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Theorem splfv2a 12396
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 12391 . . . 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 1220 . . 3  |-  ( ph  ->  ( S splice  <. F ,  T ,  R >. )  =  ( ( ( S substr  <. 0 ,  F >. ) concat  R ) concat  ( S substr  <. T ,  (
# `  S ) >. ) ) )
7 elfznn0 11479 . . . . . . 7  |-  ( F  e.  ( 0 ... T )  ->  F  e.  NN0 )
82, 7syl 16 . . . . . 6  |-  ( ph  ->  F  e.  NN0 )
98nn0cnd 10636 . . . . 5  |-  ( ph  ->  F  e.  CC )
10 splfv2a.x . . . . . . 7  |-  ( ph  ->  X  e.  ( 0..^ ( # `  R
) ) )
11 elfzoelz 11551 . . . . . . 7  |-  ( X  e.  ( 0..^ (
# `  R )
)  ->  X  e.  ZZ )
1210, 11syl 16 . . . . . 6  |-  ( ph  ->  X  e.  ZZ )
1312zcnd 10746 . . . . 5  |-  ( ph  ->  X  e.  CC )
149, 13addcomd 9569 . . . 4  |-  ( ph  ->  ( F  +  X
)  =  ( X  +  F ) )
15 nn0uz 10893 . . . . . . . . 9  |-  NN0  =  ( ZZ>= `  0 )
168, 15syl6eleq 2531 . . . . . . . 8  |-  ( ph  ->  F  e.  ( ZZ>= ` 
0 ) )
17 eluzfz1 11456 . . . . . . . 8  |-  ( F  e.  ( ZZ>= `  0
)  ->  0  e.  ( 0 ... F
) )
1816, 17syl 16 . . . . . . 7  |-  ( ph  ->  0  e.  ( 0 ... F ) )
19 elfzuz3 11448 . . . . . . . . . 10  |-  ( T  e.  ( 0 ... ( # `  S
) )  ->  ( # `
 S )  e.  ( ZZ>= `  T )
)
203, 19syl 16 . . . . . . . . 9  |-  ( ph  ->  ( # `  S
)  e.  ( ZZ>= `  T ) )
21 elfzuz3 11448 . . . . . . . . . 10  |-  ( F  e.  ( 0 ... T )  ->  T  e.  ( ZZ>= `  F )
)
222, 21syl 16 . . . . . . . . 9  |-  ( ph  ->  T  e.  ( ZZ>= `  F ) )
23 uztrn 10875 . . . . . . . . 9  |-  ( ( ( # `  S
)  e.  ( ZZ>= `  T )  /\  T  e.  ( ZZ>= `  F )
)  ->  ( # `  S
)  e.  ( ZZ>= `  F ) )
2420, 22, 23syl2anc 661 . . . . . . . 8  |-  ( ph  ->  ( # `  S
)  e.  ( ZZ>= `  F ) )
25 elfzuzb 11445 . . . . . . . 8  |-  ( F  e.  ( 0 ... ( # `  S
) )  <->  ( F  e.  ( ZZ>= `  0 )  /\  ( # `  S
)  e.  ( ZZ>= `  F ) ) )
2616, 24, 25sylanbrc 664 . . . . . . 7  |-  ( ph  ->  F  e.  ( 0 ... ( # `  S
) ) )
27 swrdlen 12317 . . . . . . 7  |-  ( ( S  e. Word  A  /\  0  e.  ( 0 ... F )  /\  F  e.  ( 0 ... ( # `  S
) ) )  -> 
( # `  ( S substr  <. 0 ,  F >. ) )  =  ( F  -  0 ) )
281, 18, 26, 27syl3anc 1218 . . . . . 6  |-  ( ph  ->  ( # `  ( S substr  <. 0 ,  F >. ) )  =  ( F  -  0 ) )
299subid1d 9706 . . . . . 6  |-  ( ph  ->  ( F  -  0 )  =  F )
3028, 29eqtrd 2473 . . . . 5  |-  ( ph  ->  ( # `  ( S substr  <. 0 ,  F >. ) )  =  F )
3130oveq2d 6105 . . . 4  |-  ( ph  ->  ( X  +  (
# `  ( S substr  <.
0 ,  F >. ) ) )  =  ( X  +  F ) )
3214, 31eqtr4d 2476 . . 3  |-  ( ph  ->  ( F  +  X
)  =  ( X  +  ( # `  ( S substr  <. 0 ,  F >. ) ) ) )
336, 32fveq12d 5695 . 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 12313 . . . . 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 12272 . . . 4  |-  ( ( ( S substr  <. 0 ,  F >. )  e. Word  A  /\  R  e. Word  A )  ->  ( ( S substr  <. 0 ,  F >. ) concat  R )  e. Word  A
)
3735, 4, 36syl2anc 661 . . 3  |-  ( ph  ->  ( ( S substr  <. 0 ,  F >. ) concat  R )  e. Word  A )
38 swrdcl 12313 . . . 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 10592 . . . . . . . 8  |-  0  e.  NN0
41 nn0addcl 10613 . . . . . . . 8  |-  ( ( 0  e.  NN0  /\  F  e.  NN0 )  -> 
( 0  +  F
)  e.  NN0 )
4240, 8, 41sylancr 663 . . . . . . 7  |-  ( ph  ->  ( 0  +  F
)  e.  NN0 )
43 fzoss1 11574 . . . . . . . 8  |-  ( ( 0  +  F )  e.  ( ZZ>= `  0
)  ->  ( (
0  +  F )..^ ( ( # `  R
)  +  F ) )  C_  ( 0..^ ( ( # `  R
)  +  F ) ) )
4443, 15eleq2s 2533 . . . . . . 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 12273 . . . . . . . . 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 661 . . . . . . . 8  |-  ( ph  ->  ( # `  (
( S substr  <. 0 ,  F >. ) concat  R )
