Theorem numclwlk1lem2foa 27224

Description: Going forth and back form the end of a (closed) walk: W represents the closed walk p₀, ..., p_n-3, p₀. With X = p₀ and Y = p_n-1, ( ( W ++ <" X "> ) ++ <" Y "> ) represents the closed walk p₀, ..., p_n-3, p₀, p_n-1, p₀. (Contributed by Alexander van der Vekens, 22-Sep-2018.) (Revised by AV, 29-May-2021.) (Proof shortened by AV, 30-Jun-2022.)

Hypotheses

Ref	Expression
extwwlkfab.v	|- V = (Vtx ` G )
extwwlkfab.f	|- F = ( v e. V , n e. NN |-> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } )
extwwlkfab.c	|- C = ( v e. V , n e. ( ZZ>= ` 2 ) |-> { w e. ( n ClWWalksN G ) | ( ( w ` 0 ) = v /\ ( w ` ( n - 2 ) ) = ( w ` 0 ) ) } )

Assertion

Ref	Expression
numclwlk1lem2foa	|- ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( ( W e. ( X F ( N - 2 ) ) /\ Y e. ( G NeighbVtx X ) ) -> ( ( W ++ <" X "> ) ++ <" Y "> ) e. ( X C N ) ) )

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

Allowed substitution hints:   C( w, v, n)   F( v, n)   W( v, n)   Y( v, n)

Proof of Theorem numclwlk1lem2foa

Dummy variable i is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1061	. . . . . . . . . 10 |- ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> G e. USGraph )
2		uz3m2nn 11731	. . . . . . . . . . 11 |- ( N e. ( ZZ>= ` 3 ) -> ( N - 2 ) e. NN )
3	2	3ad2ant3 1084	. . . . . . . . . 10 |- ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( N - 2 ) e. NN )
4		simp2 1062	. . . . . . . . . 10 |- ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> X e. V )
5		extwwlkfab.f	. . . . . . . . . . 11 |- F = ( v e. V , n e. NN |-> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } )
6		extwwlkfab.v	. . . . . . . . . . 11 |- V = (Vtx ` G )
7		eqid 2622	. . . . . . . . . . 11 |- (Edg ` G ) = (Edg ` G )
8	5, 6, 7	numclwwlkovfel2 27216	. . . . . . . . . 10 |- ( ( G e. USGraph /\ ( N - 2 ) e. NN /\ X e. V ) -> ( W e. ( X F ( N - 2 ) ) <-> ( ( W e. Word V /\ A. i e. ( 0..^ ( ( # ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) /\ { ( lastS ` W ) , ( W ` 0 ) } e. (Edg ` G ) ) /\ ( # ` W ) = ( N - 2 ) /\ ( W ` 0 ) = X ) ) )
9	1, 3, 4, 8	syl3anc 1326	. . . . . . . . 9 |- ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( W e. ( X F ( N - 2 ) ) <-> ( ( W e. Word V /\ A. i e. ( 0..^ ( ( # ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) /\ { ( lastS ` W ) , ( W ` 0 ) } e. (Edg ` G ) ) /\ ( # ` W ) = ( N - 2 ) /\ ( W ` 0 ) = X ) ) )
10	5, 6, 7	numclwwlkovf2ex 27219	. . . . . . . . . . . . . . . 16 |- ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ Y e. ( G NeighbVtx X ) /\ W e. ( X F ( N - 2 ) ) ) -> ( ( W ++ <" X "> ) ++ <" Y "> ) e. ( N ClWWalksN G ) )
11	10	ad4ant134 1296	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ ( W e. Word V /\ ( # ` W ) = ( N - 2 ) ) ) /\ Y e. ( G NeighbVtx X ) ) /\ W e. ( X F ( N - 2 ) ) ) -> ( ( W ++ <" X "> ) ++ <" Y "> ) e. ( N ClWWalksN G ) )
12		simplr 792	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ ( W e. Word V /\ ( # ` W ) = ( N - 2 ) ) ) /\ Y e. ( G NeighbVtx X ) ) -> ( W e. Word V /\ ( # ` W ) = ( N - 2 ) ) )
13		simplr 792	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( ( G e. USGraph /\ X e. V ) /\ Y e. ( G NeighbVtx X ) ) -> X e. V )
