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Theorem splfv2a 11786
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 11781 . . . 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 ) >. ) ) )
7 elfznn0 11084 . . . . . . 7  |-  ( F  e.  ( 0 ... T )  ->  F  e.  NN0 )
82, 7syl 16 . . . . . 6  |-  ( ph  ->  F  e.  NN0 )
98nn0cnd 10277 . . . . 5  |-  ( ph  ->  F  e.  CC )
10 splfv2a.x . . . . . . 7  |-  ( ph  ->  X  e.  ( 0..^ ( # `  R
) ) )
11 elfzoelz 11141 . . . . . . 7  |-  ( X  e.  ( 0..^ (
# `  R )
)  ->  X  e.  ZZ )
1210, 11syl 16 . . . . . 6  |-  ( ph  ->  X  e.  ZZ )
1312zcnd 10377 . . . . 5  |-  ( ph  ->  X  e.  CC )
149, 13addcomd 9269 . . . 4  |-  ( ph  ->  ( F  +  X
)  =  ( X  +  F ) )
15 nn0uz 10521 . . . . . . . . 9  |-  NN0  =  ( ZZ>= `  0 )
168, 15syl6eleq 2527 . . . . . . . 8  |-  ( ph  ->  F  e.  ( ZZ>= ` 
0 ) )
17 eluzfz1 11065 . . . . . . . 8  |-  ( F  e.  ( ZZ>= `  0
)  ->  0  e.  ( 0 ... F
) )
1816, 17syl 16 . . . . . . 7  |-  ( ph  ->  0  e.  ( 0 ... F ) )
19 elfzuz3 11057 . . . . . . . . . 10  |-  ( T  e.  ( 0 ... ( # `  S
) )  ->  ( # `
 S )  e.  ( ZZ>= `  T )
)
203, 19syl 16 . . . . . . . . 9  |-  ( ph  ->  ( # `  S
)  e.  ( ZZ>= `  T ) )
21 elfzuz3 11057 . . . . . . . . . 10  |-  ( F  e.  ( 0 ... T )  ->  T  e.  ( ZZ>= `  F )
)
222, 21syl 16 . . . . . . . . 9  |-  ( ph  ->  T  e.  ( ZZ>= `  F ) )
23 uztrn 10503 . . . . . . . . 9  |-  ( ( ( # `  S
)  e.  ( ZZ>= `  T )  /\  T  e.  ( ZZ>= `  F )
)  ->  ( # `  S
)  e.  ( ZZ>= `  F ) )
2420, 22, 23syl2anc 644 . . . . . . . 8  |-  ( ph  ->  ( # `  S
)  e.  ( ZZ>= `  F ) )
25 elfzuzb 11054 . . . . . . . 8  |-  ( F  e.  ( 0 ... ( # `  S
) )  <->  ( F  e.  ( ZZ>= `  0 )  /\  ( # `  S
)  e.  ( ZZ>= `  F ) ) )
2616, 24, 25sylanbrc 647 . . . . . . 7  |-  ( ph  ->  F  e.  ( 0 ... ( # `  S
) ) )
27 swrdlen 11771 . . . . . . 7  |-  ( ( S  e. Word  A  /\  0  e.  ( 0 ... F )  /\  F  e.  ( 0 ... ( # `  S
) ) )  -> 
( # `  ( S substr  <. 0 ,  F >. ) )  =  ( F  -  0 ) )
281, 18, 26, 27syl3anc 1185 . . . . . 6  |-  ( ph  ->  ( # `  ( S substr  <. 0 ,  F >. ) )  =  ( F  -  0 ) )
299subid1d 9401 . . . . . 6  |-  ( ph  ->  ( F  -  0 )  =  F )
3028, 29eqtrd 2469 . . . . 5  |-  ( ph  ->  ( # `  ( S substr  <. 0 ,  F >. ) )  =  F )
