Theorem clwwisshclwws 26928

Description: Cyclically shifting a closed walk as word results in a closed walk as word (in an undirected graph). (Contributed by Alexander van der Vekens, 24-Mar-2018.) (Revised by AV, 28-Apr-2021.)

Assertion

Ref	Expression
clwwisshclwws	|- ( ( W e. (ClWWalks ` G ) /\ N e. ( 0..^ ( # ` W ) ) ) -> ( W cyclShift N ) e. (ClWWalks ` G ) )

Proof of Theorem clwwisshclwws

Dummy variables i j are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . . 6 |- (Vtx ` G ) = (Vtx ` G )
2	1	clwwlkbp 26883	. . . . 5 |- ( W e. (ClWWalks ` G ) -> ( G e. _V /\ W e. Word (Vtx ` G ) /\ W =/= (/) ) )
3		cshw0 13540	. . . . . . . 8 |- ( W e. Word (Vtx ` G ) -> ( W cyclShift 0 ) = W )
4	3	3ad2ant2 1083	. . . . . . 7 |- ( ( G e. _V /\ W e. Word (Vtx ` G ) /\ W =/= (/) ) -> ( W cyclShift 0 ) = W )
5	4	eleq1d 2686	. . . . . 6 |- ( ( G e. _V /\ W e. Word (Vtx ` G ) /\ W =/= (/) ) -> ( ( W cyclShift 0 ) e. (ClWWalks ` G ) <-> W e. (ClWWalks ` G ) ) )
6	5	biimprd 238	. . . . 5 |- ( ( G e. _V /\ W e. Word (Vtx ` G ) /\ W =/= (/) ) -> ( W e. (ClWWalks ` G ) -> ( W cyclShift 0 ) e. (ClWWalks ` G ) ) )
7	2, 6	mpcom 38	. . . 4 |- ( W e. (ClWWalks ` G ) -> ( W cyclShift 0 ) e. (ClWWalks ` G ) )
8		oveq2 6658	. . . . 5 |- ( N = 0 -> ( W cyclShift N ) = ( W cyclShift 0 ) )
9	8	eleq1d 2686	. . . 4 |- ( N = 0 -> ( ( W cyclShift N ) e. (ClWWalks ` G ) <-> ( W cyclShift 0 ) e. (ClWWalks ` G ) ) )
10	7, 9	syl5ibrcom 237	. . 3 |- ( W e. (ClWWalks ` G ) -> ( N = 0 -> ( W cyclShift N ) e. (ClWWalks ` G ) ) )
11	10	adantr 481	. 2 |- ( ( W e. (ClWWalks ` G ) /\ N e. ( 0..^ ( # ` W ) ) ) -> ( N = 0 -> ( W cyclShift N ) e. (ClWWalks ` G ) ) )
12		fzo1fzo0n0 12518	. . . . . 6 |- ( N e. ( 1..^ ( # ` W ) ) <-> ( N e. ( 0..^ ( # ` W ) ) /\ N =/= 0 ) )
13		cshwcl 13544	. . . . . . . . . . . . 13 |- ( W e. Word (Vtx ` G ) -> ( W cyclShift N ) e. Word (Vtx ` G ) )
14	13	adantr 481	. . . . . . . . . . . 12 |- ( ( W e. Word (Vtx ` G ) /\ W =/= (/) ) -> ( W cyclShift N ) e. Word (Vtx ` G ) )
15	14	3ad2ant1 1082	. . . . . . . . . . 11 |- ( ( ( W e. Word (Vtx ` G ) /\ W =/= (/) ) /\ A. i e. ( 0..^ ( ( # ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) /\ { ( lastS ` W ) , ( W ` 0 ) } e. (Edg ` G ) ) -> ( W cyclShift N ) e. Word (Vtx ` G ) )
16	15	adantr 481	. . . . . . . . . 10 |- ( ( ( ( W e. Word (Vtx ` G ) /\ W =/= (/) ) /\ A. i e. ( 0..^ ( ( # ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) /\ { ( lastS ` W ) , ( W ` 0 ) } e. (Edg ` G ) ) /\ N e. ( 1..^ ( # ` W ) ) ) -> ( W cyclShift N ) e. Word (Vtx ` G ) )
