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Theorem cshword2 41437
Description: Perform a cyclical shift for a word. (Contributed by AV, 11-May-2020.)
Assertion
Ref Expression
cshword2 |- ( ( W e. Word V /\ N e. ZZ ) -> ( W cyclShift N ) = ( ( W substr <. ( N mod ( # ` W ) ) , ( # ` W ) >. ) ++ ( W prefix ( N mod ( # ` W ) ) ) ) )
Proof of Theorem cshword2
Dummy variables l w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iswrd 13307 . . . . 5 |- ( W e. Word V <-> E. l e. NN0 W : ( 0..^ l ) --> V )
2 ffn 6045 . . . . . 6 |- ( W : ( 0..^ l ) --> V -> W Fn ( 0..^ l ) )
32reximi 3011 . . . . 5 |- ( E. l e. NN0 W : ( 0..^ l ) --> V -> E. l e. NN0 W Fn ( 0..^ l ) )
41, 3sylbi 207 . . . 4 |- ( W e. Word V -> E. l e. NN0 W Fn ( 0..^ l ) )
5 fneq1 5979 . . . . . 6 |- ( w = W -> ( w Fn ( 0..^ l ) <-> W Fn ( 0..^ l ) ) )
65rexbidv 3052 . . . . 5 |- ( w = W -> ( E. l e. NN0 w Fn ( 0..^ l ) <-> E. l e. NN0 W Fn ( 0..^ l ) ) )
76elabg 3351 . . . 4 |- ( W e. Word V -> ( W e. { w | E. l e. NN0 w Fn ( 0..^ l ) } <-> E. l e. NN0 W Fn ( 0..^ l ) ) )
84, 7mpbird 247 . . 3 |- ( W e. Word V -> W e. { w | E. l e. NN0 w Fn ( 0..^ l ) } )
9 cshfn 13536 . . 3 |- ( ( W e. { w | E. l e. NN0 w Fn ( 0..^ l ) } /\ N e. ZZ ) -> ( W cyclShift N ) = if ( W = (/) , (/) , ( ( W substr <. ( N mod ( # ` W ) ) , ( # ` W ) >. ) ++ ( W substr <. 0 , ( N mod ( # ` W ) ) >. ) ) ) )
108, 9sylan 488 . 2 |- ( ( W e. Word V /\ N e. ZZ ) -> ( W cyclShift N ) = if ( W = (/) , (/) , ( ( W substr <. ( N mod ( # ` W ) ) , ( # ` W ) >. ) ++ ( W substr <. 0 , ( N mod ( # ` W ) ) >. ) ) ) )
11 iftrue 4092 . . . . 5 |- ( W = (/) -> if ( W = (/) , (/) , ( ( W substr <. ( N mod ( # ` W ) ) , ( # ` W ) >. ) ++ ( W substr <. 0 , ( N mod ( # ` W ) ) >. ) ) ) = (/) )
1211adantr 481 . . . 4 |- ( ( W = (/) /\ ( W e. Word V /\ N e. ZZ ) ) -> if ( W = (/) , (/) , ( ( W substr <. ( N mod ( # ` W ) ) , ( # ` W ) >. ) ++ ( W substr <. 0 , ( N mod ( # ` W ) ) >. ) ) ) = (/) )
13 oveq1 6657 . . . . . . . 8 |- ( W = (/) -> ( W substr <. ( N mod ( # ` W ) ) , ( # ` W ) >. ) = ( (/) substr <. ( N mod ( # ` W ) ) , ( # ` W ) >. ) )
14 swrd0 13434 . . . . . . . 8 |- ( (/) substr <. ( N mod ( # ` W ) ) , ( # ` W ) >. ) = (/)
1513, 14syl6eq 2672 . . . . . . 7 |- ( W = (/) -> ( W substr <. ( N mod ( # ` W ) ) , ( # ` W ) >. ) = (/) )
16 oveq1 6657 . . . . . . . 8 |- ( W = (/) -> ( W prefix ( N mod ( # ` W ) ) ) = ( (/) prefix ( N mod ( # ` W ) ) ) )
17 pfx0 41385 . . . . . . . 8 |- ( (/) prefix ( N mod ( # ` W ) ) ) = (/)
