Theorem umgrhashecclwwlk 26955

Description: The size of every equivalence class of the equivalence relation over the set of closed walks (defined as words) with a fixed length which is a prime number equals this length (in an undirected simple graph). (Contributed by Alexander van der Vekens, 17-Jun-2018.) (Revised by AV, 1-May-2021.)

Hypotheses

Ref	Expression
erclwwlksn.w	|- W = ( N ClWWalksN G )
erclwwlksn.r	|- .~ = { <. t , u >. | ( t e. W /\ u e. W /\ E. n e. ( 0 ... N ) t = ( u cyclShift n ) ) }

Assertion

Ref	Expression
umgrhashecclwwlk	|- ( ( G e. UMGraph /\ N e. Prime ) -> ( U e. ( W /. .~ ) -> ( # ` U ) = N ) )

Distinct variable groups:   t, W, u   n, N, u, t   n, W   n, G, u   U, n, u

Allowed substitution hints:   .~ ( u, t, n)   U( t)   G( t)

Proof of Theorem umgrhashecclwwlk

Dummy variables x y m i are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		erclwwlksn.w	. . . . 5 |- W = ( N ClWWalksN G )
2		erclwwlksn.r	. . . . 5 |- .~ = { <. t , u >. | ( t e. W /\ u e. W /\ E. n e. ( 0 ... N ) t = ( u cyclShift n ) ) }
3	1, 2	eclclwwlksn1 26952	. . . 4 |- ( U e. ( W /. .~ ) -> ( U e. ( W /. .~ ) <-> E. x e. W U = { y e. W | E. n e. ( 0 ... N ) y = ( x cyclShift n ) } ) )
4		rabeq 3192	. . . . . . . . . 10 |- ( W = ( N ClWWalksN G ) -> { y e. W | E. n e. ( 0 ... N ) y = ( x cyclShift n ) } = { y e. ( N ClWWalksN G ) | E. n e. ( 0 ... N ) y = ( x cyclShift n ) } )
5	1, 4	mp1i 13	. . . . . . . . 9 |- ( ( ( G e. UMGraph /\ N e. Prime ) /\ x e. W ) -> { y e. W | E. n e. ( 0 ... N ) y = ( x cyclShift n ) } = { y e. ( N ClWWalksN G ) | E. n e. ( 0 ... N ) y = ( x cyclShift n ) } )
6		prmnn 15388	. . . . . . . . . . . 12 |- ( N e. Prime -> N e. NN )
7	6	nnnn0d 11351	. . . . . . . . . . 11 |- ( N e. Prime -> N e. NN0 )
8	7	adantl 482	. . . . . . . . . 10 |- ( ( G e. UMGraph /\ N e. Prime ) -> N e. NN0 )
9	1	eleq2i 2693	. . . . . . . . . . 11 |- ( x e. W <-> x e. ( N ClWWalksN G ) )
10	9	biimpi 206	. . . . . . . . . 10 |- ( x e. W -> x e. ( N ClWWalksN G ) )
11		clwwlksnscsh 26940	. . . . . . . . . 10 |- ( ( N e. NN0 /\ x e. ( N ClWWalksN G ) ) -> { y e. ( N ClWWalksN G ) | E. n e. ( 0 ... N ) y = ( x cyclShift n ) } = { y e. Word (Vtx ` G ) | E. n e. ( 0 ... N ) y = ( x cyclShift n ) } )
12	8, 10, 11	syl2an 494	. . . . . . . . 9 |- ( ( ( G e. UMGraph /\ N e. Prime ) /\ x e. W ) -> { y e. ( N ClWWalksN G ) | E. n e. ( 0 ... N ) y = ( x cyclShift n ) } = { y e. Word (Vtx ` G ) | E. n e. ( 0 ... N ) y = ( x cyclShift n ) } )
