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Theorem spllen 12709
Description: The length of a splice. (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 )
Assertion
Ref Expression
spllen  |-  ( ph  ->  ( # `  ( S splice  <. F ,  T ,  R >. ) )  =  ( ( # `  S
)  +  ( (
# `  R )  -  ( T  -  F ) ) ) )

Proof of Theorem spllen
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 12706 . . . 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 1231 . . 3  |-  ( ph  ->  ( S splice  <. F ,  T ,  R >. )  =  ( ( ( S substr  <. 0 ,  F >. ) concat  R ) concat  ( S substr  <. T ,  (
# `  S ) >. ) ) )
76fveq2d 5860 . 2  |-  ( ph  ->  ( # `  ( S splice  <. F ,  T ,  R >. ) )  =  ( # `  (
( ( S substr  <. 0 ,  F >. ) concat  R ) concat  ( S substr  <. T ,  (
# `  S ) >. ) ) ) )
8 swrdcl 12625 . . . . 5  |-  ( S  e. Word  A  ->  ( S substr  <. 0 ,  F >. )  e. Word  A )
91, 8syl 16 . . . 4  |-  ( ph  ->  ( S substr  <. 0 ,  F >. )  e. Word  A
)
10 ccatcl 12572 . . . 4  |-  ( ( ( S substr  <. 0 ,  F >. )  e. Word  A  /\  R  e. Word  A )  ->  ( ( S substr  <. 0 ,  F >. ) concat  R )  e. Word  A
)
119, 4, 10syl2anc 661 . . 3  |-  ( ph  ->  ( ( S substr  <. 0 ,  F >. ) concat  R )  e. Word  A )
12 swrdcl 12625 . . . 4  |-  ( S  e. Word  A  ->  ( S substr  <. T ,  (
# `  S ) >. )  e. Word  A )
131, 12syl 16 . . 3  |-  ( ph  ->  ( S substr  <. T , 
( # `  S )
>. )  e. Word  A )
14 ccatlen 12573 . . 3  |-  ( ( ( ( S substr  <. 0 ,  F >. ) concat  R )  e. Word  A  /\  ( S substr  <. T ,  ( # `  S ) >. )  e. Word  A )  ->  ( # `
 ( ( ( S substr  <. 0 ,  F >. ) concat  R ) concat  ( S substr  <. T ,  (
# `  S ) >. ) ) )  =  ( ( # `  (
( S substr  <. 0 ,  F >. ) concat  R )
)  +  ( # `  ( S substr  <. T , 
( # `  S )
>. ) ) ) )
1511, 13, 14syl2anc 661 . 2  |-  ( ph  ->  ( # `  (
( ( S substr  <. 0 ,  F >. ) concat  R ) concat  ( S substr  <. T ,  (
# `  S ) >. ) ) )  =  ( ( # `  (
( S substr  <. 0 ,  F >. ) concat  R )
)  +  ( # `  ( S substr  <. T , 
( # `  S )
>. ) ) ) )
16 lencl 12541 . . . . . . 7  |-  ( R  e. Word  A  ->  ( # `
 R )  e. 