17		simpl 473	. . . . . . . . . . . . . 14 |- ( ( W e. Word (Vtx ` G ) /\ W =/= (/) ) -> W e. Word (Vtx ` G ) )
18		elfzoelz 12470	. . . . . . . . . . . . . 14 |- ( N e. ( 1..^ ( # ` W ) ) -> N e. ZZ )
19		cshwlen 13545	. . . . . . . . . . . . . 14 |- ( ( W e. Word (Vtx ` G ) /\ N e. ZZ ) -> ( # ` ( W cyclShift N ) ) = ( # ` W ) )
20	17, 18, 19	syl2an 494	. . . . . . . . . . . . 13 |- ( ( ( W e. Word (Vtx ` G ) /\ W =/= (/) ) /\ N e. ( 1..^ ( # ` W ) ) ) -> ( # ` ( W cyclShift N ) ) = ( # ` W ) )
21		hasheq0 13154	. . . . . . . . . . . . . . . . 17 |- ( W e. Word (Vtx ` G ) -> ( ( # ` W ) = 0 <-> W = (/) ) )
22	21	bicomd 213	. . . . . . . . . . . . . . . 16 |- ( W e. Word (Vtx ` G ) -> ( W = (/) <-> ( # ` W ) = 0 ) )
23	22	necon3bid 2838	. . . . . . . . . . . . . . 15 |- ( W e. Word (Vtx ` G ) -> ( W =/= (/) <-> ( # ` W ) =/= 0 ) )
24	23	biimpa 501	. . . . . . . . . . . . . 14 |- ( ( W e. Word (Vtx ` G ) /\ W =/= (/) ) -> ( # ` W ) =/= 0 )
25	24	adantr 481	. . . . . . . . . . . . 13 |- ( ( ( W e. Word (Vtx ` G ) /\ W =/= (/) ) /\ N e. ( 1..^ ( # ` W ) ) ) -> ( # ` W ) =/= 0 )
26	20, 25	eqnetrd 2861	. . . . . . . . . . . 12 |- ( ( ( W e. Word (Vtx ` G ) /\ W =/= (/) ) /\ N e. ( 1..^ ( # ` W ) ) ) -> ( # ` ( W cyclShift N ) ) =/= 0 )
27	14	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ( W e. Word (Vtx ` G ) /\ W =/= (/) ) /\ N e. ( 1..^ ( # ` W ) ) ) -> ( W cyclShift N ) e. Word (Vtx ` G ) )
28		hasheq0 13154	. . . . . . . . . . . . . 14 |- ( ( W cyclShift N ) e. Word (Vtx ` G ) -> ( ( # ` ( W cyclShift N ) ) = 0 <-> ( W cyclShift N ) = (/) ) )
29	27, 28	syl 17	. . . . . . . . . . . . 13 |- ( ( ( W e. Word (Vtx ` G ) /\ W =/= (/) ) /\ N e. ( 1..^ ( # ` W ) ) ) -> ( ( # ` ( W cyclShift N ) ) = 0 <-> ( W cyclShift N ) = (/) ) )
30	29	necon3bid 2838	. . . . . . . . . . . 12 |- ( ( ( W e. Word (Vtx ` G ) /\ W =/= (/) ) /\ N e. ( 1..^ ( # ` W ) ) ) -> ( ( # ` ( W cyclShift N ) ) =/= 0 <-> ( W cyclShift N ) =/= (/) ) )
31	26, 30	mpbid 222	. . . . . . . . . . 11 |- ( ( ( W e. Word (Vtx ` G ) /\ W =/= (/) ) /\ N e. ( 1..^ ( # ` W ) ) ) -> ( W cyclShift N ) =/= (/) )
32	31	3ad2antl1 1223	. . . . . . . . . 10 |- ( ( ( ( W e. Word (Vtx ` G ) /\ W =/= (/) ) /\ A. i e. ( 0..^ ( ( # ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) /\ { ( lastS ` W ) , ( W ` 0 ) } e. (Edg ` G ) ) /\ N e. ( 1..^ ( # ` W ) ) ) -> ( W cyclShift N ) =/= (/) )
