Theorem numclwwlk2lem1 27235

Description: In a friendship graph, for each walk of length n starting at a fixed vertex v and ending not at this vertex, there is a unique vertex so that the walk extended by an edge to this vertex and an edge from this vertex to the first vertex of the walk is a value of operation H. If the walk is represented as a word, it is sufficient to add one vertex to the word to obtain the closed walk contained in the value of operation H, since in a word representing a closed walk the starting vertex is not repeated at the end. This theorem generally holds only for Friendship Graphs, because these guarantee that for the first and last vertex there is a (unique) third vertex "in between". (Contributed by Alexander van der Vekens, 3-Oct-2018.) (Revised by AV, 30-May-2021.)

Hypotheses

Ref	Expression
numclwwlk.v	|- V = (Vtx ` G )
numclwwlk.q	|- Q = ( v e. V , n e. NN |-> { w e. ( n WWalksN G ) | ( ( w ` 0 ) = v /\ ( lastS ` w ) =/= v ) } )
numclwwlk.f	|- F = ( v e. V , n e. NN |-> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } )
numclwwlk.h	|- H = ( v e. V , n e. NN |-> { w e. ( n ClWWalksN G ) | ( ( w ` 0 ) = v /\ ( w ` ( n - 2 ) ) =/= ( w ` 0 ) ) } )

Assertion

Ref	Expression
numclwwlk2lem1	|- ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) -> ( W e. ( X Q N ) -> E! v e. V ( W ++ <" v "> ) e. ( X H ( N + 2 ) ) ) )

Distinct variable groups:   n, G, v, w   n, N, v, w   n, V, v   n, X, v, w   w, V   v, W, w

Allowed substitution hints:   Q( w, v, n)   F( w, v, n)   H( w, v, n)   W( n)

Proof of Theorem numclwwlk2lem1

Dummy variable i is distinct from all other variables.

Step	Hyp	Ref	Expression
1		numclwwlk.v	. . . . . 6 |- V = (Vtx ` G )
2		numclwwlk.q	. . . . . 6 |- Q = ( v e. V , n e. NN |-> { w e. ( n WWalksN G ) | ( ( w ` 0 ) = v /\ ( lastS ` w ) =/= v ) } )
3	1, 2	numclwwlkovq 27232	. . . . 5 |- ( ( X e. V /\ N e. NN ) -> ( X Q N ) = { w e. ( N WWalksN G ) | ( ( w ` 0 ) = X /\ ( lastS ` w ) =/= X ) } )
4	3	3adant1 1079	. . . 4 |- ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) -> ( X Q N ) = { w e. ( N WWalksN G ) | ( ( w ` 0 ) = X /\ ( lastS ` w ) =/= X ) } )
5	4	eleq2d 2687	. . 3 |- ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) -> ( W e. ( X Q N ) <-> W e. { w e. ( N WWalksN G ) | ( ( w ` 0 ) = X /\ ( lastS ` w ) =/= X ) } ) )
6		fveq1 6190	. . . . . 6 |- ( w = W -> ( w ` 0 ) = ( W ` 0 ) )
7	6	eqeq1d 2624	. . . . 5 |- ( w = W -> ( ( w ` 0 ) = X <-> ( W ` 0 ) = X ) )
8		fveq2 6191	. . . . . 6 |- ( w = W -> ( lastS ` w ) = ( lastS ` W ) )
9	8	neeq1d 2853	. . . . 5 |- ( w = W -> ( ( lastS ` w ) =/= X <-> ( lastS ` W ) =/= X ) )
10	7, 9	anbi12d 747	. . . 4 |- ( w = W -> ( ( ( w ` 0 ) = X /\ ( lastS ` w ) =/= X ) <-> ( ( W ` 0 ) = X /\ ( lastS ` W ) =/= X ) ) )
11	10	elrab 3363	. . 3 |- ( W e. { w e. ( N WWalksN G ) | ( ( w ` 0 ) = X /\ ( lastS ` w ) =/= X ) } <-> ( W e. ( N WWalksN G ) /\ ( ( W ` 0 ) = X /\ ( lastS ` W ) =/= X ) ) )
