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Theorem swrdccatwrd 13468
Description: Reconstruct a nonempty word from its prefix and last symbol. (Contributed by Alexander van der Vekens, 5-Aug-2018.)
Assertion
Ref Expression
swrdccatwrd |- ( ( W e. Word V /\ W =/= (/) ) -> ( ( W substr <. 0 , ( ( # ` W ) - 1 ) >. ) ++ <" ( lastS ` W ) "> ) = W )
Proof of Theorem swrdccatwrd
StepHypRef Expression
1 lennncl 13325 . . . . . 6 |- ( ( W e. Word V /\ W =/= (/) ) -> ( # ` W ) e. NN )
2 fzo0end 12560 . . . . . 6 |- ( ( # ` W ) e. NN -> ( ( # ` W ) - 1 ) e. ( 0..^ ( # ` W ) ) )
31, 2syl 17 . . . . 5 |- ( ( W e. Word V /\ W =/= (/) ) -> ( ( # ` W ) - 1 ) e. ( 0..^ ( # ` W ) ) )
4 swrds1 13451 . . . . 5 |- ( ( W e. Word V /\ ( ( # ` W ) - 1 ) e. ( 0..^ ( # ` W ) ) ) -> ( W substr <. ( ( # ` W ) - 1 ) , ( ( ( # ` W ) - 1 ) + 1 ) >. ) = <" ( W ` ( ( # ` W ) - 1 ) ) "> )
53, 4syldan 487 . . . 4 |- ( ( W e. Word V /\ W =/= (/) ) -> ( W substr <. ( ( # ` W ) - 1 ) , ( ( ( # ` W ) - 1 ) + 1 ) >. ) = <" ( W ` ( ( # ` W ) - 1 ) ) "> )
6 nncn 11028 . . . . . . . . 9 |- ( ( # ` W ) e. NN -> ( # ` W ) e. CC )
7 1cnd 10056 . . . . . . . . 9 |- ( ( # ` W ) e. NN -> 1 e. CC )
86, 7npcand 10396 . . . . . . . 8 |- ( ( # ` W ) e. NN -> ( ( ( # ` W ) - 1 ) + 1 ) = ( # ` W ) )
98eqcomd 2628 . . . . . . 7 |- ( ( # ` W ) e. NN -> ( # ` W ) = ( ( ( # ` W ) - 1 ) + 1 ) )
101, 9syl 17 . . . . . 6 |- ( ( W e. Word V /\ W =/= (/) ) -> ( # ` W ) = ( ( ( # ` W ) - 1 ) + 1 ) )
1110opeq2d 4409 . . . . 5 |- ( ( W e. Word V /\ W =/= (/) ) -> <. ( ( # ` W ) - 1 ) , ( # ` W ) >. = <. ( ( # ` W ) - 1 ) , ( ( ( # ` W ) - 1 ) + 1 ) >. )
1211oveq2d 6666 . . . 4 |- ( ( W e. Word V /\ W =/= (/) ) -> ( W substr <. ( ( # ` W ) - 1 ) , ( # ` W ) >. ) = ( W substr <. ( ( # ` W ) - 1 ) , ( ( ( # ` W ) - 1 ) + 1 ) >. ) )
13 lsw 13351 . . . . . 6 |- ( W e. Word V -> ( lastS ` W ) = ( W ` ( ( # ` W ) - 1 ) ) )
1413adantr 481 . . . . 5 |- ( ( W e. Word V /\ W =/= (/) ) -> ( lastS ` W ) = ( W ` ( ( # ` W ) - 1 ) ) )
1514s1eqd 13381 . . . 4 |- ( ( W e. Word V /\ W =/= (/) ) -> <" ( lastS ` W ) "> = <" ( W ` ( ( # ` W ) - 1 ) ) "> )
165, 12, 153eqtr4rd 2667 . . 3 |- ( ( W e. Word V /\ W =/= (/) ) -> <" ( lastS ` W ) "> = ( W substr <. ( ( # ` W ) - 1 ) , ( # ` W ) >. ) )
1716oveq2d 6666 . 2 |- ( ( W e. Word V /\ W =/= (/) ) -> ( ( W substr <. 0 , ( ( # ` W ) - 1 ) >. ) ++ <" ( lastS ` W ) "> ) = ( ( W substr <. 0 , ( ( # ` W ) - 1 ) >. ) ++ ( W substr <. ( ( # ` W ) - 1 ) , ( # ` W ) >. ) ) )
18 nnm1nn0 11334 . . . . . 6 |- ( ( # ` W ) e. NN -> ( ( # ` W ) - 1 ) e. NN0 )
19 0elfz 12436 . . . . . 6 |- ( ( ( # ` W ) - 1 ) e. NN0 -> 0 e. ( 0 ... ( ( # ` W ) - 1 ) ) )
2018, 19syl 17 . . . . 5 |- ( ( # ` W ) e. NN -> 0 e. ( 0 ... ( ( # ` W ) - 1 ) ) )
21 1nn0 11308 . . . . . . . 8 |- 1 e. NN0
2221a1i 11 . . . . . . 7 |- ( ( # ` W ) e. NN -> 1 e. NN0 )
