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Theorem splfv2a 12788
Description: Symbols within the replacement region of a splice, expressed using the coordinates of the replacement region. (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 )
splfv2a.x  |-  ( ph  ->  X  e.  ( 0..^ ( # `  R
) ) )
Assertion
Ref Expression
splfv2a  |-  ( ph  ->  ( ( S splice  <. F ,  T ,  R >. ) `
 ( F  +  X ) )  =  ( R `  X
) )

Proof of Theorem splfv2a
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 12783 . . . 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 ) >. ) ) )
7 elfznn0 11826 . . . . . . 7  |-  ( F  e.  ( 0 ... T )  ->  F  e.  NN0 )
82, 7syl 17 . . . . . 6  |-  ( ph  ->  F  e.  NN0 )
98nn0cnd 10895 . . . . 5  |-  ( ph  ->  F  e.  CC )
10 splfv2a.x . . . . . . 7  |-  ( ph  ->  X  e.  ( 0..^ ( # `  R
) ) )
11 elfzoelz 11859 . . . . . . 7  |-  ( X  e.  ( 0..^ (
# `  R )
)  ->  X  e.  ZZ )
1210, 11syl 17 . . . . . 6  |-  ( ph  ->  X  e.  ZZ )
1312zcnd 11009 . . . . 5  |-  ( ph  ->  X  e.  CC )
149, 13addcomd 9816 . . . 4  |-  ( ph  ->  ( F  +  X
)  =  ( X  +  F ) )
15 nn0uz 11161 . . . . . . . . 9  |-  NN0  =  ( ZZ>= `  0 )
168, 15syl6eleq 2500 . . . . . . . 8  |-  ( ph  ->  F  e.  ( ZZ>= ` 
0 ) )
17 eluzfz1 11747 . . . . . . . 8  |-  ( F  e.  ( ZZ>= `  0
)  ->  0  e.  ( 0 ... F
) )
1816, 17syl 17 . . . . . . 7  |-  ( ph  ->  0  e.  ( 0 ... F ) )
19 elfzuz3 11739 . . . . . . . . . 10  |-  ( T  e.  ( 0 ... ( # `  S
) )  ->  ( # `
 S )  e.  ( ZZ>= `  T )
)
203, 19syl 17 . . . . . . . . 9  |-  ( ph  ->  ( # `  S
)  e.  ( ZZ>= `  T ) )
21 elfzuz3 11739 . . . . . . . . . 10  |-  ( F  e.  ( 0 ... T )  ->  T  e.  ( ZZ>= `  F )
)
222, 21syl 17 . . . . . . . . 9  |-  ( ph  ->  T  e.  ( ZZ>= `  F ) )
23 uztrn 11143 . . . . . . . . 9  |-  ( ( ( # `  S
)  e.  ( ZZ>= `  T )  /\  T  e.  ( ZZ>= `  F )
)  ->  ( # `  S
)  e.  ( ZZ>= `  F ) )
2420, 22, 23syl2anc 659 . . . . . . . 8  |-  ( ph  ->  ( # `  S
)  e.  ( ZZ>= `  F ) )
25 elfzuzb 11736 . . . . . . . 8  |-  ( F  e.  ( 0 ... ( # `  S
) )  <->  ( F  e.  ( ZZ>= `  0 )  /\  ( # `  S
)  e.  ( ZZ>= `  F ) ) )
2616, 24, 25sylanbrc 662 . . . . . . 7  |-  ( ph  ->  F  e.  ( 0 ... ( # `  S
) ) )
27 swrdlen 12704 . . . . . . 7  |-  ( ( S  e. Word  A  /\  0  e.  ( 0 ... F )  /\  F  e.  ( 0 ... ( # `  S
) ) )  -> 
( # `  ( S substr  <. 0 ,  F >. ) )  =  ( F  -  0 ) )
281, 18, 26, 27syl3anc 1230 . . . . . 6  |-  ( ph  ->  ( # `  ( S substr  <. 0 ,  F >. ) )  =  ( F  -  0 ) )
299subid1d 9956 . . . . . 6  |-  ( ph  ->  ( F  -  0 )  =  F )
3028, 29eqtrd 2443 . . . . 5  |-  ( ph  ->  ( # `  ( S substr  <. 0 ,  F >. ) )  =  F )
