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Theorem wrdeqs1cat 11717
Description: Decompose a non-empty word by separating off the first symbol. (Contributed by Stefan O'Rear, 25-Aug-2015.) (Revised by Mario Carneiro, 1-Oct-2015.)
Assertion
Ref Expression
wrdeqs1cat  |-  ( ( W  e. Word  A  /\  W  =/=  (/) )  ->  W  =  ( <" ( W `  0 ) "> concat  ( W substr  <. 1 ,  ( # `  W
) >. ) ) )

Proof of Theorem wrdeqs1cat
StepHypRef Expression
1 simpl 444 . . 3  |-  ( ( W  e. Word  A  /\  W  =/=  (/) )  ->  W  e. Word  A )
2 1nn0 10170 . . . 4  |-  1  e.  NN0
3 eluzfz1 10997 . . . . 5  |-  ( 1  e.  ( ZZ>= `  0
)  ->  0  e.  ( 0 ... 1
) )
4 nn0uz 10453 . . . . 5  |-  NN0  =  ( ZZ>= `  0 )
53, 4eleq2s 2480 . . . 4  |-  ( 1  e.  NN0  ->  0  e.  ( 0 ... 1
) )
62, 5mp1i 12 . . 3  |-  ( ( W  e. Word  A  /\  W  =/=  (/) )  ->  0  e.  ( 0 ... 1
) )
72a1i 11 . . . 4  |-  ( ( W  e. Word  A  /\  W  =/=  (/) )  ->  1  e.  NN0 )
8 lennncl 11664 . . . . 5  |-  ( ( W  e. Word  A  /\  W  =/=  (/) )  ->  ( # `
 W )  e.  NN )
98nnnn0d 10207 . . . 4  |-  ( ( W  e. Word  A  /\  W  =/=  (/) )  ->  ( # `
 W )  e. 
NN0 )
10 hashge1 11591 . . . 4  |-  ( ( W  e. Word  A  /\  W  =/=  (/) )  ->  1  <_  ( # `  W
) )
11 elfz2nn0 11015 . . . 4  |-  ( 1  e.  ( 0 ... ( # `  W
) )  <->  ( 1  e.  NN0  /\  ( # `
 W )  e. 
NN0  /\  1  <_  (
# `  W )
) )
127, 9, 10, 11syl3anbrc 1138 . . 3  |-  ( ( W  e. Word  A  /\  W  =/=  (/) )  ->  1  e.  ( 0 ... ( # `
 W ) ) )
13 eluzfz2 10998 . . . . 5  |-  ( (
# `  W )  e.  ( ZZ>= `  0 )  ->  ( # `  W
)  e.  ( 0 ... ( # `  W
) ) )
1413, 4eleq2s 2480 . . . 4  |-  ( (
# `  W )  e.  NN0  ->  ( # `  W
)  e.  ( 0 ... ( # `  W
) ) )
159, 14syl 16 . . 3  |-  ( ( W  e. Word  A  /\  W  =/=  (/) )  ->  ( # `
 W )  e.  ( 0 ... ( # `
 W ) ) )
16 ccatswrd 11701 . . 3  |-  ( ( W  e. Word  A  /\  ( 0  e.  ( 0 ... 1 )  /\  1  e.  ( 0 ... ( # `  W ) )  /\  ( # `  W )  e.  ( 0 ... ( # `  W
) ) ) )  ->  ( ( W substr  <. 0 ,  1 >.
) concat  ( W substr  <. 1 ,  ( # `  W
) >. ) )  =  ( W substr  <. 0 ,  ( # `  W
) >. ) )
171, 6, 12, 15, 16syl13anc 1186 . 2  |-  ( ( W  e. Word  A  /\  W  =/=  (/) )  ->  (
( W substr  <. 0 ,  1 >. ) concat  ( W substr  <.
1 ,  ( # `  W ) >. )
)  =  ( W substr  <. 0 ,  ( # `  W ) >. )
)
18 0p1e1 10026 . . . . . 6  |-  ( 0  +  1 )  =  1
1918opeq2i 3931 . . . . 5  |-  <. 0 ,  ( 0  +  1 ) >.  =  <. 0 ,  1 >.
2019oveq2i 6032 . . . 4  |-  ( W substr  <. 0 ,  ( 0  +  1 ) >.
)  =  ( W substr  <. 0 ,  1 >.
)
21 0nn0 10169 . . . . . . 7  |-  0  e.  NN0
2221a1i 11 . . . . . 6  |-  ( ( W  e. Word  A  /\  W  =/=  (/) )  ->  0  e.  NN0 )
23 hashgt0 11590 . . . . . 6  |-  ( ( W  e. Word  A  /\  W  =/=  (/) )  ->  0  <  ( # `  W
) )
24 elfzo0 11102 . . . . . 6  |-  ( 0  e.  ( 0..^ (
# `  W )
)  <->  ( 0  e. 
NN0  /\  ( # `  W
)  e.  NN  /\  0  <  ( # `  W
) ) )
2522, 8, 23, 24syl3anbrc 1138 . . . . 5  |-  ( ( W  e. Word  A  /\  W  =/=  (/) )  ->  0  e.  ( 0..^ ( # `  W ) ) )
26 swrds1 11715 . . . . 5  |-  ( ( W  e. Word  A  /\  0  e.  ( 0..^ ( # `  W
) ) )  -> 
( W substr  <. 0 ,  ( 0  +  1 ) >. )  =  <" ( W `  0
) "> )
2725, 26syldan 457 . . . 4  |-  ( ( W  e. Word  A  /\  W  =/=  (/) )  ->  ( W substr  <. 0 ,  ( 0  +  1 )
>. )  =  <" ( W `  0
) "> )
2820, 27syl5eqr 2434 . . 3  |-  ( ( W  e. Word  A  /\  W  =/=  (/) )  ->  ( W substr  <. 0 ,  1
>. )  =  <" ( W `  0
) "> )
2928oveq1d 6036 . 2  |-  ( ( W  e. Word  A  /\  W  =/=  (/) )  ->  (
( W substr  <. 0 ,  1 >. ) concat  ( W substr  <.
1 ,  ( # `  W ) >. )
)  =  ( <" ( W ` 
0 ) "> concat  ( W substr  <. 1 ,  (
# `  W ) >. ) ) )
30 swrdid 11700 . . 3  |-  ( W  e. Word  A  ->  ( W substr  <. 0 ,  (
# `  W ) >. )  =  W )
3130adantr 452 . 2  |-  ( ( W  e. Word  A  /\  W  =/=  (/) )  ->  ( W substr  <. 0 ,  (
# `  W ) >. )  =  W )
3217, 29, 313eqtr3rd 2429 1  |-  ( ( W  e. Word  A  /\  W  =/=  (/) )  ->  W  =  ( <" ( W `  0 ) "> concat  ( W substr  <. 1 ,  ( # `  W
) >. ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2551   (/)c0 3572   <.cop 3761   class class class wbr 4154   ` cfv 5395  (class class class)co 6021   0cc0 8924   1c1 8925    + caddc 8927    < clt 9054    <_ cle 9055   NNcn 9933   NN0cn0 10154   ZZ>=cuz 10421   ...cfz 10976  ..^cfzo 11066   #chash 11546  Word cword 11645   concat cconcat 11646   <"cs1 11647   substr csubstr 11648
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-1o 6661  df-oadd 6665  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-card 7760  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-nn 9934  df-n0 10155  df-z 10216  df-uz 10422  df-fz 10977  df-fzo 11067  df-hash 11547  df-word 11651  df-concat 11652  df-s1 11653  df-substr 11654
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