12	5, 11	syl6bb 276	. 2 |- ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) -> ( W e. ( X Q N ) <-> ( W e. ( N WWalksN G ) /\ ( ( W ` 0 ) = X /\ ( lastS ` W ) =/= X ) ) ) )
13		simpl1 1064	. . . . 5 |- ( ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) /\ ( W e. ( N WWalksN G ) /\ ( ( W ` 0 ) = X /\ ( lastS ` W ) =/= X ) ) ) -> G e. FriendGraph )
14		eqid 2622	. . . . . . . . . . . . 13 |- (Edg ` G ) = (Edg ` G )
15	1, 14	wwlknp 26734	. . . . . . . . . . . 12 |- ( W e. ( N WWalksN G ) -> ( W e. Word V /\ ( # ` W ) = ( N + 1 ) /\ A. i e. ( 0..^ N ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) ) )
16		peano2nn 11032	. . . . . . . . . . . . . . . 16 |- ( N e. NN -> ( N + 1 ) e. NN )
17	16	adantl 482	. . . . . . . . . . . . . . 15 |- ( ( ( W e. Word V /\ ( # ` W ) = ( N + 1 ) ) /\ N e. NN ) -> ( N + 1 ) e. NN )
18		simpl 473	. . . . . . . . . . . . . . 15 |- ( ( ( W e. Word V /\ ( # ` W ) = ( N + 1 ) ) /\ N e. NN ) -> ( W e. Word V /\ ( # ` W ) = ( N + 1 ) ) )
19	17, 18	jca 554	. . . . . . . . . . . . . 14 |- ( ( ( W e. Word V /\ ( # ` W ) = ( N + 1 ) ) /\ N e. NN ) -> ( ( N + 1 ) e. NN /\ ( W e. Word V /\ ( # ` W ) = ( N + 1 ) ) ) )
20	19	ex 450	. . . . . . . . . . . . 13 |- ( ( W e. Word V /\ ( # ` W ) = ( N + 1 ) ) -> ( N e. NN -> ( ( N + 1 ) e. NN /\ ( W e. Word V /\ ( # ` W ) = ( N + 1 ) ) ) ) )
21	20	3adant3 1081	. . . . . . . . . . . 12 |- ( ( W e. Word V /\ ( # ` W ) = ( N + 1 ) /\ A. i e. ( 0..^ N ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) ) -> ( N e. NN -> ( ( N + 1 ) e. NN /\ ( W e. Word V /\ ( # ` W ) = ( N + 1 ) ) ) ) )
22	15, 21	syl 17	. . . . . . . . . . 11 |- ( W e. ( N WWalksN G ) -> ( N e. NN -> ( ( N + 1 ) e. NN /\ ( W e. Word V /\ ( # ` W ) = ( N + 1 ) ) ) ) )
23		lswlgt0cl 13356	. . . . . . . . . . 11 |- ( ( ( N + 1 ) e. NN /\ ( W e. Word V /\ ( # ` W ) = ( N + 1 ) ) ) -> ( lastS ` W ) e. V )
24	22, 23	syl6 35	. . . . . . . . . 10 |- ( W e. ( N WWalksN G ) -> ( N e. NN -> ( lastS ` W ) e. V ) )
25	24	adantr 481	. . . . . . . . 9 |- ( ( W e. ( N WWalksN G ) /\ ( ( W ` 0 ) = X /\ ( lastS ` W ) =/= X ) ) -> ( N e. NN -> ( lastS ` W ) e. V ) )
26	25	com12 32	. . . . . . . 8 |- ( N e. NN -> ( ( W e. ( N WWalksN G ) /\ ( ( W ` 0 ) = X /\ ( lastS ` W ) =/= X ) ) -> ( lastS ` W ) e. V ) )
27	26	3ad2ant3 1084	. . . . . . 7 |- ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) -> ( ( W e. ( N WWalksN G ) /\ ( ( W ` 0 ) = X /\ ( lastS ` W ) =/= X ) ) -> ( lastS ` W ) e. V ) )
28	27	imp 445	. . . . . 6 |- ( ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) /\ ( W e. ( N WWalksN G ) /\ ( ( W ` 0 ) = X /\ ( lastS ` W ) =/= X ) ) ) -> ( lastS ` W ) e. V )
29		eleq1 2689	. . . . . . . . . . 11 |- ( ( W ` 0 ) = X -> ( ( W ` 0 ) e. V <-> X e. V ) )
30	29	biimprd 238	. . . . . . . . . 10 |- ( ( W ` 0 ) = X -> ( X e. V -> ( W ` 0 ) e. V ) )
31	30	ad2antrl 764	. . . . . . . . 9 |- ( ( W e. ( N WWalksN G ) /\ ( ( W ` 0 ) = X /\ ( lastS ` W ) =/= X ) ) -> ( X e. V -> ( W ` 0 ) e. V ) )