NN0 )
174, 16syl 16 . . . . . 6  |-  ( ph  ->  ( # `  R
)  e.  NN0 )
1817nn0cnd 10860 . . . . 5  |-  ( ph  ->  ( # `  R
)  e.  CC )
19 elfzelz 11697 . . . . . . 7  |-  ( F  e.  ( 0 ... T )  ->  F  e.  ZZ )
202, 19syl 16 . . . . . 6  |-  ( ph  ->  F  e.  ZZ )
2120zcnd 10975 . . . . 5  |-  ( ph  ->  F  e.  CC )
2218, 21addcld 9618 . . . 4  |-  ( ph  ->  ( ( # `  R
)  +  F )  e.  CC )
23 elfzel2 11695 . . . . . 6  |-  ( T  e.  ( 0 ... ( # `  S
) )  ->  ( # `
 S )  e.  ZZ )
243, 23syl 16 . . . . 5  |-  ( ph  ->  ( # `  S
)  e.  ZZ )
2524zcnd 10975 . . . 4  |-  ( ph  ->  ( # `  S
)  e.  CC )
26 elfzelz 11697 . . . . . 6  |-  ( T  e.  ( 0 ... ( # `  S
) )  ->  T  e.  ZZ )
273, 26syl 16 . . . . 5  |-  ( ph  ->  T  e.  ZZ )
2827zcnd 10975 . . . 4  |-  ( ph  ->  T  e.  CC )
2922, 25, 28addsub12d 9959 . . 3  |-  ( ph  ->  ( ( ( # `  R )  +  F
)  +  ( (
# `  S )  -  T ) )  =  ( ( # `  S
)  +  ( ( ( # `  R
)  +  F )  -  T ) ) )
30 ccatlen 12573 . . . . . 6  |-  ( ( ( S substr  <. 0 ,  F >. )  e. Word  A  /\  R  e. Word  A )  ->  ( # `  (
( S substr  <. 0 ,  F >. ) concat  R )
)  =  ( (
# `  ( S substr  <.
0 ,  F >. ) )  +  ( # `  R ) ) )
319, 4, 30syl2anc 661 . . . . 5  |-  ( ph  ->  ( # `  (
( S substr  <. 0 ,  F >. ) concat  R )
)  =  ( (
# `  ( S substr  <.
0 ,  F >. ) )  +  ( # `  R ) ) )
32 elfzuz 11693 . . . . . . . . . 10  |-  ( F  e.  ( 0 ... T )  ->  F  e.  ( ZZ>= `  0 )
)
332, 32syl 16 . . . . . . . . 9  |-  ( ph  ->  F  e.  ( ZZ>= ` 
0 ) )
34 eluzfz1 11702 . . . . . . . . 9  |-  ( F  e.  ( ZZ>= `  0
)  ->  0  e.  ( 0 ... F
) )
3533, 34syl 16 . . . . . . . 8  |-  ( ph  ->  0  e.  ( 0 ... F ) )
36 elfzuz3 11694 . . . . . . . . . . 11  |-  ( T  e.  ( 0 ... ( # `  S
) )  ->  ( # `
 S )  e.  ( ZZ>= `  T )
)
373, 36syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( # `  S
)  e.  ( ZZ>= `  T ) )
38 elfzuz3 11694 . . . . . . . . . . 11  |-  ( F  e.  ( 0 ... T )  ->  T  e.  ( ZZ>= `  F )
)
392, 38syl 16 . . . . . . . . . 10  |-  ( ph  ->  T  e.  ( ZZ>= `  F ) )
40 uztrn 11106 . . . . . . . . . 10  |-  ( ( ( # `  S
)  e.  ( ZZ>= `  T )  /\  T  e.  ( ZZ>= `  F )
)  ->  ( # `  S
)  e.  ( ZZ>= `  F ) )
4137, 39, 40syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  ( # `  S
)  e.  ( ZZ>= `  F ) )
42 elfzuzb 11691 . . . . . . . . 9  |-  ( F  e.  ( 0 ... ( # `  S
) )  <->  ( F  e.  ( ZZ>= `  0 )  /\  ( # `  S
)  e.  ( ZZ>= `  F ) ) )
4333, 41, 42sylanbrc 664 . . . . . . . 8  |-  ( ph  ->  F  e.  ( 0 ... ( # `  S
) ) )
44 swrdlen 12629 . . . . . . . 8  |-  ( ( S  e. Word  A  /\  0  e.  ( 0 ... F )  /\  F  e.  ( 0 ... ( # `  S
) ) )  -> 
( # `  ( S substr  <. 0 ,  F >. ) )  =  ( F  -  0 ) )
451, 35, 43, 44syl3anc 1229 . . . . . . 7  |-  ( ph  ->  ( # `  ( S substr  <. 0 ,  F >. ) )  =  ( F  -  0 ) )
