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Theorem splval2 12695
Description: Value of a splice, assuming the input word  S has already been decomposed into its pieces. (Contributed by Mario Carneiro, 1-Oct-2015.)
Hypotheses
Ref Expression
splval2.a  |-  ( ph  ->  A  e. Word  X )
splval2.b  |-  ( ph  ->  B  e. Word  X )
splval2.c  |-  ( ph  ->  C  e. Word  X )
splval2.r  |-  ( ph  ->  R  e. Word  X )
splval2.s  |-  ( ph  ->  S  =  ( ( A concat  B ) concat  C
) )
splval2.f  |-  ( ph  ->  F  =  ( # `  A ) )
splval2.t  |-  ( ph  ->  T  =  ( F  +  ( # `  B
) ) )
Assertion
Ref Expression
splval2  |-  ( ph  ->  ( S splice  <. F ,  T ,  R >. )  =  ( ( A concat  R ) concat  C ) )

Proof of Theorem splval2
StepHypRef Expression
1 splval2.s . . . 4  |-  ( ph  ->  S  =  ( ( A concat  B ) concat  C
) )
2 splval2.a . . . . . 6  |-  ( ph  ->  A  e. Word  X )
3 splval2.b . . . . . 6  |-  ( ph  ->  B  e. Word  X )
4 ccatcl 12557 . . . . . 6  |-  ( ( A  e. Word  X  /\  B  e. Word  X )  ->  ( A concat  B )  e. Word  X )
52, 3, 4syl2anc 661 . . . . 5  |-  ( ph  ->  ( A concat  B )  e. Word  X )
6 splval2.c . . . . 5  |-  ( ph  ->  C  e. Word  X )
7 ccatcl 12557 . . . . 5  |-  ( ( ( A concat  B )  e. Word  X  /\  C  e. Word  X )  ->  (
( A concat  B ) concat  C )  e. Word  X )
85, 6, 7syl2anc 661 . . . 4  |-  ( ph  ->  ( ( A concat  B
) concat  C )  e. Word  X
)
91, 8eqeltrd 2555 . . 3  |-  ( ph  ->  S  e. Word  X )
10 splval2.f . . . 4  |-  ( ph  ->  F  =  ( # `  A ) )
11 lencl 12527 . . . . 5  |-  ( A  e. Word  X  ->  ( # `
 A )  e. 
NN0 )
122, 11syl 16 . . . 4  |-  ( ph  ->  ( # `  A
)  e.  NN0 )
1310, 12eqeltrd 2555 . . 3  |-  ( ph  ->  F  e.  NN0 )
14 splval2.t . . . 4  |-  ( ph  ->  T  =  ( F  +  ( # `  B
) ) )
15 lencl 12527 . . . . . 6  |-  ( B  e. Word  X  ->  ( # `
 B )  e. 
NN0 )
163, 15syl 16 . . . . 5  |-  ( ph  ->  ( # `  B
)  e.  NN0 )
1713, 16nn0addcld 10855 . . . 4  |-  ( ph  ->  ( F  +  (
# `  B )
)  e.  NN0 )
1814, 17eqeltrd 2555 . . 3  |-  ( ph  ->  T  e.  NN0 )
19 splval2.r . . 3  |-  ( ph  ->  R  e. Word  X )
20 splval 12689 . . 3  |-  ( ( S  e. Word  X  /\  ( F  e.  NN0  /\  T  e.  NN0  /\  R  e. Word  X )
)  ->  ( S splice  <. F ,  T ,  R >. )  =  ( ( ( S substr  <. 0 ,  F >. ) concat  R ) concat  ( S substr  <. T ,  (
# `  S ) >. ) ) )
219, 13, 18, 19, 20syl13anc 1230 . 2  |-  ( ph  ->  ( S splice  <. F ,  T ,  R >. )  =  ( ( ( S substr  <. 0 ,  F >. ) concat  R ) concat  ( S substr  <. T ,  (
# `  S ) >. ) ) )
22 nn0uz 11115 . . . . . . . . . 10  |-  NN0  =  ( ZZ>= `  0 )
2313, 22syl6eleq 2565 . . . . . . . . 9  |-  ( ph  ->  F  e.  ( ZZ>= ` 
0 ) )
24 eluzfz1 11692 . . . . . . . . 9  |-  ( F  e.  ( ZZ>= `  0
)  ->  0  e.  ( 0 ... F
) )
2523, 24syl 16 . . . . . . . 8  |-  ( ph  ->  0  e.  ( 0 ... F ) )
2613nn0zd 10963 . . . . . . . . . . . 12  |-  ( ph  ->  F  e.  ZZ )
27 uzid 11095 . . . . . . . . . . . 12  |-  ( F  e.  ZZ  ->  F  e.  ( ZZ>= `  F )
)
2826, 27syl 16 . . . . . . . . . . 11  |-  ( ph  ->  F  e.  ( ZZ>= `  F ) )
29 uzaddcl 11136 . . . . . . . . . . 11  |-  ( ( F  e.  ( ZZ>= `  F )  /\  ( # `
 B )  e. 
