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Theorem splfv1 12683
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 12679 . . . 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 1225 . . 3  |-  ( ph  ->  ( S splice  <. F ,  T ,  R >. )  =  ( ( ( S substr  <. 0 ,  F >. ) concat  R ) concat  ( S substr  <. T ,  (
# `  S ) >. ) ) )
76fveq1d 5861 . 2  |-  ( ph  ->  ( ( S splice  <. F ,  T ,  R >. ) `
 X )  =  ( ( ( ( S substr  <. 0 ,  F >. ) concat  R ) concat  ( S substr  <. T ,  (
# `  S ) >. ) ) `  X
) )
8 swrdcl 12598 . . . . 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 12547 . . . 4  |-  ( ( ( S substr  <. 0 ,  F >. )  e. Word  A  /\  R  e. Word  A )  ->  ( ( S substr  <. 0 ,  F >. ) concat  R )  e. Word  A
)
119, 4, 10syl2anc 661 . . 3  |-  ( ph  ->  ( ( S substr  <. 0 ,  F >. ) concat  R )  e. Word  A )
12 swrdcl 12598 . . . 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 11679 . . . . . . . 8  |-  ( F  e.  ( 0 ... T )  ->  F  e.  ZZ )
15 uzid 11087 . . . . . . . 8  |-  ( F  e.  ZZ  ->  F  e.  ( ZZ>= `  F )
)
162, 14, 153syl 20 . . . . . . 7  |-  ( ph  ->  F  e.  ( ZZ>= `  F ) )
17 wrdfin 12516 . . . . . . . 8  |-  ( R  e. Word  A  ->  R  e.  Fin )
18 hashcl 12385 . . . . . . . 8  |-  ( R  e.  Fin  ->  ( # `
 R )  e. 
NN0 )
194, 17, 183syl 20 . . . . . . 7  |-  ( ph  ->  ( # `  R
)  e.  NN0 )
20 uzaddcl 11128 . . . . . . 7  |-  ( ( F  e.  ( ZZ>= `  F )  /\  ( # `
 R )  e. 
NN0 )  ->  ( F  +  ( # `  R
) )  e.  (
ZZ>= `  F ) )
2116, 19, 20syl2anc 661 . . . . . 6  |-  ( ph  ->  ( F  +  (
# `  R )
)  e.  ( ZZ>= `  F ) )
22 fzoss2 11812 . . . . . 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 3500 . . . 4  |-  ( ph  ->  X  e.  ( 0..^ ( F  +  (
# `  R )
) ) )
26 ccatlen 12548 . . . . . . 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 661 . . . . . 6  |-  ( ph  ->  ( # `  (
( S substr  <. 0 ,  F >. ) concat  R )
)  =  ( (
# `  ( S substr  <.
0 ,  F >. ) )  +  ( # `  R ) ) )
28 elfzuz 11675 . . . . . . . . . 10  |-  ( F  e.  ( 0 ... T )  ->  F  e.  ( ZZ>= `  0 )
)
29 eluzfz1 11684 . . . . . . . . . 10  |-  ( F  e.  ( ZZ>= `  0
)  ->  0  e.  ( 0 ... F
) )
302, 28, 293syl 20 . . . . . . . . 9  |-  ( ph  ->  0  e.  ( 0 ... F ) )
31 fzass4 11712 . . . . . . . . . . . 12  |-  ( ( F  e.  ( 0 ... ( # `  S
) )  /\  T  e.  ( F ... ( # `
 S ) ) )  <->  ( F  e.  ( 0 ... T
)  /\  T  e.  ( 0 ... ( # `
 S ) ) ) )
3231bicomi 202 . . . . . . . . . . 11  |-  ( ( F  e.  ( 0 ... T )  /\  T  e.  ( 0 ... ( # `  S
) ) )  <->  ( F  e.  ( 0 ... ( # `
 S ) )  /\  T  e.  ( F ... ( # `  S ) ) ) )
3332simplbi 460 . . . . . . . . . 10  |-  ( ( F  e.  ( 0 ... T )  /\  T  e.  ( 0 ... ( # `  S
) ) )  ->  F  e.  ( 0 ... ( # `  S
) ) )
342, 3, 33syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  F  e.  ( 0 ... ( # `  S
) ) )
35 swrdlen 12602 . . . . . . . . 9  |-  ( ( S  e. Word  A  /\  0  e.  ( 0 ... F )  /\  F  e.  ( 0 ... ( # `  S
) ) )  -> 
( # `  ( S substr  <. 0 ,  F >. ) )  =  ( F  -  0 ) )
361, 30, 34, 35syl3anc 1223 . . . . . . . 8  |-  ( ph  ->  ( # `  ( S substr  <. 0 ,  F >. ) )  =  ( F  -  0 ) )
372, 14syl 16 . . . . . . . . . 10  |-  ( ph  ->  F  e.  ZZ )