3130oveq2d 6098 . . . 4  |-  ( ph  ->  ( X  +  (
# `  ( S substr  <.
0 ,  F >. ) ) )  =  ( X  +  F ) )
3214, 31eqtr4d 2472 . . 3  |-  ( ph  ->  ( F  +  X
)  =  ( X  +  ( # `  ( S substr  <. 0 ,  F >. ) ) ) )
336, 32fveq12d 5735 . 2  |-  ( ph  ->  ( ( S splice  <. F ,  T ,  R >. ) `
 ( F  +  X ) )  =  ( ( ( ( S substr  <. 0 ,  F >. ) concat  R ) concat  ( S substr  <. T ,  (
# `  S ) >. ) ) `  ( X  +  ( # `  ( S substr  <. 0 ,  F >. ) ) ) ) )
34 swrdcl 11767 . . . . 5  |-  ( S  e. Word  A  ->  ( S substr  <. 0 ,  F >. )  e. Word  A )
351, 34syl 16 . . . 4  |-  ( ph  ->  ( S substr  <. 0 ,  F >. )  e. Word  A
)
36 ccatcl 11744 . . . 4  |-  ( ( ( S substr  <. 0 ,  F >. )  e. Word  A  /\  R  e. Word  A )  ->  ( ( S substr  <. 0 ,  F >. ) concat  R )  e. Word  A
)
3735, 4, 36syl2anc 644 . . 3  |-  ( ph  ->  ( ( S substr  <. 0 ,  F >. ) concat  R )  e. Word  A )
38 swrdcl 11767 . . . 4  |-  ( S  e. Word  A  ->  ( S substr  <. T ,  (
# `  S ) >. )  e. Word  A )
391, 38syl 16 . . 3  |-  ( ph  ->  ( S substr  <. T , 
( # `  S )
>. )  e. Word  A )
40 0nn0 10237 . . . . . . . 8  |-  0  e.  NN0
41 nn0addcl 10256 . . . . . . . 8  |-  ( ( 0  e.  NN0  /\  F  e.  NN0 )  -> 
( 0  +  F
)  e.  NN0 )
4240, 8, 41sylancr 646 . . . . . . 7  |-  ( ph  ->  ( 0  +  F
)  e.  NN0 )
43 fzoss1 11163 . . . . . . . 8  |-  ( ( 0  +  F )  e.  ( ZZ>= `  0
)  ->  ( (
0  +  F )..^ ( ( # `  R
)  +  F ) )  C_  ( 0..^ ( ( # `  R
)  +  F ) ) )
4443, 15eleq2s 2529 . . . . . . 7  |-  ( ( 0  +  F )  e.  NN0  ->  ( ( 0  +  F )..^ ( ( # `  R
)  +  F ) )  C_  ( 0..^ ( ( # `  R
)  +  F ) ) )
4542, 44syl 16 . . . . . 6  |-  ( ph  ->  ( ( 0  +  F )..^ ( (
# `  R )  +  F ) )  C_  ( 0..^ ( ( # `  R )  +  F
) ) )
46 ccatlen 11745 . . . . . . . . 9  |-  ( ( ( S substr  <. 0 ,  F >. )  e. Word  A  /\  R  e. Word  A )  ->  ( # `  (
( S substr  <. 0 ,  F >. ) concat  R )
)  =  ( (
# `  ( S substr  <.
0 ,  F >. ) )  +  ( # `  R ) ) )
4735, 4, 46syl2anc 644 . . . . . . . 8  |-  ( ph  ->  ( # `  (
( S substr  <. 0 ,  F >. ) concat  R )
)  =  ( (
# `  ( S substr  <.
0 ,  F >. ) )  +  ( # `  R ) ) )
4830oveq1d 6097 . . . . . . . 8  |-  ( ph  ->  ( ( # `  ( S substr  <. 0 ,  F >. ) )  +  (
# `  R )
)  =  ( F  +  ( # `  R
) ) )
49 wrdfin 11735 . . . . . . . . . . . 12  |-  ( R  e. Word  A  ->  R  e.  Fin )
504, 49syl 16 . . . . . . . . . . 11  |-  ( ph  ->  R  e.  Fin )
51 hashcl 11640 . . . . . . . . . . 11  |-  ( R  e.  Fin  ->  ( # `
 R )  e. 