NN0 )  ->  ( F  +  ( # `  B
) )  e.  (
ZZ>= `  F ) )
3028, 16, 29syl2anc 661 . . . . . . . . . 10  |-  ( ph  ->  ( F  +  (
# `  B )
)  e.  ( ZZ>= `  F ) )
3114, 30eqeltrd 2555 . . . . . . . . 9  |-  ( ph  ->  T  e.  ( ZZ>= `  F ) )
32 elfzuzb 11681 . . . . . . . . 9  |-  ( F  e.  ( 0 ... T )  <->  ( F  e.  ( ZZ>= `  0 )  /\  T  e.  ( ZZ>=
`  F ) ) )
3323, 31, 32sylanbrc 664 . . . . . . . 8  |-  ( ph  ->  F  e.  ( 0 ... T ) )
3418, 22syl6eleq 2565 . . . . . . . . 9  |-  ( ph  ->  T  e.  ( ZZ>= ` 
0 ) )
35 ccatlen 12558 . . . . . . . . . . . 12  |-  ( ( ( A concat  B )  e. Word  X  /\  C  e. Word  X )  ->  ( # `
 ( ( A concat  B ) concat  C ) )  =  ( ( # `  ( A concat  B ) )  +  ( # `  C ) ) )
365, 6, 35syl2anc 661 . . . . . . . . . . 11  |-  ( ph  ->  ( # `  (
( A concat  B ) concat  C ) )  =  ( ( # `  ( A concat  B ) )  +  ( # `  C
) ) )
371fveq2d 5869 . . . . . . . . . . 11  |-  ( ph  ->  ( # `  S
)  =  ( # `  ( ( A concat  B
) concat  C ) ) )
3810oveq1d 6298 . . . . . . . . . . . . 13  |-  ( ph  ->  ( F  +  (
# `  B )
)  =  ( (
# `  A )  +  ( # `  B
) ) )
39 ccatlen 12558 . . . . . . . . . . . . . 14  |-  ( ( A  e. Word  X  /\  B  e. Word  X )  ->  ( # `  ( A concat  B ) )  =  ( ( # `  A
)  +  ( # `  B ) ) )
402, 3, 39syl2anc 661 . . . . . . . . . . . . 13  |-  ( ph  ->  ( # `  ( A concat  B ) )  =  ( ( # `  A
)  +  ( # `  B ) ) )
4138, 14, 403eqtr4d 2518 . . . . . . . . . . . 12  |-  ( ph  ->  T  =  ( # `  ( A concat  B ) ) )
4241oveq1d 6298 . . . . . . . . . . 11  |-  ( ph  ->  ( T  +  (
# `  C )
)  =  ( (
# `  ( A concat  B ) )  +  (
# `  C )
) )
4336, 37, 423eqtr4d 2518 . . . . . . . . . 10  |-  ( ph  ->  ( # `  S
)  =  ( T  +  ( # `  C
) ) )
4418nn0zd 10963 . . . . . . . . . . . 12  |-  ( ph  ->  T  e.  ZZ )
45 uzid 11095 . . . . . . . . . . . 12  |-  ( T  e.  ZZ  ->  T  e.  ( ZZ>= `  T )
)
4644, 45syl 16 . . . . . . . . . . 11  |-  ( ph  ->  T  e.  ( ZZ>= `  T ) )
47 lencl 12527 . . . . . . . . . . . 12  |-  ( C  e. Word  X  ->  ( # `
 C )  e. 
