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Theorem spllen 12682
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 12679 . . . 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 ) >. ) ) )
76fveq2d 5863 . 2  |-  ( ph  ->  ( # `  ( S splice  <. F ,  T ,  R >. ) )  =  ( # `  (
( ( S substr  <. 0 ,  F >. ) concat  R ) concat  ( S substr  <. T ,  (
# `  S ) >. ) ) ) )
8 swrdcl 12598 . . . . 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 12547 . . . 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 12598 . . . 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 12548 . . 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 12517 . . . . . . 7  |-  ( R  e. Word  A  ->  ( # `
 R )  e. 
NN0 )
174, 16syl 16 . . . . . 6  |-  ( ph  ->  ( # `  R
)  e.  NN0 )
1817nn0cnd 10845 . . . . 5  |-  ( ph  ->  ( # `  R
)  e.  CC )
19 elfzelz 11679 . . . . . . 7  |-  ( F  e.  ( 0 ... T )  ->  F  e.  ZZ )
202, 19syl 16 . . . . . 6  |-  ( ph  ->  F  e.  ZZ )
2120zcnd 10958 . . . . 5  |-  ( ph  ->  F  e.  CC )
2218, 21addcld 9606 . . . 4  |-  ( ph  ->  ( ( # `  R
)  +  F )  e.  CC )
23 elfzel2 11677 . . . . . 6  |-  ( T  e.  ( 0 ... ( # `  S
) )  ->  ( # `
 S )  e.  ZZ )
243, 23syl 16 . . . . 5  |-  ( ph  ->  ( # `  S
)  e.  ZZ )
2524zcnd 10958 . . . 4  |-  ( ph  ->  ( # `  S
)  e.  CC )
26 elfzelz 11679 . . . . . 6  |-  ( T  e.  ( 0 ... ( # `  S
) )  ->  T  e.  ZZ )
273, 26syl 16 . . . . 5  |-  ( ph  ->  T  e.  ZZ )
2827zcnd 10958 . . . 4  |-  ( ph  ->  T  e.  CC )
2922, 25, 28addsub12d 9944 . . 3  |-  ( ph  ->  ( ( ( # `  R )  +  F
)  +  ( (
# `  S )  -  T ) )  =  ( ( # `  S
)  +  ( ( ( # `  R
)  +  F )  -  T ) ) )
30 ccatlen 12548 . . . . . 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 11675 . . . . . . . . . 10  |-  ( F  e.  ( 0 ... T )  ->  F  e.  ( ZZ>= `  0 )
)
332, 32syl 16 . . . . . . . . 9  |-  ( ph  ->  F  e.  ( ZZ>= ` 
0 ) )
34 eluzfz1 11684 . . . . . . . . 9  |-  ( F  e.  ( ZZ>= `  0
)  ->  0  e.  ( 0 ... F
) )
3533, 34syl 16 . . . . . . . 8  |-  ( ph  ->  0  e.  ( 0 ... F ) )
36 elfzuz3 11676 . . . . . . . . . . 11  |-  ( T  e.  ( 0 ... ( # `  S
) )  ->  ( # `
 S )  e.  ( ZZ>= `  T )
)
373, 36syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( # `  S
)  e.  ( ZZ>= `  T ) )
38 elfzuz3 11676 . . . . . . . . . . 11  |-  ( F  e.  ( 0 ... T )  ->  T  e.  ( ZZ>= `  F )
)
392, 38syl 16 . . . . . . . . . 10  |-  ( ph  ->  T  e.  ( ZZ>= `  F ) )
40 uztrn 11089 . . . . . . . . . 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 11673 . . . . . . . . 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 12602 . . . . . . . 8  |-  ( ( S  e. Word  A  /\  0  e.  ( 0 ... F )  /\  F  e.  ( 0 ... ( # `  S
) ) )  -> 
( # `  ( S substr  <. 0 ,  F >. ) )  =  ( F  -  0 ) )
451, 35, 43, 44syl3anc 1223 . . . . . . 7  |-  ( ph  ->  ( # `  ( S substr  <. 0 ,  F >. ) )  =  ( F  -  0 ) )
4621subid1d 9910 . . . . . . 7  |-  ( ph  ->  ( F  -  0 )  =  F )