33	16, 32	jca 554	. . . . . . . . 9 |- ( ( ( ( W e. Word (Vtx ` G ) /\ W =/= (/) ) /\ A. i e. ( 0..^ ( ( # ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) /\ { ( lastS ` W ) , ( W ` 0 ) } e. (Edg ` G ) ) /\ N e. ( 1..^ ( # ` W ) ) ) -> ( ( W cyclShift N ) e. Word (Vtx ` G ) /\ ( W cyclShift N ) =/= (/) ) )
34	17	3ad2ant1 1082	. . . . . . . . . . 11 |- ( ( ( W e. Word (Vtx ` G ) /\ W =/= (/) ) /\ A. i e. ( 0..^ ( ( # ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) /\ { ( lastS ` W ) , ( W ` 0 ) } e. (Edg ` G ) ) -> W e. Word (Vtx ` G ) )
35	34	anim1i 592	. . . . . . . . . 10 |- ( ( ( ( W e. Word (Vtx ` G ) /\ W =/= (/) ) /\ A. i e. ( 0..^ ( ( # ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) /\ { ( lastS ` W ) , ( W ` 0 ) } e. (Edg ` G ) ) /\ N e. ( 1..^ ( # ` W ) ) ) -> ( W e. Word (Vtx ` G ) /\ N e. ( 1..^ ( # ` W ) ) ) )
36		3simpc 1060	. . . . . . . . . . 11 |- ( ( ( W e. Word (Vtx ` G ) /\ W =/= (/) ) /\ A. i e. ( 0..^ ( ( # ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) /\ { ( lastS ` W ) , ( W ` 0 ) } e. (Edg ` G ) ) -> ( A. i e. ( 0..^ ( ( # ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) /\ { ( lastS ` W ) , ( W ` 0 ) } e. (Edg ` G ) ) )
37	36	adantr 481	. . . . . . . . . 10 |- ( ( ( ( W e. Word (Vtx ` G ) /\ W =/= (/) ) /\ A. i e. ( 0..^ ( ( # ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) /\ { ( lastS ` W ) , ( W ` 0 ) } e. (Edg ` G ) ) /\ N e. ( 1..^ ( # ` W ) ) ) -> ( A. i e. ( 0..^ ( ( # ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) /\ { ( lastS ` W ) , ( W ` 0 ) } e. (Edg ` G ) ) )
38		clwwisshclwwslem 26927	. . . . . . . . . 10 |- ( ( W e. Word (Vtx ` G ) /\ N e. ( 1..^ ( # ` W ) ) ) -> ( ( A. i e. ( 0..^ ( ( # ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) /\ { ( lastS ` W ) , ( W ` 0 ) } e. (Edg ` G ) ) -> A. j e. ( 0..^ ( ( # ` ( W cyclShift N ) ) - 1 ) ) { ( ( W cyclShift N ) ` j ) , ( ( W cyclShift N ) ` ( j + 1 ) ) } e. (Edg ` G ) ) )
39	35, 37, 38	sylc 65	. . . . . . . . 9 |- ( ( ( ( W e. Word (Vtx ` G ) /\ W =/= (/) ) /\ A. i e. ( 0..^ ( ( # ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) /\ { ( lastS ` W ) , ( W ` 0 ) } e. (Edg ` G ) ) /\ N e. ( 1..^ ( # ` W ) ) ) -> A. j e. ( 0..^ ( ( # ` ( W cyclShift N ) ) - 1 ) ) { ( ( W cyclShift N ) ` j ) , ( ( W cyclShift N ) ` ( j + 1 ) ) } e. (Edg ` G ) )