32	31	com12 32	. . . . . . . 8 |- ( X e. V -> ( ( W e. ( N WWalksN G ) /\ ( ( W ` 0 ) = X /\ ( lastS ` W ) =/= X ) ) -> ( W ` 0 ) e. V ) )
33	32	3ad2ant2 1083	. . . . . . 7 |- ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) -> ( ( W e. ( N WWalksN G ) /\ ( ( W ` 0 ) = X /\ ( lastS ` W ) =/= X ) ) -> ( W ` 0 ) e. V ) )
34	33	imp 445	. . . . . 6 |- ( ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) /\ ( W e. ( N WWalksN G ) /\ ( ( W ` 0 ) = X /\ ( lastS ` W ) =/= X ) ) ) -> ( W ` 0 ) e. V )
35		neeq2 2857	. . . . . . . . . 10 |- ( X = ( W ` 0 ) -> ( ( lastS ` W ) =/= X <-> ( lastS ` W ) =/= ( W ` 0 ) ) )
36	35	eqcoms 2630	. . . . . . . . 9 |- ( ( W ` 0 ) = X -> ( ( lastS ` W ) =/= X <-> ( lastS ` W ) =/= ( W ` 0 ) ) )
37	36	biimpa 501	. . . . . . . 8 |- ( ( ( W ` 0 ) = X /\ ( lastS ` W ) =/= X ) -> ( lastS ` W ) =/= ( W ` 0 ) )
38	37	adantl 482	. . . . . . 7 |- ( ( W e. ( N WWalksN G ) /\ ( ( W ` 0 ) = X /\ ( lastS ` W ) =/= X ) ) -> ( lastS ` W ) =/= ( W ` 0 ) )
39	38	adantl 482	. . . . . 6 |- ( ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) /\ ( W e. ( N WWalksN G ) /\ ( ( W ` 0 ) = X /\ ( lastS ` W ) =/= X ) ) ) -> ( lastS ` W ) =/= ( W ` 0 ) )
40	28, 34, 39	3jca 1242	. . . . 5 |- ( ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) /\ ( W e. ( N WWalksN G ) /\ ( ( W ` 0 ) = X /\ ( lastS ` W ) =/= X ) ) ) -> ( ( lastS ` W ) e. V /\ ( W ` 0 ) e. V /\ ( lastS ` W ) =/= ( W ` 0 ) ) )
41	1, 14	frcond2 27131	. . . . 5 |- ( G e. FriendGraph -> ( ( ( lastS ` W ) e. V /\ ( W ` 0 ) e. V /\ ( lastS ` W ) =/= ( W ` 0 ) ) -> E! v e. V ( { ( lastS ` W ) , v } e. (Edg ` G ) /\ { v , ( W ` 0 ) } e. (Edg ` G ) ) ) )
42	13, 40, 41	sylc 65	. . . 4 |- ( ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) /\ ( W e. ( N WWalksN G ) /\ ( ( W ` 0 ) = X /\ ( lastS ` W ) =/= X ) ) ) -> E! v e. V ( { ( lastS ` W ) , v } e. (Edg ` G ) /\ { v , ( W ` 0 ) } e. (Edg ` G ) ) )
43		simpl 473	. . . . . . . . . 10 |- ( ( W e. ( N WWalksN G ) /\ ( ( W ` 0 ) = X /\ ( lastS ` W ) =/= X ) ) -> W e. ( N WWalksN G ) )
44	43	ad2antlr 763	. . . . . . . . 9 |- ( ( ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) /\ ( W e. ( N WWalksN G ) /\ ( ( W ` 0 ) = X /\ ( lastS ` W ) =/= X ) ) ) /\ v e. V ) -> W e. ( N WWalksN G ) )
45		simpr 477	. . . . . . . . 9 |- ( ( ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) /\ ( W e. ( N WWalksN G ) /\ ( ( W ` 0 ) = X /\ ( lastS ` W ) =/= X ) ) ) /\ v e. V ) -> v e. V )
46		nnnn0 11299	. . . . . . . . . . 11 |- ( N e. NN -> N e. NN0 )
47	46	3ad2ant3 1084	. . . . . . . . . 10 |- ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) -> N e. NN0 )
48	47	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) /\ ( W e. ( N WWalksN G ) /\ ( ( W ` 0 ) = X /\ ( lastS ` W ) =/= X ) ) ) /\ v e. V ) -> N e. NN0 )