14	6	nbgrisvtx 26255	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( G e. USGraph /\ Y e. ( G NeighbVtx X ) ) -> Y e. V )
15	14	adantlr 751	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( ( G e. USGraph /\ X e. V ) /\ Y e. ( G NeighbVtx X ) ) -> Y e. V )
16	13, 15	jca 554	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( G e. USGraph /\ X e. V ) /\ Y e. ( G NeighbVtx X ) ) -> ( X e. V /\ Y e. V ) )
17	16	ex 450	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( G e. USGraph /\ X e. V ) -> ( Y e. ( G NeighbVtx X ) -> ( X e. V /\ Y e. V ) ) )
18	17	3adant3 1081	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( Y e. ( G NeighbVtx X ) -> ( X e. V /\ Y e. V ) ) )
19	18	adantr 481	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ ( W e. Word V /\ ( # ` W ) = ( N - 2 ) ) ) -> ( Y e. ( G NeighbVtx X ) -> ( X e. V /\ Y e. V ) ) )
20	19	imp 445	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ ( W e. Word V /\ ( # ` W ) = ( N - 2 ) ) ) /\ Y e. ( G NeighbVtx X ) ) -> ( X e. V /\ Y e. V ) )
21		simpll3 1102	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ ( W e. Word V /\ ( # ` W ) = ( N - 2 ) ) ) /\ Y e. ( G NeighbVtx X ) ) -> N e. ( ZZ>= ` 3 ) )
22	12, 20, 21	3jca 1242	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ ( W e. Word V /\ ( # ` W ) = ( N - 2 ) ) ) /\ Y e. ( G NeighbVtx X ) ) -> ( ( W e. Word V /\ ( # ` W ) = ( N - 2 ) ) /\ ( X e. V /\ Y e. V ) /\ N e. ( ZZ>= ` 3 ) ) )
23	22	adantr 481	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ ( W e. Word V /\ ( # ` W ) = ( N - 2 ) ) ) /\ Y e. ( G NeighbVtx X ) ) /\ W e. ( X F ( N - 2 ) ) ) -> ( ( W e. Word V /\ ( # ` W ) = ( N - 2 ) ) /\ ( X e. V /\ Y e. V ) /\ N e. ( ZZ>= ` 3 ) ) )
24		numclwlk1lem2foalem 27222	. . . . . . . . . . . . . . . . 17 |- ( ( ( W e. Word V /\ ( # ` W ) = ( N - 2 ) ) /\ ( X e. V /\ Y e. V ) /\ N e. ( ZZ>= ` 3 ) ) -> ( ( ( ( W ++ <" X "> ) ++ <" Y "> ) substr <. 0 , ( N - 2 ) >. ) = W /\ ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( N - 1 ) ) = Y /\ ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( N - 2 ) ) = X ) )
25	23, 24	syl 17	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ ( W e. Word V /\ ( # ` W ) = ( N - 2 ) ) ) /\ Y e. ( G NeighbVtx X ) ) /\ W e. ( X F ( N - 2 ) ) ) -> ( ( ( ( W ++ <" X "> ) ++ <" Y "> ) substr <. 0 , ( N - 2 ) >. ) = W /\ ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( N - 1 ) ) = Y /\ ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( N - 2 ) ) = X ) )
26		eleq1a 2696	. . . . . . . . . . . . . . . . . 18 |- ( W e. ( X F ( N - 2 ) ) -> ( ( ( ( W ++ <" X "> ) ++ <" Y "> ) substr <. 0 , ( N - 2 ) >. ) = W -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) substr <. 0 , ( N - 2 ) >. ) e. ( X F ( N - 2 ) ) ) )
27	26	adantl 482	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ ( W e. Word V /\ ( # ` W ) = ( N - 2 ) ) ) /\ Y e. ( G NeighbVtx X ) ) /\ W e. ( X F ( N - 2 ) ) ) -> ( ( ( ( W ++ <" X "> ) ++ <" Y "> ) substr <. 0 , ( N - 2 ) >. ) = W -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) substr <. 0 , ( N - 2 ) >. ) e. ( X F ( N - 2 ) ) ) )