4745, 46eqtrd 2503 . . . . . 6  |-  ( ph  ->  ( # `  ( S substr  <. 0 ,  F >. ) )  =  F )
4847oveq1d 6292 . . . . 5  |-  ( ph  ->  ( ( # `  ( S substr  <. 0 ,  F >. ) )  +  (
# `  R )
)  =  ( F  +  ( # `  R
) ) )
4921, 18addcomd 9772 . . . . 5  |-  ( ph  ->  ( F  +  (
# `  R )
)  =  ( (
# `  R )  +  F ) )
5031, 48, 493eqtrd 2507 . . . 4  |-  ( ph  ->  ( # `  (
( S substr  <. 0 ,  F >. ) concat  R )
)  =  ( (
# `  R )  +  F ) )
51 elfzuz2 11682 . . . . . . 7  |-  ( T  e.  ( 0 ... ( # `  S
) )  ->  ( # `
 S )  e.  ( ZZ>= `  0 )
)
523, 51syl 16 . . . . . 6  |-  ( ph  ->  ( # `  S
)  e.  ( ZZ>= ` 
0 ) )
53 eluzfz2 11685 . . . . . 6  |-  ( (
# `  S )  e.  ( ZZ>= `  0 )  ->  ( # `  S
)  e.  ( 0 ... ( # `  S
) ) )
5452, 53syl 16 . . . . 5  |-  ( ph  ->  ( # `  S
)  e.  ( 0 ... ( # `  S
) ) )
55 swrdlen 12602 . . . . 5  |-  ( ( S  e. Word  A  /\  T  e.  ( 0 ... ( # `  S
) )  /\  ( # `
 S )  e.  ( 0 ... ( # `
 S ) ) )  ->  ( # `  ( S substr  <. T ,  (
# `  S ) >. ) )  =  ( ( # `  S
)  -  T ) )
561, 3, 54, 55syl3anc 1223 . . . 4  |-  ( ph  ->  ( # `  ( S substr  <. T ,  (
# `  S ) >. ) )  =  ( ( # `  S
)  -  T ) )
5750, 56oveq12d 6295 . . 3  |-  ( ph  ->  ( ( # `  (
( S substr  <. 0 ,  F >. ) concat  R )
)  +  ( # `  ( S substr  <. T , 
( # `  S )
>. ) ) )  =  ( ( ( # `  R )  +  F
)  +  ( (
# `  S )  -  T ) ) )
5818, 28, 21subsub3d 9951 . . . 4  |-  ( ph  ->  ( ( # `  R
)  -  ( T  -  F ) )  =  ( ( (
# `  R )  +  F )  -  T
) )
5958oveq2d 6293 . . 3  |-  ( ph  ->  ( ( # `  S
)  +  ( (
# `  R )  -  ( T  -  F ) ) )  =  ( ( # `  S )  +  ( ( ( # `  R
)  +  F )  -  T ) ) )
6029, 57, 593eqtr4d 2513 . 2  |-  ( ph  ->  ( ( # `  (
( S substr  <. 0 ,  F >. ) concat  R )
)  +  ( # `  ( S substr  <. T , 
( # `  S )
>. ) ) )  =  ( ( # `  S
)  +  ( (
# `  R )  -  ( T  -  F ) ) ) )
617, 15, 603eqtrd 2507 1  |-  ( ph  ->  ( # `  ( S splice  <. F ,  T ,  R >. ) )  =  ( ( # `  S
)  +  ( (
# `  R )  -  ( T  -  F ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1374    e. wcel 1762   <.cop 4028   <.cotp 4030   ` cfv 5581  (class class class)co 6277   0cc0 9483    + caddc 9486    - cmin 9796   NN0cn0 10786   ZZcz 10855   ZZ>=cuz 11073   ...cfz 11663   #chash 12362  Word cword 12489   concat cconcat 12491   substr csubstr 12493   splice csplice 12494
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 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560
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 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-ot 4031  df-uni 4241  df-int 4278  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-1st 6776  df-2nd 6777  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  df-card 8311  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-nn 10528  df-n0 10787  df-z 10856  df-uz 11074  df-fz 11664  df-fzo 11784  df-hash 12363  df-word 12497  df-concat 12499  df-substr 12501  df-splice 12502
This theorem is referenced by:  psgnunilem2  16311  efgtlen  16535
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