40		elfzofz 12485	. . . . . . . . . . . . . . . 16 |- ( N e. ( 1..^ ( # ` W ) ) -> N e. ( 1 ... ( # ` W ) ) )
41		lswcshw 13561	. . . . . . . . . . . . . . . 16 |- ( ( W e. Word (Vtx ` G ) /\ N e. ( 1 ... ( # ` W ) ) ) -> ( lastS ` ( W cyclShift N ) ) = ( W ` ( N - 1 ) ) )
42	40, 41	sylan2 491	. . . . . . . . . . . . . . 15 |- ( ( W e. Word (Vtx ` G ) /\ N e. ( 1..^ ( # ` W ) ) ) -> ( lastS ` ( W cyclShift N ) ) = ( W ` ( N - 1 ) ) )
43		fzo0ss1 12498	. . . . . . . . . . . . . . . . 17 |- ( 1..^ ( # ` W ) ) C_ ( 0..^ ( # ` W ) )
44	43	sseli 3599	. . . . . . . . . . . . . . . 16 |- ( N e. ( 1..^ ( # ` W ) ) -> N e. ( 0..^ ( # ` W ) ) )
45		cshwidx0 13552	. . . . . . . . . . . . . . . 16 |- ( ( W e. Word (Vtx ` G ) /\ N e. ( 0..^ ( # ` W ) ) ) -> ( ( W cyclShift N ) ` 0 ) = ( W ` N ) )
46	44, 45	sylan2 491	. . . . . . . . . . . . . . 15 |- ( ( W e. Word (Vtx ` G ) /\ N e. ( 1..^ ( # ` W ) ) ) -> ( ( W cyclShift N ) ` 0 ) = ( W ` N ) )
47	42, 46	preq12d 4276	. . . . . . . . . . . . . 14 |- ( ( W e. Word (Vtx ` G ) /\ N e. ( 1..^ ( # ` W ) ) ) -> { ( lastS ` ( W cyclShift N ) ) , ( ( W cyclShift N ) ` 0 ) } = { ( W ` ( N - 1 ) ) , ( W ` N ) } )
48	47	ex 450	. . . . . . . . . . . . 13 |- ( W e. Word (Vtx ` G ) -> ( N e. ( 1..^ ( # ` W ) ) -> { ( lastS ` ( W cyclShift N ) ) , ( ( W cyclShift N ) ` 0 ) } = { ( W ` ( N - 1 ) ) , ( W ` N ) } ) )
49	48	adantr 481	. . . . . . . . . . . 12 |- ( ( W e. Word (Vtx ` G ) /\ W =/= (/) ) -> ( N e. ( 1..^ ( # ` W ) ) -> { ( lastS ` ( W cyclShift N ) ) , ( ( W cyclShift N ) ` 0 ) } = { ( W ` ( N - 1 ) ) , ( W ` N ) } ) )
50	49	3ad2ant1 1082	. . . . . . . . . . 11 |- ( ( ( W e. Word (Vtx ` G ) /\ W =/= (/) ) /\ A. i e. ( 0..^ ( ( # ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) /\ { ( lastS ` W ) , ( W ` 0 ) } e. (Edg ` G ) ) -> ( N e. ( 1..^ ( # ` W ) ) -> { ( lastS ` ( W cyclShift N ) ) , ( ( W cyclShift N ) ` 0 ) } = { ( W ` ( N - 1 ) ) , ( W ` N ) } ) )
51	50	imp 445	. . . . . . . . . 10 |- ( ( ( ( W e. Word (Vtx ` G ) /\ W =/= (/) ) /\ A. i e. ( 0..^ ( ( # ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) /\ { ( lastS ` W ) , ( W ` 0 ) } e. (Edg ` G ) ) /\ N e. ( 1..^ ( # ` W ) ) ) -> { ( lastS ` ( W cyclShift N ) ) , ( ( W cyclShift N ) ` 0 ) } = { ( W ` ( N - 1 ) ) , ( W ` N ) } )
52		elfzo1elm1fzo0 12569	. . . . . . . . . . . . . . . . 17 |- ( N e. ( 1..^ ( # ` W ) ) -> ( N - 1 ) e. ( 0..^ ( ( # ` W ) - 1 ) ) )
