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Theorem splfv2a 12682
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 12677 . . . 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 ) >. ) ) )
7 elfznn0 11759 . . . . . . 7  |-  ( F  e.  ( 0 ... T )  ->  F  e.  NN0 )
82, 7syl 16 . . . . . 6  |-  ( ph  ->  F  e.  NN0 )
98nn0cnd 10843 . . . . 5  |-  ( ph  ->  F  e.  CC )
10 splfv2a.x . . . . . . 7  |-  ( ph  ->  X  e.  ( 0..^ ( # `  R
) ) )
11 elfzoelz 11786 . . . . . . 7  |-  ( X  e.  ( 0..^ (
# `  R )
)  ->  X  e.  ZZ )
1210, 11syl 16 . . . . . 6  |-  ( ph  ->  X  e.  ZZ )
1312zcnd 10956 . . . . 5  |-  ( ph  ->  X  e.  CC )
149, 13addcomd 9770 . . . 4  |-  ( ph  ->  ( F  +  X
)  =  ( X  +  F ) )
15 nn0uz 11105 . . . . . . . . 9  |-  NN0  =  ( ZZ>= `  0 )
168, 15syl6eleq 2558 . . . . . . . 8  |-  ( ph  ->  F  e.  ( ZZ>= ` 
0 ) )
17 eluzfz1 11682 . . . . . . . 8  |-  ( F  e.  ( ZZ>= `  0
)  ->  0  e.  ( 0 ... F
) )
1816, 17syl 16 . . . . . . 7  |-  ( ph  ->  0  e.  ( 0 ... F ) )
19 elfzuz3 11674 . . . . . . . . . 10  |-  ( T  e.  ( 0 ... ( # `  S
) )  ->  ( # `
 S )  e.  ( ZZ>= `  T )
)
203, 19syl 16 . . . . . . . . 9  |-  ( ph  ->  ( # `  S
)  e.  ( ZZ>= `  T ) )
21 elfzuz3 11674 . . . . . . . . . 10  |-  ( F  e.  ( 0 ... T )  ->  T  e.  ( ZZ>= `  F )
)
222, 21syl 16 . . . . . . . . 9  |-  ( ph  ->  T  e.  ( ZZ>= `  F ) )
23 uztrn 11087 . . . . . . . . 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 11671 . . . . . . . 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 12600 . . . . . . 7  |-  ( ( S  e. Word  A  /\  0  e.  ( 0 ... F )  /\  F  e.  ( 0 ... ( # `  S
) ) )  -> 
( # `  ( S substr  <. 0 ,  F >. ) )  =  ( F  -  0 ) )
281, 18, 26, 27syl3anc 1223 . . . . . 6  |-  ( ph  ->  ( # `  ( S substr  <. 0 ,  F >. ) )  =  ( F  -  0 ) )
299subid1d 9908 . . . . . 6  |-  ( ph  ->  ( F  -  0 )  =  F )
3028, 29eqtrd 2501 . . . . 5  |-  ( ph  ->  ( # `  ( S substr  <. 0 ,  F >. ) )  =  F )