13	5, 12	eqtrd 2656	. . . . . . . 8 |- ( ( ( G e. UMGraph /\ N e. Prime ) /\ x e. W ) -> { y e. W | E. n e. ( 0 ... N ) y = ( x cyclShift n ) } = { y e. Word (Vtx ` G ) | E. n e. ( 0 ... N ) y = ( x cyclShift n ) } )
14	13	eqeq2d 2632	. . . . . . 7 |- ( ( ( G e. UMGraph /\ N e. Prime ) /\ x e. W ) -> ( U = { y e. W | E. n e. ( 0 ... N ) y = ( x cyclShift n ) } <-> U = { y e. Word (Vtx ` G ) | E. n e. ( 0 ... N ) y = ( x cyclShift n ) } ) )
15	6	adantl 482	. . . . . . . . . . . 12 |- ( ( G e. UMGraph /\ N e. Prime ) -> N e. NN )
16		simpll 790	. . . . . . . . . . . . . . . 16 |- ( ( ( x e. Word (Vtx ` G ) /\ ( # ` x ) = N ) /\ N e. NN ) -> x e. Word (Vtx ` G ) )
17		elnnne0 11306	. . . . . . . . . . . . . . . . . 18 |- ( N e. NN <-> ( N e. NN0 /\ N =/= 0 ) )
18		eqeq1 2626	. . . . . . . . . . . . . . . . . . . . . 22 |- ( N = ( # ` x ) -> ( N = 0 <-> ( # ` x ) = 0 ) )
19	18	eqcoms 2630	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( # ` x ) = N -> ( N = 0 <-> ( # ` x ) = 0 ) )
20		hasheq0 13154	. . . . . . . . . . . . . . . . . . . . 21 |- ( x e. Word (Vtx ` G ) -> ( ( # ` x ) = 0 <-> x = (/) ) )
21	19, 20	sylan9bbr 737	. . . . . . . . . . . . . . . . . . . 20 |- ( ( x e. Word (Vtx ` G ) /\ ( # ` x ) = N ) -> ( N = 0 <-> x = (/) ) )
22	21	necon3bid 2838	. . . . . . . . . . . . . . . . . . 19 |- ( ( x e. Word (Vtx ` G ) /\ ( # ` x ) = N ) -> ( N =/= 0 <-> x =/= (/) ) )
23	22	biimpcd 239	. . . . . . . . . . . . . . . . . 18 |- ( N =/= 0 -> ( ( x e. Word (Vtx ` G ) /\ ( # ` x ) = N ) -> x =/= (/) ) )
24	17, 23	simplbiim 659	. . . . . . . . . . . . . . . . 17 |- ( N e. NN -> ( ( x e. Word (Vtx ` G ) /\ ( # ` x ) = N ) -> x =/= (/) ) )
25	24	impcom 446	. . . . . . . . . . . . . . . 16 |- ( ( ( x e. Word (Vtx ` G ) /\ ( # ` x ) = N ) /\ N e. NN ) -> x =/= (/) )
26		simplr 792	. . . . . . . . . . . . . . . . 17 |- ( ( ( x e. Word (Vtx ` G ) /\ ( # ` x ) = N ) /\ N e. NN ) -> ( # ` x ) = N )
27	26	eqcomd 2628	. . . . . . . . . . . . . . . 16 |- ( ( ( x e. Word (Vtx ` G ) /\ ( # ` x ) = N ) /\ N e. NN ) -> N = ( # ` x ) )
28	16, 25, 27	3jca 1242	. . . . . . . . . . . . . . 15 |- ( ( ( x e. Word (Vtx ` G ) /\ ( # ` x ) = N ) /\ N e. NN ) -> ( x e. Word (Vtx ` G ) /\ x =/= (/) /\ N = ( # ` x ) ) )
29	28	ex 450	. . . . . . . . . . . . . 14 |- ( ( x e. Word (Vtx ` G ) /\ ( # ` x ) = N ) -> ( N e. NN -> ( x e. Word (Vtx ` G ) /\ x =/= (/) /\ N = ( # ` x ) ) ) )
30		eqid 2622	. . . . . . . . . . . . . . 15 |- (Vtx ` G ) = (Vtx ` G )