49	44, 45, 48	3jca 1242	. . . . . . . 8 |- ( ( ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) /\ ( W e. ( N WWalksN G ) /\ ( ( W ` 0 ) = X /\ ( lastS ` W ) =/= X ) ) ) /\ v e. V ) -> ( W e. ( N WWalksN G ) /\ v e. V /\ N e. NN0 ) )
50	1, 14	wwlksext2clwwlk 26924	. . . . . . . . 9 |- ( ( W e. ( N WWalksN G ) /\ v e. V /\ N e. NN0 ) -> ( ( { ( lastS ` W ) , v } e. (Edg ` G ) /\ { v , ( W ` 0 ) } e. (Edg ` G ) ) -> ( W ++ <" v "> ) e. ( ( N + 2 ) ClWWalksN G ) ) )
51	50	imp 445	. . . . . . . 8 |- ( ( ( W e. ( N WWalksN G ) /\ v e. V /\ N e. NN0 ) /\ ( { ( lastS ` W ) , v } e. (Edg ` G ) /\ { v , ( W ` 0 ) } e. (Edg ` G ) ) ) -> ( W ++ <" v "> ) e. ( ( N + 2 ) ClWWalksN G ) )
52	49, 51	sylan 488	. . . . . . 7 |- ( ( ( ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) /\ ( W e. ( N WWalksN G ) /\ ( ( W ` 0 ) = X /\ ( lastS ` W ) =/= X ) ) ) /\ v e. V ) /\ ( { ( lastS ` W ) , v } e. (Edg ` G ) /\ { v , ( W ` 0 ) } e. (Edg ` G ) ) ) -> ( W ++ <" v "> ) e. ( ( N + 2 ) ClWWalksN G ) )
53	1	wwlknbp 26733	. . . . . . . . . . 11 |- ( W e. ( N WWalksN G ) -> ( G e. _V /\ N e. NN0 /\ W e. Word V ) )
54	53	simp3d 1075	. . . . . . . . . 10 |- ( W e. ( N WWalksN G ) -> W e. Word V )
55	54	ad2antrl 764	. . . . . . . . 9 |- ( ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) /\ ( W e. ( N WWalksN G ) /\ ( ( W ` 0 ) = X /\ ( lastS ` W ) =/= X ) ) ) -> W e. Word V )
56	55	ad2antrr 762	. . . . . . . 8 |- ( ( ( ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) /\ ( W e. ( N WWalksN G ) /\ ( ( W ` 0 ) = X /\ ( lastS ` W ) =/= X ) ) ) /\ v e. V ) /\ ( W ++ <" v "> ) e. ( ( N + 2 ) ClWWalksN G ) ) -> W e. Word V )
57	45	adantr 481	. . . . . . . 8 |- ( ( ( ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) /\ ( W e. ( N WWalksN G ) /\ ( ( W ` 0 ) = X /\ ( lastS ` W ) =/= X ) ) ) /\ v e. V ) /\ ( W ++ <" v "> ) e. ( ( N + 2 ) ClWWalksN G ) ) -> v e. V )
58		2z 11409	. . . . . . . . . . 11 |- 2 e. ZZ
59		nn0pzuz 11745	. . . . . . . . . . 11 |- ( ( N e. NN0 /\ 2 e. ZZ ) -> ( N + 2 ) e. ( ZZ>= ` 2 ) )
60	46, 58, 59	sylancl 694	. . . . . . . . . 10 |- ( N e. NN -> ( N + 2 ) e. ( ZZ>= ` 2 ) )
61	60	3ad2ant3 1084	. . . . . . . . 9 |- ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) -> ( N + 2 ) e. ( ZZ>= ` 2 ) )
62	61	ad3antrrr 766	. . . . . . . 8 |- ( ( ( ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) /\ ( W e. ( N WWalksN G ) /\ ( ( W ` 0 ) = X /\ ( lastS ` W ) =/= X ) ) ) /\ v e. V ) /\ ( W ++ <" v "> ) e. ( ( N + 2 ) ClWWalksN G ) ) -> ( N + 2 ) e. ( ZZ>= ` 2 ) )
63		simpr 477	. . . . . . . 8 |- ( ( ( ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) /\ ( W e. ( N WWalksN G ) /\ ( ( W ` 0 ) = X /\ ( lastS ` W ) =/= X ) ) ) /\ v e. V ) /\ ( W ++ <" v "> ) e. ( ( N + 2 ) ClWWalksN G ) ) -> ( W ++ <" v "> ) e. ( ( N + 2 ) ClWWalksN G ) )
64	1, 14	clwwlksext2edg 26923	. . . . . . . 8 |- ( ( ( W e. Word V /\ v e. V /\ ( N + 2 ) e. ( ZZ>= ` 2 ) ) /\ ( W ++ <" v "> ) e. ( ( N + 2 ) ClWWalksN G ) ) -> ( { ( lastS ` W ) , v } e. (Edg ` G ) /\ { v , ( W ` 0 ) } e. (Edg ` G ) ) )