3130oveq2d 6294 . . . 4  |-  ( ph  ->  ( X  +  (
# `  ( S substr  <.
0 ,  F >. ) ) )  =  ( X  +  F ) )
3214, 31eqtr4d 2446 . . 3  |-  ( ph  ->  ( F  +  X
)  =  ( X  +  ( # `  ( S substr  <. 0 ,  F >. ) ) ) )
336, 32fveq12d 5855 . 2  |-  ( ph  ->  ( ( S splice  <. F ,  T ,  R >. ) `
 ( F  +  X ) )  =  ( ( ( ( S substr  <. 0 ,  F >. ) ++  R ) ++  ( S substr  <. T ,  (
# `  S ) >. ) ) `  ( X  +  ( # `  ( S substr  <. 0 ,  F >. ) ) ) ) )
34 swrdcl 12700 . . . . 5  |-  ( S  e. Word  A  ->  ( S substr  <. 0 ,  F >. )  e. Word  A )
351, 34syl 17 . . . 4  |-  ( ph  ->  ( S substr  <. 0 ,  F >. )  e. Word  A
)
36 ccatcl 12647 . . . 4  |-  ( ( ( S substr  <. 0 ,  F >. )  e. Word  A  /\  R  e. Word  A )  ->  ( ( S substr  <. 0 ,  F >. ) ++  R )  e. Word  A
)
3735, 4, 36syl2anc 659 . . 3  |-  ( ph  ->  ( ( S substr  <. 0 ,  F >. ) ++  R )  e. Word  A )
38 swrdcl 12700 . . . 4  |-  ( S  e. Word  A  ->  ( S substr  <. T ,  (
# `  S ) >. )  e. Word  A )
391, 38syl 17 . . 3  |-  ( ph  ->  ( S substr  <. T , 
( # `  S )
>. )  e. Word  A )
40 0nn0 10851 . . . . . . . 8  |-  0  e.  NN0
41 nn0addcl 10872 . . . . . . . 8  |-  ( ( 0  e.  NN0  /\  F  e.  NN0 )  -> 
( 0  +  F
)  e.  NN0 )
4240, 8, 41sylancr 661 . . . . . . 7  |-  ( ph  ->  ( 0  +  F
)  e.  NN0 )
43 fzoss1 11884 . . . . . . . 8  |-  ( ( 0  +  F )  e.  ( ZZ>= `  0
)  ->  ( (
0  +  F )..^ ( ( # `  R
)  +  F ) )  C_  ( 0..^ ( ( # `  R
)  +  F ) ) )
4443, 15eleq2s 2510 . . . . . . 7  |-  ( ( 0  +  F )  e.  NN0  ->  ( ( 0  +  F )..^ ( ( # `  R
)  +  F ) )  C_  ( 0..^ ( ( # `  R
)  +  F ) ) )
4542, 44syl 17 . . . . . 6  |-  ( ph  ->  ( ( 0  +  F )..^ ( (
# `  R )  +  F ) )  C_  ( 0..^ ( ( # `  R )  +  F
) ) )
46 ccatlen 12648 . . . . . . . . 9  |-  ( ( ( S substr  <. 0 ,  F >. )  e. Word  A  /\  R  e. Word  A )  ->  ( # `  (
( S substr  <. 0 ,  F >. ) ++  R ) )  =  ( (
# `  ( S substr  <.
0 ,  F >. ) )  +  ( # `  R ) ) )
4735, 4, 46syl2anc 659 . . . . . . . 8  |-  ( ph  ->  ( # `  (
( S substr  <. 0 ,  F >. ) ++  R ) )  =  ( (
# `  ( S substr  <.
0 ,  F >. ) )  +  ( # `  R ) ) )
4830oveq1d 6293 . . . . . . . 8  |-  ( ph  ->  ( ( # `  ( S substr  <. 0 ,  F >. ) )  +  (
# `  R )
)  =  ( F  +  ( # `  R
) ) )
49 wrdfin 12613 . . . . . . . . . . . 12  |-  ( R  e. Word  A  ->  R  e.  Fin )
504, 49syl 17 . . . . . . . . . . 11  |-  ( ph  ->  R  e.  Fin )
51 hashcl 12475 . . . . . . . . . . 11  |-  ( R  e.  Fin  ->  ( # `
 R )  e. 
