Theorem numclwwlkovf2ex 27219

Description: Extending a closed walk starting at a fixed vertex by an additional edge (forth and back). (Contributed by AV, 22-Sep-2018.) (Proof shortened by AV, 25-Jan-2022.)

Hypotheses

Ref	Expression
numclwwlkovf.f	|- F = ( v e. V , n e. NN |-> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } )
numclwwlkffin.v	|- V = (Vtx ` G )
numclwwlkovfel2.e	|- E = (Edg ` G )

Assertion

Ref	Expression
numclwwlkovf2ex	|- ( ( ( 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 ) )

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

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

Proof of Theorem numclwwlkovf2ex

Dummy variable i is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1061	. . . . . 6 |- ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> G e. USGraph )
2		uz3m2nn 11731	. . . . . . 7 |- ( N e. ( ZZ>= ` 3 ) -> ( N - 2 ) e. NN )
3	2	3ad2ant3 1084	. . . . . 6 |- ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( N - 2 ) e. NN )
4		simp2 1062	. . . . . 6 |- ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> X e. V )
5		numclwwlkovf.f	. . . . . . 7 |- F = ( v e. V , n e. NN |-> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } )
6		numclwwlkffin.v	. . . . . . 7 |- V = (Vtx ` G )
7		numclwwlkovfel2.e	. . . . . . 7 |- E = (Edg ` G )
8	5, 6, 7	numclwwlkovfel2 27216	. . . . . 6 |- ( ( 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. E /\ { ( lastS ` W ) , ( W ` 0 ) } e. E ) /\ ( # ` W ) = ( N - 2 ) /\ ( W ` 0 ) = X ) ) )
9	1, 3, 4, 8	syl3anc 1326	. . . . 5 |- ( ( 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. E /\ { ( lastS ` W ) , ( W ` 0 ) } e. E ) /\ ( # ` W ) = ( N - 2 ) /\ ( W ` 0 ) = X ) ) )
10		simpr11 1145	. . . . . . . . . 10 |- ( ( ( 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. E /\ { ( lastS ` W ) , ( W ` 0 ) } e. E ) /\ ( # ` W ) = ( N - 2 ) /\ ( W ` 0 ) = X ) ) -> W e. Word V )
11	10	adantr 481	. . . . . . . . 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. E /\ { ( lastS ` W ) , ( W ` 0 ) } e. E ) /\ ( # ` W ) = ( N - 2 ) /\ ( W ` 0 ) = X ) ) /\ Y e. ( G NeighbVtx X ) ) -> W e. Word V )
12		simpll2 1101	. . . . . . . . 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. E /\ { ( lastS ` W ) , ( W ` 0 ) } e. E ) /\ ( # ` W ) = ( N - 2 ) /\ ( W ` 0 ) = X ) ) /\ Y e. ( G NeighbVtx X ) ) -> X e. V )
13	6	nbgrisvtx 26255	. . . . . . . . . . . . 13 |- ( ( G e. USGraph /\ Y e. ( G NeighbVtx X ) ) -> Y e. V )
14	13	ex 450	. . . . . . . . . . . 12 |- ( G e. USGraph -> ( Y e. ( G NeighbVtx X ) -> Y e. V ) )
15	14	3ad2ant1 1082	. . . . . . . . . . 11 |- ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( Y e. ( G NeighbVtx X ) -> Y e. V ) )
16	15	adantr 481	. . . . . . . . . 10 |- ( ( ( 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. E /\ { ( lastS ` W ) , ( W ` 0 ) } e. E ) /\ ( # ` W ) = ( N - 2 ) /\ ( W ` 0 ) = X ) ) -> ( Y e. ( G NeighbVtx X ) -> Y e. V ) )
17	16	imp 445	. . . . . . . . 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. E /\ { ( lastS ` W ) , ( W ` 0 ) } e. E ) /\ ( # ` W ) = ( N - 2 ) /\ ( W ` 0 ) = X ) ) /\ Y e. ( G NeighbVtx X ) ) -> Y e. V )
18		ccatw2s1cl 13401	. . . . . . . . 9 |- ( ( W e. Word V /\ X e. V /\ Y e. V ) -> ( ( W ++ <" X "> ) ++ <" Y "> ) e. Word V )