65	56, 57, 62, 63, 64	syl31anc 1329	. . . . . . 7 |- ( ( ( ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) /\ ( W e. ( N WWalksN G ) /\ ( ( W ` 0 ) = X /\ ( lastS ` W ) =/= X ) ) ) /\ v e. V ) /\ ( W ++ <" v "> ) e. ( ( N + 2 ) ClWWalksN G ) ) -> ( { ( lastS ` W ) , v } e. (Edg ` G ) /\ { v , ( W ` 0 ) } e. (Edg ` G ) ) )
66	52, 65	impbida 877	. . . . . 6 |- ( ( ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) /\ ( W e. ( N WWalksN G ) /\ ( ( W ` 0 ) = X /\ ( lastS ` W ) =/= X ) ) ) /\ v e. V ) -> ( ( { ( lastS ` W ) , v } e. (Edg ` G ) /\ { v , ( W ` 0 ) } e. (Edg ` G ) ) <-> ( W ++ <" v "> ) e. ( ( N + 2 ) ClWWalksN G ) ) )
67	47	adantr 481	. . . . . . . . . . 11 |- ( ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) /\ ( W e. ( N WWalksN G ) /\ ( ( W ` 0 ) = X /\ ( lastS ` W ) =/= X ) ) ) -> N e. NN0 )
68	1	eleq2i 2693	. . . . . . . . . . . 12 |- ( v e. V <-> v e. (Vtx ` G ) )
69	68	biimpi 206	. . . . . . . . . . 11 |- ( v e. V -> v e. (Vtx ` G ) )
70	67, 69	anim12i 590	. . . . . . . . . 10 |- ( ( ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) /\ ( W e. ( N WWalksN G ) /\ ( ( W ` 0 ) = X /\ ( lastS ` W ) =/= X ) ) ) /\ v e. V ) -> ( N e. NN0 /\ v e. (Vtx ` G ) ) )
71	37	anim2i 593	. . . . . . . . . . 11 |- ( ( W e. ( N WWalksN G ) /\ ( ( W ` 0 ) = X /\ ( lastS ` W ) =/= X ) ) -> ( W e. ( N WWalksN G ) /\ ( lastS ` W ) =/= ( W ` 0 ) ) )
72	71	ad2antlr 763	. . . . . . . . . 10 |- ( ( ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) /\ ( W e. ( N WWalksN G ) /\ ( ( W ` 0 ) = X /\ ( lastS ` W ) =/= X ) ) ) /\ v e. V ) -> ( W e. ( N WWalksN G ) /\ ( lastS ` W ) =/= ( W ` 0 ) ) )
73		clwwlkextfrlem1 27208	. . . . . . . . . 10 |- ( ( ( N e. NN0 /\ v e. (Vtx ` G ) ) /\ ( W e. ( N WWalksN G ) /\ ( lastS ` W ) =/= ( W ` 0 ) ) ) -> ( ( ( W ++ <" v "> ) ` 0 ) = ( W ` 0 ) /\ ( ( W ++ <" v "> ) ` N ) =/= ( W ` 0 ) ) )
74	70, 72, 73	syl2anc 693	. . . . . . . . 9 |- ( ( ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) /\ ( W e. ( N WWalksN G ) /\ ( ( W ` 0 ) = X /\ ( lastS ` W ) =/= X ) ) ) /\ v e. V ) -> ( ( ( W ++ <" v "> ) ` 0 ) = ( W ` 0 ) /\ ( ( W ++ <" v "> ) ` N ) =/= ( W ` 0 ) ) )
75		eqeq2 2633	. . . . . . . . . . . . 13 |- ( X = ( W ` 0 ) -> ( ( ( W ++ <" v "> ) ` 0 ) = X <-> ( ( W ++ <" v "> ) ` 0 ) = ( W ` 0 ) ) )
76	75	eqcoms 2630	. . . . . . . . . . . 12 |- ( ( W ` 0 ) = X -> ( ( ( W ++ <" v "> ) ` 0 ) = X <-> ( ( W ++ <" v "> ) ` 0 ) = ( W ` 0 ) ) )
77	76	ad2antrl 764	. . . . . . . . . . 11 |- ( ( W e. ( N WWalksN G ) /\ ( ( W ` 0 ) = X /\ ( lastS ` W ) =/= X ) ) -> ( ( ( W ++ <" v "> ) ` 0 ) = X <-> ( ( W ++ <" v "> ) ` 0 ) = ( W ` 0 ) ) )
78	77	ad2antlr 763	. . . . . . . . . 10 |- ( ( ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) /\ ( W e. ( N WWalksN G ) /\ ( ( W ` 0 ) = X /\ ( lastS ` W ) =/= X ) ) ) /\ v e. V ) -> ( ( ( W ++ <" v "> ) ` 0 ) = X <-> ( ( W ++ <" v "> ) ` 0 ) = ( W ` 0 ) ) )