23 nnnn0 11299 . . . . . . 7 |- ( ( # ` W ) e. NN -> ( # ` W ) e. NN0 )
24 nnge1 11046 . . . . . . 7 |- ( ( # ` W ) e. NN -> 1 <_ ( # ` W ) )
25 elfz2nn0 12431 . . . . . . 7 |- ( 1 e. ( 0 ... ( # ` W ) ) <-> ( 1 e. NN0 /\ ( # ` W ) e. NN0 /\ 1 <_ ( # ` W ) ) )
2622, 23, 24, 25syl3anbrc 1246 . . . . . 6 |- ( ( # ` W ) e. NN -> 1 e. ( 0 ... ( # ` W ) ) )
27 elfz1end 12371 . . . . . . 7 |- ( ( # ` W ) e. NN <-> ( # ` W ) e. ( 1 ... ( # ` W ) ) )
2827biimpi 206 . . . . . 6 |- ( ( # ` W ) e. NN -> ( # ` W ) e. ( 1 ... ( # ` W ) ) )
29 fz0fzdiffz0 12448 . . . . . 6 |- ( ( 1 e. ( 0 ... ( # ` W ) ) /\ ( # ` W ) e. ( 1 ... ( # ` W ) ) ) -> ( ( # ` W ) - 1 ) e. ( 0 ... ( # ` W ) ) )
3026, 28, 29syl2anc 693 . . . . 5 |- ( ( # ` W ) e. NN -> ( ( # ` W ) - 1 ) e. ( 0 ... ( # ` W ) ) )
31 nn0fz0 12437 . . . . . . 7 |- ( ( # ` W ) e. NN0 <-> ( # ` W ) e. ( 0 ... ( # ` W ) ) )
3231biimpi 206 . . . . . 6 |- ( ( # ` W ) e. NN0 -> ( # ` W ) e. ( 0 ... ( # ` W ) ) )
3323, 32syl 17 . . . . 5 |- ( ( # ` W ) e. NN -> ( # ` W ) e. ( 0 ... ( # ` W ) ) )
3420, 30, 333jca 1242 . . . 4 |- ( ( # ` W ) e. NN -> ( 0 e. ( 0 ... ( ( # ` W ) - 1 ) ) /\ ( ( # ` W ) - 1 ) e. ( 0 ... ( # ` W ) ) /\ ( # ` W ) e. ( 0 ... ( # ` W ) ) ) )
351, 34syl 17 . . 3 |- ( ( W e. Word V /\ W =/= (/) ) -> ( 0 e. ( 0 ... ( ( # ` W ) - 1 ) ) /\ ( ( # ` W ) - 1 ) e. ( 0 ... ( # ` W ) ) /\ ( # ` W ) e. ( 0 ... ( # ` W ) ) ) )
36 ccatswrd 13456 . . 3 |- ( ( W e. Word V /\ ( 0 e. ( 0 ... ( ( # ` W ) - 1 ) ) /\ ( ( # ` W ) - 1 ) e. ( 0 ... ( # ` W ) ) /\ ( # ` W ) e. ( 0 ... ( # ` W ) ) ) ) -> ( ( W substr <. 0 , ( ( # ` W ) - 1 ) >. ) ++ ( W substr <. ( ( # ` W ) - 1 ) , ( # ` W ) >. ) ) = ( W substr <. 0 , ( # ` W ) >. ) )
3735, 36syldan 487 . 2 |- ( ( W e. Word V /\ W =/= (/) ) -> ( ( W substr <. 0 , ( ( # ` W ) - 1 ) >. ) ++ ( W substr <. ( ( # ` W ) - 1 ) , ( # ` W ) >. ) ) = ( W substr <. 0 , ( # ` W ) >. ) )
38 swrdid 13428 . . 3 |- ( W e. Word V -> ( W substr <. 0 , ( # ` W ) >. ) = W )
3938adantr 481 . 2 |- ( ( W e. Word V /\ W =/= (/) ) -> ( W substr <. 0 , ( # ` W ) >. ) = W )
4017, 37, 393eqtrd 2660 1 |- ( ( W e. Word V /\ W =/= (/) ) -> ( ( W substr <. 0 , ( ( # ` W ) - 1 ) >. ) ++ <" ( lastS ` W ) "> ) = W )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   (/)c0 3915   <.cop 4183   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   <_ cle 10075   - cmin 10266   NNcn 11020   NN0cn0 11292   ...cfz 12326  ..^cfzo 12465   #chash 13117  Word cword 13291   lastS clsw 13292   ++ cconcat 13293   <"cs1 13294   substr csubstr 13295
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-lsw 13300  df-concat 13301  df-s1 13302  df-substr 13303
This theorem is referenced by:  ccats1swrdeq  13469  wrdind  13476  wrd2ind  13477  psgnunilem5  17914  wwlksnextwrd  26792  iwrdsplit  30449  signsvtn0  30647  signstfveq0  30654
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