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Theorem swrdid 12602
Description: A word is a subword of itself. (Contributed by Stefan O'Rear, 16-Aug-2015.)
Assertion
Ref Expression
swrdid  |-  ( S  e. Word  A  ->  ( S substr  <. 0 ,  (
# `  S ) >. )  =  S )

Proof of Theorem swrdid
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 swrdcl 12596 . . . 4  |-  ( S  e. Word  A  ->  ( S substr  <. 0 ,  (
# `  S ) >. )  e. Word  A )
2 wrdf 12506 . . . 4  |-  ( ( S substr  <. 0 ,  (
# `  S ) >. )  e. Word  A  -> 
( S substr  <. 0 ,  ( # `  S
) >. ) : ( 0..^ ( # `  ( S substr  <. 0 ,  (
# `  S ) >. ) ) ) --> A )
3 ffn 5722 . . . 4  |-  ( ( S substr  <. 0 ,  (
# `  S ) >. ) : ( 0..^ ( # `  ( S substr  <. 0 ,  (
# `  S ) >. ) ) ) --> A  ->  ( S substr  <. 0 ,  ( # `  S
) >. )  Fn  (
0..^ ( # `  ( S substr  <. 0 ,  (
# `  S ) >. ) ) ) )
41, 2, 33syl 20 . . 3  |-  ( S  e. Word  A  ->  ( S substr  <. 0 ,  (
# `  S ) >. )  Fn  ( 0..^ ( # `  ( S substr  <. 0 ,  (
# `  S ) >. ) ) ) )
5 lencl 12515 . . . . . . . 8  |-  ( S  e. Word  A  ->  ( # `
 S )  e. 
NN0 )
6 eluzfz1 11682 . . . . . . . . 9  |-  ( (
# `  S )  e.  ( ZZ>= `  0 )  ->  0  e.  ( 0 ... ( # `  S
) ) )
7 nn0uz 11105 . . . . . . . . 9  |-  NN0  =  ( ZZ>= `  0 )
86, 7eleq2s 2568 . . . . . . . 8  |-  ( (
# `  S )  e.  NN0  ->  0  e.  ( 0 ... ( # `
 S ) ) )
95, 8syl 16 . . . . . . 7  |-  ( S  e. Word  A  ->  0  e.  ( 0 ... ( # `
 S ) ) )
10 eluzfz2 11683 . . . . . . . . 9  |-  ( (
# `  S )  e.  ( ZZ>= `  0 )  ->  ( # `  S
)  e.  ( 0 ... ( # `  S
) ) )
1110, 7eleq2s 2568 . . . . . . . 8  |-  ( (
# `  S )  e.  NN0  ->  ( # `  S
)  e.  ( 0 ... ( # `  S
) ) )
125, 11syl 16 . . . . . . 7  |-  ( S  e. Word  A  ->  ( # `
 S )  e.  ( 0 ... ( # `
 S ) ) )
13 swrdlen 12600 . . . . . . 7  |-  ( ( S  e. Word  A  /\  0  e.  ( 0 ... ( # `  S
) )  /\  ( # `
 S )  e.  ( 0 ... ( # `
 S ) ) )  ->  ( # `  ( S substr  <. 0 ,  (
# `  S ) >. ) )  =  ( ( # `  S
)  -  0 ) )
149, 12, 13mpd3an23 1321 . . . . . 6  |-  ( S  e. Word  A  ->  ( # `
 ( S substr  <. 0 ,  ( # `  S
) >. ) )  =  ( ( # `  S
)  -  0 ) )
155nn0cnd 10843 . . . . . . 7  |-  ( S  e. Word  A  ->  ( # `
 S )  e.  CC )
1615subid1d 9908 . . . . . 6  |-  ( S  e. Word  A  ->  (
( # `  S )  -  0 )  =  ( # `  S
) )
1714, 16eqtrd 2501 . . . . 5  |-  ( S  e. Word  A  ->  ( # `
 ( S substr  <. 0 ,  ( # `  S
) >. ) )  =  ( # `  S
) )
