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Theorem splid 12679
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 6300 . . 3  |-  ( S substr  <. X ,  Y >. )  e.  _V
2 splval 12677 . . 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 1318 . 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 457 . . . . 5  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... ( # `  S
) ) ) )  ->  S  e. Word  A
)
5 elfzuz 11673 . . . . . . 7  |-  ( X  e.  ( 0 ... Y )  ->  X  e.  ( ZZ>= `  0 )
)
65ad2antrl 727 . . . . . 6  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... ( # `  S
) ) ) )  ->  X  e.  (
ZZ>= `  0 ) )
7 eluzfz1 11682 . . . . . 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 755 . . . . 5  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... ( # `  S
) ) ) )  ->  X  e.  ( 0 ... Y ) )
10 simprr 756 . . . . 5  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... ( # `  S
) ) ) )  ->  Y  e.  ( 0 ... ( # `  S ) ) )
11 ccatswrd 12631 . . . . 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 1225 . . . 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 6290 . . 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 11673 . . . . . . 7  |-  ( Y  e.  ( 0 ... ( # `  S
) )  ->  Y  e.  ( ZZ>= `  0 )
)
1514ad2antll 728 . . . . . 6  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... ( # `  S
) ) ) )  ->  Y  e.  (
ZZ>= `  0 ) )
16 eluzfz1 11682 . . . . . 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 11680 . . . . . . 7  |-  ( Y  e.  ( 0 ... ( # `  S
) )  ->  ( # `
 S )  e.  ( ZZ>= `  0 )
)
1918ad2antll 728 . . . . . 6  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( # `  S
)  e.  ( ZZ>= ` 
0 ) )
20 eluzfz2 11683 . . . . . 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 12631 . . . . 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 1225 . . . 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 12602 . . . . 5  |-  ( S  e. Word  A  ->  ( S substr  <. 0 ,  (
# `  S ) >. )  =  S )
2524adantr 465 . . . 4  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( S substr  <. 0 ,  ( # `  S
) >. )  =  S )
2623, 25eqtrd 2501 . . 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 2501 . 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 2501 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762   _Vcvv 3106   <.cop 4026   <.cotp 4028   ` cfv 5579  (class class class)co 6275   0cc0 9481   ZZ>=cuz 11071   ...cfz 11661   #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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