1816, 17syl6eq 2672 . . . . . . 7 |- ( W = (/) -> ( W prefix ( N mod ( # ` W ) ) ) = (/) )
1915, 18oveq12d 6668 . . . . . 6 |- ( W = (/) -> ( ( W substr <. ( N mod ( # ` W ) ) , ( # ` W ) >. ) ++ ( W prefix ( N mod ( # ` W ) ) ) ) = ( (/) ++ (/) ) )
2019adantr 481 . . . . 5 |- ( ( W = (/) /\ ( W e. Word V /\ N e. ZZ ) ) -> ( ( W substr <. ( N mod ( # ` W ) ) , ( # ` W ) >. ) ++ ( W prefix ( N mod ( # ` W ) ) ) ) = ( (/) ++ (/) ) )
21 wrd0 13330 . . . . . 6 |- (/) e. Word _V
22 ccatrid 13370 . . . . . 6 |- ( (/) e. Word _V -> ( (/) ++ (/) ) = (/) )
2321, 22ax-mp 5 . . . . 5 |- ( (/) ++ (/) ) = (/)
2420, 23syl6req 2673 . . . 4 |- ( ( W = (/) /\ ( W e. Word V /\ N e. ZZ ) ) -> (/) = ( ( W substr <. ( N mod ( # ` W ) ) , ( # ` W ) >. ) ++ ( W prefix ( N mod ( # ` W ) ) ) ) )
2512, 24eqtrd 2656 . . 3 |- ( ( W = (/) /\ ( W e. Word V /\ N e. ZZ ) ) -> if ( W = (/) , (/) , ( ( W substr <. ( N mod ( # ` W ) ) , ( # ` W ) >. ) ++ ( W substr <. 0 , ( N mod ( # ` W ) ) >. ) ) ) = ( ( W substr <. ( N mod ( # ` W ) ) , ( # ` W ) >. ) ++ ( W prefix ( N mod ( # ` W ) ) ) ) )
26 iffalse 4095 . . . . 5 |- ( -. W = (/) -> if ( W = (/) , (/) , ( ( W substr <. ( N mod ( # ` W ) ) , ( # ` W ) >. ) ++ ( W substr <. 0 , ( N mod ( # ` W ) ) >. ) ) ) = ( ( W substr <. ( N mod ( # ` W ) ) , ( # ` W ) >. ) ++ ( W substr <. 0 , ( N mod ( # ` W ) ) >. ) ) )
2726adantr 481 . . . 4 |- ( ( -. W = (/) /\ ( W e. Word V /\ N e. ZZ ) ) -> if ( W = (/) , (/) , ( ( W substr <. ( N mod ( # ` W ) ) , ( # ` W ) >. ) ++ ( W substr <. 0 , ( N mod ( # ` W ) ) >. ) ) ) = ( ( W substr <. ( N mod ( # ` W ) ) , ( # ` W ) >. ) ++ ( W substr <. 0 , ( N mod ( # ` W ) ) >. ) ) )
28 simprl 794 . . . . . . 7 |- ( ( -. W = (/) /\ ( W e. Word V /\ N e. ZZ ) ) -> W e. Word V )
29 simprr 796 . . . . . . . 8 |- ( ( -. W = (/) /\ ( W e. Word V /\ N e. ZZ ) ) -> N e. ZZ )
30 df-ne 2795 . . . . . . . . . 10 |- ( W =/= (/) <-> -. W = (/) )
31 lennncl 13325 . . . . . . . . . . . 12 |- ( ( W e. Word V /\ W =/= (/) ) -> ( # ` W ) e. NN )
3231ex 450 . . . . . . . . . . 11 |- ( W e. Word V -> ( W =/= (/) -> ( # ` W ) e. NN ) )
3332adantr 481 . . . . . . . . . 10 |- ( ( W e. Word V /\ N e. ZZ ) -> ( W =/= (/) -> ( # ` W ) e. NN ) )
3430, 33syl5bir 233 . . . . . . . . 9 |- ( ( W e. Word V /\ N e. ZZ ) -> ( -. W = (/) -> ( # ` W ) e. NN ) )
3534impcom 446 . . . . . . . 8 |- ( ( -. W = (/) /\ ( W e. Word V /\ N e. ZZ ) ) -> ( # ` W ) e. NN )