53	52	adantl 482	. . . . . . . . . . . . . . . 16 |- ( ( W e. Word (Vtx ` G ) /\ N e. ( 1..^ ( # ` W ) ) ) -> ( N - 1 ) e. ( 0..^ ( ( # ` W ) - 1 ) ) )
54		fveq2 6191	. . . . . . . . . . . . . . . . . . 19 |- ( i = ( N - 1 ) -> ( W ` i ) = ( W ` ( N - 1 ) ) )
55	54	adantl 482	. . . . . . . . . . . . . . . . . 18 |- ( ( ( W e. Word (Vtx ` G ) /\ N e. ( 1..^ ( # ` W ) ) ) /\ i = ( N - 1 ) ) -> ( W ` i ) = ( W ` ( N - 1 ) ) )
56		oveq1 6657	. . . . . . . . . . . . . . . . . . . 20 |- ( i = ( N - 1 ) -> ( i + 1 ) = ( ( N - 1 ) + 1 ) )
57	56	fveq2d 6195	. . . . . . . . . . . . . . . . . . 19 |- ( i = ( N - 1 ) -> ( W ` ( i + 1 ) ) = ( W ` ( ( N - 1 ) + 1 ) ) )
58	18	zcnd 11483	. . . . . . . . . . . . . . . . . . . . . 22 |- ( N e. ( 1..^ ( # ` W ) ) -> N e. CC )
59	58	adantl 482	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( W e. Word (Vtx ` G ) /\ N e. ( 1..^ ( # ` W ) ) ) -> N e. CC )
60		1cnd 10056	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( W e. Word (Vtx ` G ) /\ N e. ( 1..^ ( # ` W ) ) ) -> 1 e. CC )
61	59, 60	npcand 10396	. . . . . . . . . . . . . . . . . . . 20 |- ( ( W e. Word (Vtx ` G ) /\ N e. ( 1..^ ( # ` W ) ) ) -> ( ( N - 1 ) + 1 ) = N )
62	61	fveq2d 6195	. . . . . . . . . . . . . . . . . . 19 |- ( ( W e. Word (Vtx ` G ) /\ N e. ( 1..^ ( # ` W ) ) ) -> ( W ` ( ( N - 1 ) + 1 ) ) = ( W ` N ) )
63	57, 62	sylan9eqr 2678	. . . . . . . . . . . . . . . . . 18 |- ( ( ( W e. Word (Vtx ` G ) /\ N e. ( 1..^ ( # ` W ) ) ) /\ i = ( N - 1 ) ) -> ( W ` ( i + 1 ) ) = ( W ` N ) )
64	55, 63	preq12d 4276	. . . . . . . . . . . . . . . . 17 |- ( ( ( W e. Word (Vtx ` G ) /\ N e. ( 1..^ ( # ` W ) ) ) /\ i = ( N - 1 ) ) -> { ( W ` i ) , ( W ` ( i + 1 ) ) } = { ( W ` ( N - 1 ) ) , ( W ` N ) } )
65	64	eleq1d 2686	. . . . . . . . . . . . . . . 16 |- ( ( ( W e. Word (Vtx ` G ) /\ N e. ( 1..^ ( # ` W ) ) ) /\ i = ( N - 1 ) ) -> ( { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) <-> { ( W ` ( N - 1 ) ) , ( W ` N ) } e. (Edg ` G ) ) )
66	53, 65	rspcdv 3312	. . . . . . . . . . . . . . 15 |- ( ( W e. Word (Vtx ` G ) /\ N e. ( 1..^ ( # ` W ) ) ) -> ( A. i e. ( 0..^ ( ( # ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) -> { ( W ` ( N - 1 ) ) , ( W ` N ) } e. (Edg ` G ) ) )
67	66	a1d 25	. . . . . . . . . . . . . 14 |- ( ( W e. Word (Vtx ` G ) /\ N e. ( 1..^ ( # ` W ) ) ) -> ( { ( lastS ` W ) , ( W ` 0 ) } e. (Edg ` G ) -> ( A. i e. ( 0..^ ( ( # ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) -> { ( W ` ( N - 1 ) ) , ( W ` N ) } e. (Edg ` G ) ) ) )
