Theorem cshwlen 13545

Description: The length of a cyclically shifted word is the same as the length of the original word. (Contributed by AV, 16-May-2018.) (Revised by AV, 20-May-2018.) (Revised by AV, 27-Oct-2018.)

Assertion

Ref	Expression
cshwlen	|- ( ( W e. Word V /\ N e. ZZ ) -> ( # ` ( W cyclShift N ) ) = ( # ` W ) )

Proof of Theorem cshwlen

Step	Hyp	Ref	Expression
1		oveq1 6657	. . . . 5 |- ( W = (/) -> ( W cyclShift N ) = ( (/) cyclShift N ) )
2		0csh0 13539	. . . . . 6 |- ( (/) cyclShift N ) = (/)
3	2	a1i 11	. . . . 5 |- ( W = (/) -> ( (/) cyclShift N ) = (/) )
4		eqcom 2629	. . . . . 6 |- ( W = (/) <-> (/) = W )
5	4	biimpi 206	. . . . 5 |- ( W = (/) -> (/) = W )
6	1, 3, 5	3eqtrd 2660	. . . 4 |- ( W = (/) -> ( W cyclShift N ) = W )
7	6	fveq2d 6195	. . 3 |- ( W = (/) -> ( # ` ( W cyclShift N ) ) = ( # ` W ) )
8	7	a1d 25	. 2 |- ( W = (/) -> ( ( W e. Word V /\ N e. ZZ ) -> ( # ` ( W cyclShift N ) ) = ( # ` W ) ) )
9		cshword 13537	. . . . . 6 |- ( ( W e. Word V /\ N e. ZZ ) -> ( W cyclShift N ) = ( ( W substr <. ( N mod ( # ` W ) ) , ( # ` W ) >. ) ++ ( W substr <. 0 , ( N mod ( # ` W ) ) >. ) ) )
10	9	fveq2d 6195	. . . . 5 |- ( ( W e. Word V /\ N e. ZZ ) -> ( # ` ( W cyclShift N ) ) = ( # ` ( ( W substr <. ( N mod ( # ` W ) ) , ( # ` W ) >. ) ++ ( W substr <. 0 , ( N mod ( # ` W ) ) >. ) ) ) )
11	10	adantr 481	. . . 4 |- ( ( ( W e. Word V /\ N e. ZZ ) /\ W =/= (/) ) -> ( # ` ( W cyclShift N ) ) = ( # ` ( ( W substr <. ( N mod ( # ` W ) ) , ( # ` W ) >. ) ++ ( W substr <. 0 , ( N mod ( # ` W ) ) >. ) ) ) )
12		swrdcl 13419	. . . . . . 7 |- ( W e. Word V -> ( W substr <. ( N mod ( # ` W ) ) , ( # ` W ) >. ) e. Word V )
13		swrdcl 13419	. . . . . . 7 |- ( W e. Word V -> ( W substr <. 0 , ( N mod ( # ` W ) ) >. ) e. Word V )
14		ccatlen 13360	. . . . . . 7 |- ( ( ( W substr <. ( N mod ( # ` W ) ) , ( # ` W ) >. ) e. Word V /\ ( W substr <. 0 , ( N mod ( # ` W ) ) >. ) e. Word V ) -> ( # ` ( ( W substr <. ( N mod ( # ` W ) ) , ( # ` W ) >. ) ++ ( W substr <. 0 , ( N mod ( # ` W ) ) >. ) ) ) = ( ( # ` ( W substr <. ( N mod ( # ` W ) ) , ( # ` W ) >. ) ) + ( # ` ( W substr <. 0 , ( N mod ( # ` W ) ) >. ) ) ) )