79	74	simpld 475	. . . . . . . . . . 11 |- ( ( ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) /\ ( W e. ( N WWalksN G ) /\ ( ( W ` 0 ) = X /\ ( lastS ` W ) =/= X ) ) ) /\ v e. V ) -> ( ( W ++ <" v "> ) ` 0 ) = ( W ` 0 ) )
80	79	neeq2d 2854	. . . . . . . . . 10 |- ( ( ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) /\ ( W e. ( N WWalksN G ) /\ ( ( W ` 0 ) = X /\ ( lastS ` W ) =/= X ) ) ) /\ v e. V ) -> ( ( ( W ++ <" v "> ) ` N ) =/= ( ( W ++ <" v "> ) ` 0 ) <-> ( ( W ++ <" v "> ) ` N ) =/= ( W ` 0 ) ) )
81	78, 80	anbi12d 747	. . . . . . . . 9 |- ( ( ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) /\ ( W e. ( N WWalksN G ) /\ ( ( W ` 0 ) = X /\ ( lastS ` W ) =/= X ) ) ) /\ v e. V ) -> ( ( ( ( W ++ <" v "> ) ` 0 ) = X /\ ( ( W ++ <" v "> ) ` N ) =/= ( ( W ++ <" v "> ) ` 0 ) ) <-> ( ( ( W ++ <" v "> ) ` 0 ) = ( W ` 0 ) /\ ( ( W ++ <" v "> ) ` N ) =/= ( W ` 0 ) ) ) )
82	74, 81	mpbird 247	. . . . . . . 8 |- ( ( ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) /\ ( W e. ( N WWalksN G ) /\ ( ( W ` 0 ) = X /\ ( lastS ` W ) =/= X ) ) ) /\ v e. V ) -> ( ( ( W ++ <" v "> ) ` 0 ) = X /\ ( ( W ++ <" v "> ) ` N ) =/= ( ( W ++ <" v "> ) ` 0 ) ) )
83		nncn 11028	. . . . . . . . . . . . . 14 |- ( N e. NN -> N e. CC )
84		2cnd 11093	. . . . . . . . . . . . . 14 |- ( N e. NN -> 2 e. CC )
85	83, 84	pncand 10393	. . . . . . . . . . . . 13 |- ( N e. NN -> ( ( N + 2 ) - 2 ) = N )
86	85	3ad2ant3 1084	. . . . . . . . . . . 12 |- ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) -> ( ( N + 2 ) - 2 ) = N )
87	86	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) /\ ( W e. ( N WWalksN G ) /\ ( ( W ` 0 ) = X /\ ( lastS ` W ) =/= X ) ) ) /\ v e. V ) -> ( ( N + 2 ) - 2 ) = N )
88	87	fveq2d 6195	. . . . . . . . . 10 |- ( ( ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) /\ ( W e. ( N WWalksN G ) /\ ( ( W ` 0 ) = X /\ ( lastS ` W ) =/= X ) ) ) /\ v e. V ) -> ( ( W ++ <" v "> ) ` ( ( N + 2 ) - 2 ) ) = ( ( W ++ <" v "> ) ` N ) )
89	88	neeq1d 2853	. . . . . . . . 9 |- ( ( ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) /\ ( W e. ( N WWalksN G ) /\ ( ( W ` 0 ) = X /\ ( lastS ` W ) =/= X ) ) ) /\ v e. V ) -> ( ( ( W ++ <" v "> ) ` ( ( N + 2 ) - 2 ) ) =/= ( ( W ++ <" v "> ) ` 0 ) <-> ( ( W ++ <" v "> ) ` N ) =/= ( ( W ++ <" v "> ) ` 0 ) ) )
90	89	anbi2d 740	. . . . . . . 8 |- ( ( ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) /\ ( W e. ( N WWalksN G ) /\ ( ( W ` 0 ) = X /\ ( lastS ` W ) =/= X ) ) ) /\ v e. V ) -> ( ( ( ( W ++ <" v "> ) ` 0 ) = X /\ ( ( W ++ <" v "> ) ` ( ( N + 2 ) - 2 ) ) =/= ( ( W ++ <" v "> ) ` 0 ) ) <-> ( ( ( W ++ <" v "> ) ` 0 ) = X /\ ( ( W ++ <" v "> ) ` N ) =/= ( ( W ++ <" v "> ) ` 0 ) ) ) )
91	82, 90	mpbird 247	. . . . . . 7 |- ( ( ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) /\ ( W e. ( N WWalksN G ) /\ ( ( W ` 0 ) = X /\ ( lastS ` W ) =/= X ) ) ) /\ v e. V ) -> ( ( ( W ++ <" v "> ) ` 0 ) = X /\ ( ( W ++ <" v "> ) ` ( ( N + 2 ) - 2 ) ) =/= ( ( W ++ <" v "> ) ` 0 ) ) )