31	30	clwwlknbp 26885	. . . . . . . . . . . . . 14 |- ( x e. ( N ClWWalksN G ) -> ( x e. Word (Vtx ` G ) /\ ( # ` x ) = N ) )
32	29, 31	syl11 33	. . . . . . . . . . . . 13 |- ( N e. NN -> ( x e. ( N ClWWalksN G ) -> ( x e. Word (Vtx ` G ) /\ x =/= (/) /\ N = ( # ` x ) ) ) )
33	9, 32	syl5bi 232	. . . . . . . . . . . 12 |- ( N e. NN -> ( x e. W -> ( x e. Word (Vtx ` G ) /\ x =/= (/) /\ N = ( # ` x ) ) ) )
34	15, 33	syl 17	. . . . . . . . . . 11 |- ( ( G e. UMGraph /\ N e. Prime ) -> ( x e. W -> ( x e. Word (Vtx ` G ) /\ x =/= (/) /\ N = ( # ` x ) ) ) )
35	34	imp 445	. . . . . . . . . 10 |- ( ( ( G e. UMGraph /\ N e. Prime ) /\ x e. W ) -> ( x e. Word (Vtx ` G ) /\ x =/= (/) /\ N = ( # ` x ) ) )
36		scshwfzeqfzo 13572	. . . . . . . . . 10 |- ( ( x e. Word (Vtx ` G ) /\ x =/= (/) /\ N = ( # ` x ) ) -> { y e. Word (Vtx ` G ) | E. n e. ( 0 ... N ) y = ( x cyclShift n ) } = { y e. Word (Vtx ` G ) | E. n e. ( 0..^ N ) y = ( x cyclShift n ) } )
37	35, 36	syl 17	. . . . . . . . 9 |- ( ( ( G e. UMGraph /\ N e. Prime ) /\ x e. W ) -> { y e. Word (Vtx ` G ) | E. n e. ( 0 ... N ) y = ( x cyclShift n ) } = { y e. Word (Vtx ` G ) | E. n e. ( 0..^ N ) y = ( x cyclShift n ) } )
38	37	eqeq2d 2632	. . . . . . . 8 |- ( ( ( G e. UMGraph /\ N e. Prime ) /\ x e. W ) -> ( U = { y e. Word (Vtx ` G ) | E. n e. ( 0 ... N ) y = ( x cyclShift n ) } <-> U = { y e. Word (Vtx ` G ) | E. n e. ( 0..^ N ) y = ( x cyclShift n ) } ) )
39		fveq2 6191	. . . . . . . . . . . . . . 15 |- ( U = { y e. Word (Vtx ` G ) | E. n e. ( 0..^ ( # ` x ) ) y = ( x cyclShift n ) } -> ( # ` U ) = ( # ` { y e. Word (Vtx ` G ) | E. n e. ( 0..^ ( # ` x ) ) y = ( x cyclShift n ) } ) )
40		simprl 794	. . . . . . . . . . . . . . . . 17 |- ( ( ( x e. Word (Vtx ` G ) /\ x e. ( ( # ` x ) ClWWalksN G ) ) /\ ( G e. UMGraph /\ ( # ` x ) e. Prime ) ) -> G e. UMGraph )
41		prmuz2 15408	. . . . . . . . . . . . . . . . . . 19 |- ( ( # ` x ) e. Prime -> ( # ` x ) e. ( ZZ>= ` 2 ) )
42	41	adantl 482	. . . . . . . . . . . . . . . . . 18 |- ( ( G e. UMGraph /\ ( # ` x ) e. Prime ) -> ( # ` x ) e. ( ZZ>= ` 2 ) )
43	42	adantl 482	. . . . . . . . . . . . . . . . 17 |- ( ( ( x e. Word (Vtx ` G ) /\ x e. ( ( # ` x ) ClWWalksN G ) ) /\ ( G e. UMGraph /\ ( # ` x ) e. Prime ) ) -> ( # ` x ) e. ( ZZ>= ` 2 ) )
44		simplr 792	. . . . . . . . . . . . . . . . 17 |- ( ( ( x e. Word (Vtx ` G ) /\ x e. ( ( # ` x ) ClWWalksN G ) ) /\ ( G e. UMGraph /\ ( # ` x ) e. Prime ) ) -> x e. ( ( # ` x ) ClWWalksN G ) )