NN0 )
5250, 51syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( # `  R
)  e.  NN0 )
5352nn0cnd 10277 . . . . . . . . 9  |-  ( ph  ->  ( # `  R
)  e.  CC )
549, 53addcomd 9269 . . . . . . . 8  |-  ( ph  ->  ( F  +  (
# `  R )
)  =  ( (
# `  R )  +  F ) )
5547, 48, 543eqtrd 2473 . . . . . . 7  |-  ( ph  ->  ( # `  (
( S substr  <. 0 ,  F >. ) concat  R )
)  =  ( (
# `  R )  +  F ) )
5655oveq2d 6098 . . . . . 6  |-  ( ph  ->  ( 0..^ ( # `  ( ( S substr  <. 0 ,  F >. ) concat  R )
) )  =  ( 0..^ ( ( # `  R )  +  F
) ) )
5745, 56sseqtr4d 3386 . . . . 5  |-  ( ph  ->  ( ( 0  +  F )..^ ( (
# `  R )  +  F ) )  C_  ( 0..^ ( # `  (
( S substr  <. 0 ,  F >. ) concat  R )
) ) )
588nn0zd 10374 . . . . . 6  |-  ( ph  ->  F  e.  ZZ )
59 fzoaddel 11176 . . . . . 6  |-  ( ( X  e.  ( 0..^ ( # `  R
) )  /\  F  e.  ZZ )  ->  ( X  +  F )  e.  ( ( 0  +  F )..^ ( (
# `  R )  +  F ) ) )
6010, 58, 59syl2anc 644 . . . . 5  |-  ( ph  ->  ( X  +  F
)  e.  ( ( 0  +  F )..^ ( ( # `  R
)  +  F ) ) )
6157, 60sseldd 3350 . . . 4  |-  ( ph  ->  ( X  +  F
)  e.  ( 0..^ ( # `  (
( S substr  <. 0 ,  F >. ) concat  R )
) ) )
6231, 61eqeltrd 2511 . . 3  |-  ( ph  ->  ( X  +  (
# `  ( S substr  <.
0 ,  F >. ) ) )  e.  ( 0..^ ( # `  (
( S substr  <. 0 ,  F >. ) concat  R )
) ) )
63 ccatval1 11746 . . 3  |-  ( ( ( ( S substr  <. 0 ,  F >. ) concat  R )  e. Word  A  /\  ( S substr  <. T ,  ( # `  S ) >. )  e. Word  A  /\  ( X  +  ( # `  ( S substr  <. 0 ,  F >. ) ) )  e.  ( 0..^ ( # `  ( ( S substr  <. 0 ,  F >. ) concat  R )
) ) )  -> 
( ( ( ( S substr  <. 0 ,  F >. ) concat  R ) concat  ( S substr  <. T ,  (
# `  S ) >. ) ) `  ( X  +  ( # `  ( S substr  <. 0 ,  F >. ) ) ) )  =  ( ( ( S substr  <. 0 ,  F >. ) concat  R ) `  ( X  +  ( # `
 ( S substr  <. 0 ,  F >. ) ) ) ) )
6437, 39, 62, 63syl3anc 1185 . 2  |-  ( ph  ->  ( ( ( ( S substr  <. 0 ,  F >. ) concat  R ) concat  ( S substr  <. T ,  (
# `  S ) >. ) ) `  ( X  +  ( # `  ( S substr  <. 0 ,  F >. ) ) ) )  =  ( ( ( S substr  <. 0 ,  F >. ) concat  R ) `  ( X  +  ( # `
 ( S substr  <. 0 ,  F >. ) ) ) ) )
65 ccatval3 11748 . . 3  |-  ( ( ( S substr  <. 0 ,  F >. )  e. Word  A  /\  R  e. Word  A  /\  X  e.  ( 0..^ ( # `  R
) ) )  -> 
( ( ( S substr  <. 0 ,  F >. ) concat  R ) `  ( X  +  ( # `  ( S substr  <. 0 ,  F >. ) ) ) )  =  ( R `  X ) )
6635, 4, 10, 65syl3anc 1185 . 2  |-  ( ph  ->  ( ( ( S substr  <. 0 ,  F >. ) concat  R ) `  ( X  +  ( # `  ( S substr  <. 0 ,  F >. ) ) ) )  =  ( R `  X ) )
6733, 64, 663eqtrd 2473 1  |-  ( ph  ->  ( ( S splice  <. F ,  T ,  R >. ) `
 ( F  +  X ) )  =  ( R `  X
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726    C_ wss 3321   <.cop 3818   <.cotp 3819   ` cfv 5455  (class class class)co 6082   Fincfn 7110   0cc0 8991    + caddc 8994    - cmin 9292   NN0cn0 10222   ZZcz 10283   ZZ>=cuz 10489   ...cfz 11044  ..^cfzo 11136   #chash 11619  Word cword 11718   concat cconcat 11719   substr csubstr 11721   splice csplice 11722
This theorem is referenced by:  psgnunilem2  27396
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068
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 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-ot 3825  df-uni 4017  df-int 4052  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-riota 6550  df-recs 6634  df-rdg 6669  df-1o 6725  df-oadd 6729  df-er 6906  df-en 7111  df-dom 7112  df-sdom 7113  df-fin 7114  df-card 7827  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-nn 10002  df-n0 10223  df-z 10284  df-uz 10490  df-fz 11045  df-fzo 11137  df-hash 11620  df-word 11724  df-concat 11725  df-substr 11727  df-splice 11728
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