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Theorem cshwmodn 13541
Description: Cyclically shifting a word is invariant regarding modulo the word's length. (Contributed by AV, 26-Oct-2018.)
Assertion
Ref Expression
cshwmodn |- ( ( W e. Word V /\ N e. ZZ ) -> ( W cyclShift N ) = ( W cyclShift ( N mod ( # ` W ) ) ) )
Proof of Theorem cshwmodn
StepHypRef Expression
1 0csh0 13539 . . . 4 |- ( (/) cyclShift N ) = (/)
2 oveq1 6657 . . . 4 |- ( W = (/) -> ( W cyclShift N ) = ( (/) cyclShift N ) )
3 oveq1 6657 . . . . 5 |- ( W = (/) -> ( W cyclShift ( N mod ( # ` W ) ) ) = ( (/) cyclShift ( N mod ( # ` W ) ) ) )
4 0csh0 13539 . . . . 5 |- ( (/) cyclShift ( N mod ( # ` W ) ) ) = (/)
53, 4syl6eq 2672 . . . 4 |- ( W = (/) -> ( W cyclShift ( N mod ( # ` W ) ) ) = (/) )
61, 2, 53eqtr4a 2682 . . 3 |- ( W = (/) -> ( W cyclShift N ) = ( W cyclShift ( N mod ( # ` W ) ) ) )
76a1d 25 . 2 |- ( W = (/) -> ( ( W e. Word V /\ N e. ZZ ) -> ( W cyclShift N ) = ( W cyclShift ( N mod ( # ` W ) ) ) ) )
8 lennncl 13325 . . . . . . 7 |- ( ( W e. Word V /\ W =/= (/) ) -> ( # ` W ) e. NN )
9 zre 11381 . . . . . . . . . . . 12 |- ( N e. ZZ -> N e. RR )
10 nnrp 11842 . . . . . . . . . . . 12 |- ( ( # ` W ) e. NN -> ( # ` W ) e. RR+ )
11 modabs2 12704 . . . . . . . . . . . 12 |- ( ( N e. RR /\ ( # ` W ) e. RR+ ) -> ( ( N mod ( # ` W ) ) mod ( # ` W ) ) = ( N mod ( # ` W ) ) )
129, 10, 11syl2anr 495 . . . . . . . . . . 11 |- ( ( ( # ` W ) e. NN /\ N e. ZZ ) -> ( ( N mod ( # ` W ) ) mod ( # ` W ) ) = ( N mod ( # ` W ) ) )
1312opeq1d 4408 . . . . . . . . . 10 |- ( ( ( # ` W ) e. NN /\ N e. ZZ ) -> <. ( ( N mod ( # ` W ) ) mod ( # ` W ) ) , ( # ` W ) >. = <. ( N mod ( # ` W ) ) , ( # ` W ) >. )
1413oveq2d 6666 . . . . . . . . 9 |- ( ( ( # ` W ) e. NN /\ N e. ZZ ) -> ( W substr <. ( ( N mod ( # ` W ) ) mod ( # ` W ) ) , ( # ` W ) >. ) = ( W substr <. ( N mod ( # ` W ) ) , ( # ` W ) >. ) )
1512opeq2d 4409 . . . . . . . . . 10 |- ( ( ( # ` W ) e. NN /\ N e. ZZ ) -> <. 0 , ( ( N mod ( # ` W ) ) mod ( # ` W ) ) >. = <. 0 , ( N mod ( # ` W ) ) >. )
1615oveq2d 6666 . . . . . . . . 9 |- ( ( ( # ` W ) e. NN /\ N e. ZZ ) -> ( W substr <. 0 , ( ( N mod ( # ` W ) ) mod ( # ` W ) ) >. ) = ( W substr <. 0 , ( N mod ( # ` W ) ) >. ) )
1714, 16oveq12d 6668 . . . . . . . 8 |- ( ( ( # ` W ) e. NN /\ N e. ZZ ) -> ( ( W substr <. ( ( N mod ( # ` W ) ) mod ( # ` W ) ) , ( # ` W ) >. ) ++ ( W substr <. 0 , ( ( N mod ( # ` W ) ) mod ( # ` W ) ) >. ) ) = ( ( W substr <. ( N mod ( # ` W ) ) , ( # ` W ) >. ) ++ ( W substr <. 0 , ( N mod ( # ` W ) ) >. ) ) )
1817ex 450 . . . . . . 7 |- ( ( # ` W ) e. NN -> ( N e. ZZ -> ( ( W substr <. ( ( N mod ( # ` W ) ) mod ( # ` W ) ) , ( # ` W ) >. ) ++ ( W substr <. 0 , ( ( N mod ( # ` W ) ) mod ( # ` W ) ) >. ) ) = ( ( W substr <. ( N mod ( # ` W ) ) , ( # ` W ) >. ) ++ ( W substr <. 0 , ( N mod ( # ` W ) ) >. ) ) ) )
198, 18syl 17 . . . . . 6 |- ( ( W e. Word V /\ W =/= (/) ) -> ( N e. ZZ -> ( ( W substr <. ( ( N mod ( # ` W ) ) mod ( # ` W ) ) , ( # ` W ) >. ) ++ ( W substr <. 0 , ( ( N mod ( # ` W ) ) mod ( # ` W ) ) >. ) ) = ( ( W substr <. ( N mod ( # ` W ) ) , ( # ` W ) >. ) ++ ( W substr <. 0 , ( N mod ( # ` W ) ) >. ) ) ) )