28		eleq1a 2696	. . . . . . . . . . . . . . . . . . 19 |- ( Y e. ( G NeighbVtx X ) -> ( ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( N - 1 ) ) = Y -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( N - 1 ) ) e. ( G NeighbVtx X ) ) )
29	28	adantl 482	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ ( W e. Word V /\ ( # ` W ) = ( N - 2 ) ) ) /\ Y e. ( G NeighbVtx X ) ) -> ( ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( N - 1 ) ) = Y -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( N - 1 ) ) e. ( G NeighbVtx X ) ) )
30	29	adantr 481	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ ( W e. Word V /\ ( # ` W ) = ( N - 2 ) ) ) /\ Y e. ( G NeighbVtx X ) ) /\ W e. ( X F ( N - 2 ) ) ) -> ( ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( N - 1 ) ) = Y -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( N - 1 ) ) e. ( G NeighbVtx X ) ) )
31		idd 24	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ ( W e. Word V /\ ( # ` W ) = ( N - 2 ) ) ) /\ Y e. ( G NeighbVtx X ) ) /\ W e. ( X F ( N - 2 ) ) ) -> ( ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( N - 2 ) ) = X -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( N - 2 ) ) = X ) )
32	27, 30, 31	3anim123d 1406	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ ( W e. Word V /\ ( # ` W ) = ( N - 2 ) ) ) /\ Y e. ( G NeighbVtx X ) ) /\ W e. ( X F ( N - 2 ) ) ) -> ( ( ( ( ( W ++ <" X "> ) ++ <" Y "> ) substr <. 0 , ( N - 2 ) >. ) = W /\ ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( N - 1 ) ) = Y /\ ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( N - 2 ) ) = X ) -> ( ( ( ( W ++ <" X "> ) ++ <" Y "> ) substr <. 0 , ( N - 2 ) >. ) e. ( X F ( N - 2 ) ) /\ ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( N - 1 ) ) e. ( G NeighbVtx X ) /\ ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( N - 2 ) ) = X ) ) )
33	25, 32	mpd 15	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ ( W e. Word V /\ ( # ` W ) = ( N - 2 ) ) ) /\ Y e. ( G NeighbVtx X ) ) /\ W e. ( X F ( N - 2 ) ) ) -> ( ( ( ( W ++ <" X "> ) ++ <" Y "> ) substr <. 0 , ( N - 2 ) >. ) e. ( X F ( N - 2 ) ) /\ ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( N - 1 ) ) e. ( G NeighbVtx X ) /\ ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( N - 2 ) ) = X ) )
34	11, 33	jca 554	. . . . . . . . . . . . . 14 |- ( ( ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ ( W e. Word V /\ ( # ` W ) = ( N - 2 ) ) ) /\ Y e. ( G NeighbVtx X ) ) /\ W e. ( X F ( N - 2 ) ) ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) e. ( N ClWWalksN G ) /\ ( ( ( ( W ++ <" X "> ) ++ <" Y "> ) substr <. 0 , ( N - 2 ) >. ) e. ( X F ( N - 2 ) ) /\ ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( N - 1 ) ) e. ( G NeighbVtx X ) /\ ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( N - 2 ) ) = X ) ) )
35	34	exp31 630	. . . . . . . . . . . . 13 |- ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ ( W e. Word V /\ ( # ` W ) = ( N - 2 ) ) ) -> ( Y e. ( G NeighbVtx X ) -> ( W e. ( X F ( N - 2 ) ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) e. ( N ClWWalksN G ) /\ ( ( ( ( W ++ <" X "> ) ++ <" Y "> ) substr <. 0 , ( N - 2 ) >. ) e. ( X F ( N - 2 ) ) /\ ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( N - 1 ) ) e. ( G NeighbVtx X ) /\ ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( N - 2 ) ) = X ) ) ) ) )