68	67	ex 450	. . . . . . . . . . . . 13 |- ( W e. Word (Vtx ` G ) -> ( N e. ( 1..^ ( # ` W ) ) -> ( { ( lastS ` W ) , ( W ` 0 ) } e. (Edg ` G ) -> ( A. i e. ( 0..^ ( ( # ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) -> { ( W ` ( N - 1 ) ) , ( W ` N ) } e. (Edg ` G ) ) ) ) )
69	68	adantr 481	. . . . . . . . . . . 12 |- ( ( W e. Word (Vtx ` G ) /\ W =/= (/) ) -> ( N e. ( 1..^ ( # ` W ) ) -> ( { ( lastS ` W ) , ( W ` 0 ) } e. (Edg ` G ) -> ( A. i e. ( 0..^ ( ( # ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) -> { ( W ` ( N - 1 ) ) , ( W ` N ) } e. (Edg ` G ) ) ) ) )
70	69	com24 95	. . . . . . . . . . 11 |- ( ( W e. Word (Vtx ` G ) /\ W =/= (/) ) -> ( A. i e. ( 0..^ ( ( # ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) -> ( { ( lastS ` W ) , ( W ` 0 ) } e. (Edg ` G ) -> ( N e. ( 1..^ ( # ` W ) ) -> { ( W ` ( N - 1 ) ) , ( W ` N ) } e. (Edg ` G ) ) ) ) )
71	70	3imp1 1280	. . . . . . . . . 10 |- ( ( ( ( W e. Word (Vtx ` G ) /\ W =/= (/) ) /\ A. i e. ( 0..^ ( ( # ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) /\ { ( lastS ` W ) , ( W ` 0 ) } e. (Edg ` G ) ) /\ N e. ( 1..^ ( # ` W ) ) ) -> { ( W ` ( N - 1 ) ) , ( W ` N ) } e. (Edg ` G ) )
72	51, 71	eqeltrd 2701	. . . . . . . . 9 |- ( ( ( ( W e. Word (Vtx ` G ) /\ W =/= (/) ) /\ A. i e. ( 0..^ ( ( # ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) /\ { ( lastS ` W ) , ( W ` 0 ) } e. (Edg ` G ) ) /\ N e. ( 1..^ ( # ` W ) ) ) -> { ( lastS ` ( W cyclShift N ) ) , ( ( W cyclShift N ) ` 0 ) } e. (Edg ` G ) )
73	33, 39, 72	3jca 1242	. . . . . . . 8 |- ( ( ( ( W e. Word (Vtx ` G ) /\ W =/= (/) ) /\ A. i e. ( 0..^ ( ( # ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) /\ { ( lastS ` W ) , ( W ` 0 ) } e. (Edg ` G ) ) /\ N e. ( 1..^ ( # ` W ) ) ) -> ( ( ( W cyclShift N ) e. Word (Vtx ` G ) /\ ( W cyclShift N ) =/= (/) ) /\ A. j e. ( 0..^ ( ( # ` ( W cyclShift N ) ) - 1 ) ) { ( ( W cyclShift N ) ` j ) , ( ( W cyclShift N ) ` ( j + 1 ) ) } e. (Edg ` G ) /\ { ( lastS ` ( W cyclShift N ) ) , ( ( W cyclShift N ) ` 0 ) } e. (Edg ` G ) ) )
74	73	expcom 451	. . . . . . 7 |- ( N e. ( 1..^ ( # ` W ) ) -> ( ( ( W e. Word (Vtx ` G ) /\ W =/= (/) ) /\ A. i e. ( 0..^ ( ( # ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) /\ { ( lastS ` W ) , ( W ` 0 ) } e. (Edg ` G ) ) -> ( ( ( W cyclShift N ) e. Word (Vtx ` G ) /\ ( W cyclShift N ) =/= (/) ) /\ A. j e. ( 0..^ ( ( # ` ( W cyclShift N ) ) - 1 ) ) { ( ( W cyclShift N ) ` j ) , ( ( W cyclShift N ) ` ( j + 1 ) ) } e. (Edg ` G ) /\ { ( lastS ` ( W cyclShift N ) ) , ( ( W cyclShift N ) ` 0 ) } e. (Edg ` G ) ) ) )