NN0 )
5250, 51syl 17 . . . . . . . . . 10  |-  ( ph  ->  ( # `  R
)  e.  NN0 )
5352nn0cnd 10895 . . . . . . . . 9  |-  ( ph  ->  ( # `  R
)  e.  CC )
549, 53addcomd 9816 . . . . . . . 8  |-  ( ph  ->  ( F  +  (
# `  R )
)  =  ( (
# `  R )  +  F ) )
5547, 48, 543eqtrd 2447 . . . . . . 7  |-  ( ph  ->  ( # `  (
( S substr  <. 0 ,  F >. ) ++  R ) )  =  ( (
# `  R )  +  F ) )
5655oveq2d 6294 . . . . . 6  |-  ( ph  ->  ( 0..^ ( # `  ( ( S substr  <. 0 ,  F >. ) ++  R ) ) )  =  ( 0..^ ( ( # `  R )  +  F
) ) )
5745, 56sseqtr4d 3479 . . . . 5  |-  ( ph  ->  ( ( 0  +  F )..^ ( (
# `  R )  +  F ) )  C_  ( 0..^ ( # `  (
( S substr  <. 0 ,  F >. ) ++  R ) ) ) )
588nn0zd 11006 . . . . . 6  |-  ( ph  ->  F  e.  ZZ )
59 fzoaddel 11905 . . . . . 6  |-  ( ( X  e.  ( 0..^ ( # `  R
) )  /\  F  e.  ZZ )  ->  ( X  +  F )  e.  ( ( 0  +  F )..^ ( (
# `  R )  +  F ) ) )
6010, 58, 59syl2anc 659 . . . . 5  |-  ( ph  ->  ( X  +  F
)  e.  ( ( 0  +  F )..^ ( ( # `  R
)  +  F ) ) )
6157, 60sseldd 3443 . . . 4  |-  ( ph  ->  ( X  +  F
)  e.  ( 0..^ ( # `  (
( S substr  <. 0 ,  F >. ) ++  R ) ) ) )
6231, 61eqeltrd 2490 . . 3  |-  ( ph  ->  ( X  +  (
# `  ( S substr  <.
0 ,  F >. ) ) )  e.  ( 0..^ ( # `  (
( S substr  <. 0 ,  F >. ) ++  R ) ) ) )
63 ccatval1 12649 . . 3  |-  ( ( ( ( S substr  <. 0 ,  F >. ) ++  R )  e. Word  A  /\  ( S substr  <. T ,  (
# `  S ) >. )  e. Word  A  /\  ( X  +  ( # `
 ( S substr  <. 0 ,  F >. ) ) )  e.  ( 0..^ (
# `  ( ( S substr  <. 0 ,  F >. ) ++  R ) ) ) )  ->  (
( ( ( S substr  <. 0 ,  F >. ) ++  R ) ++  ( S substr  <. T ,  ( # `  S ) >. )
) `  ( X  +  ( # `  ( S substr  <. 0 ,  F >. ) ) ) )  =  ( ( ( S substr  <. 0 ,  F >. ) ++  R ) `  ( X  +  ( # `
 ( S substr  <. 0 ,  F >. ) ) ) ) )
6437, 39, 62, 63syl3anc 1230 . 2  |-  ( ph  ->  ( ( ( ( S substr  <. 0 ,  F >. ) ++  R ) ++  ( S substr  <. T ,  (
# `  S ) >. ) ) `  ( X  +  ( # `  ( S substr  <. 0 ,  F >. ) ) ) )  =  ( ( ( S substr  <. 0 ,  F >. ) ++  R ) `  ( X  +  ( # `
 ( S substr  <. 0 ,  F >. ) ) ) ) )
65 ccatval3 12651 . . 3  |-  ( ( ( S substr  <. 0 ,  F >. )  e. Word  A  /\  R  e. Word  A  /\  X  e.  ( 0..^ ( # `  R
) ) )  -> 
( ( ( S substr  <. 0 ,  F >. ) ++  R ) `  ( X  +  ( # `  ( S substr  <. 0 ,  F >. ) ) ) )  =  ( R `  X ) )
6635, 4, 10, 65syl3anc 1230 . 2  |-  ( ph  ->  ( ( ( S substr  <. 0 ,  F >. ) ++  R ) `  ( X  +  ( # `  ( S substr  <. 0 ,  F >. ) ) ) )  =  ( R `  X ) )
6733, 64, 663eqtrd 2447 1  |-  ( ph  ->  ( ( S splice  <. F ,  T ,  R >. ) `
 ( F  +  X ) )  =  ( R `  X
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1405    e. wcel 1842    C_ wss 3414   <.cop 3978   <.cotp 3980   ` cfv 5569  (class class class)co 6278   Fincfn 7554   0cc0 9522    + caddc 9525    - cmin 9841   NN0cn0 10836   ZZcz 10905   ZZ>=cuz 11127   ...cfz 11726  ..^cfzo 11854   #chash 12452  Word cword 12583   ++ cconcat 12585   substr csubstr 12587   splice csplice 12588
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 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599
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 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-ot 3981  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-oadd 7171  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-card 8352  df-cda 8580  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-nn 10577  df-2 10635  df-n0 10837  df-z 10906  df-uz 11128  df-fz 11727  df-fzo 11855  df-hash 12453  df-word 12591  df-concat 12593  df-substr 12595  df-splice 12596
This theorem is referenced by:  psgnunilem2  16844
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