36	35	expcom 451	. . . . . . . . . . . 12 |- ( ( W e. Word V /\ ( # ` W ) = ( N - 2 ) ) -> ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( Y e. ( G NeighbVtx X ) -> ( W e. ( X F ( N - 2 ) ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) e. ( N ClWWalksN G ) /\ ( ( ( ( W ++ <" X "> ) ++ <" Y "> ) substr <. 0 , ( N - 2 ) >. ) e. ( X F ( N - 2 ) ) /\ ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( N - 1 ) ) e. ( G NeighbVtx X ) /\ ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( N - 2 ) ) = X ) ) ) ) ) )
37	36	3ad2antl1 1223	. . . . . . . . . . 11 |- ( ( ( W e. Word V /\ A. i e. ( 0..^ ( ( # ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) /\ { ( lastS ` W ) , ( W ` 0 ) } e. (Edg ` G ) ) /\ ( # ` W ) = ( N - 2 ) ) -> ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( Y e. ( G NeighbVtx X ) -> ( W e. ( X F ( N - 2 ) ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) e. ( N ClWWalksN G ) /\ ( ( ( ( W ++ <" X "> ) ++ <" Y "> ) substr <. 0 , ( N - 2 ) >. ) e. ( X F ( N - 2 ) ) /\ ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( N - 1 ) ) e. ( G NeighbVtx X ) /\ ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( N - 2 ) ) = X ) ) ) ) ) )
38	37	3adant3 1081	. . . . . . . . . 10 |- ( ( ( W e. Word V /\ A. i e. ( 0..^ ( ( # ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) /\ { ( lastS ` W ) , ( W ` 0 ) } e. (Edg ` G ) ) /\ ( # ` W ) = ( N - 2 ) /\ ( W ` 0 ) = X ) -> ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( Y e. ( G NeighbVtx X ) -> ( W e. ( X F ( N - 2 ) ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) e. ( N ClWWalksN G ) /\ ( ( ( ( W ++ <" X "> ) ++ <" Y "> ) substr <. 0 , ( N - 2 ) >. ) e. ( X F ( N - 2 ) ) /\ ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( N - 1 ) ) e. ( G NeighbVtx X ) /\ ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( N - 2 ) ) = X ) ) ) ) ) )
39	38	com12 32	. . . . . . . . 9 |- ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( ( ( W e. Word V /\ A. i e. ( 0..^ ( ( # ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) /\ { ( lastS ` W ) , ( W ` 0 ) } e. (Edg ` G ) ) /\ ( # ` W ) = ( N - 2 ) /\ ( W ` 0 ) = X ) -> ( Y e. ( G NeighbVtx X ) -> ( W e. ( X F ( N - 2 ) ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) e. ( N ClWWalksN G ) /\ ( ( ( ( W ++ <" X "> ) ++ <" Y "> ) substr <. 0 , ( N - 2 ) >. ) e. ( X F ( N - 2 ) ) /\ ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( N - 1 ) ) e. ( G NeighbVtx X ) /\ ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( N - 2 ) ) = X ) ) ) ) ) )
40	9, 39	sylbid 230	. . . . . . . 8 |- ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( W e. ( X F ( N - 2 ) ) -> ( Y e. ( G NeighbVtx X ) -> ( W e. ( X F ( N - 2 ) ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) e. ( N ClWWalksN G ) /\ ( ( ( ( W ++ <" X "> ) ++ <" Y "> ) substr <. 0 , ( N - 2 ) >. ) e. ( X F ( N - 2 ) ) /\ ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( N - 1 ) ) e. ( G NeighbVtx X ) /\ ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( N - 2 ) ) = X ) ) ) ) ) )