15	12, 13, 14	syl2anc 693	. . . . . 6 |- ( W e. Word V -> ( # ` ( ( W substr <. ( N mod ( # ` W ) ) , ( # ` W ) >. ) ++ ( W substr <. 0 , ( N mod ( # ` W ) ) >. ) ) ) = ( ( # ` ( W substr <. ( N mod ( # ` W ) ) , ( # ` W ) >. ) ) + ( # ` ( W substr <. 0 , ( N mod ( # ` W ) ) >. ) ) ) )
16	15	adantr 481	. . . . 5 |- ( ( 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 substr <. 0 , ( N mod ( # ` W ) ) >. ) ) ) )
17	16	adantr 481	. . . 4 |- ( ( ( W e. Word V /\ N e. ZZ ) /\ 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 ) ) >. ) ) ) )
18		lennncl 13325	. . . . . . . . . 10 |- ( ( W e. Word V /\ W =/= (/) ) -> ( # ` W ) e. NN )
19		pm3.21 464	. . . . . . . . . . 11 |- ( ( ( # ` W ) e. NN /\ N e. ZZ ) -> ( W e. Word V -> ( W e. Word V /\ ( ( # ` W ) e. NN /\ N e. ZZ ) ) ) )
20	19	ex 450	. . . . . . . . . 10 |- ( ( # ` W ) e. NN -> ( N e. ZZ -> ( W e. Word V -> ( W e. Word V /\ ( ( # ` W ) e. NN /\ N e. ZZ ) ) ) ) )
21	18, 20	syl 17	. . . . . . . . 9 |- ( ( W e. Word V /\ W =/= (/) ) -> ( N e. ZZ -> ( W e. Word V -> ( W e. Word V /\ ( ( # ` W ) e. NN /\ N e. ZZ ) ) ) ) )
22	21	ex 450	. . . . . . . 8 |- ( W e. Word V -> ( W =/= (/) -> ( N e. ZZ -> ( W e. Word V -> ( W e. Word V /\ ( ( # ` W ) e. NN /\ N e. ZZ ) ) ) ) ) )
23	22	com24 95	. . . . . . 7 |- ( W e. Word V -> ( W e. Word V -> ( N e. ZZ -> ( W =/= (/) -> ( W e. Word V /\ ( ( # ` W ) e. NN /\ N e. ZZ ) ) ) ) ) )
24	23	pm2.43i 52	. . . . . 6 |- ( W e. Word V -> ( N e. ZZ -> ( W =/= (/) -> ( W e. Word V /\ ( ( # ` W ) e. NN /\ N e. ZZ ) ) ) ) )
25	24	imp31 448	. . . . 5 |- ( ( ( W e. Word V /\ N e. ZZ ) /\ W =/= (/) ) -> ( W e. Word V /\ ( ( # ` W ) e. NN /\ N e. ZZ ) ) )
26		simpl 473	. . . . . . . 8 |- ( ( W e. Word V /\ ( ( # ` W ) e. NN /\ N e. ZZ ) ) -> W e. Word V )
27		pm3.22 465	. . . . . . . . . 10 |- ( ( ( # ` W ) e. NN /\ N e. ZZ ) -> ( N e. ZZ /\ ( # ` W ) e. NN ) )
28	27	adantl 482	. . . . . . . . 9 |- ( ( W e. Word V /\ ( ( # ` W ) e. NN /\ N e. ZZ ) ) -> ( N e. ZZ /\ ( # ` W ) e. NN ) )
29		zmodfzp1 12694	. . . . . . . . 9 |- ( ( N e. ZZ /\ ( # ` W ) e. NN ) -> ( N mod ( # ` W ) ) e. ( 0 ... ( # ` W ) ) )
30	28, 29	syl 17	. . . . . . . 8 |- ( ( W e. Word V /\ ( ( # ` W ) e. NN /\ N e. ZZ ) ) -> ( N mod ( # ` W ) ) e. ( 0 ... ( # ` W ) ) )