3837zcnd 10958 . . . . . . . . 9  |-  ( ph  ->  F  e.  CC )
3938subid1d 9910 . . . . . . . 8  |-  ( ph  ->  ( F  -  0 )  =  F )
4036, 39eqtrd 2503 . . . . . . 7  |-  ( ph  ->  ( # `  ( S substr  <. 0 ,  F >. ) )  =  F )
4140oveq1d 6292 . . . . . 6  |-  ( ph  ->  ( ( # `  ( S substr  <. 0 ,  F >. ) )  +  (
# `  R )
)  =  ( F  +  ( # `  R
) ) )
4227, 41eqtrd 2503 . . . . 5  |-  ( ph  ->  ( # `  (
( S substr  <. 0 ,  F >. ) concat  R )
)  =  ( F  +  ( # `  R
) ) )
4342oveq2d 6293 . . . 4  |-  ( ph  ->  ( 0..^ ( # `  ( ( S substr  <. 0 ,  F >. ) concat  R )
) )  =  ( 0..^ ( F  +  ( # `  R ) ) ) )
4425, 43eleqtrrd 2553 . . 3  |-  ( ph  ->  X  e.  ( 0..^ ( # `  (
( S substr  <. 0 ,  F >. ) concat  R )
) ) )
45 ccatval1 12549 . . 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 1223 . 2  |-  ( ph  ->  ( ( ( ( S substr  <. 0 ,  F >. ) concat  R ) concat  ( S substr  <. T ,  (
# `  S ) >. ) ) `  X
)  =  ( ( ( S substr  <. 0 ,  F >. ) concat  R ) `  X ) )
4740oveq2d 6293 . . . . 5  |-  ( ph  ->  ( 0..^ ( # `  ( S substr  <. 0 ,  F >. ) ) )  =  ( 0..^ F ) )
4824, 47eleqtrrd 2553 . . . 4  |-  ( ph  ->  X  e.  ( 0..^ ( # `  ( S substr  <. 0 ,  F >. ) ) ) )
49 ccatval1 12549 . . . 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 1223 . . 3  |-  ( ph  ->  ( ( ( S substr  <. 0 ,  F >. ) concat  R ) `  X
)  =  ( ( S substr  <. 0 ,  F >. ) `  X ) )
5139oveq2d 6293 . . . . 5  |-  ( ph  ->  ( 0..^ ( F  -  0 ) )  =  ( 0..^ F ) )
5224, 51eleqtrrd 2553 . . . 4  |-  ( ph  ->  X  e.  ( 0..^ ( F  -  0 ) ) )
53 swrdfv 12603 . . . 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 1226 . . 3  |-  ( ph  ->  ( ( S substr  <. 0 ,  F >. ) `  X
)  =  ( S `
 ( X  + 
0 ) ) )
55 elfzoelz 11788 . . . . . . 7  |-  ( X  e.  ( 0..^ F )  ->  X  e.  ZZ )
5655zcnd 10958 . . . . . 6  |-  ( X  e.  ( 0..^ F )  ->  X  e.  CC )
5724, 56syl 16 . . . . 5  |-  ( ph  ->  X  e.  CC )
5857addid1d 9770 . . . 4  |-  ( ph  ->  ( X  +  0 )  =  X )
5958fveq2d 5863 . . 3  |-  ( ph  ->  ( S `  ( X  +  0 ) )  =  ( S `
 X ) )
6050, 54, 593eqtrd 2507 . 2  |-  ( ph  ->  ( ( ( S substr  <. 0 ,  F >. ) concat  R ) `  X
)  =  ( S `
 X ) )
617, 46, 603eqtrd 2507 1  |-  ( ph  ->  ( ( S splice  <. F ,  T ,  R >. ) `
 X )  =  ( S `  X
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762    C_ wss 3471   <.cop 4028   <.cotp 4030   ` cfv 5581  (class class class)co 6277   Fincfn 7508   CCcc 9481   0cc0 9483    + caddc 9486    - cmin 9796   NN0cn0 10786   ZZcz 10855   ZZ>=cuz 11073   ...cfz 11663  ..^cfzo 11783   #chash 12362  Word cword 12489   concat cconcat 12491   substr csubstr 12493   splice csplice 12494
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-ot 4031  df-uni 4241  df-int 4278  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-1st 6776  df-2nd 6777  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  df-card 8311  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-nn 10528  df-n0 10787  df-z 10856  df-uz 11074  df-fz 11664  df-fzo 11784  df-hash 12363  df-word 12497  df-concat 12499  df-substr 12501  df-splice 12502
This theorem is referenced by:  psgnunilem2  16311
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