1817oveq2d 6291 . . . 4  |-  ( S  e. Word  A  ->  (
0..^ ( # `  ( S substr  <. 0 ,  (
# `  S ) >. ) ) )  =  ( 0..^ ( # `  S ) ) )
1918fneq2d 5663 . . 3  |-  ( S  e. Word  A  ->  (
( S substr  <. 0 ,  ( # `  S
) >. )  Fn  (
0..^ ( # `  ( S substr  <. 0 ,  (
# `  S ) >. ) ) )  <->  ( S substr  <.
0 ,  ( # `  S ) >. )  Fn  ( 0..^ ( # `  S ) ) ) )
204, 19mpbid 210 . 2  |-  ( S  e. Word  A  ->  ( S substr  <. 0 ,  (
# `  S ) >. )  Fn  ( 0..^ ( # `  S
) ) )
21 wrdf 12506 . . 3  |-  ( S  e. Word  A  ->  S : ( 0..^ (
# `  S )
) --> A )
22 ffn 5722 . . 3  |-  ( S : ( 0..^ (
# `  S )
) --> A  ->  S  Fn  ( 0..^ ( # `  S ) ) )
2321, 22syl 16 . 2  |-  ( S  e. Word  A  ->  S  Fn  ( 0..^ ( # `  S ) ) )
24 simpl 457 . . . 4  |-  ( ( S  e. Word  A  /\  x  e.  ( 0..^ ( # `  S
) ) )  ->  S  e. Word  A )
259adantr 465 . . . 4  |-  ( ( S  e. Word  A  /\  x  e.  ( 0..^ ( # `  S
) ) )  -> 
0  e.  ( 0 ... ( # `  S
) ) )
2612adantr 465 . . . 4  |-  ( ( S  e. Word  A  /\  x  e.  ( 0..^ ( # `  S
) ) )  -> 
( # `  S )  e.  ( 0 ... ( # `  S
) ) )
2716oveq2d 6291 . . . . . 6  |-  ( S  e. Word  A  ->  (
0..^ ( ( # `  S )  -  0 ) )  =  ( 0..^ ( # `  S
) ) )
2827eleq2d 2530 . . . . 5  |-  ( S  e. Word  A  ->  (
x  e.  ( 0..^ ( ( # `  S
)  -  0 ) )  <->  x  e.  (
0..^ ( # `  S
) ) ) )
2928biimpar 485 . . . 4  |-  ( ( S  e. Word  A  /\  x  e.  ( 0..^ ( # `  S
) ) )  ->  x  e.  ( 0..^ ( ( # `  S
)  -  0 ) ) )
30 swrdfv 12601 . . . 4  |-  ( ( ( S  e. Word  A  /\  0  e.  (
0 ... ( # `  S
) )  /\  ( # `
 S )  e.  ( 0 ... ( # `
 S ) ) )  /\  x  e.  ( 0..^ ( (
# `  S )  -  0 ) ) )  ->  ( ( S substr  <. 0 ,  (
# `  S ) >. ) `  x )  =  ( S `  ( x  +  0
) ) )
3124, 25, 26, 29, 30syl31anc 1226 . . 3  |-  ( ( S  e. Word  A  /\  x  e.  ( 0..^ ( # `  S
) ) )  -> 
( ( S substr  <. 0 ,  ( # `  S
) >. ) `  x
)  =  ( S `
 ( x  + 
0 ) ) )
32 elfzoelz 11786 . . . . . . 7  |-  ( x  e.  ( 0..^ (
# `  S )
)  ->  x  e.  ZZ )
3332zcnd 10956 . . . . . 6  |-  ( x  e.  ( 0..^ (
# `  S )
)  ->  x  e.  CC )
3433addid1d 9768 . . . . 5  |-  ( x  e.  ( 0..^ (
# `  S )
)  ->  ( x  +  0 )  =  x )
3534adantl 466 . . . 4  |-  ( ( S  e. Word  A  /\  x  e.  ( 0..^ ( # `  S
) ) )  -> 
( x  +  0 )  =  x )
3635fveq2d 5861 . . 3  |-  ( ( S  e. Word  A  /\  x  e.  ( 0..^ ( # `  S
) ) )  -> 
( S `  (
x  +  0 ) )  =  ( S `
 x ) )
3731, 36eqtrd 2501 . 2  |-  ( ( S  e. Word  A  /\  x  e.  ( 0..^ ( # `  S
) ) )  -> 
( ( S substr  <. 0 ,  ( # `  S
) >. ) `  x
)  =  ( S `
 x ) )
3820, 23, 37eqfnfvd 5969 1  |-  ( S  e. Word  A  ->  ( S substr  <. 0 ,  (
# `  S ) >. )  =  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762   <.cop 4026    Fn wfn 5574   -->wf 5575   ` cfv 5579  (class class class)co 6275   0cc0 9481    + caddc 9484    - cmin 9794   NN0cn0 10784   ZZ>=cuz 11071   ...cfz 11661  ..^cfzo 11781   #chash 12360  Word cword 12487   substr csubstr 12491
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-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-substr 12499
This theorem is referenced by:  wrdcctswrd  12640  swrdccatwrd  12643  wrdeqcats1  12649  wrdeqs1cat  12650  swrdccat3a  12669  swrdccat3b  12671  swrdccatid  12672  splid  12679  splval2  12683  cshw0  12715  efgredleme  16550  efgredlemc  16552  efgcpbllemb  16562  frgpuplem  16579  wrdsplex  27985
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