31		lencl 13324	. . . . . . . . . 10 |- ( W e. Word V -> ( # ` W ) e. NN0 )
32		nn0fz0 12437	. . . . . . . . . 10 |- ( ( # ` W ) e. NN0 <-> ( # ` W ) e. ( 0 ... ( # ` W ) ) )
33	31, 32	sylib 208	. . . . . . . . 9 |- ( W e. Word V -> ( # ` W ) e. ( 0 ... ( # ` W ) ) )
34	33	adantr 481	. . . . . . . 8 |- ( ( W e. Word V /\ ( ( # ` W ) e. NN /\ N e. ZZ ) ) -> ( # ` W ) e. ( 0 ... ( # ` W ) ) )
35		swrdlen 13423	. . . . . . . 8 |- ( ( W e. Word V /\ ( N mod ( # ` W ) ) e. ( 0 ... ( # ` W ) ) /\ ( # ` W ) e. ( 0 ... ( # ` W ) ) ) -> ( # ` ( W substr <. ( N mod ( # ` W ) ) , ( # ` W ) >. ) ) = ( ( # ` W ) - ( N mod ( # ` W ) ) ) )
36	26, 30, 34, 35	syl3anc 1326	. . . . . . 7 |- ( ( W e. Word V /\ ( ( # ` W ) e. NN /\ N e. ZZ ) ) -> ( # ` ( W substr <. ( N mod ( # ` W ) ) , ( # ` W ) >. ) ) = ( ( # ` W ) - ( N mod ( # ` W ) ) ) )
37		zmodcl 12690	. . . . . . . . . . 11 |- ( ( N e. ZZ /\ ( # ` W ) e. NN ) -> ( N mod ( # ` W ) ) e. NN0 )
38	37	ancoms 469	. . . . . . . . . 10 |- ( ( ( # ` W ) e. NN /\ N e. ZZ ) -> ( N mod ( # ` W ) ) e. NN0 )
39	38	adantl 482	. . . . . . . . 9 |- ( ( W e. Word V /\ ( ( # ` W ) e. NN /\ N e. ZZ ) ) -> ( N mod ( # ` W ) ) e. NN0 )
40		0elfz 12436	. . . . . . . . 9 |- ( ( N mod ( # ` W ) ) e. NN0 -> 0 e. ( 0 ... ( N mod ( # ` W ) ) ) )
41	39, 40	syl 17	. . . . . . . 8 |- ( ( W e. Word V /\ ( ( # ` W ) e. NN /\ N e. ZZ ) ) -> 0 e. ( 0 ... ( N mod ( # ` W ) ) ) )
42		swrdlen 13423	. . . . . . . 8 |- ( ( W e. Word V /\ 0 e. ( 0 ... ( N mod ( # ` W ) ) ) /\ ( N mod ( # ` W ) ) e. ( 0 ... ( # ` W ) ) ) -> ( # ` ( W substr <. 0 , ( N mod ( # ` W ) ) >. ) ) = ( ( N mod ( # ` W ) ) - 0 ) )
43	26, 41, 30, 42	syl3anc 1326	. . . . . . 7 |- ( ( W e. Word V /\ ( ( # ` W ) e. NN /\ N e. ZZ ) ) -> ( # ` ( W substr <. 0 , ( N mod ( # ` W ) ) >. ) ) = ( ( N mod ( # ` W ) ) - 0 ) )
44	36, 43	oveq12d 6668	. . . . . 6 |- ( ( W e. Word V /\ ( ( # ` W ) e. NN /\ N e. ZZ ) ) -> ( ( # ` ( W substr <. ( N mod ( # ` W ) ) , ( # ` W ) >. ) ) + ( # ` ( W substr <. 0 , ( N mod ( # ` W ) ) >. ) ) ) = ( ( ( # ` W ) - ( N mod ( # ` W ) ) ) + ( ( N mod ( # ` W ) ) - 0 ) ) )
45	37	nn0cnd 11353	. . . . . . . . . 10 |- ( ( N e. ZZ /\ ( # ` W ) e. NN ) -> ( N mod ( # ` W ) ) e. CC )