)  =  ( (
# `  ( S substr  <.
0 ,  F >. ) )  +  ( # `  R ) ) )
4830oveq1d 6104 . . . . . . . 8  |-  ( ph  ->  ( ( # `  ( S substr  <. 0 ,  F >. ) )  +  (
# `  R )
)  =  ( F  +  ( # `  R
) ) )
49 wrdfin 12246 . . . . . . . . . . . 12  |-  ( R  e. Word  A  ->  R  e.  Fin )
504, 49syl 16 . . . . . . . . . . 11  |-  ( ph  ->  R  e.  Fin )
51 hashcl 12124 . . . . . . . . . . 11  |-  ( R  e.  Fin  ->  ( # `
 R )  e. 
NN0 )
5250, 51syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( # `  R
)  e.  NN0 )
5352nn0cnd 10636 . . . . . . . . 9  |-  ( ph  ->  ( # `  R
)  e.  CC )
549, 53addcomd 9569 . . . . . . . 8  |-  ( ph  ->  ( F  +  (
# `  R )
)  =  ( (
# `  R )  +  F ) )
5547, 48, 543eqtrd 2477 . . . . . . 7  |-  ( ph  ->  ( # `  (
( S substr  <. 0 ,  F >. ) concat  R )
)  =  ( (
# `  R )  +  F ) )
5655oveq2d 6105 . . . . . 6  |-  ( ph  ->  ( 0..^ ( # `  ( ( S substr  <. 0 ,  F >. ) concat  R )
) )  =  ( 0..^ ( ( # `  R )  +  F
) ) )
5745, 56sseqtr4d 3391 . . . . 5  |-  ( ph  ->  ( ( 0  +  F )..^ ( (
# `  R )  +  F ) )  C_  ( 0..^ ( # `  (
( S substr  <. 0 ,  F >. ) concat  R )
) ) )
588nn0zd 10743 . . . . . 6  |-  ( ph  ->  F  e.  ZZ )
59 fzoaddel 11595 . . . . . 6  |-  ( ( X  e.  ( 0..^ ( # `  R
) )  /\  F  e.  ZZ )  ->  ( X  +  F )  e.  ( ( 0  +  F )..^ ( (
# `  R )  +  F ) ) )
6010, 58, 59syl2anc 661 . . . . 5  |-  ( ph  ->  ( X  +  F
)  e.  ( ( 0  +  F )..^ ( ( # `  R
)  +  F ) ) )
6157, 60sseldd 3355 . . . 4  |-  ( ph  ->  ( X  +  F
)  e.  ( 0..^ ( # `  (
( S substr  <. 0 ,  F >. ) concat  R )
) ) )
6231, 61eqeltrd 2515 . . 3  |-  ( ph  ->  ( X  +  (
# `  ( S substr  <.
0 ,  F >. ) ) )  e.  ( 0..^ ( # `  (
( S substr  <. 0 ,  F >. ) concat  R )
) ) )
63 ccatval1 12274 . . 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 1218 . 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 12276 . . 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 1218 . 2  |-  ( ph  ->  ( ( ( S substr  <. 0 ,  F >. ) concat  R ) `  ( X  +  ( # `  ( S substr  <. 0 ,  F >. ) ) ) )  =  ( R `  X ) )
6733, 64, 663eqtrd 2477 1  |-  ( ph  ->  ( ( S splice  <. F ,  T ,  R >. ) `
 ( F  +  X ) )  =  ( R `  X
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756    C_ wss 3326   <.cop 3881   <.cotp 3883   ` cfv 5416  (class class class)co 6089   Fincfn 7308   0cc0 9280    + caddc 9283    - cmin 9593   NN0cn0 10577   ZZcz 10644   ZZ>=cuz 10859   ...cfz 11435  ..^cfzo 11546   #chash 12101  Word cword 12219   concat cconcat 12221   substr csubstr 12223   splice csplice 12224
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-ot 3884  df-uni 4090  df-int 4127  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-om 6475  df-1st 6575  df-2nd 6576  df-recs 6830  df-rdg 6864  df-1o 6918  df-oadd 6922  df-er 7099  df-en 7309  df-dom 7310  df-sdom 7311  df-fin 7312  df-card 8107  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-nn 10321  df-n0 10578  df-z 10645  df-uz 10860  df-fz 11436  df-fzo 11547  df-hash 12102  df-word 12227  df-concat 12229  df-substr 12231  df-splice 12232
This theorem is referenced by:  psgnunilem2  15999
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