41	40	com14 96	. . . . . . 7 |- ( W e. ( X F ( N - 2 ) ) -> ( W e. ( X F ( N - 2 ) ) -> ( Y e. ( G NeighbVtx X ) -> ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) e. ( N ClWWalksN G ) /\ ( ( ( ( W ++ <" X "> ) ++ <" Y "> ) substr <. 0 , ( N - 2 ) >. ) e. ( X F ( N - 2 ) ) /\ ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( N - 1 ) ) e. ( G NeighbVtx X ) /\ ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( N - 2 ) ) = X ) ) ) ) ) )
42	41	pm2.43i 52	. . . . . 6 |- ( W e. ( X F ( N - 2 ) ) -> ( Y e. ( G NeighbVtx X ) -> ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) e. ( N ClWWalksN G ) /\ ( ( ( ( W ++ <" X "> ) ++ <" Y "> ) substr <. 0 , ( N - 2 ) >. ) e. ( X F ( N - 2 ) ) /\ ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( N - 1 ) ) e. ( G NeighbVtx X ) /\ ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( N - 2 ) ) = X ) ) ) ) )
43	42	imp 445	. . . . 5 |- ( ( W e. ( X F ( N - 2 ) ) /\ Y e. ( G NeighbVtx X ) ) -> ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) e. ( N ClWWalksN G ) /\ ( ( ( ( W ++ <" X "> ) ++ <" Y "> ) substr <. 0 , ( N - 2 ) >. ) e. ( X F ( N - 2 ) ) /\ ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( N - 1 ) ) e. ( G NeighbVtx X ) /\ ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( N - 2 ) ) = X ) ) ) )
44	43	impcom 446	. . . 4 |- ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ ( W e. ( X F ( N - 2 ) ) /\ Y e. ( G NeighbVtx X ) ) ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) e. ( N ClWWalksN G ) /\ ( ( ( ( W ++ <" X "> ) ++ <" Y "> ) substr <. 0 , ( N - 2 ) >. ) e. ( X F ( N - 2 ) ) /\ ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( N - 1 ) ) e. ( G NeighbVtx X ) /\ ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( N - 2 ) ) = X ) ) )
45		oveq1 6657	. . . . . . 7 |- ( w = ( ( W ++ <" X "> ) ++ <" Y "> ) -> ( w substr <. 0 , ( N - 2 ) >. ) = ( ( ( W ++ <" X "> ) ++ <" Y "> ) substr <. 0 , ( N - 2 ) >. ) )
46	45	eleq1d 2686	. . . . . 6 |- ( w = ( ( W ++ <" X "> ) ++ <" Y "> ) -> ( ( w substr <. 0 , ( N - 2 ) >. ) e. ( X F ( N - 2 ) ) <-> ( ( ( W ++ <" X "> ) ++ <" Y "> ) substr <. 0 , ( N - 2 ) >. ) e. ( X F ( N - 2 ) ) ) )
47		fveq1 6190	. . . . . . 7 |- ( w = ( ( W ++ <" X "> ) ++ <" Y "> ) -> ( w ` ( N - 1 ) ) = ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( N - 1 ) ) )
48	47	eleq1d 2686	. . . . . 6 |- ( w = ( ( W ++ <" X "> ) ++ <" Y "> ) -> ( ( w ` ( N - 1 ) ) e. ( G NeighbVtx X ) <-> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( N - 1 ) ) e. ( G NeighbVtx X ) ) )
49		fveq1 6190	. . . . . . 7 |- ( w = ( ( W ++ <" X "> ) ++ <" Y "> ) -> ( w ` ( N - 2 ) ) = ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( N - 2 ) ) )
50	49	eqeq1d 2624	. . . . . 6 |- ( w = ( ( W ++ <" X "> ) ++ <" Y "> ) -> ( ( w ` ( N - 2 ) ) = X <-> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( N - 2 ) ) = X ) )
51	46, 48, 50	3anbi123d 1399	. . . . 5 |- ( w = ( ( W ++ <" X "> ) ++ <" Y "> ) -> ( ( ( w substr <. 0 , ( N - 2 ) >. ) e. ( X F ( N - 2 ) ) /\ ( w ` ( N - 1 ) ) e. ( G NeighbVtx X ) /\ ( w ` ( N - 2 ) ) = X ) <-> ( ( ( ( W ++ <" X "> ) ++ <" Y "> ) substr <. 0 , ( N - 2 ) >. ) e. ( X F ( N - 2 ) ) /\ ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( N - 1 ) ) e. ( G NeighbVtx X ) /\ ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( N - 2 ) ) = X ) ) )