45		umgr2cwwkdifex 26942	. . . . . . . . . . . . . . . . 17 |- ( ( G e. UMGraph /\ ( # ` x ) e. ( ZZ>= ` 2 ) /\ x e. ( ( # ` x ) ClWWalksN G ) ) -> E. i e. ( 0..^ ( # ` x ) ) ( x ` i ) =/= ( x ` 0 ) )
46	40, 43, 44, 45	syl3anc 1326	. . . . . . . . . . . . . . . 16 |- ( ( ( x e. Word (Vtx ` G ) /\ x e. ( ( # ` x ) ClWWalksN G ) ) /\ ( G e. UMGraph /\ ( # ` x ) e. Prime ) ) -> E. i e. ( 0..^ ( # ` x ) ) ( x ` i ) =/= ( x ` 0 ) )
47		oveq2 6658	. . . . . . . . . . . . . . . . . . . . . 22 |- ( n = m -> ( x cyclShift n ) = ( x cyclShift m ) )
48	47	eqeq2d 2632	. . . . . . . . . . . . . . . . . . . . 21 |- ( n = m -> ( y = ( x cyclShift n ) <-> y = ( x cyclShift m ) ) )
49	48	cbvrexv 3172	. . . . . . . . . . . . . . . . . . . 20 |- ( E. n e. ( 0..^ ( # ` x ) ) y = ( x cyclShift n ) <-> E. m e. ( 0..^ ( # ` x ) ) y = ( x cyclShift m ) )
50		eqeq1 2626	. . . . . . . . . . . . . . . . . . . . . 22 |- ( y = u -> ( y = ( x cyclShift m ) <-> u = ( x cyclShift m ) ) )
51		eqcom 2629	. . . . . . . . . . . . . . . . . . . . . 22 |- ( u = ( x cyclShift m ) <-> ( x cyclShift m ) = u )
52	50, 51	syl6bb 276	. . . . . . . . . . . . . . . . . . . . 21 |- ( y = u -> ( y = ( x cyclShift m ) <-> ( x cyclShift m ) = u ) )
53	52	rexbidv 3052	. . . . . . . . . . . . . . . . . . . 20 |- ( y = u -> ( E. m e. ( 0..^ ( # ` x ) ) y = ( x cyclShift m ) <-> E. m e. ( 0..^ ( # ` x ) ) ( x cyclShift m ) = u ) )
54	49, 53	syl5bb 272	. . . . . . . . . . . . . . . . . . 19 |- ( y = u -> ( E. n e. ( 0..^ ( # ` x ) ) y = ( x cyclShift n ) <-> E. m e. ( 0..^ ( # ` x ) ) ( x cyclShift m ) = u ) )
55	54	cbvrabv 3199	. . . . . . . . . . . . . . . . . 18 |- { y e. Word (Vtx ` G ) | E. n e. ( 0..^ ( # ` x ) ) y = ( x cyclShift n ) } = { u e. Word (Vtx ` G ) | E. m e. ( 0..^ ( # ` x ) ) ( x cyclShift m ) = u }
56	55	cshwshashnsame 15810	. . . . . . . . . . . . . . . . 17 |- ( ( x e. Word (Vtx ` G ) /\ ( # ` x ) e. Prime ) -> ( E. i e. ( 0..^ ( # ` x ) ) ( x ` i ) =/= ( x ` 0 ) -> ( # ` { y e. Word (Vtx ` G ) | E. n e. ( 0..^ ( # ` x ) ) y = ( x cyclShift n ) } ) = ( # ` x ) ) )
57	56	ad2ant2rl 785	. . . . . . . . . . . . . . . 16 |- ( ( ( x e. Word (Vtx ` G ) /\ x e. ( ( # ` x ) ClWWalksN G ) ) /\ ( G e. UMGraph /\ ( # ` x ) e. Prime ) ) -> ( E. i e. ( 0..^ ( # ` x ) ) ( x ` i ) =/= ( x ` 0 ) -> ( # ` { y e. Word (Vtx ` G ) | E. n e. ( 0..^ ( # ` x ) ) y = ( x cyclShift n ) } ) = ( # ` x ) ) )
58	46, 57	mpd 15	. . . . . . . . . . . . . . 15 |- ( ( ( x e. Word (Vtx ` G ) /\ x e. ( ( # ` x ) ClWWalksN G ) ) /\ ( G e. UMGraph /\ ( # ` x ) e. Prime ) ) -> ( # ` { y e. Word (Vtx ` G ) | E. n e. ( 0..^ ( # ` x ) ) y = ( x cyclShift n ) } ) = ( # ` x ) )