46	45	ancoms 469	. . . . . . . . 9 |- ( ( ( # ` W ) e. NN /\ N e. ZZ ) -> ( N mod ( # ` W ) ) e. CC )
47	46	adantl 482	. . . . . . . 8 |- ( ( W e. Word V /\ ( ( # ` W ) e. NN /\ N e. ZZ ) ) -> ( N mod ( # ` W ) ) e. CC )
48	47	subid1d 10381	. . . . . . 7 |- ( ( W e. Word V /\ ( ( # ` W ) e. NN /\ N e. ZZ ) ) -> ( ( N mod ( # ` W ) ) - 0 ) = ( N mod ( # ` W ) ) )
49	48	oveq2d 6666	. . . . . 6 |- ( ( W e. Word V /\ ( ( # ` W ) e. NN /\ N e. ZZ ) ) -> ( ( ( # ` W ) - ( N mod ( # ` W ) ) ) + ( ( N mod ( # ` W ) ) - 0 ) ) = ( ( ( # ` W ) - ( N mod ( # ` W ) ) ) + ( N mod ( # ` W ) ) ) )
50	31	nn0cnd 11353	. . . . . . 7 |- ( W e. Word V -> ( # ` W ) e. CC )
51		npcan 10290	. . . . . . 7 |- ( ( ( # ` W ) e. CC /\ ( N mod ( # ` W ) ) e. CC ) -> ( ( ( # ` W ) - ( N mod ( # ` W ) ) ) + ( N mod ( # ` W ) ) ) = ( # ` W ) )
52	50, 46, 51	syl2an 494	. . . . . 6 |- ( ( W e. Word V /\ ( ( # ` W ) e. NN /\ N e. ZZ ) ) -> ( ( ( # ` W ) - ( N mod ( # ` W ) ) ) + ( N mod ( # ` W ) ) ) = ( # ` W ) )
53	44, 49, 52	3eqtrd 2660	. . . . 5 |- ( ( W e. Word V /\ ( ( # ` W ) e. NN /\ N e. ZZ ) ) -> ( ( # ` ( W substr <. ( N mod ( # ` W ) ) , ( # ` W ) >. ) ) + ( # ` ( W substr <. 0 , ( N mod ( # ` W ) ) >. ) ) ) = ( # ` W ) )
54	25, 53	syl 17	. . . 4 |- ( ( ( W e. Word V /\ N e. ZZ ) /\ W =/= (/) ) -> ( ( # ` ( W substr <. ( N mod ( # ` W ) ) , ( # ` W ) >. ) ) + ( # ` ( W substr <. 0 , ( N mod ( # ` W ) ) >. ) ) ) = ( # ` W ) )
55	11, 17, 54	3eqtrd 2660	. . 3 |- ( ( ( W e. Word V /\ N e. ZZ ) /\ W =/= (/) ) -> ( # ` ( W cyclShift N ) ) = ( # ` W ) )
56	55	expcom 451	. 2 |- ( W =/= (/) -> ( ( W e. Word V /\ N e. ZZ ) -> ( # ` ( W cyclShift N ) ) = ( # ` W ) ) )
57	8, 56	pm2.61ine 2877	1 |- ( ( W e. Word V /\ N e. ZZ ) -> ( # ` ( W cyclShift N ) ) = ( # ` W ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   (/)c0 3915   <.cop 4183   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   + caddc 9939   - cmin 10266   NNcn 11020   NN0cn0 11292   ZZcz 11377   ...cfz 12326   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:  cshwf  13546  2cshw  13559  lswcshw  13561  cshwleneq  13563  crctcshlem2  26710  clwwisshclwwslem  26927  clwwisshclwws  26928  clwwnisshclwwsn  26930  erclwwlkseqlen  26933  erclwwlksneqlen  26945  eucrct2eupth  27105