92	91	biantrud 528	. . . . . 6 |- ( ( ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) /\ ( W e. ( N WWalksN G ) /\ ( ( W ` 0 ) = X /\ ( lastS ` W ) =/= X ) ) ) /\ v e. V ) -> ( ( W ++ <" v "> ) e. ( ( N + 2 ) ClWWalksN G ) <-> ( ( W ++ <" v "> ) e. ( ( N + 2 ) ClWWalksN G ) /\ ( ( ( W ++ <" v "> ) ` 0 ) = X /\ ( ( W ++ <" v "> ) ` ( ( N + 2 ) - 2 ) ) =/= ( ( W ++ <" v "> ) ` 0 ) ) ) ) )
93		2nn 11185	. . . . . . . . . . . . 13 |- 2 e. NN
94		nnaddcl 11042	. . . . . . . . . . . . 13 |- ( ( N e. NN /\ 2 e. NN ) -> ( N + 2 ) e. NN )
95	93, 94	mpan2 707	. . . . . . . . . . . 12 |- ( N e. NN -> ( N + 2 ) e. NN )
96	95	anim2i 593	. . . . . . . . . . 11 |- ( ( X e. V /\ N e. NN ) -> ( X e. V /\ ( N + 2 ) e. NN ) )
97	96	3adant1 1079	. . . . . . . . . 10 |- ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) -> ( X e. V /\ ( N + 2 ) e. NN ) )
98	97	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) /\ ( W e. ( N WWalksN G ) /\ ( ( W ` 0 ) = X /\ ( lastS ` W ) =/= X ) ) ) /\ v e. V ) -> ( X e. V /\ ( N + 2 ) e. NN ) )
99		numclwwlk.f	. . . . . . . . . 10 |- F = ( v e. V , n e. NN |-> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } )
100		numclwwlk.h	. . . . . . . . . 10 |- H = ( v e. V , n e. NN |-> { w e. ( n ClWWalksN G ) | ( ( w ` 0 ) = v /\ ( w ` ( n - 2 ) ) =/= ( w ` 0 ) ) } )
101	1, 2, 99, 100	numclwwlkovh 27234	. . . . . . . . 9 |- ( ( X e. V /\ ( N + 2 ) e. NN ) -> ( X H ( N + 2 ) ) = { w e. ( ( N + 2 ) ClWWalksN G ) | ( ( w ` 0 ) = X /\ ( w ` ( ( N + 2 ) - 2 ) ) =/= ( w ` 0 ) ) } )
102	98, 101	syl 17	. . . . . . . 8 |- ( ( ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) /\ ( W e. ( N WWalksN G ) /\ ( ( W ` 0 ) = X /\ ( lastS ` W ) =/= X ) ) ) /\ v e. V ) -> ( X H ( N + 2 ) ) = { w e. ( ( N + 2 ) ClWWalksN G ) | ( ( w ` 0 ) = X /\ ( w ` ( ( N + 2 ) - 2 ) ) =/= ( w ` 0 ) ) } )
103	102	eleq2d 2687	. . . . . . 7 |- ( ( ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) /\ ( W e. ( N WWalksN G ) /\ ( ( W ` 0 ) = X /\ ( lastS ` W ) =/= X ) ) ) /\ v e. V ) -> ( ( W ++ <" v "> ) e. ( X H ( N + 2 ) ) <-> ( W ++ <" v "> ) e. { w e. ( ( N + 2 ) ClWWalksN G ) | ( ( w ` 0 ) = X /\ ( w ` ( ( N + 2 ) - 2 ) ) =/= ( w ` 0 ) ) } ) )
104		fveq1 6190	. . . . . . . . . 10 |- ( w = ( W ++ <" v "> ) -> ( w ` 0 ) = ( ( W ++ <" v "> ) ` 0 ) )
105	104	eqeq1d 2624	. . . . . . . . 9 |- ( w = ( W ++ <" v "> ) -> ( ( w ` 0 ) = X <-> ( ( W ++ <" v "> ) ` 0 ) = X ) )
106		fveq1 6190	. . . . . . . . . 10 |- ( w = ( W ++ <" v "> ) -> ( w ` ( ( N + 2 ) - 2 ) ) = ( ( W ++ <" v "> ) ` ( ( N + 2 ) - 2 ) ) )
107	106, 104	neeq12d 2855	. . . . . . . . 9 |- ( w = ( W ++ <" v "> ) -> ( ( w ` ( ( N + 2 ) - 2 ) ) =/= ( w ` 0 ) <-> ( ( W ++ <" v "> ) ` ( ( N + 2 ) - 2 ) ) =/= ( ( W ++ <" v "> ) ` 0 ) ) )