59	39, 58	sylan9eqr 2678	. . . . . . . . . . . . . 14 |- ( ( ( ( x e. Word (Vtx ` G ) /\ x e. ( ( # ` x ) ClWWalksN G ) ) /\ ( G e. UMGraph /\ ( # ` x ) e. Prime ) ) /\ U = { y e. Word (Vtx ` G ) | E. n e. ( 0..^ ( # ` x ) ) y = ( x cyclShift n ) } ) -> ( # ` U ) = ( # ` x ) )
60	59	exp41 638	. . . . . . . . . . . . 13 |- ( x e. Word (Vtx ` G ) -> ( x e. ( ( # ` x ) ClWWalksN G ) -> ( ( G e. UMGraph /\ ( # ` x ) e. Prime ) -> ( U = { y e. Word (Vtx ` G ) | E. n e. ( 0..^ ( # ` x ) ) y = ( x cyclShift n ) } -> ( # ` U ) = ( # ` x ) ) ) ) )
61	60	adantr 481	. . . . . . . . . . . 12 |- ( ( x e. Word (Vtx ` G ) /\ ( # ` x ) = N ) -> ( x e. ( ( # ` x ) ClWWalksN G ) -> ( ( G e. UMGraph /\ ( # ` x ) e. Prime ) -> ( U = { y e. Word (Vtx ` G ) | E. n e. ( 0..^ ( # ` x ) ) y = ( x cyclShift n ) } -> ( # ` U ) = ( # ` x ) ) ) ) )
62		oveq1 6657	. . . . . . . . . . . . . . . 16 |- ( N = ( # ` x ) -> ( N ClWWalksN G ) = ( ( # ` x ) ClWWalksN G ) )
63	62	eleq2d 2687	. . . . . . . . . . . . . . 15 |- ( N = ( # ` x ) -> ( x e. ( N ClWWalksN G ) <-> x e. ( ( # ` x ) ClWWalksN G ) ) )
64		eleq1 2689	. . . . . . . . . . . . . . . . 17 |- ( N = ( # ` x ) -> ( N e. Prime <-> ( # ` x ) e. Prime ) )
65	64	anbi2d 740	. . . . . . . . . . . . . . . 16 |- ( N = ( # ` x ) -> ( ( G e. UMGraph /\ N e. Prime ) <-> ( G e. UMGraph /\ ( # ` x ) e. Prime ) ) )
66		oveq2 6658	. . . . . . . . . . . . . . . . . . . 20 |- ( N = ( # ` x ) -> ( 0..^ N ) = ( 0..^ ( # ` x ) ) )
67	66	rexeqdv 3145	. . . . . . . . . . . . . . . . . . 19 |- ( N = ( # ` x ) -> ( E. n e. ( 0..^ N ) y = ( x cyclShift n ) <-> E. n e. ( 0..^ ( # ` x ) ) y = ( x cyclShift n ) ) )
68	67	rabbidv 3189	. . . . . . . . . . . . . . . . . 18 |- ( N = ( # ` x ) -> { y e. Word (Vtx ` G ) | E. n e. ( 0..^ N ) y = ( x cyclShift n ) } = { y e. Word (Vtx ` G ) | E. n e. ( 0..^ ( # ` x ) ) y = ( x cyclShift n ) } )
69	68	eqeq2d 2632	. . . . . . . . . . . . . . . . 17 |- ( N = ( # ` x ) -> ( U = { y e. Word (Vtx ` G ) | E. n e. ( 0..^ N ) y = ( x cyclShift n ) } <-> U = { y e. Word (Vtx ` G ) | E. n e. ( 0..^ ( # ` x ) ) y = ( x cyclShift n ) } ) )
70		eqeq2 2633	. . . . . . . . . . . . . . . . 17 |- ( N = ( # ` x ) -> ( ( # ` U ) = N <-> ( # ` U ) = ( # ` x ) ) )
71	69, 70	imbi12d 334	. . . . . . . . . . . . . . . 16 |- ( N = ( # ` x ) -> ( ( U = { y e. Word (Vtx ` G ) | E. n e. ( 0..^ N ) y = ( x cyclShift n ) } -> ( # ` U ) = N ) <-> ( U = { y e. Word (Vtx ` G ) | E. n e. ( 0..^ ( # ` x ) ) y = ( x cyclShift n ) } -> ( # ` U ) = ( # ` x ) ) ) )
