
5139oveq2d 6064 . . . . 5  |-  ( ph  ->  ( 0..^ ( F  -  0 ) )  =  ( 0..^ F ) )
5224, 51eleqtrrd 2489 . . . 4  |-  ( ph  ->  X  e.  ( 0..^ ( F  -  0 ) ) )
53 swrdfv 11734 . . . 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 1187 . . 3  |-  ( ph  ->  ( ( S substr  <. 0 ,  F >. ) `  X
)  =  ( S `
 ( X  + 
0 ) ) )
55 elfzoelz 11103 . . . . . . 7  |-  ( X  e.  ( 0..^ F )  ->  X  e.  ZZ )
5655zcnd 10340 . . . . . 6  |-  ( X  e.  ( 0..^ F )  ->  X  e.  CC )
5724, 56syl 16 . . . . 5  |-  ( ph  ->  X  e.  CC )
5857addid1d 9230 . . . 4  |-  ( ph  ->  ( X  +  0 )  =  X )
5958fveq2d 5699 . . 3  |-  ( ph  ->  ( S `  ( X  +  0 ) )  =  ( S `
 X ) )
6050, 54, 593eqtrd 2448 . 2  |-  ( ph  ->  ( ( ( S substr  <. 0 ,  F >. ) concat  R ) `  X
)  =  ( S `
 X ) )
617, 46, 603eqtrd 2448 1  |-  ( ph  ->  ( ( S splice  <. F ,  T ,  R >. ) `
 X )  =  ( S `  X
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721    C_ wss 3288   <.cop 3785   <.cotp 3786   ` cfv 5421  (class class class)co 6048   Fincfn 7076   CCcc 8952   0cc0 8954    + caddc 8957    - cmin 9255   NN0cn0 10185   ZZcz 10246   ZZ>=cuz 10452   ...cfz 11007  ..^cfzo 11098   #chash 11581  Word cword 11680   concat cconcat 11681   substr csubstr 11683   splice csplice 11684
This theorem is referenced by:  psgnunilem2  27294
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-ot 3792  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-oadd 6695  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-card 7790  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-nn 9965  df-n0 10186  df-z 10247  df-uz 10453  df-fz 11008  df-fzo 11099  df-hash 11582  df-word 11686  df-concat 11687  df-substr 11689  df-splice 11690
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