Theorem cshweqdif2 13565

Description: If cyclically shifting two words (of the same length) results in the same word, cyclically shifting one of the words by the difference of the numbers of shifts results in the other word. (Contributed by AV, 21-Apr-2018.) (Revised by AV, 6-Jun-2018.) (Revised by AV, 1-Nov-2018.)

Assertion

Ref	Expression
cshweqdif2	|- ( ( ( W e. Word V /\ U e. Word V ) /\ ( N e. ZZ /\ M e. ZZ ) ) -> ( ( W cyclShift N ) = ( U cyclShift M ) -> ( U cyclShift ( M - N ) ) = W ) )

Proof of Theorem cshweqdif2

Step	Hyp	Ref	Expression
1		simpr 477	. . . . . . . . 9 |- ( ( W e. Word V /\ U e. Word V ) -> U e. Word V )
2	1	adantr 481	. . . . . . . 8 |- ( ( ( W e. Word V /\ U e. Word V ) /\ ( N e. ZZ /\ M e. ZZ ) ) -> U e. Word V )
3		zsubcl 11419	. . . . . . . . . 10 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M - N ) e. ZZ )
4	3	ancoms 469	. . . . . . . . 9 |- ( ( N e. ZZ /\ M e. ZZ ) -> ( M - N ) e. ZZ )
5	4	adantl 482	. . . . . . . 8 |- ( ( ( W e. Word V /\ U e. Word V ) /\ ( N e. ZZ /\ M e. ZZ ) ) -> ( M - N ) e. ZZ )
6		simpr 477	. . . . . . . . 9 |- ( ( N e. ZZ /\ M e. ZZ ) -> M e. ZZ )
7	6	adantl 482	. . . . . . . 8 |- ( ( ( W e. Word V /\ U e. Word V ) /\ ( N e. ZZ /\ M e. ZZ ) ) -> M e. ZZ )
8	2, 5, 7	3jca 1242	. . . . . . 7 |- ( ( ( W e. Word V /\ U e. Word V ) /\ ( N e. ZZ /\ M e. ZZ ) ) -> ( U e. Word V /\ ( M - N ) e. ZZ /\ M e. ZZ ) )
9	8	adantr 481	. . . . . 6 |- ( ( ( ( W e. Word V /\ U e. Word V ) /\ ( N e. ZZ /\ M e. ZZ ) ) /\ ( W cyclShift N ) = ( U cyclShift M ) ) -> ( U e. Word V /\ ( M - N ) e. ZZ /\ M e. ZZ ) )
10		3cshw 13564	. . . . . 6 |- ( ( U e. Word V /\ ( M - N ) e. ZZ /\ M e. ZZ ) -> ( U cyclShift ( M - N ) ) = ( ( ( U cyclShift M ) cyclShift ( M - N ) ) cyclShift ( ( # ` U ) - M ) ) )
11	9, 10	syl 17	. . . . 5 |- ( ( ( ( W e. Word V /\ U e. Word V ) /\ ( N e. ZZ /\ M e. ZZ ) ) /\ ( W cyclShift N ) = ( U cyclShift M ) ) -> ( U cyclShift ( M - N ) ) = ( ( ( U cyclShift M ) cyclShift ( M - N ) ) cyclShift ( ( # ` U ) - M ) ) )
12		simpl 473	. . . . . . . . . 10 |- ( ( ( W e. Word V /\ U e. Word V ) /\ ( N e. ZZ /\ M e. ZZ ) ) -> ( W e. Word V /\ U e. Word V ) )
13	12	ancomd 467	. . . . . . . . 9 |- ( ( ( W e. Word V /\ U e. Word V ) /\ ( N e. ZZ /\ M e. ZZ ) ) -> ( U e. Word V /\ W e. Word V ) )
14	13	adantr 481	. . . . . . . 8 |- ( ( ( ( W e. Word V /\ U e. Word V ) /\ ( N e. ZZ /\ M e. ZZ ) ) /\ ( W cyclShift N ) = ( U cyclShift M ) ) -> ( U e. Word V /\ W e. Word V ) )
15		simpr 477	. . . . . . . . . 10 |- ( ( ( W e. Word V /\ U e. Word V ) /\ ( N e. ZZ /\ M e. ZZ ) ) -> ( N e. ZZ /\ M e. ZZ ) )
16	15	ancomd 467	. . . . . . . . 9 |- ( ( ( W e. Word V /\ U e. Word V ) /\ ( N e. ZZ /\ M e. ZZ ) ) -> ( M e. ZZ /\ N e. ZZ ) )