NN0 )
486, 47syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ( # `  C
)  e.  NN0 )
49 uzaddcl 11136 . . . . . . . . . . 11  |-  ( ( T  e.  ( ZZ>= `  T )  /\  ( # `
 C )  e. 
NN0 )  ->  ( T  +  ( # `  C
) )  e.  (
ZZ>= `  T ) )
5046, 48, 49syl2anc 661 . . . . . . . . . 10  |-  ( ph  ->  ( T  +  (
# `  C )
)  e.  ( ZZ>= `  T ) )
5143, 50eqeltrd 2555 . . . . . . . . 9  |-  ( ph  ->  ( # `  S
)  e.  ( ZZ>= `  T ) )
52 elfzuzb 11681 . . . . . . . . 9  |-  ( T  e.  ( 0 ... ( # `  S
) )  <->  ( T  e.  ( ZZ>= `  0 )  /\  ( # `  S
)  e.  ( ZZ>= `  T ) ) )
5334, 51, 52sylanbrc 664 . . . . . . . 8  |-  ( ph  ->  T  e.  ( 0 ... ( # `  S
) ) )
54 ccatswrd 12643 . . . . . . . 8  |-  ( ( S  e. Word  X  /\  ( 0  e.  ( 0 ... F )  /\  F  e.  ( 0 ... T )  /\  T  e.  ( 0 ... ( # `  S ) ) ) )  ->  ( ( S substr  <. 0 ,  F >. ) concat  ( S substr  <. F ,  T >. ) )  =  ( S substr  <. 0 ,  T >. ) )
559, 25, 33, 53, 54syl13anc 1230 . . . . . . 7  |-  ( ph  ->  ( ( S substr  <. 0 ,  F >. ) concat  ( S substr  <. F ,  T >. ) )  =  ( S substr  <. 0 ,  T >. ) )
56 eluzfz1 11692 . . . . . . . . . . . 12  |-  ( T  e.  ( ZZ>= `  0
)  ->  0  e.  ( 0 ... T
) )
5734, 56syl 16 . . . . . . . . . . 11  |-  ( ph  ->  0  e.  ( 0 ... T ) )
58 lencl 12527 . . . . . . . . . . . . . 14  |-  ( S  e. Word  X  ->  ( # `
 S )  e. 
NN0 )
599, 58syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  ( # `  S
)  e.  NN0 )
6059, 22syl6eleq 2565 . . . . . . . . . . . 12  |-  ( ph  ->  ( # `  S
)  e.  ( ZZ>= ` 
0 ) )
61 eluzfz2 11693 . . . . . . . . . . . 12  |-  ( (
# `  S )  e.  ( ZZ>= `  0 )  ->  ( # `  S
)  e.  ( 0 ... ( # `  S
) ) )
6260, 61syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ( # `  S
)  e.  ( 0 ... ( # `  S
) ) )
63 ccatswrd 12643 . . . . . . . . . . 11  |-  ( ( S  e. Word  X  /\  ( 0  e.  ( 0 ... T )  /\  T  e.  ( 0 ... ( # `  S ) )  /\  ( # `  S )  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( ( S substr  <. 0 ,  T >. ) concat 
( S substr  <. T , 
( # `  S )
>. ) )  =  ( S substr  <. 0 ,  (
# `  S ) >. ) )
649, 57, 53, 62, 63syl13anc 1230 . . . . . . . . . 10  |-  ( ph  ->  ( ( S substr  <. 0 ,  T >. ) concat  ( S substr  <. T ,  ( # `  S
) >. ) )  =  ( S substr  <. 0 ,  ( # `  S
) >. ) )
65 swrdid 12614 . . . . . . . . . . 11  |-  ( S  e. Word  X  ->  ( S substr  <. 0 ,  (
# `  S ) >. )  =  S )
669, 65syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( S substr  <. 0 ,  ( # `  S
) >. )  =  S )
6764, 66, 13eqtrd 2512 . . . . . . . . 9  |-  ( ph  ->  ( ( S substr  <. 0 ,  T >. ) concat  ( S substr  <. T ,  ( # `  S
) >. ) )  =  ( ( A concat  B
) concat  C ) )
68 swrdcl 12608 . . . . . . . . . . 11  |-  ( S  e. Word  X  ->  ( S substr  <. 0 ,  T >. )  e. Word  X )