4621subid1d 9925 . . . . . . 7  |-  ( ph  ->  ( F  -  0 )  =  F )
4745, 46eqtrd 2484 . . . . . 6  |-  ( ph  ->  ( # `  ( S substr  <. 0 ,  F >. ) )  =  F )
4847oveq1d 6296 . . . . 5  |-  ( ph  ->  ( ( # `  ( S substr  <. 0 ,  F >. ) )  +  (
# `  R )
)  =  ( F  +  ( # `  R
) ) )
4921, 18addcomd 9785 . . . . 5  |-  ( ph  ->  ( F  +  (
# `  R )
)  =  ( (
# `  R )  +  F ) )
5031, 48, 493eqtrd 2488 . . . 4  |-  ( ph  ->  ( # `  (
( S substr  <. 0 ,  F >. ) concat  R )
)  =  ( (
# `  R )  +  F ) )
51 elfzuz2 11700 . . . . . . 7  |-  ( T  e.  ( 0 ... ( # `  S
) )  ->  ( # `
 S )  e.  ( ZZ>= `  0 )
)
523, 51syl 16 . . . . . 6  |-  ( ph  ->  ( # `  S
)  e.  ( ZZ>= ` 
0 ) )
53 eluzfz2 11703 . . . . . 6  |-  ( (
# `  S )  e.  ( ZZ>= `  0 )  ->  ( # `  S
)  e.  ( 0 ... ( # `  S
) ) )
5452, 53syl 16 . . . . 5  |-  ( ph  ->  ( # `  S
)  e.  ( 0 ... ( # `  S
) ) )
55 swrdlen 12629 . . . . 5  |-  ( ( S  e. Word  A  /\  T  e.  ( 0 ... ( # `  S
) )  /\  ( # `
 S )  e.  ( 0 ... ( # `
 S ) ) )  ->  ( # `  ( S substr  <. T ,  (
# `  S ) >. ) )  =  ( ( # `  S
)  -  T ) )
561, 3, 54, 55syl3anc 1229 . . . 4  |-  ( ph  ->  ( # `  ( S substr  <. T ,  (
# `  S ) >. ) )  =  ( ( # `  S
)  -  T ) )
5750, 56oveq12d 6299 . . 3  |-  ( ph  ->  ( ( # `  (
( S substr  <. 0 ,  F >. ) concat  R )
)  +  ( # `  ( S substr  <. T , 
( # `  S )
>. ) ) )  =  ( ( ( # `  R )  +  F
)  +  ( (
# `  S )  -  T ) ) )
5818, 28, 21subsub3d 9966 . . . 4  |-  ( ph  ->  ( ( # `  R
)  -  ( T  -  F ) )  =  ( ( (
# `  R )  +  F )  -  T
) )
5958oveq2d 6297 . . 3  |-  ( ph  ->  ( ( # `  S
)  +  ( (
# `  R )  -  ( T  -  F ) ) )  =  ( ( # `  S )  +  ( ( ( # `  R
)  +  F )  -  T ) ) )
6029, 57, 593eqtr4d 2494 . 2  |-  ( ph  ->  ( ( # `  (
( S substr  <. 0 ,  F >. ) concat  R )
)  +  ( # `  ( S substr  <. T , 
( # `  S )
>. ) ) )  =  ( ( # `  S
)  +  ( (
# `  R )  -  ( T  -  F ) ) ) )
617, 15, 603eqtrd 2488 1  |-  ( ph  ->  ( # `  ( S splice  <. F ,  T ,  R >. ) )  =  ( ( # `  S
)  +  ( (
# `  R )  -  ( T  -  F ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1383    e. wcel 1804   <.cop 4020   <.cotp 4022   ` cfv 5578  (class class class)co 6281   0cc0 9495    + caddc 9498    - cmin 9810   NN0cn0 10801   ZZcz 10870   ZZ>=cuz 11090   ...cfz 11681   #chash 12384  Word cword 12513   concat cconcat 12515   substr csubstr 12517   splice csplice 12518
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-ot 4023  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-card 8323  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10543  df-2 10600  df-n0 10802  df-z 10871  df-uz 11091  df-fz 11682  df-fzo 11804  df-hash 12385  df-word 12521  df-concat 12523  df-substr 12525  df-splice 12526
This theorem is referenced by:  psgnunilem2  16394  efgtlen  16618
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