72	65, 71	imbi12d 334	. . . . . . . . . . . . . . 15 |- ( N = ( # ` x ) -> ( ( ( G e. UMGraph /\ N e. Prime ) -> ( U = { y e. Word (Vtx ` G ) | E. n e. ( 0..^ N ) y = ( x cyclShift n ) } -> ( # ` U ) = N ) ) <-> ( ( G e. UMGraph /\ ( # ` x ) e. Prime ) -> ( U = { y e. Word (Vtx ` G ) | E. n e. ( 0..^ ( # ` x ) ) y = ( x cyclShift n ) } -> ( # ` U ) = ( # ` x ) ) ) ) )
73	63, 72	imbi12d 334	. . . . . . . . . . . . . 14 |- ( N = ( # ` x ) -> ( ( x e. ( N ClWWalksN G ) -> ( ( G e. UMGraph /\ N e. Prime ) -> ( U = { y e. Word (Vtx ` G ) | E. n e. ( 0..^ N ) y = ( x cyclShift n ) } -> ( # ` U ) = N ) ) ) <-> ( x e. ( ( # ` x ) ClWWalksN G ) -> ( ( G e. UMGraph /\ ( # ` x ) e. Prime ) -> ( U = { y e. Word (Vtx ` G ) | E. n e. ( 0..^ ( # ` x ) ) y = ( x cyclShift n ) } -> ( # ` U ) = ( # ` x ) ) ) ) ) )
74	73	eqcoms 2630	. . . . . . . . . . . . 13 |- ( ( # ` x ) = N -> ( ( x e. ( N ClWWalksN G ) -> ( ( G e. UMGraph /\ N e. Prime ) -> ( U = { y e. Word (Vtx ` G ) | E. n e. ( 0..^ N ) y = ( x cyclShift n ) } -> ( # ` U ) = N ) ) ) <-> ( x e. ( ( # ` x ) ClWWalksN G ) -> ( ( G e. UMGraph /\ ( # ` x ) e. Prime ) -> ( U = { y e. Word (Vtx ` G ) | E. n e. ( 0..^ ( # ` x ) ) y = ( x cyclShift n ) } -> ( # ` U ) = ( # ` x ) ) ) ) ) )
75	74	adantl 482	. . . . . . . . . . . 12 |- ( ( x e. Word (Vtx ` G ) /\ ( # ` x ) = N ) -> ( ( x e. ( N ClWWalksN G ) -> ( ( G e. UMGraph /\ N e. Prime ) -> ( U = { y e. Word (Vtx ` G ) | E. n e. ( 0..^ N ) y = ( x cyclShift n ) } -> ( # ` U ) = N ) ) ) <-> ( x e. ( ( # ` x ) ClWWalksN G ) -> ( ( G e. UMGraph /\ ( # ` x ) e. Prime ) -> ( U = { y e. Word (Vtx ` G ) | E. n e. ( 0..^ ( # ` x ) ) y = ( x cyclShift n ) } -> ( # ` U ) = ( # ` x ) ) ) ) ) )
76	61, 75	mpbird 247	. . . . . . . . . . 11 |- ( ( x e. Word (Vtx ` G ) /\ ( # ` x ) = N ) -> ( x e. ( N ClWWalksN G ) -> ( ( G e. UMGraph /\ N e. Prime ) -> ( U = { y e. Word (Vtx ` G ) | E. n e. ( 0..^ N ) y = ( x cyclShift n ) } -> ( # ` U ) = N ) ) ) )
77	31, 76	mpcom 38	. . . . . . . . . 10 |- ( x e. ( N ClWWalksN G ) -> ( ( G e. UMGraph /\ N e. Prime ) -> ( U = { y e. Word (Vtx ` G ) | E. n e. ( 0..^ N ) y = ( x cyclShift n ) } -> ( # ` U ) = N ) ) )
78	77, 1	eleq2s 2719	. . . . . . . . 9 |- ( x e. W -> ( ( G e. UMGraph /\ N e. Prime ) -> ( U = { y e. Word (Vtx ` G ) | E. n e. ( 0..^ N ) y = ( x cyclShift n ) } -> ( # ` U ) = N ) ) )