3629, 35zmodcld 12691 . . . . . . 7 |- ( ( -. W = (/) /\ ( W e. Word V /\ N e. ZZ ) ) -> ( N mod ( # ` W ) ) e. NN0 )
37 pfxval 41383 . . . . . . 7 |- ( ( W e. Word V /\ ( N mod ( # ` W ) ) e. NN0 ) -> ( W prefix ( N mod ( # ` W ) ) ) = ( W substr <. 0 , ( N mod ( # ` W ) ) >. ) )
3828, 36, 37syl2anc 693 . . . . . 6 |- ( ( -. W = (/) /\ ( W e. Word V /\ N e. ZZ ) ) -> ( W prefix ( N mod ( # ` W ) ) ) = ( W substr <. 0 , ( N mod ( # ` W ) ) >. ) )
3938eqcomd 2628 . . . . 5 |- ( ( -. W = (/) /\ ( W e. Word V /\ N e. ZZ ) ) -> ( W substr <. 0 , ( N mod ( # ` W ) ) >. ) = ( W prefix ( N mod ( # ` W ) ) ) )
4039oveq2d 6666 . . . 4 |- ( ( -. W = (/) /\ ( W e. Word V /\ N e. ZZ ) ) -> ( ( W substr <. ( N mod ( # ` W ) ) , ( # ` W ) >. ) ++ ( W substr <. 0 , ( N mod ( # ` W ) ) >. ) ) = ( ( W substr <. ( N mod ( # ` W ) ) , ( # ` W ) >. ) ++ ( W prefix ( N mod ( # ` W ) ) ) ) )
4127, 40eqtrd 2656 . . 3 |- ( ( -. W = (/) /\ ( W e. Word V /\ N e. ZZ ) ) -> if ( W = (/) , (/) , ( ( W substr <. ( N mod ( # ` W ) ) , ( # ` W ) >. ) ++ ( W substr <. 0 , ( N mod ( # ` W ) ) >. ) ) ) = ( ( W substr <. ( N mod ( # ` W ) ) , ( # ` W ) >. ) ++ ( W prefix ( N mod ( # ` W ) ) ) ) )
4225, 41pm2.61ian 831 . 2 |- ( ( W e. Word V /\ N e. ZZ ) -> if ( W = (/) , (/) , ( ( W substr <. ( N mod ( # ` W ) ) , ( # ` W ) >. ) ++ ( W substr <. 0 , ( N mod ( # ` W ) ) >. ) ) ) = ( ( W substr <. ( N mod ( # ` W ) ) , ( # ` W ) >. ) ++ ( W prefix ( N mod ( # ` W ) ) ) ) )
4310, 42eqtrd 2656 1 |- ( ( W e. Word V /\ N e. ZZ ) -> ( W cyclShift N ) = ( ( W substr <. ( N mod ( # ` W ) ) , ( # ` W ) >. ) ++ ( W prefix ( N mod ( # ` W ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   =/= wne 2794   E.wrex 2913   _Vcvv 3200   (/)c0 3915   ifcif 4086   <.cop 4183   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   0cc0 9936   NNcn 11020   NN0cn0 11292   ZZcz 11377  ..^cfzo 12465   mod cmo 12668   #chash 13117  Word cword 13291   ++ cconcat 13293   substr csubstr 13295   cyclShift ccsh 13534   prefix cpfx 41381
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  ax-pre-sup 10014
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-rmo 2920  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-sup 8348  df-inf 8349  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-div 10685  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-hash 13118  df-word 13299  df-concat 13301  df-substr 13303  df-csh 13535  df-pfx 41382
This theorem is referenced by: (None)
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