108	105, 107	anbi12d 747	. . . . . . . 8 |- ( w = ( W ++ <" v "> ) -> ( ( ( w ` 0 ) = X /\ ( w ` ( ( N + 2 ) - 2 ) ) =/= ( w ` 0 ) ) <-> ( ( ( W ++ <" v "> ) ` 0 ) = X /\ ( ( W ++ <" v "> ) ` ( ( N + 2 ) - 2 ) ) =/= ( ( W ++ <" v "> ) ` 0 ) ) ) )
109	108	elrab 3363	. . . . . . 7 |- ( ( W ++ <" v "> ) e. { w e. ( ( N + 2 ) ClWWalksN G ) | ( ( w ` 0 ) = X /\ ( w ` ( ( N + 2 ) - 2 ) ) =/= ( w ` 0 ) ) } <-> ( ( W ++ <" v "> ) e. ( ( N + 2 ) ClWWalksN G ) /\ ( ( ( W ++ <" v "> ) ` 0 ) = X /\ ( ( W ++ <" v "> ) ` ( ( N + 2 ) - 2 ) ) =/= ( ( W ++ <" v "> ) ` 0 ) ) ) )
110	103, 109	syl6rbb 277	. . . . . 6 |- ( ( ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) /\ ( W e. ( N WWalksN G ) /\ ( ( W ` 0 ) = X /\ ( lastS ` W ) =/= X ) ) ) /\ v e. V ) -> ( ( ( W ++ <" v "> ) e. ( ( N + 2 ) ClWWalksN G ) /\ ( ( ( W ++ <" v "> ) ` 0 ) = X /\ ( ( W ++ <" v "> ) ` ( ( N + 2 ) - 2 ) ) =/= ( ( W ++ <" v "> ) ` 0 ) ) ) <-> ( W ++ <" v "> ) e. ( X H ( N + 2 ) ) ) )
111	66, 92, 110	3bitrd 294	. . . . 5 |- ( ( ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) /\ ( W e. ( N WWalksN G ) /\ ( ( W ` 0 ) = X /\ ( lastS ` W ) =/= X ) ) ) /\ v e. V ) -> ( ( { ( lastS ` W ) , v } e. (Edg ` G ) /\ { v , ( W ` 0 ) } e. (Edg ` G ) ) <-> ( W ++ <" v "> ) e. ( X H ( N + 2 ) ) ) )
112	111	reubidva 3125	. . . 4 |- ( ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) /\ ( W e. ( N WWalksN G ) /\ ( ( W ` 0 ) = X /\ ( lastS ` W ) =/= X ) ) ) -> ( E! v e. V ( { ( lastS ` W ) , v } e. (Edg ` G ) /\ { v , ( W ` 0 ) } e. (Edg ` G ) ) <-> E! v e. V ( W ++ <" v "> ) e. ( X H ( N + 2 ) ) ) )
113	42, 112	mpbid 222	. . 3 |- ( ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) /\ ( W e. ( N WWalksN G ) /\ ( ( W ` 0 ) = X /\ ( lastS ` W ) =/= X ) ) ) -> E! v e. V ( W ++ <" v "> ) e. ( X H ( N + 2 ) ) )
114	113	ex 450	. 2 |- ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) -> ( ( W e. ( N WWalksN G ) /\ ( ( W ` 0 ) = X /\ ( lastS ` W ) =/= X ) ) -> E! v e. V ( W ++ <" v "> ) e. ( X H ( N + 2 ) ) ) )
115	12, 114	sylbid 230	1 |- ( ( G e. FriendGraph /\ X e. V /\ N e. NN ) -> ( W e. ( X Q N ) -> E! v e. V ( W ++ <" v "> ) e. ( X H ( N + 2 ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E!wreu 2914   {crab 2916   _Vcvv 3200   {cpr 4179   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   0cc0 9936   1c1 9937   + caddc 9939   - cmin 10266   NNcn 11020   2c2 11070   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687  ..^cfzo 12465   #chash 13117  Word cword 13291   lastS clsw 13292   ++ cconcat 13293   <"cs1 13294  Vtxcvtx 25874  Edgcedg 25939   WWalksN cwwlksn 26718   ClWWalksN cclwwlksn 26876   FriendGraph cfrgr 27120

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

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-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-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-lsw 13300  df-concat 13301  df-s1 13302  df-wwlks 26722  df-wwlksn 26723  df-clwwlks 26877  df-clwwlksn 26878  df-frgr 27121

This theorem is referenced by:  numclwlk2lem2f1o  27238