17	16	adantr 481	. . . . . . . 8 |- ( ( ( ( W e. Word V /\ U e. Word V ) /\ ( N e. ZZ /\ M e. ZZ ) ) /\ ( W cyclShift N ) = ( U cyclShift M ) ) -> ( M e. ZZ /\ N e. ZZ ) )
18		simpr 477	. . . . . . . . 9 |- ( ( ( ( W e. Word V /\ U e. Word V ) /\ ( N e. ZZ /\ M e. ZZ ) ) /\ ( W cyclShift N ) = ( U cyclShift M ) ) -> ( W cyclShift N ) = ( U cyclShift M ) )
19	18	eqcomd 2628	. . . . . . . 8 |- ( ( ( ( W e. Word V /\ U e. Word V ) /\ ( N e. ZZ /\ M e. ZZ ) ) /\ ( W cyclShift N ) = ( U cyclShift M ) ) -> ( U cyclShift M ) = ( W cyclShift N ) )
20		cshwleneq 13563	. . . . . . . 8 |- ( ( ( U e. Word V /\ W e. Word V ) /\ ( M e. ZZ /\ N e. ZZ ) /\ ( U cyclShift M ) = ( W cyclShift N ) ) -> ( # ` U ) = ( # ` W ) )
21	14, 17, 19, 20	syl3anc 1326	. . . . . . 7 |- ( ( ( ( W e. Word V /\ U e. Word V ) /\ ( N e. ZZ /\ M e. ZZ ) ) /\ ( W cyclShift N ) = ( U cyclShift M ) ) -> ( # ` U ) = ( # ` W ) )
22	21	oveq1d 6665	. . . . . 6 |- ( ( ( ( W e. Word V /\ U e. Word V ) /\ ( N e. ZZ /\ M e. ZZ ) ) /\ ( W cyclShift N ) = ( U cyclShift M ) ) -> ( ( # ` U ) - M ) = ( ( # ` W ) - M ) )
23	22	oveq2d 6666	. . . . 5 |- ( ( ( ( W e. Word V /\ U e. Word V ) /\ ( N e. ZZ /\ M e. ZZ ) ) /\ ( W cyclShift N ) = ( U cyclShift M ) ) -> ( ( ( U cyclShift M ) cyclShift ( M - N ) ) cyclShift ( ( # ` U ) - M ) ) = ( ( ( U cyclShift M ) cyclShift ( M - N ) ) cyclShift ( ( # ` W ) - M ) ) )
24	11, 23	eqtrd 2656	. . . 4 |- ( ( ( ( W e. Word V /\ U e. Word V ) /\ ( N e. ZZ /\ M e. ZZ ) ) /\ ( W cyclShift N ) = ( U cyclShift M ) ) -> ( U cyclShift ( M - N ) ) = ( ( ( U cyclShift M ) cyclShift ( M - N ) ) cyclShift ( ( # ` W ) - M ) ) )
25	19	oveq1d 6665	. . . . . 6 |- ( ( ( ( W e. Word V /\ U e. Word V ) /\ ( N e. ZZ /\ M e. ZZ ) ) /\ ( W cyclShift N ) = ( U cyclShift M ) ) -> ( ( U cyclShift M ) cyclShift ( M - N ) ) = ( ( W cyclShift N ) cyclShift ( M - N ) ) )
26		simpl 473	. . . . . . . . . 10 |- ( ( W e. Word V /\ U e. Word V ) -> W e. Word V )
27	26	adantr 481	. . . . . . . . 9 |- ( ( ( W e. Word V /\ U e. Word V ) /\ ( N e. ZZ /\ M e. ZZ ) ) -> W e. Word V )
28		simpl 473	. . . . . . . . . 10 |- ( ( N e. ZZ /\ M e. ZZ ) -> N e. ZZ )
29	28	adantl 482	. . . . . . . . 9 |- ( ( ( W e. Word V /\ U e. Word V ) /\ ( N e. ZZ /\ M e. ZZ ) ) -> N e. ZZ )
30	27, 29, 5	3jca 1242	. . . . . . . 8 |- ( ( ( W e. Word V /\ U e. Word V ) /\ ( N e. ZZ /\ M e. ZZ ) ) -> ( W e. Word V /\ N e. ZZ /\ ( M - N ) e. ZZ ) )
31	30	adantr 481	. . . . . . 7 |- ( ( ( ( W e. Word V /\ U e. Word V ) /\ ( N e. ZZ /\ M e. ZZ ) ) /\ ( W cyclShift N ) = ( U cyclShift M ) ) -> ( W e. Word V /\ N e. ZZ /\ ( M - N ) e. ZZ ) )
