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Theorem splid 11468
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 5883 . . 3  |-  ( S substr  <. X ,  Y >. )  e.  _V
2 splval 11466 . . 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 1276 . 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 443 . . . . 5  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... ( # `  S
) ) ) )  ->  S  e. Word  A
)
5 elfzuz 10794 . . . . . . 7  |-  ( X  e.  ( 0 ... Y )  ->  X  e.  ( ZZ>= `  0 )
)
65ad2antrl 708 . . . . . 6  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... ( # `  S
) ) ) )  ->  X  e.  (
ZZ>= `  0 ) )
7 eluzfz1 10803 . . . . . 6  |-  ( X  e.  ( ZZ>= `  0
)  ->  0  e.  ( 0 ... X
) )
86, 7syl 15 . . . . 5  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... ( # `  S
) ) ) )  ->  0  e.  ( 0 ... X ) )
9 simprl 732 . . . . 5  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... ( # `  S
) ) ) )  ->  X  e.  ( 0 ... Y ) )
10 simprr 733 . . . . 5  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... ( # `  S
) ) ) )  ->  Y  e.  ( 0 ... ( # `  S ) ) )
11 ccatswrd 11459 . . . . 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 1184 . . . 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 5873 . . 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 10794 . . . . . . 7  |-  ( Y  e.  ( 0 ... ( # `  S
) )  ->  Y  e.  ( ZZ>= `  0 )
)
1514ad2antll 709 . . . . . 6  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... ( # `  S
) ) ) )  ->  Y  e.  (
ZZ>= `  0 ) )
16 eluzfz1 10803 . . . . . 6  |-  ( Y  e.  ( ZZ>= `  0
)  ->  0  e.  ( 0 ... Y
) )
1715, 16syl 15 . . . . 5  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... ( # `  S
) ) ) )  ->  0  e.  ( 0 ... Y ) )
18 elfzuz2 10801 . . . . . . 7  |-  ( Y  e.  ( 0 ... ( # `  S
) )  ->  ( # `
 S )  e.  ( ZZ>= `  0 )
)
1918ad2antll 709 . . . . . 6  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( # `  S
)  e.  ( ZZ>= ` 
0 ) )
20 eluzfz2 10804 . . . . . 6  |-  ( (
# `  S )  e.  ( ZZ>= `  0 )  ->  ( # `  S
)  e.  ( 0 ... ( # `  S
) ) )
2119, 20syl 15 . . . . 5  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( # `  S
)  e.  ( 0 ... ( # `  S
) ) )
22 ccatswrd 11459 . . . . 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 1184 . . . 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 11458 . . . . 5  |-  ( S  e. Word  A  ->  ( S substr  <. 0 ,  (
# `  S ) >. )  =  S )
2524adantr 451 . . . 4  |-  ( ( S  e. Word  A  /\  ( X  e.  (
0 ... Y )  /\  Y  e.  ( 0 ... ( # `  S
) ) ) )  ->  ( S substr  <. 0 ,  ( # `  S
) >. )  =  S )
2623, 25eqtrd 2315 . . 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 2315 . 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 2315 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 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   <.cop 3643   <.cotp 3644   ` cfv 5255  (class class class)co 5858   0cc0 8737   ZZ>=cuz 10230   ...cfz 10782   #chash 11337  Word cword 11403   concat cconcat 11404   substr csubstr 11406   splice csplice 11407
This theorem is referenced by:  psgnunilem2  27418
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-ot 3650  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-fzo 10871  df-hash 11338  df-word 11409  df-concat 11410  df-substr 11412  df-splice 11413
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