699, 68syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( S substr  <. 0 ,  T >. )  e. Word  X
)
70 swrdcl 12608 . . . . . . . . . . 11  |-  ( S  e. Word  X  ->  ( S substr  <. T ,  (
# `  S ) >. )  e. Word  X )
719, 70syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( S substr  <. T , 
( # `  S )
>. )  e. Word  X )
72 swrd0len 12611 . . . . . . . . . . . 12  |-  ( ( S  e. Word  X  /\  T  e.  ( 0 ... ( # `  S
) ) )  -> 
( # `  ( S substr  <. 0 ,  T >. ) )  =  T )
739, 53, 72syl2anc 661 . . . . . . . . . . 11  |-  ( ph  ->  ( # `  ( S substr  <. 0 ,  T >. ) )  =  T )
7473, 41eqtrd 2508 . . . . . . . . . 10  |-  ( ph  ->  ( # `  ( S substr  <. 0 ,  T >. ) )  =  (
# `  ( A concat  B ) ) )
75 ccatopth 12657 . . . . . . . . . 10  |-  ( ( ( ( S substr  <. 0 ,  T >. )  e. Word  X  /\  ( S substr  <. T , 
( # `  S )
>. )  e. Word  X )  /\  ( ( A concat  B )  e. Word  X  /\  C  e. Word  X )  /\  ( # `  ( S substr  <. 0 ,  T >. ) )  =  (
# `  ( A concat  B ) ) )  -> 
( ( ( S substr  <. 0 ,  T >. ) concat 
( S substr  <. T , 
( # `  S )
>. ) )  =  ( ( A concat  B ) concat  C )  <->  ( ( S substr  <. 0 ,  T >. )  =  ( A concat  B )  /\  ( S substr  <. T ,  (
# `  S ) >. )  =  C ) ) )
7669, 71, 5, 6, 74, 75syl221anc 1239 . . . . . . . . 9  |-  ( ph  ->  ( ( ( S substr  <. 0 ,  T >. ) concat 
( S substr  <. T , 
( # `  S )
>. ) )  =  ( ( A concat  B ) concat  C )  <->  ( ( S substr  <. 0 ,  T >. )  =  ( A concat  B )  /\  ( S substr  <. T ,  (
# `  S ) >. )  =  C ) ) )
7767, 76mpbid 210 . . . . . . . 8  |-  ( ph  ->  ( ( S substr  <. 0 ,  T >. )  =  ( A concat  B )  /\  ( S substr  <. T , 
( # `  S )
>. )  =  C
) )
7877simpld 459 . . . . . . 7  |-  ( ph  ->  ( S substr  <. 0 ,  T >. )  =  ( A concat  B ) )
7955, 78eqtrd 2508 . . . . . 6  |-  ( ph  ->  ( ( S substr  <. 0 ,  F >. ) concat  ( S substr  <. F ,  T >. ) )  =  ( A concat  B ) )
80 swrdcl 12608 . . . . . . . 8  |-  ( S  e. Word  X  ->  ( S substr  <. 0 ,  F >. )  e. Word  X )
819, 80syl 16 . . . . . . 7  |-  ( ph  ->  ( S substr  <. 0 ,  F >. )  e. Word  X
)
82 swrdcl 12608 . . . . . . . 8  |-  ( S  e. Word  X  ->  ( S substr  <. F ,  T >. )  e. Word  X )
839, 82syl 16 . . . . . . 7  |-  ( ph  ->  ( S substr  <. F ,  T >. )  e. Word  X
)
84 uztrn 11097 . . . . . . . . . . 11  |-  ( ( ( # `  S
)  e.  ( ZZ>= `  T )  /\  T  e.  ( ZZ>= `  F )
)  ->  ( # `  S
)  e.  ( ZZ>= `  F ) )
8551, 31, 84syl2anc 661 . . . . . . . . . 10  |-  ( ph  ->  ( # `  S
)  e.  ( ZZ>= `  F ) )
86 elfzuzb 11681 . . . . . . . . . 10  |-  ( F  e.  ( 0 ... ( # `  S
) )  <->  ( F  e.  ( ZZ>= `  0 )  /\  ( # `  S
)  e.  ( ZZ>= `  F ) ) )
