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Theorem splfv1 12785
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 12781 . . . 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 >. ) ++  R ) ++  ( S substr  <. T ,  (
# `  S ) >. ) ) )
61, 2, 3, 4, 5syl13anc 1232 . . 3  |-  ( ph  ->  ( S splice  <. F ,  T ,  R >. )  =  ( ( ( S substr  <. 0 ,  F >. ) ++  R ) ++  ( S substr  <. T ,  (
# `  S ) >. ) ) )
76fveq1d 5850 . 2  |-  ( ph  ->  ( ( S splice  <. F ,  T ,  R >. ) `
 X )  =  ( ( ( ( S substr  <. 0 ,  F >. ) ++  R ) ++  ( S substr  <. T ,  (
# `  S ) >. ) ) `  X
) )
8 swrdcl 12698 . . . . 5  |-  ( S  e. Word  A  ->  ( S substr  <. 0 ,  F >. )  e. Word  A )
91, 8syl 17 . . . 4  |-  ( ph  ->  ( S substr  <. 0 ,  F >. )  e. Word  A
)
10 ccatcl 12645 . . . 4  |-  ( ( ( S substr  <. 0 ,  F >. )  e. Word  A  /\  R  e. Word  A )  ->  ( ( S substr  <. 0 ,  F >. ) ++  R )  e. Word  A
)
119, 4, 10syl2anc 659 . . 3  |-  ( ph  ->  ( ( S substr  <. 0 ,  F >. ) ++  R )  e. Word  A )
12 swrdcl 12698 . . . 4  |-  ( S  e. Word  A  ->  ( S substr  <. T ,  (
# `  S ) >. )  e. Word  A )
131, 12syl 17 . . 3  |-  ( ph  ->  ( S substr  <. T , 
( # `  S )
>. )  e. Word  A )
14 elfzelz 11740 . . . . . . . 8  |-  ( F  e.  ( 0 ... T )  ->  F  e.  ZZ )
15 uzid 11140 . . . . . . . 8  |-  ( F  e.  ZZ  ->  F  e.  ( ZZ>= `  F )
)
162, 14, 153syl 20 . . . . . . 7  |-  ( ph  ->  F  e.  ( ZZ>= `  F ) )
17 wrdfin 12611 . . . . . . . 8  |-  ( R  e. Word  A  ->  R  e.  Fin )
18 hashcl 12473 . . . . . . . 8  |-  ( R  e.  Fin  ->  ( # `
 R )  e. 
NN0 )
194, 17, 183syl 20 . . . . . . 7  |-  ( ph  ->  ( # `  R
)  e.  NN0 )
20 uzaddcl 11182 . . . . . . 7  |-  ( ( F  e.  ( ZZ>= `  F )  /\  ( # `
 R )  e. 
NN0 )  ->  ( F  +  ( # `  R
) )  e.  (
ZZ>= `  F ) )
2116, 19, 20syl2anc 659 . . . . . 6  |-  ( ph  ->  ( F  +  (
# `  R )
)  e.  ( ZZ>= `  F ) )
22 fzoss2 11883 . . . . . 6  |-  ( ( F  +  ( # `  R ) )  e.  ( ZZ>= `  F )  ->  ( 0..^ F ) 
C_  ( 0..^ ( F  +  ( # `  R ) ) ) )
2321, 22syl 17 . . . . 5  |-  ( ph  ->  ( 0..^ F ) 
C_  ( 0..^ ( F  +  ( # `  R ) ) ) )
24 splfv1.x . . . . 5  |-  ( ph  ->  X  e.  ( 0..^ F ) )
2523, 24sseldd 3442 . . . 4  |-  ( ph  ->  X  e.  ( 0..^ ( F  +  (
# `  R )
) ) )
26 ccatlen 12646 . . . . . . 7  |-  ( ( ( S substr  <. 0 ,  F >. )  e. Word  A  /\  R  e. Word  A )  ->  ( # `  (