32		2cshw 13559	. . . . . . 7 |- ( ( W e. Word V /\ N e. ZZ /\ ( M - N ) e. ZZ ) -> ( ( W cyclShift N ) cyclShift ( M - N ) ) = ( W cyclShift ( N + ( M - N ) ) ) )
33	31, 32	syl 17	. . . . . 6 |- ( ( ( ( W e. Word V /\ U e. Word V ) /\ ( N e. ZZ /\ M e. ZZ ) ) /\ ( W cyclShift N ) = ( U cyclShift M ) ) -> ( ( W cyclShift N ) cyclShift ( M - N ) ) = ( W cyclShift ( N + ( M - N ) ) ) )
34		zcn 11382	. . . . . . . . . . 11 |- ( N e. ZZ -> N e. CC )
35		zcn 11382	. . . . . . . . . . 11 |- ( M e. ZZ -> M e. CC )
36	34, 35	anim12i 590	. . . . . . . . . 10 |- ( ( N e. ZZ /\ M e. ZZ ) -> ( N e. CC /\ M e. CC ) )
37	36	adantl 482	. . . . . . . . 9 |- ( ( ( W e. Word V /\ U e. Word V ) /\ ( N e. ZZ /\ M e. ZZ ) ) -> ( N e. CC /\ M e. CC ) )
38	37	adantr 481	. . . . . . . 8 |- ( ( ( ( W e. Word V /\ U e. Word V ) /\ ( N e. ZZ /\ M e. ZZ ) ) /\ ( W cyclShift N ) = ( U cyclShift M ) ) -> ( N e. CC /\ M e. CC ) )
39		pncan3 10289	. . . . . . . 8 |- ( ( N e. CC /\ M e. CC ) -> ( N + ( M - N ) ) = M )
40	38, 39	syl 17	. . . . . . 7 |- ( ( ( ( W e. Word V /\ U e. Word V ) /\ ( N e. ZZ /\ M e. ZZ ) ) /\ ( W cyclShift N ) = ( U cyclShift M ) ) -> ( N + ( M - N ) ) = M )
41	40	oveq2d 6666	. . . . . 6 |- ( ( ( ( W e. Word V /\ U e. Word V ) /\ ( N e. ZZ /\ M e. ZZ ) ) /\ ( W cyclShift N ) = ( U cyclShift M ) ) -> ( W cyclShift ( N + ( M - N ) ) ) = ( W cyclShift M ) )
42	25, 33, 41	3eqtrd 2660	. . . . 5 |- ( ( ( ( W e. Word V /\ U e. Word V ) /\ ( N e. ZZ /\ M e. ZZ ) ) /\ ( W cyclShift N ) = ( U cyclShift M ) ) -> ( ( U cyclShift M ) cyclShift ( M - N ) ) = ( W cyclShift M ) )
43	42	oveq1d 6665	. . . 4 |- ( ( ( ( W e. Word V /\ U e. Word V ) /\ ( N e. ZZ /\ M e. ZZ ) ) /\ ( W cyclShift N ) = ( U cyclShift M ) ) -> ( ( ( U cyclShift M ) cyclShift ( M - N ) ) cyclShift ( ( # ` W ) - M ) ) = ( ( W cyclShift M ) cyclShift ( ( # ` W ) - M ) ) )
44		lencl 13324	. . . . . . . . . 10 |- ( W e. Word V -> ( # ` W ) e. NN0 )
45	44	nn0zd 11480	. . . . . . . . 9 |- ( W e. Word V -> ( # ` W ) e. ZZ )
46	45	adantr 481	. . . . . . . 8 |- ( ( W e. Word V /\ U e. Word V ) -> ( # ` W ) e. ZZ )
47		zsubcl 11419	. . . . . . . 8 |- ( ( ( # ` W ) e. ZZ /\ M e. ZZ ) -> ( ( # ` W ) - M ) e. ZZ )
48	46, 6, 47	syl2an 494	. . . . . . 7 |- ( ( ( W e. Word V /\ U e. Word V ) /\ ( N e. ZZ /\ M e. ZZ ) ) -> ( ( # ` W ) - M ) e. ZZ )
49	27, 7, 48	3jca 1242	. . . . . 6 |- ( ( ( W e. Word V /\ U e. Word V ) /\ ( N e. ZZ /\ M e. ZZ ) ) -> ( W e. Word V /\ M e. ZZ /\ ( ( # ` W ) - M ) e. ZZ ) )
50	49	adantr 481	. . . . 5 |- ( ( ( ( W e. Word V /\ U e. Word V ) /\ ( N e. ZZ /\ M e. ZZ ) ) /\ ( W cyclShift N ) = ( U cyclShift M ) ) -> ( W e. Word V /\ M e. ZZ /\ ( ( # ` W ) - M ) e. ZZ ) )
51		2cshw 13559	. . . . 5 |- ( ( W e. Word V /\ M e. ZZ /\ ( ( # ` W ) - M ) e. ZZ ) -> ( ( W cyclShift M ) cyclShift ( ( # ` W ) - M ) ) = ( W cyclShift ( M + ( ( # ` W ) - M ) ) ) )