79	78	impcom 446	. . . . . . . 8 |- ( ( ( G e. UMGraph /\ N e. Prime ) /\ x e. W ) -> ( U = { y e. Word (Vtx ` G ) | E. n e. ( 0..^ N ) y = ( x cyclShift n ) } -> ( # ` U ) = N ) )
80	38, 79	sylbid 230	. . . . . . 7 |- ( ( ( G e. UMGraph /\ N e. Prime ) /\ x e. W ) -> ( U = { y e. Word (Vtx ` G ) | E. n e. ( 0 ... N ) y = ( x cyclShift n ) } -> ( # ` U ) = N ) )
81	14, 80	sylbid 230	. . . . . 6 |- ( ( ( G e. UMGraph /\ N e. Prime ) /\ x e. W ) -> ( U = { y e. W | E. n e. ( 0 ... N ) y = ( x cyclShift n ) } -> ( # ` U ) = N ) )
82	81	rexlimdva 3031	. . . . 5 |- ( ( G e. UMGraph /\ N e. Prime ) -> ( E. x e. W U = { y e. W | E. n e. ( 0 ... N ) y = ( x cyclShift n ) } -> ( # ` U ) = N ) )
83	82	com12 32	. . . 4 |- ( E. x e. W U = { y e. W | E. n e. ( 0 ... N ) y = ( x cyclShift n ) } -> ( ( G e. UMGraph /\ N e. Prime ) -> ( # ` U ) = N ) )
84	3, 83	syl6bi 243	. . 3 |- ( U e. ( W /. .~ ) -> ( U e. ( W /. .~ ) -> ( ( G e. UMGraph /\ N e. Prime ) -> ( # ` U ) = N ) ) )
85	84	pm2.43i 52	. 2 |- ( U e. ( W /. .~ ) -> ( ( G e. UMGraph /\ N e. Prime ) -> ( # ` U ) = N ) )
86	85	com12 32	1 |- ( ( G e. UMGraph /\ N e. Prime ) -> ( U e. ( W /. .~ ) -> ( # ` U ) = N ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   {crab 2916   (/)c0 3915   {copab 4712   ` cfv 5888  (class class class)co 6650   /.cqs 7741   0cc0 9936   NNcn 11020   2c2 11070   NN0cn0 11292   ZZ>=cuz 11687   ...cfz 12326  ..^cfzo 12465   #chash 13117  Word cword 13291   cyclShift ccsh 13534   Primecprime 15385  Vtxcvtx 25874   UMGraph cumgr 25976   ClWWalksN cclwwlksn 26876

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-inf2 8538  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-fal 1489  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-disj 4621  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-se 5074  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-isom 5897  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-2o 7561  df-oadd 7564  df-er 7742  df-ec 7744  df-qs 7748  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-oi 8415  df-card 8765  df-cda 8990  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-3 11080  df-n0 11293  df-xnn0 11364  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-seq 12802  df-exp 12861  df-hash 13118  df-word 13299  df-lsw 13300  df-concat 13301  df-substr 13303  df-reps 13306  df-csh 13535  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-dvds 14984  df-gcd 15217  df-prm 15386  df-phi 15471  df-edg 25940  df-umgr 25978  df-clwwlks 26877  df-clwwlksn 26878

This theorem is referenced by:  fusgrhashclwwlkn  26956