( S substr  <. 0 ,  F >. ) ++  R ) )  =  ( (
# `  ( S substr  <.
0 ,  F >. ) )  +  ( # `  R ) ) )
279, 4, 26syl2anc 659 . . . . . 6  |-  ( ph  ->  ( # `  (
( S substr  <. 0 ,  F >. ) ++  R ) )  =  ( (
# `  ( S substr  <.
0 ,  F >. ) )  +  ( # `  R ) ) )
28 elfzuz 11736 . . . . . . . . . 10  |-  ( F  e.  ( 0 ... T )  ->  F  e.  ( ZZ>= `  0 )
)
29 eluzfz1 11745 . . . . . . . . . 10  |-  ( F  e.  ( ZZ>= `  0
)  ->  0  e.  ( 0 ... F
) )
302, 28, 293syl 20 . . . . . . . . 9  |-  ( ph  ->  0  e.  ( 0 ... F ) )
31 fzass4 11774 . . . . . . . . . . . 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 458 . . . . . . . . . 10  |-  ( ( F  e.  ( 0 ... T )  /\  T  e.  ( 0 ... ( # `  S
) ) )  ->  F  e.  ( 0 ... ( # `  S
) ) )
342, 3, 33syl2anc 659 . . . . . . . . 9  |-  ( ph  ->  F  e.  ( 0 ... ( # `  S
) ) )
35 swrdlen 12702 . . . . . . . . 9  |-  ( ( S  e. Word  A  /\  0  e.  ( 0 ... F )  /\  F  e.  ( 0 ... ( # `  S
) ) )  -> 
( # `  ( S substr  <. 0 ,  F >. ) )  =  ( F  -  0 ) )
361, 30, 34, 35syl3anc 1230 . . . . . . . 8  |-  ( ph  ->  ( # `  ( S substr  <. 0 ,  F >. ) )  =  ( F  -  0 ) )
372, 14syl 17 . . . . . . . . . 10  |-  ( ph  ->  F  e.  ZZ )
3837zcnd 11008 . . . . . . . . 9  |-  ( ph  ->  F  e.  CC )
3938subid1d 9955 . . . . . . . 8  |-  ( ph  ->  ( F  -  0 )  =  F )
4036, 39eqtrd 2443 . . . . . . 7  |-  ( ph  ->  ( # `  ( S substr  <. 0 ,  F >. ) )  =  F )
4140oveq1d 6292 . . . . . 6  |-  ( ph  ->  ( ( # `  ( S substr  <. 0 ,  F >. ) )  +  (
# `  R )
)  =  ( F  +  ( # `  R
) ) )
4227, 41eqtrd 2443 . . . . 5  |-  ( ph  ->  ( # `  (
( S substr  <. 0 ,  F >. ) ++  R ) )  =  ( F  +  ( # `  R
) ) )
4342oveq2d 6293 . . . 4  |-  ( ph  ->  ( 0..^ ( # `  ( ( S substr  <. 0 ,  F >. ) ++  R ) ) )  =  ( 0..^ ( F  +  ( # `  R ) ) ) )
4425, 43eleqtrrd 2493 . . 3  |-  ( ph  ->  X  e.  ( 0..^ ( # `  (
( S substr  <. 0 ,  F >. ) ++  R ) ) ) )
45 ccatval1 12647 . . 3  |-  ( ( ( ( S substr  <. 0 ,  F >. ) ++  R )  e. Word  A  /\  ( S substr  <. T ,  (
# `  S ) >. )  e. Word  A  /\  X  e.  ( 0..^ ( # `  (
( S substr  <. 0 ,  F >. ) ++  R ) ) ) )  -> 
( ( ( ( S substr  <. 0 ,  F >. ) ++  R ) ++  ( S substr  <. T ,  (
# `  S ) >. ) ) `  X
)  =  ( ( ( S substr  <. 0 ,  F >. ) ++  R ) `
 X ) )
4611, 13, 44, 45syl3anc 1230 . 2  |-  ( ph  ->  ( ( ( ( S substr  <. 0 ,  F >. ) ++  R ) ++  ( S substr  <. T ,  (
# `  S ) >. ) ) `  X
)  =  ( ( ( S substr  <. 0 ,  F >. ) ++  R ) `
 X ) )
