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Theorem swrdid 11772
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 11766 . . . 4  |-  ( S  e. Word  A  ->  ( S substr  <. 0 ,  (
# `  S ) >. )  e. Word  A )
2 wrdf 11733 . . . 4  |-  ( ( S substr  <. 0 ,  (
# `  S ) >. )  e. Word  A  -> 
( S substr  <. 0 ,  ( # `  S
) >. ) : ( 0..^ ( # `  ( S substr  <. 0 ,  (
# `  S ) >. ) ) ) --> A )
3 ffn 5591 . . . 4  |-  ( ( S substr  <. 0 ,  (
# `  S ) >. ) : ( 0..^ ( # `  ( S substr  <. 0 ,  (
# `  S ) >. ) ) ) --> A  ->  ( S substr  <. 0 ,  ( # `  S
) >. )  Fn  (
0..^ ( # `  ( S substr  <. 0 ,  (
# `  S ) >. ) ) ) )
41, 2, 33syl 19 . . 3  |-  ( S  e. Word  A  ->  ( S substr  <. 0 ,  (
# `  S ) >. )  Fn  ( 0..^ ( # `  ( S substr  <. 0 ,  (
# `  S ) >. ) ) ) )
5 lencl 11735 . . . . . . . 8  |-  ( S  e. Word  A  ->  ( # `
 S )  e. 
NN0 )
6 eluzfz1 11064 . . . . . . . . 9  |-  ( (
# `  S )  e.  ( ZZ>= `  0 )  ->  0  e.  ( 0 ... ( # `  S
) ) )
7 nn0uz 10520 . . . . . . . . 9  |-  NN0  =  ( ZZ>= `  0 )
86, 7eleq2s 2528 . . . . . . . 8  |-  ( (
# `  S )  e.  NN0  ->  0  e.  ( 0 ... ( # `
 S ) ) )
95, 8syl 16 . . . . . . 7  |-  ( S  e. Word  A  ->  0  e.  ( 0 ... ( # `
 S ) ) )
10 eluzfz2 11065 . . . . . . . . 9  |-  ( (
# `  S )  e.  ( ZZ>= `  0 )  ->  ( # `  S
)  e.  ( 0 ... ( # `  S
) ) )
1110, 7eleq2s 2528 . . . . . . . 8  |-  ( (
# `  S )  e.  NN0  ->  ( # `  S
)  e.  ( 0 ... ( # `  S
) ) )
125, 11syl 16 . . . . . . 7  |-  ( S  e. Word  A  ->  ( # `
 S )  e.  ( 0 ... ( # `
 S ) ) )
13 swrdlen 11770 . . . . . . 7  |-  ( ( S  e. Word  A  /\  0  e.  ( 0 ... ( # `  S
) )  /\  ( # `
 S )  e.  ( 0 ... ( # `
 S ) ) )  ->  ( # `  ( S substr  <. 0 ,  (
# `  S ) >. ) )  =  ( ( # `  S
)  -  0 ) )
149, 12, 13mpd3an23 1281 . . . . . 6  |-  ( S  e. Word  A  ->  ( # `
 ( S substr  <. 0 ,  ( # `  S
) >. ) )  =  ( ( # `  S
)  -  0 ) )
155nn0cnd 10276 . . . . . . 7  |-  ( S  e. Word  A  ->  ( # `
 S )  e.  CC )
1615subid1d 9400 . . . . . 6  |-  ( S  e. Word  A  ->  (
( # `  S )  -  0 )  =  ( # `  S
) )
1714, 16eqtrd 2468 . . . . 5  |-  ( S  e. Word  A  ->  ( # `
 ( S substr  <. 0 ,  ( # `  S
) >. ) )  =  ( # `  S
) )
