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Theorem wrdeqs1cat 12650
Description: Decompose a nonempty 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 457 . . 3  |-  ( ( W  e. Word  A  /\  W  =/=  (/) )  ->  W  e. Word  A )
2 1nn0 10800 . . . 4  |-  1  e.  NN0
3 eluzfz1 11682 . . . . 5  |-  ( 1  e.  ( ZZ>= `  0
)  ->  0  e.  ( 0 ... 1
) )
4 nn0uz 11105 . . . . 5  |-  NN0  =  ( ZZ>= `  0 )
53, 4eleq2s 2568 . . . 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 12516 . . . . 5  |-  ( ( W  e. Word  A  /\  W  =/=  (/) )  ->  ( # `
 W )  e.  NN )
98nnnn0d 10841 . . . 4  |-  ( ( W  e. Word  A  /\  W  =/=  (/) )  ->  ( # `
 W )  e. 
NN0 )
10 hashge1 12412 . . . 4  |-  ( ( W  e. Word  A  /\  W  =/=  (/) )  ->  1  <_  ( # `  W
) )
11 elfz2nn0 11757 . . . 4  |-  ( 1  e.  ( 0 ... ( # `  W
) )  <->  ( 1  e.  NN0  /\  ( # `
 W )  e. 
NN0  /\  1  <_  (
# `  W )
) )
127, 9, 10, 11syl3anbrc 1175 . . 3  |-  ( ( W  e. Word  A  /\  W  =/=  (/) )  ->  1  e.  ( 0 ... ( # `
 W ) ) )
13 eluzfz2 11683 . . . . 5  |-  ( (
# `  W )  e.  ( ZZ>= `  0 )  ->  ( # `  W
)  e.  ( 0 ... ( # `  W
) ) )
1413, 4eleq2s 2568 . . . 4  |-  ( (
# `  W )  e.  NN0  ->  ( # `  W
)  e.  ( 0 ... ( # `  W
) ) )
159, 14syl 16 . . 3  |-  ( ( W  e. Word  A  /\  W  =/=  (/) )  ->  ( # `
 W )  e.  ( 0 ... ( # `
 W ) ) )
16 ccatswrd 12631 . . 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 1225 . 2  |-  ( ( W  e. Word  A  /\  W  =/=  (/) )  ->  (
( W substr  <. 0 ,  1 >. ) concat  ( W substr  <.
1 ,  ( # `  W ) >. )
)  =  ( W substr  <. 0 ,  ( # `  W ) >. )
)
18 0p1e1 10636 . . . . . 6  |-  ( 0  +  1 )  =  1
1918opeq2i 4210 . . . . 5  |-  <. 0 ,  ( 0  +  1 ) >.  =  <. 0 ,  1 >.
2019oveq2i 6286 . . . 4  |-  ( W substr  <. 0 ,  ( 0  +  1 ) >.
)  =  ( W substr  <. 0 ,  1 >.
)
21 0nn0 10799 . . . . . . 7  |-  0  e.  NN0
2221a1i 11 . . . . . 6  |-  ( ( W  e. Word  A  /\  W  =/=  (/) )  ->  0  e.  NN0 )
23 hashgt0 12411 . . . . . 6  |-  ( ( W  e. Word  A  /\  W  =/=  (/) )  ->  0  <  ( # `  W
) )
24 elfzo0 11820 . . . . . 6  |-  ( 0  e.  ( 0..^ (
# `  W )
)  <->  ( 0  e. 
NN0  /\  ( # `  W
)  e.  NN  /\  0  <  ( # `  W
) ) )
2522, 8, 23, 24syl3anbrc 1175 . . . . 5  |-  ( ( W  e. Word  A  /\  W  =/=  (/) )  ->  0  e.  ( 0..^ ( # `  W ) ) )
26 swrds1 12626 . . . . 5  |-  ( ( W  e. Word  A  /\  0  e.  ( 0..^ ( # `  W
) ) )  -> 
( W substr  <. 0 ,  ( 0  +  1 ) >. )  =  <" ( W `  0
) "> )
2725, 26syldan 470 . . . 4  |-  ( ( W  e. Word  A  /\  W  =/=  (/) )  ->  ( W substr  <. 0 ,  ( 0  +  1 )
>. )  =  <" ( W `  0
) "> )
2820, 27syl5eqr 2515 . . 3  |-  ( ( W  e. Word  A  /\  W  =/=  (/) )  ->  ( W substr  <. 0 ,  1
>. )  =  <" ( W `  0
) "> )
2928oveq1d 6290 . 2  |-  ( ( W  e. Word  A  /\  W  =/=  (/) )  ->  (
( W substr  <. 0 ,  1 >. ) concat  ( W substr  <.
1 ,  ( # `  W ) >. )
)  =  ( <" ( W ` 
0 ) "> concat  ( W substr  <. 1 ,  (
# `  W ) >. ) ) )
30 swrdid 12602 . . 3  |-  ( W  e. Word  A  ->  ( W substr  <. 0 ,  (
# `  W ) >. )  =  W )
3130adantr 465 . 2  |-  ( ( W  e. Word  A  /\  W  =/=  (/) )  ->  ( W substr  <. 0 ,  (
# `  W ) >. )  =  W )
3217, 29, 313eqtr3rd 2510 1  |-  ( ( W  e. Word  A  /\  W  =/=  (/) )  ->  W  =  ( <" ( W `  0 ) "> concat  ( W substr  <. 1 ,  ( # `  W
) >. ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762    =/= wne 2655   (/)c0 3778   <.cop 4026   class class class wbr 4440   ` cfv 5579  (class class class)co 6275   0cc0 9481   1c1 9482    + caddc 9484    < clt 9617    <_ cle 9618   NNcn 10525   NN0cn0 10784   ZZ>=cuz 11071   ...cfz 11661  ..^cfzo 11781   #chash 12360  Word cword 12487   concat cconcat 12489   <"cs1 12490   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-concat 12497  df-s1 12498  df-substr 12499
This theorem is referenced by: (None)
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