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Theorem wrdeqs1cat 11477
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 443 . . 3  |-  ( ( W  e. Word  A  /\  W  =/=  (/) )  ->  W  e. Word  A )
2 1nn0 9983 . . . 4  |-  1  e.  NN0
3 eluzfz1 10805 . . . . 5  |-  ( 1  e.  ( ZZ>= `  0
)  ->  0  e.  ( 0 ... 1
) )
4 nn0uz 10264 . . . . 5  |-  NN0  =  ( ZZ>= `  0 )
53, 4eleq2s 2377 . . . 4  |-  ( 1  e.  NN0  ->  0  e.  ( 0 ... 1
) )
62, 5mp1i 11 . . 3  |-  ( ( W  e. Word  A  /\  W  =/=  (/) )  ->  0  e.  ( 0 ... 1
) )
72a1i 10 . . . 4  |-  ( ( W  e. Word  A  /\  W  =/=  (/) )  ->  1  e.  NN0 )
8 lennncl 11424 . . . . 5  |-  ( ( W  e. Word  A  /\  W  =/=  (/) )  ->  ( # `
 W )  e.  NN )
98nnnn0d 10020 . . . 4  |-  ( ( W  e. Word  A  /\  W  =/=  (/) )  ->  ( # `
 W )  e. 
NN0 )
108nnge1d 9790 . . . 4  |-  ( ( W  e. Word  A  /\  W  =/=  (/) )  ->  1  <_  ( # `  W
) )
11 elfz2nn0 10823 . . . 4  |-  ( 1  e.  ( 0 ... ( # `  W
) )  <->  ( 1  e.  NN0  /\  ( # `
 W )  e. 
NN0  /\  1  <_  (
# `  W )
) )
127, 9, 10, 11syl3anbrc 1136 . . 3  |-  ( ( W  e. Word  A  /\  W  =/=  (/) )  ->  1  e.  ( 0 ... ( # `
 W ) ) )
13 eluzfz2 10806 . . . . 5  |-  ( (
# `  W )  e.  ( ZZ>= `  0 )  ->  ( # `  W
)  e.  ( 0 ... ( # `  W
) ) )
1413, 4eleq2s 2377 . . . 4  |-  ( (
# `  W )  e.  NN0  ->  ( # `  W
)  e.  ( 0 ... ( # `  W
) ) )
159, 14syl 15 . . 3  |-  ( ( W  e. Word  A  /\  W  =/=  (/) )  ->  ( # `
 W )  e.  ( 0 ... ( # `
 W ) ) )
16 ccatswrd 11461 . . 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 1184 . 2  |-  ( ( W  e. Word  A  /\  W  =/=  (/) )  ->  (
( W substr  <. 0 ,  1 >. ) concat  ( W substr  <.
1 ,  ( # `  W ) >. )
)  =  ( W substr  <. 0 ,  ( # `  W ) >. )
)
18 0p1e1 9841 . . . . . 6  |-  ( 0  +  1 )  =  1
1918opeq2i 3802 . . . . 5  |-  <. 0 ,  ( 0  +  1 ) >.  =  <. 0 ,  1 >.
2019oveq2i 5871 . . . 4  |-  ( W substr  <. 0 ,  ( 0  +  1 ) >.
)  =  ( W substr  <. 0 ,  1 >.
)
21 0nn0 9982 . . . . . . 7  |-  0  e.  NN0
2221a1i 10 . . . . . 6  |-  ( ( W  e. Word  A  /\  W  =/=  (/) )  ->  0  e.  NN0 )
238nngt0d 9791 . . . . . 6  |-  ( ( W  e. Word  A  /\  W  =/=  (/) )  ->  0  <  ( # `  W
) )
24 elfzo0 10906 . . . . . 6  |-  ( 0  e.  ( 0..^ (
# `  W )
)  <->  ( 0  e. 
NN0  /\  ( # `  W
)  e.  NN  /\  0  <  ( # `  W
) ) )
2522, 8, 23, 24syl3anbrc 1136 . . . . 5  |-  ( ( W  e. Word  A  /\  W  =/=  (/) )  ->  0  e.  ( 0..^ ( # `  W ) ) )
26 swrds1 11475 . . . . 5  |-  ( ( W  e. Word  A  /\  0  e.  ( 0..^ ( # `  W
) ) )  -> 
( W substr  <. 0 ,  ( 0  +  1 ) >. )  =  <" ( W `  0
) "> )
2725, 26syldan 456 . . . 4  |-  ( ( W  e. Word  A  /\  W  =/=  (/) )  ->  ( W substr  <. 0 ,  ( 0  +  1 )
>. )  =  <" ( W `  0
) "> )
2820, 27syl5eqr 2331 . . 3  |-  ( ( W  e. Word  A  /\  W  =/=  (/) )  ->  ( W substr  <. 0 ,  1
>. )  =  <" ( W `  0
) "> )
2928oveq1d 5875 . 2  |-  ( ( W  e. Word  A  /\  W  =/=  (/) )  ->  (
( W substr  <. 0 ,  1 >. ) concat  ( W substr  <.
1 ,  ( # `  W ) >. )
)  =  ( <" ( W ` 
0 ) "> concat  ( W substr  <. 1 ,  (
# `  W ) >. ) ) )
30 swrdid 11460 . . 3  |-  ( W  e. Word  A  ->  ( W substr  <. 0 ,  (
# `  W ) >. )  =  W )
3130adantr 451 . 2  |-  ( ( W  e. Word  A  /\  W  =/=  (/) )  ->  ( W substr  <. 0 ,  (
# `  W ) >. )  =  W )
3217, 29, 313eqtr3rd 2326 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 358    = wceq 1625    e. wcel 1686    =/= wne 2448   (/)c0 3457   <.cop 3645   class class class wbr 4025   ` cfv 5257  (class class class)co 5860   0cc0 8739   1c1 8740    + caddc 8742    < clt 8869    <_ cle 8870   NNcn 9748   NN0cn0 9967   ZZ>=cuz 10232   ...cfz 10784  ..^cfzo 10872   #chash 11339  Word cword 11405   concat cconcat 11406   <"cs1 11407   substr csubstr 11408
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-1st 6124  df-2nd 6125  df-riota 6306  df-recs 6390  df-rdg 6425  df-1o 6481  df-oadd 6485  df-er 6662  df-en 6866  df-dom 6867  df-sdom 6868  df-fin 6869  df-card 7574  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-nn 9749  df-n0 9968  df-z 10027  df-uz 10233  df-fz 10785  df-fzo 10873  df-hash 11340  df-word 11411  df-concat 11412  df-s1 11413  df-substr 11414
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