3130oveq2d 6291 . . . 4  |-  ( ph  ->  ( X  +  (
# `  ( S substr  <.
0 ,  F >. ) ) )  =  ( X  +  F ) )
3214, 31eqtr4d 2504 . . 3  |-  ( ph  ->  ( F  +  X
)  =  ( X  +  ( # `  ( S substr  <. 0 ,  F >. ) ) ) )
336, 32fveq12d 5863 . 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 12596 . . . . 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 12545 . . . 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 12596 . . . 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 10799 . . . . . . . 8  |-  0  e.  NN0
41 nn0addcl 10820 . . . . . . . 8  |-  ( ( 0  e.  NN0  /\  F  e.  NN0 )  -> 
( 0  +  F
)  e.  NN0 )
4240, 8, 41sylancr 663 . . . . . . 7  |-  ( ph  ->  ( 0  +  F
)  e.  NN0 )
43 fzoss1 11809 . . . . . . . 8  |-  ( ( 0  +  F )  e.  ( ZZ>= `  0
)  ->  ( (
0  +  F )..^ ( ( # `  R
)  +  F ) )  C_  ( 0..^ ( ( # `  R
)  +  F ) ) )
4443, 15eleq2s 2568 . . . . . . 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 12546 . . . . . . . . 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 6290 . . . . . . . 8  |-  ( ph  ->  ( ( # `  ( S substr  <. 0 ,  F >. ) )  +  (
# `  R )
)  =  ( F  +  ( # `  R
) ) )
49 wrdfin 12514 . . . . . . . . . . . 12  |-  ( R  e. Word  A  ->  R  e.  Fin )
504, 49syl 16 . . . . . . . . . . 11  |-  ( ph  ->  R  e.  Fin )
51 hashcl 12383 . . . . . . . . . . 11  |-  ( R  e.  Fin  ->  ( # `
 R )  e. 
NN0 )
5250, 51syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( # `  R
)  e.  NN0 )
5352nn0cnd 10843 . . . . . . . . 9  |-  ( ph  ->  ( # `  R
)  e.  CC )
549, 53addcomd 9770 . . . . . . . 8  |-  ( ph  ->  ( F  +  (
# `  R )
)  =  ( (
# `  R )  +  F ) )
5547, 48, 543eqtrd 2505 . . . . . . 7  |-  ( ph  ->  ( # `  (
( S substr  <. 0 ,  F >. ) concat  R )
)  =  ( (
# `  R )  +  F ) )
5655oveq2d 6291 . . . . . 6  |-  ( ph  ->  ( 0..^ ( # `  ( ( S substr  <. 0 ,  F >. ) concat  R )
) )  =  ( 0..^ ( ( # `  R )  +  F
) ) )
5745, 56sseqtr4d 3534 . . . . 5  |-  ( ph  ->  ( ( 0  +  F )..^ ( (
# `  R )  +  F ) )  C_  ( 0..^ ( # `  (
( S substr  <. 0 ,  F >. ) concat  R )
) ) )
588nn0zd 10953 . . . . . 6  |-  ( ph  ->  F  e.  ZZ )
59 fzoaddel 11830 . . . . . 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 3498 . . . 4  |-  ( ph  ->  ( X  +  F
)  e.  ( 0..^ ( # `  (
( S substr  <. 0 ,  F >. ) concat  R )
) ) )
6231, 61eqeltrd 2548 . . 3  |-  ( ph  ->  ( X  +  (
# `  ( S substr  <.
0 ,  F >. ) ) )  e.  ( 0..^ ( # `  (
( S substr  <. 0 ,  F >. ) concat  R )
) ) )
63 ccatval1 12547 . . 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 1223 . 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 12549 . . 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 1223 . 2  |-  ( ph  ->  ( ( ( S substr  <. 0 ,  F >. ) concat  R ) `  ( X  +  ( # `  ( S substr  <. 0 ,  F >. ) ) ) )  =  ( R `  X ) )
6733, 64, 663eqtrd 2505 1  |-  ( ph  ->  ( ( S splice  <. F ,  T ,  R >. ) `
 ( F  +  X ) )  =  ( R `  X
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1374    e. wcel 1762    C_ wss 3469   <.cop 4026   <.cotp 4028   ` cfv 5579  (class class class)co 6275   Fincfn 7506   0cc0 9481    + caddc 9484    - cmin 9794   NN0cn0 10784   ZZcz 10853   ZZ>=cuz 11071   ...cfz 11661  ..^cfzo 11781   #chash 12360  Word cword 12487   concat cconcat 12489   substr csubstr 12491   splice csplice 12492
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 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
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 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-ot 4029  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-card 8309  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-n0 10785  df-z 10854  df-uz 11072  df-fz 11662  df-fzo 11782  df-hash 12361  df-word 12495  df-concat 12497  df-substr 12499  df-splice 12500
This theorem is referenced by:  psgnunilem2  16309
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