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Theorem splid 11774
Description: Splicing a subword for the same subword makes no difference. (Contributed by Stefan O'Rear, 20-Aug-2015.)
Assertion
Ref Expression
splid  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( S splice  <. X ,  Y ,  ( S substr  <. X ,  Y >. )
>. )  =  S
)

Proof of Theorem splid
StepHypRef Expression
1 ovex 6098 . . 3  |-  ( S substr  <. X ,  Y >. )  e.  _V
2 splval 11772 . . 3  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... ( # `  S
) )  /\  ( S substr  <. X ,  Y >. )  e.  _V )
)  ->  ( S splice  <. X ,  Y , 
( S substr  <. X ,  Y >. ) >. )  =  ( ( ( S substr  <. 0 ,  X >. ) concat  ( S substr  <. X ,  Y >. ) ) concat  ( S substr  <. Y ,  (
# `  S ) >. ) ) )
31, 2mp3anr3 1278 . 2  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( S splice  <. X ,  Y ,  ( S substr  <. X ,  Y >. )
>. )  =  (
( ( S substr  <. 0 ,  X >. ) concat  ( S substr  <. X ,  Y >. ) ) concat  ( S substr  <. Y , 
( # `  S )
>. ) ) )
4 simpl 444 . . . . 5  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... ( # `  S
) ) ) )  ->  S  e. Word  A
)
5 elfzuz 11047 . . . . . . 7  |-  ( X  e.  ( 0 ... Y )  ->  X  e.  ( ZZ>= `  0 )
)
65ad2antrl 709 . . . . . 6  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... ( # `  S
) ) ) )  ->  X  e.  (
ZZ>= `  0 ) )
7 eluzfz1 11056 . . . . . 6  |-  ( X  e.  ( ZZ>= `  0
)  ->  0  e.  ( 0 ... X
) )
86, 7syl 16 . . . . 5  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... ( # `  S
) ) ) )  ->  0  e.  ( 0 ... X ) )
9 simprl 733 . . . . 5  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... ( # `  S
) ) ) )  ->  X  e.  ( 0 ... Y ) )
10 simprr 734 . . . . 5  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... ( # `  S
) ) ) )  ->  Y  e.  ( 0 ... ( # `  S ) ) )
11 ccatswrd 11765 . . . . 5  |-  ( ( S  e. Word  A  /\  ( 0  e.  ( 0 ... X )  /\  X  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... ( # `  S ) ) ) )  ->  ( ( S substr  <. 0 ,  X >. ) concat  ( S substr  <. X ,  Y >. ) )  =  ( S substr  <. 0 ,  Y >. ) )
124, 8, 9, 10, 11syl13anc 1186 . . . 4  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( ( S substr  <. 0 ,  X >. ) concat 
( S substr  <. X ,  Y >. ) )  =  ( S substr  <. 0 ,  Y >. ) )
1312oveq1d 6088 . . 3  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( ( ( S substr  <. 0 ,  X >. ) concat  ( S substr  <. X ,  Y >. ) ) concat  ( S substr  <. Y ,  (
# `  S ) >. ) )  =  ( ( S substr  <. 0 ,  Y >. ) concat  ( S substr  <. Y ,  ( # `  S
) >. ) ) )
14 elfzuz 11047 . . . . . . 7  |-  ( Y  e.  ( 0 ... ( # `  S
) )  ->  Y  e.  ( ZZ>= `  0 )
)
1514ad2antll 710 . . . . . 6  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... ( # `  S
) ) ) )  ->  Y  e.  (
ZZ>= `  0 ) )
16 eluzfz1 11056 . . . . . 6  |-  ( Y  e.  ( ZZ>= `  0
)  ->  0  e.  ( 0 ... Y
) )
1715, 16syl 16 . . . . 5  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... ( # `  S
) ) ) )  ->  0  e.  ( 0 ... Y ) )
18 elfzuz2 11054 . . . . . . 7  |-  ( Y  e.  ( 0 ... ( # `  S
) )  ->  ( # `
 S )  e.  ( ZZ>= `  0 )
)
1918ad2antll 710 . . . . . 6  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( # `  S
)  e.  ( ZZ>= ` 
0 ) )
20 eluzfz2 11057 . . . . . 6  |-  ( (
# `  S )  e.  ( ZZ>= `  0 )  ->  ( # `  S
)  e.  ( 0 ... ( # `  S
) ) )
2119, 20syl 16 . . . . 5  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( # `  S
)  e.  ( 0 ... ( # `  S
) ) )
22 ccatswrd 11765 . . . . 5  |-  ( ( S  e. Word  A  /\  ( 0  e.  ( 0 ... Y )  /\  Y  e.  ( 0 ... ( # `  S ) )  /\  ( # `  S )  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( ( S substr  <. 0 ,  Y >. ) concat 
( S substr  <. Y , 
( # `  S )
>. ) )  =  ( S substr  <. 0 ,  (
# `  S ) >. ) )
234, 17, 10, 21, 22syl13anc 1186 . . . 4  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( ( S substr  <. 0 ,  Y >. ) concat 
( S substr  <. Y , 
( # `  S )
>. ) )  =  ( S substr  <. 0 ,  (
# `  S ) >. ) )
24 swrdid 11764 . . . . 5  |-  ( S  e. Word  A  ->  ( S substr  <. 0 ,  (
# `  S ) >. )  =  S )
2524adantr 452 . . . 4  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( S substr  <. 0 ,  ( # `  S
) >. )  =  S )
2623, 25eqtrd 2467 . . 3  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( ( S substr  <. 0 ,  Y >. ) concat 
( S substr  <. Y , 
( # `  S )
>. ) )  =  S )
2713, 26eqtrd 2467 . 2  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( ( ( S substr  <. 0 ,  X >. ) concat  ( S substr  <. X ,  Y >. ) ) concat  ( S substr  <. Y ,  (
# `  S ) >. ) )  =  S )
283, 27eqtrd 2467 1  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( S splice  <. X ,  Y ,  ( S substr  <. X ,  Y >. )
>. )  =  S
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2948   <.cop 3809   <.cotp 3810   ` cfv 5446  (class class class)co 6073   0cc0 8982   ZZ>=cuz 10480   ...cfz 11035   #chash 11610  Word cword 11709   concat cconcat 11710   substr csubstr 11712   splice csplice 11713
This theorem is referenced by:  psgnunilem2  27386
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-ot 3816  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-n0 10214  df-z 10275  df-uz 10481  df-fz 11036  df-fzo 11128  df-hash 11611  df-word 11715  df-concat 11716  df-substr 11718  df-splice 11719
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