2019impancom 456 . . . . 5 |- ( ( W e. Word V /\ N e. ZZ ) -> ( W =/= (/) -> ( ( W substr <. ( ( N mod ( # ` W ) ) mod ( # ` W ) ) , ( # ` W ) >. ) ++ ( W substr <. 0 , ( ( N mod ( # ` W ) ) mod ( # ` W ) ) >. ) ) = ( ( W substr <. ( N mod ( # ` W ) ) , ( # ` W ) >. ) ++ ( W substr <. 0 , ( N mod ( # ` W ) ) >. ) ) ) )
2120impcom 446 . . . 4 |- ( ( W =/= (/) /\ ( W e. Word V /\ N e. ZZ ) ) -> ( ( W substr <. ( ( N mod ( # ` W ) ) mod ( # ` W ) ) , ( # ` W ) >. ) ++ ( W substr <. 0 , ( ( N mod ( # ` W ) ) mod ( # ` W ) ) >. ) ) = ( ( W substr <. ( N mod ( # ` W ) ) , ( # ` W ) >. ) ++ ( W substr <. 0 , ( N mod ( # ` W ) ) >. ) ) )
22 simprl 794 . . . . 5 |- ( ( W =/= (/) /\ ( W e. Word V /\ N e. ZZ ) ) -> W e. Word V )
23 simprr 796 . . . . . . 7 |- ( ( W =/= (/) /\ ( W e. Word V /\ N e. ZZ ) ) -> N e. ZZ )
248ex 450 . . . . . . . . 9 |- ( W e. Word V -> ( W =/= (/) -> ( # ` W ) e. NN ) )
2524adantr 481 . . . . . . . 8 |- ( ( W e. Word V /\ N e. ZZ ) -> ( W =/= (/) -> ( # ` W ) e. NN ) )
2625impcom 446 . . . . . . 7 |- ( ( W =/= (/) /\ ( W e. Word V /\ N e. ZZ ) ) -> ( # ` W ) e. NN )
2723, 26zmodcld 12691 . . . . . 6 |- ( ( W =/= (/) /\ ( W e. Word V /\ N e. ZZ ) ) -> ( N mod ( # ` W ) ) e. NN0 )
2827nn0zd 11480 . . . . 5 |- ( ( W =/= (/) /\ ( W e. Word V /\ N e. ZZ ) ) -> ( N mod ( # ` W ) ) e. ZZ )
29 cshword 13537 . . . . 5 |- ( ( W e. Word V /\ ( N mod ( # ` W ) ) e. ZZ ) -> ( W cyclShift ( N mod ( # ` W ) ) ) = ( ( W substr <. ( ( N mod ( # ` W ) ) mod ( # ` W ) ) , ( # ` W ) >. ) ++ ( W substr <. 0 , ( ( N mod ( # ` W ) ) mod ( # ` W ) ) >. ) ) )
3022, 28, 29syl2anc 693 . . . 4 |- ( ( W =/= (/) /\ ( W e. Word V /\ N e. ZZ ) ) -> ( W cyclShift ( N mod ( # ` W ) ) ) = ( ( W substr <. ( ( N mod ( # ` W ) ) mod ( # ` W ) ) , ( # ` W ) >. ) ++ ( W substr <. 0 , ( ( N mod ( # ` W ) ) mod ( # ` W ) ) >. ) ) )
31 cshword 13537 . . . . 5 |- ( ( W e. Word V /\ N e. ZZ ) -> ( W cyclShift N ) = ( ( W substr <. ( N mod ( # ` W ) ) , ( # ` W ) >. ) ++ ( W substr <. 0 , ( N mod ( # ` W ) ) >. ) ) )
3231adantl 482 . . . 4 |- ( ( W =/= (/) /\ ( W e. Word V /\ N e. ZZ ) ) -> ( W cyclShift N ) = ( ( W substr <. ( N mod ( # ` W ) ) , ( # ` W ) >. ) ++ ( W substr <. 0 , ( N mod ( # ` W ) ) >. ) ) )
3321, 30, 323eqtr4rd 2667 . . 3 |- ( ( W =/= (/) /\ ( W e. Word V /\ N e. ZZ ) ) -> ( W cyclShift N ) = ( W cyclShift ( N mod ( # ` W ) ) ) )
3433ex 450 . 2 |- ( W =/= (/) -> ( ( W e. Word V /\ N e. ZZ ) -> ( W cyclShift N ) = ( W cyclShift ( N mod ( # ` W ) ) ) ) )
357, 34pm2.61ine 2877 1 |- ( ( W e. Word V /\ N e. ZZ ) -> ( W cyclShift N ) = ( W cyclShift ( N mod ( # ` W ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   (/)c0 3915   <.cop 4183   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   NNcn 11020   ZZcz 11377   RR+crp 11832   mod cmo 12668   #chash 13117  Word cword 13291   ++ cconcat 13293   substr csubstr 13295   cyclShift ccsh 13534
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
This theorem is referenced by:  cshwsublen  13542  cshwn  13543
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