8723, 85, 86sylanbrc 664 . . . . . . . . 9  |-  ( ph  ->  F  e.  ( 0 ... ( # `  S
) ) )
88 swrd0len 12611 . . . . . . . . 9  |-  ( ( S  e. Word  X  /\  F  e.  ( 0 ... ( # `  S
) ) )  -> 
( # `  ( S substr  <. 0 ,  F >. ) )  =  F )
899, 87, 88syl2anc 661 . . . . . . . 8  |-  ( ph  ->  ( # `  ( S substr  <. 0 ,  F >. ) )  =  F )
9089, 10eqtrd 2508 . . . . . . 7  |-  ( ph  ->  ( # `  ( S substr  <. 0 ,  F >. ) )  =  (
# `  A )
)
91 ccatopth 12657 . . . . . . 7  |-  ( ( ( ( S substr  <. 0 ,  F >. )  e. Word  X  /\  ( S substr  <. F ,  T >. )  e. Word  X
)  /\  ( A  e. Word  X  /\  B  e. Word  X )  /\  ( # `
 ( S substr  <. 0 ,  F >. ) )  =  ( # `  A
) )  ->  (
( ( S substr  <. 0 ,  F >. ) concat  ( S substr  <. F ,  T >. ) )  =  ( A concat  B )  <->  ( ( S substr  <. 0 ,  F >. )  =  A  /\  ( S substr  <. F ,  T >. )  =  B ) ) )
9281, 83, 2, 3, 90, 91syl221anc 1239 . . . . . 6  |-  ( ph  ->  ( ( ( S substr  <. 0 ,  F >. ) concat 
( S substr  <. F ,  T >. ) )  =  ( A concat  B )  <-> 
( ( S substr  <. 0 ,  F >. )  =  A  /\  ( S substr  <. F ,  T >. )  =  B ) ) )
9379, 92mpbid 210 . . . . 5  |-  ( ph  ->  ( ( S substr  <. 0 ,  F >. )  =  A  /\  ( S substr  <. F ,  T >. )  =  B ) )
9493simpld 459 . . . 4  |-  ( ph  ->  ( S substr  <. 0 ,  F >. )  =  A )
9594oveq1d 6298 . . 3  |-  ( ph  ->  ( ( S substr  <. 0 ,  F >. ) concat  R )  =  ( A concat  R
) )
9677simprd 463 . . 3  |-  ( ph  ->  ( S substr  <. T , 
( # `  S )
>. )  =  C
)
9795, 96oveq12d 6301 . 2  |-  ( ph  ->  ( ( ( S substr  <. 0 ,  F >. ) concat  R ) concat  ( S substr  <. T , 
( # `  S )
>. ) )  =  ( ( A concat  R ) concat  C ) )
9821, 97eqtrd 2508 1  |-  ( ph  ->  ( S splice  <. F ,  T ,  R >. )  =  ( ( A concat  R ) concat  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   <.cop 4033   <.cotp 4035   ` cfv 5587  (class class class)co 6283   0cc0 9491    + caddc 9494   NN0cn0 10794   ZZcz 10863   ZZ>=cuz 11081   ...cfz 11671   #chash 12372  Word cword 12499   concat cconcat 12501   substr csubstr 12503   splice csplice 12504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-cnex 9547  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-ot 4036  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-om 6680  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-card 8319  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-nn 10536  df-n0 10795  df-z 10864  df-uz 11082  df-fz 11672  df-fzo 11792  df-hash 12373  df-word 12507  df-concat 12509  df-substr 12511  df-splice 12512
This theorem is referenced by:  efginvrel2  16548  efgredleme  16564  efgcpbllemb  16576  frgpnabllem1  16677
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