52	50, 51	syl 17	. . . 4 |- ( ( ( ( W e. Word V /\ U e. Word V ) /\ ( N e. ZZ /\ M e. ZZ ) ) /\ ( W cyclShift N ) = ( U cyclShift M ) ) -> ( ( W cyclShift M ) cyclShift ( ( # ` W ) - M ) ) = ( W cyclShift ( M + ( ( # ` W ) - M ) ) ) )
53	24, 43, 52	3eqtrd 2660	. . 3 |- ( ( ( ( W e. Word V /\ U e. Word V ) /\ ( N e. ZZ /\ M e. ZZ ) ) /\ ( W cyclShift N ) = ( U cyclShift M ) ) -> ( U cyclShift ( M - N ) ) = ( W cyclShift ( M + ( ( # ` W ) - M ) ) ) )
54	44	nn0cnd 11353	. . . . . . . . 9 |- ( W e. Word V -> ( # ` W ) e. CC )
55	54	adantr 481	. . . . . . . 8 |- ( ( W e. Word V /\ U e. Word V ) -> ( # ` W ) e. CC )
56	35	adantl 482	. . . . . . . 8 |- ( ( N e. ZZ /\ M e. ZZ ) -> M e. CC )
57	55, 56	anim12i 590	. . . . . . 7 |- ( ( ( W e. Word V /\ U e. Word V ) /\ ( N e. ZZ /\ M e. ZZ ) ) -> ( ( # ` W ) e. CC /\ M e. CC ) )
58	57	ancomd 467	. . . . . 6 |- ( ( ( W e. Word V /\ U e. Word V ) /\ ( N e. ZZ /\ M e. ZZ ) ) -> ( M e. CC /\ ( # ` W ) e. CC ) )
59	58	adantr 481	. . . . 5 |- ( ( ( ( W e. Word V /\ U e. Word V ) /\ ( N e. ZZ /\ M e. ZZ ) ) /\ ( W cyclShift N ) = ( U cyclShift M ) ) -> ( M e. CC /\ ( # ` W ) e. CC ) )
60		pncan3 10289	. . . . 5 |- ( ( M e. CC /\ ( # ` W ) e. CC ) -> ( M + ( ( # ` W ) - M ) ) = ( # ` W ) )
61	59, 60	syl 17	. . . 4 |- ( ( ( ( W e. Word V /\ U e. Word V ) /\ ( N e. ZZ /\ M e. ZZ ) ) /\ ( W cyclShift N ) = ( U cyclShift M ) ) -> ( M + ( ( # ` W ) - M ) ) = ( # ` W ) )
62	61	oveq2d 6666	. . 3 |- ( ( ( ( W e. Word V /\ U e. Word V ) /\ ( N e. ZZ /\ M e. ZZ ) ) /\ ( W cyclShift N ) = ( U cyclShift M ) ) -> ( W cyclShift ( M + ( ( # ` W ) - M ) ) ) = ( W cyclShift ( # ` W ) ) )
63		cshwn 13543	. . . . 5 |- ( W e. Word V -> ( W cyclShift ( # ` W ) ) = W )
64	27, 63	syl 17	. . . 4 |- ( ( ( W e. Word V /\ U e. Word V ) /\ ( N e. ZZ /\ M e. ZZ ) ) -> ( W cyclShift ( # ` W ) ) = W )
65	64	adantr 481	. . 3 |- ( ( ( ( W e. Word V /\ U e. Word V ) /\ ( N e. ZZ /\ M e. ZZ ) ) /\ ( W cyclShift N ) = ( U cyclShift M ) ) -> ( W cyclShift ( # ` W ) ) = W )
66	53, 62, 65	3eqtrd 2660	. 2 |- ( ( ( ( W e. Word V /\ U e. Word V ) /\ ( N e. ZZ /\ M e. ZZ ) ) /\ ( W cyclShift N ) = ( U cyclShift M ) ) -> ( U cyclShift ( M - N ) ) = W )
67	66	ex 450	1 |- ( ( ( W e. Word V /\ U e. Word V ) /\ ( N e. ZZ /\ M e. ZZ ) ) -> ( ( W cyclShift N ) = ( U cyclShift M ) -> ( U cyclShift ( M - N ) ) = W ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   CCcc 9934   + caddc 9939   - cmin 10266   ZZcz 11377   #chash 13117  Word cword 13291   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-2 11079  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:  cshweqdifid  13566