75		eqid 2622	. . . . . . . 8 |- (Edg ` G ) = (Edg ` G )
76	1, 75	isclwwlks 26880	. . . . . . 7 |- ( W e. (ClWWalks ` G ) <-> ( ( W e. Word (Vtx ` G ) /\ W =/= (/) ) /\ A. i e. ( 0..^ ( ( # ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) /\ { ( lastS ` W ) , ( W ` 0 ) } e. (Edg ` G ) ) )
77	1, 75	isclwwlks 26880	. . . . . . 7 |- ( ( W cyclShift N ) e. (ClWWalks ` G ) <-> ( ( ( W cyclShift N ) e. Word (Vtx ` G ) /\ ( W cyclShift N ) =/= (/) ) /\ A. j e. ( 0..^ ( ( # ` ( W cyclShift N ) ) - 1 ) ) { ( ( W cyclShift N ) ` j ) , ( ( W cyclShift N ) ` ( j + 1 ) ) } e. (Edg ` G ) /\ { ( lastS ` ( W cyclShift N ) ) , ( ( W cyclShift N ) ` 0 ) } e. (Edg ` G ) ) )
78	74, 76, 77	3imtr4g 285	. . . . . 6 |- ( N e. ( 1..^ ( # ` W ) ) -> ( W e. (ClWWalks ` G ) -> ( W cyclShift N ) e. (ClWWalks ` G ) ) )
79	12, 78	sylbir 225	. . . . 5 |- ( ( N e. ( 0..^ ( # ` W ) ) /\ N =/= 0 ) -> ( W e. (ClWWalks ` G ) -> ( W cyclShift N ) e. (ClWWalks ` G ) ) )
80	79	expcom 451	. . . 4 |- ( N =/= 0 -> ( N e. ( 0..^ ( # ` W ) ) -> ( W e. (ClWWalks ` G ) -> ( W cyclShift N ) e. (ClWWalks ` G ) ) ) )
81	80	com13 88	. . 3 |- ( W e. (ClWWalks ` G ) -> ( N e. ( 0..^ ( # ` W ) ) -> ( N =/= 0 -> ( W cyclShift N ) e. (ClWWalks ` G ) ) ) )
82	81	imp 445	. 2 |- ( ( W e. (ClWWalks ` G ) /\ N e. ( 0..^ ( # ` W ) ) ) -> ( N =/= 0 -> ( W cyclShift N ) e. (ClWWalks ` G ) ) )
83	11, 82	pm2.61dne 2880	1 |- ( ( W e. (ClWWalks ` G ) /\ N e. ( 0..^ ( # ` W ) ) ) -> ( W cyclShift N ) e. (ClWWalks ` G ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   _Vcvv 3200   (/)c0 3915   {cpr 4179   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   - cmin 10266   ZZcz 11377   ...cfz 12326  ..^cfzo 12465   #chash 13117  Word cword 13291   lastS clsw 13292   cyclShift ccsh 13534  Vtxcvtx 25874  Edgcedg 25939  ClWWalkscclwwlks 26875

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-map 7859  df-pm 7860  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-ico 12181  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-hash 13118  df-word 13299  df-lsw 13300  df-concat 13301  df-substr 13303  df-csh 13535  df-clwwlks 26877

This theorem is referenced by:  clwwisshclwwsn  26929