52	51	elrab 3363	. . . 4 |- ( ( ( W ++ <" X "> ) ++ <" Y "> ) e. { w e. ( N ClWWalksN G ) | ( ( w substr <. 0 , ( N - 2 ) >. ) e. ( X F ( N - 2 ) ) /\ ( w ` ( N - 1 ) ) e. ( G NeighbVtx X ) /\ ( w ` ( N - 2 ) ) = X ) } <-> ( ( ( W ++ <" X "> ) ++ <" Y "> ) e. ( N ClWWalksN G ) /\ ( ( ( ( W ++ <" X "> ) ++ <" Y "> ) substr <. 0 , ( N - 2 ) >. ) e. ( X F ( N - 2 ) ) /\ ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( N - 1 ) ) e. ( G NeighbVtx X ) /\ ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( N - 2 ) ) = X ) ) )
53	44, 52	sylibr 224	. . 3 |- ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ ( W e. ( X F ( N - 2 ) ) /\ Y e. ( G NeighbVtx X ) ) ) -> ( ( W ++ <" X "> ) ++ <" Y "> ) e. { w e. ( N ClWWalksN G ) | ( ( w substr <. 0 , ( N - 2 ) >. ) e. ( X F ( N - 2 ) ) /\ ( w ` ( N - 1 ) ) e. ( G NeighbVtx X ) /\ ( w ` ( N - 2 ) ) = X ) } )
54		extwwlkfab.c	. . . . 5 |- C = ( v e. V , n e. ( ZZ>= ` 2 ) |-> { w e. ( n ClWWalksN G ) | ( ( w ` 0 ) = v /\ ( w ` ( n - 2 ) ) = ( w ` 0 ) ) } )
55	6, 5, 54	extwwlkfab 27223	. . . 4 |- ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( X C N ) = { w e. ( N ClWWalksN G ) | ( ( w substr <. 0 , ( N - 2 ) >. ) e. ( X F ( N - 2 ) ) /\ ( w ` ( N - 1 ) ) e. ( G NeighbVtx X ) /\ ( w ` ( N - 2 ) ) = X ) } )
56	55	adantr 481	. . 3 |- ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ ( W e. ( X F ( N - 2 ) ) /\ Y e. ( G NeighbVtx X ) ) ) -> ( X C N ) = { w e. ( N ClWWalksN G ) | ( ( w substr <. 0 , ( N - 2 ) >. ) e. ( X F ( N - 2 ) ) /\ ( w ` ( N - 1 ) ) e. ( G NeighbVtx X ) /\ ( w ` ( N - 2 ) ) = X ) } )
57	53, 56	eleqtrrd 2704	. 2 |- ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ ( W e. ( X F ( N - 2 ) ) /\ Y e. ( G NeighbVtx X ) ) ) -> ( ( W ++ <" X "> ) ++ <" Y "> ) e. ( X C N ) )
58	57	ex 450	1 |- ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( ( W e. ( X F ( N - 2 ) ) /\ Y e. ( G NeighbVtx X ) ) -> ( ( W ++ <" X "> ) ++ <" Y "> ) e. ( X C N ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   {cpr 4179   <.cop 4183   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   0cc0 9936   1c1 9937   + caddc 9939   - cmin 10266   NNcn 11020   2c2 11070   3c3 11071   ZZ>=cuz 11687  ..^cfzo 12465   #chash 13117  Word cword 13291   lastS clsw 13292   ++ cconcat 13293   <"cs1 13294   substr csubstr 13295  Vtxcvtx 25874  Edgcedg 25939   USGraph cusgr 26044   NeighbVtx cnbgr 26224   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-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-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-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-2o 7561  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-cda 8990  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-3 11080  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-substr 13303  df-edg 25940  df-upgr 25977  df-umgr 25978  df-usgr 26046  df-nbgr 26228  df-wwlks 26722  df-wwlksn 26723  df-clwwlks 26877  df-clwwlksn 26878

This theorem is referenced by:  numclwlk1lem2fo  27228