4740oveq2d 6293 . . . . 5  |-  ( ph  ->  ( 0..^ ( # `  ( S substr  <. 0 ,  F >. ) ) )  =  ( 0..^ F ) )
4824, 47eleqtrrd 2493 . . . 4  |-  ( ph  ->  X  e.  ( 0..^ ( # `  ( S substr  <. 0 ,  F >. ) ) ) )
49 ccatval1 12647 . . . 4  |-  ( ( ( S substr  <. 0 ,  F >. )  e. Word  A  /\  R  e. Word  A  /\  X  e.  ( 0..^ ( # `  ( S substr  <. 0 ,  F >. ) ) ) )  ->  ( ( ( S substr  <. 0 ,  F >. ) ++  R ) `  X )  =  ( ( S substr  <. 0 ,  F >. ) `  X
) )
509, 4, 48, 49syl3anc 1230 . . 3  |-  ( ph  ->  ( ( ( S substr  <. 0 ,  F >. ) ++  R ) `  X
)  =  ( ( S substr  <. 0 ,  F >. ) `  X ) )
5139oveq2d 6293 . . . . 5  |-  ( ph  ->  ( 0..^ ( F  -  0 ) )  =  ( 0..^ F ) )
5224, 51eleqtrrd 2493 . . . 4  |-  ( ph  ->  X  e.  ( 0..^ ( F  -  0 ) ) )
53 swrdfv 12703 . . . 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 1233 . . 3  |-  ( ph  ->  ( ( S substr  <. 0 ,  F >. ) `  X
)  =  ( S `
 ( X  + 
0 ) ) )
55 elfzoelz 11857 . . . . . . 7  |-  ( X  e.  ( 0..^ F )  ->  X  e.  ZZ )
5655zcnd 11008 . . . . . 6  |-  ( X  e.  ( 0..^ F )  ->  X  e.  CC )
5724, 56syl 17 . . . . 5  |-  ( ph  ->  X  e.  CC )
5857addid1d 9813 . . . 4  |-  ( ph  ->  ( X  +  0 )  =  X )
5958fveq2d 5852 . . 3  |-  ( ph  ->  ( S `  ( X  +  0 ) )  =  ( S `
 X ) )
6050, 54, 593eqtrd 2447 . 2  |-  ( ph  ->  ( ( ( S substr  <. 0 ,  F >. ) ++  R ) `  X
)  =  ( S `
 X ) )
617, 46, 603eqtrd 2447 1  |-  ( ph  ->  ( ( S splice  <. F ,  T ,  R >. ) `
 X )  =  ( S `  X
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842    C_ wss 3413   <.cop 3977   <.cotp 3979   ` cfv 5568  (class class class)co 6277   Fincfn 7553   CCcc 9519   0cc0 9521    + caddc 9524    - cmin 9840   NN0cn0 10835   ZZcz 10904   ZZ>=cuz 11126   ...cfz 11724  ..^cfzo 11852   #chash 12450  Word cword 12581   ++ cconcat 12583   substr csubstr 12585   splice csplice 12586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-ot 3980  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-1st 6783  df-2nd 6784  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-1o 7166  df-oadd 7170  df-er 7347  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-card 8351  df-cda 8579  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-nn 10576  df-2 10634  df-n0 10836  df-z 10905  df-uz 11127  df-fz 11725  df-fzo 11853  df-hash 12451  df-word 12589  df-concat 12591  df-substr 12593  df-splice 12594
This theorem is referenced by:  psgnunilem2  16842
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