MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  wrdeqs1cat Structured version   Unicode version

Theorem wrdeqs1cat 11781
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 10229 . . . 4  |-  1  e.  NN0
3 eluzfz1 11056 . . . . 5  |-  ( 1  e.  ( ZZ>= `  0
)  ->  0  e.  ( 0 ... 1
) )
4 nn0uz 10512 . . . . 5  |-  NN0  =  ( ZZ>= `  0 )
53, 4eleq2s 2527 . . . 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 11728 . . . . 5  |-  ( ( W  e. Word  A  /\  W  =/=  (/) )  ->  ( # `
 W )  e.  NN )
98nnnn0d 10266 . . . 4  |-  ( ( W  e. Word  A  /\  W  =/=  (/) )  ->  ( # `
 W )  e. 
NN0 )
10 hashge1 11655 . . . 4  |-  ( ( W  e. Word  A  /\  W  =/=  (/) )  ->  1  <_  ( # `  W
) )
11 elfz2nn0 11074 . . . 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 11057 . . . . 5  |-  ( (
# `  W )  e.  ( ZZ>= `  0 )  ->  ( # `  W
)  e.  ( 0 ... ( # `  W
) ) )
1413, 4eleq2s 2527 . . . 4  |-  ( (
# `  W )  e.  NN0  ->  ( # `  W
)  e.  ( 0 ... ( # `  W
) ) )
159, 14syl 16 . . 3  |-  ( ( W  e. Word  A  /\  W  =/=  (/) )  ->  ( # `
 W )  e.  ( 0 ... ( # `
 W ) ) )
16 ccatswrd 11765 . . 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 10085 . . . . . 6  |-  ( 0  +  1 )  =  1
1918opeq2i 3980 . . . . 5  |-  <. 0 ,  ( 0  +  1 ) >.  =  <. 0 ,  1 >.
2019oveq2i 6084 . . . 4  |-  ( W substr  <. 0 ,  ( 0  +  1 ) >.
)  =  ( W substr  <. 0 ,  1 >.
)
21 0nn0 10228 . . . . . . 7  |-  0  e.  NN0
2221a1i 11 . . . . . 6  |-  ( ( W  e. Word  A  /\  W  =/=  (/) )  ->  0  e.  NN0 )
23 hashgt0 11654 . . . . . 6  |-  ( ( W  e. Word  A  /\  W  =/=  (/) )  ->  0  <  ( # `  W
) )
24 elfzo0 11163 . . . . . 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 11779 . . . . 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 2481 . . 3  |-  ( ( W  e. Word  A  /\  W  =/=  (/) )  ->  ( W substr  <. 0 ,  1
>. )  =  <" ( W `  0
) "> )
2928oveq1d 6088 . 2  |-  ( ( W  e. Word  A  /\  W  =/=  (/) )  ->  (
( W substr  <. 0 ,  1 >. ) concat  ( W substr  <.
1 ,  ( # `  W ) >. )
)  =  ( <" ( W ` 
0 ) "> concat  ( W substr  <. 1 ,  (
# `  W ) >. ) ) )
30 swrdid 11764 . . 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 2476 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 1652    e. wcel 1725    =/= wne 2598   (/)c0 3620   <.cop 3809   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   0cc0 8982   1c1 8983    + caddc 8985    < clt 9112    <_ cle 9113   NNcn 9992   NN0cn0 10213   ZZ>=cuz 10480   ...cfz 11035  ..^cfzo 11127   #chash 11610  Word cword 11709   concat cconcat 11710   <"cs1 11711   substr csubstr 11712
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-n0 10214  df-z 10275  df-uz 10481  df-fz 11036  df-fzo 11128  df-hash 11611  df-word 11715  df-concat 11716  df-s1 11717  df-substr 11718
  Copyright terms: Public domain W3C validator