1817oveq2d 6097 . . . 4  |-  ( S  e. Word  A  ->  (
0..^ ( # `  ( S substr  <. 0 ,  (
# `  S ) >. ) ) )  =  ( 0..^ ( # `  S ) ) )
1918fneq2d 5537 . . 3  |-  ( S  e. Word  A  ->  (
( S substr  <. 0 ,  ( # `  S
) >. )  Fn  (
0..^ ( # `  ( S substr  <. 0 ,  (
# `  S ) >. ) ) )  <->  ( S substr  <.
0 ,  ( # `  S ) >. )  Fn  ( 0..^ ( # `  S ) ) ) )
204, 19mpbid 202 . 2  |-  ( S  e. Word  A  ->  ( S substr  <. 0 ,  (
# `  S ) >. )  Fn  ( 0..^ ( # `  S
) ) )
21 wrdf 11733 . . 3  |-  ( S  e. Word  A  ->  S : ( 0..^ (
# `  S )
) --> A )
22 ffn 5591 . . 3  |-  ( S : ( 0..^ (
# `  S )
) --> A  ->  S  Fn  ( 0..^ ( # `  S ) ) )
2321, 22syl 16 . 2  |-  ( S  e. Word  A  ->  S  Fn  ( 0..^ ( # `  S ) ) )
24 simpl 444 . . . 4  |-  ( ( S  e. Word  A  /\  x  e.  ( 0..^ ( # `  S
) ) )  ->  S  e. Word  A )
259adantr 452 . . . 4  |-  ( ( S  e. Word  A  /\  x  e.  ( 0..^ ( # `  S
) ) )  -> 
0  e.  ( 0 ... ( # `  S
) ) )
2612adantr 452 . . . 4  |-  ( ( S  e. Word  A  /\  x  e.  ( 0..^ ( # `  S
) ) )  -> 
( # `  S )  e.  ( 0 ... ( # `  S
) ) )
2716oveq2d 6097 . . . . . 6  |-  ( S  e. Word  A  ->  (
0..^ ( ( # `  S )  -  0 ) )  =  ( 0..^ ( # `  S
) ) )
2827eleq2d 2503 . . . . 5  |-  ( S  e. Word  A  ->  (
x  e.  ( 0..^ ( ( # `  S
)  -  0 ) )  <->  x  e.  (
0..^ ( # `  S
) ) ) )
2928biimpar 472 . . . 4  |-  ( ( S  e. Word  A  /\  x  e.  ( 0..^ ( # `  S
) ) )  ->  x  e.  ( 0..^ ( ( # `  S
)  -  0 ) ) )
30 swrdfv 11771 . . . 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 1187 . . 3  |-  ( ( S  e. Word  A  /\  x  e.  ( 0..^ ( # `  S
) ) )  -> 
( ( S substr  <. 0 ,  ( # `  S
) >. ) `  x
)  =  ( S `
 ( x  + 
0 ) ) )
32 elfzoelz 11140 . . . . . . 7  |-  ( x  e.  ( 0..^ (
# `  S )
)  ->  x  e.  ZZ )
3332zcnd 10376 . . . . . 6  |-  ( x  e.  ( 0..^ (
# `  S )
)  ->  x  e.  CC )
3433addid1d 9266 . . . . 5  |-  ( x  e.  ( 0..^ (
# `  S )
)  ->  ( x  +  0 )  =  x )
3534adantl 453 . . . 4  |-  ( ( S  e. Word  A  /\  x  e.  ( 0..^ ( # `  S
) ) )  -> 
( x  +  0 )  =  x )
3635fveq2d 5732 . . 3  |-  ( ( S  e. Word  A  /\  x  e.  ( 0..^ ( # `  S
) ) )  -> 
( S `  (
x  +  0 ) )  =  ( S `
 x ) )
3731, 36eqtrd 2468 . 2  |-  ( ( S  e. Word  A  /\  x  e.  ( 0..^ ( # `  S
) ) )  -> 
( ( S substr  <. 0 ,  ( # `  S
) >. ) `  x
)  =  ( S `
 x ) )
3820, 23, 37eqfnfvd 5830 1  |-  ( S  e. Word  A  ->  ( S substr  <. 0 ,  (
# `  S ) >. )  =  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   <.cop 3817    Fn wfn 5449   -->wf 5450   ` cfv 5454  (class class class)co 6081   0cc0 8990    + caddc 8993    - cmin 9291   NN0cn0 10221   ZZ>=cuz 10488   ...cfz 11043  ..^cfzo 11135   #chash 11618  Word cword 11717   substr csubstr 11720
This theorem is referenced by:  splid  11782  splval2  11786  wrdeqcats1  11788  wrdeqs1cat  11789  efgredleme  15375  efgredlemc  15377  efgcpbllemb  15387  frgpuplem  15404  swrdccat3a  28217  swrdccat3b  28219  cshw0  28238  cshwn  28239
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-n0 10222  df-z 10283  df-uz 10489  df-fz 11044  df-fzo 11136  df-hash 11619  df-word 11723  df-substr 11726
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