19	11, 12, 17, 18	syl3anc 1326	. . . . . . . 8 |- ( ( ( ( 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. E /\ { ( lastS ` W ) , ( W ` 0 ) } e. E ) /\ ( # ` W ) = ( N - 2 ) /\ ( W ` 0 ) = X ) ) /\ Y e. ( G NeighbVtx X ) ) -> ( ( W ++ <" X "> ) ++ <" Y "> ) e. Word V )
20	6, 7	numclwwlkovf2exlem2 27212	. . . . . . . . 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. E /\ { ( lastS ` W ) , ( W ` 0 ) } e. E ) /\ ( # ` W ) = ( N - 2 ) /\ ( W ` 0 ) = X ) ) /\ Y e. ( G NeighbVtx X ) ) -> A. i e. ( ( 0..^ ( ( # ` W ) - 1 ) ) u. { ( ( # ` W ) - 1 ) , ( # ` W ) } ) { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` i ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( i + 1 ) ) } e. E )
21		simp11 1091	. . . . . . . . . . . . . . 15 |- ( ( ( W e. Word V /\ A. i e. ( 0..^ ( ( # ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { ( lastS ` W ) , ( W ` 0 ) } e. E ) /\ ( # ` W ) = ( N - 2 ) /\ ( W ` 0 ) = X ) -> W e. Word V )
22	21	ad2antlr 763	. . . . . . . . . . . . . 14 |- ( ( ( ( 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. E /\ { ( lastS ` W ) , ( W ` 0 ) } e. E ) /\ ( # ` W ) = ( N - 2 ) /\ ( W ` 0 ) = X ) ) /\ Y e. ( G NeighbVtx X ) ) -> W e. Word V )
23		ccatw2s1len 13402	. . . . . . . . . . . . . 14 |- ( ( W e. Word V /\ X e. V /\ Y e. V ) -> ( # ` ( ( W ++ <" X "> ) ++ <" Y "> ) ) = ( ( # ` W ) + 2 ) )
24	22, 12, 17, 23	syl3anc 1326	. . . . . . . . . . . . 13 |- ( ( ( ( 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. E /\ { ( lastS ` W ) , ( W ` 0 ) } e. E ) /\ ( # ` W ) = ( N - 2 ) /\ ( W ` 0 ) = X ) ) /\ Y e. ( G NeighbVtx X ) ) -> ( # ` ( ( W ++ <" X "> ) ++ <" Y "> ) ) = ( ( # ` W ) + 2 ) )
25	24	oveq1d 6665	. . . . . . . . . . . 12 |- ( ( ( ( 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. E /\ { ( lastS ` W ) , ( W ` 0 ) } e. E ) /\ ( # ` W ) = ( N - 2 ) /\ ( W ` 0 ) = X ) ) /\ Y e. ( G NeighbVtx X ) ) -> ( ( # ` ( ( W ++ <" X "> ) ++ <" Y "> ) ) - 1 ) = ( ( ( # ` W ) + 2 ) - 1 ) )
26	25	oveq2d 6666	. . . . . . . . . . 11 |- ( ( ( ( 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. E /\ { ( lastS ` W ) , ( W ` 0 ) } e. E ) /\ ( # ` W ) = ( N - 2 ) /\ ( W ` 0 ) = X ) ) /\ Y e. ( G NeighbVtx X ) ) -> ( 0..^ ( ( # ` ( ( W ++ <" X "> ) ++ <" Y "> ) ) - 1 ) ) = ( 0..^ ( ( ( # ` W ) + 2 ) - 1 ) ) )
27		simp3 1063	. . . . . . . . . . . . . 14 |- ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> N e. ( ZZ>= ` 3 ) )
28		simp2 1062	. . . . . . . . . . . . . 14 |- ( ( ( W e. Word V /\ A. i e. ( 0..^ ( ( # ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { ( lastS ` W ) , ( W ` 0 ) } e. E ) /\ ( # ` W ) = ( N - 2 ) /\ ( W ` 0 ) = X ) -> ( # ` W ) = ( N - 2 ) )
29	27, 28	anim12i 590	. . . . . . . . . . . . 13 |- ( ( ( 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. E /\ { ( lastS ` W ) , ( W ` 0 ) } e. E ) /\ ( # ` W ) = ( N - 2 ) /\ ( W ` 0 ) = X ) ) -> ( N e. ( ZZ>= ` 3 ) /\ ( # ` W ) = ( N - 2 ) ) )
30	29	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ( 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. E /\ { ( lastS ` W ) , ( W ` 0 ) } e. E ) /\ ( # ` W ) = ( N - 2 ) /\ ( W ` 0 ) = X ) ) /\ Y e. ( G NeighbVtx X ) ) -> ( N e. ( ZZ>= ` 3 ) /\ ( # ` W ) = ( N - 2 ) ) )
31		numclwwlkovf2exlem1 27211	. . . . . . . . . . . 12 |- ( ( N e. ( ZZ>= ` 3 ) /\ ( # ` W ) = ( N - 2 ) ) -> ( 0..^ ( ( ( # ` W ) + 2 ) - 1 ) ) = ( ( 0..^ ( ( # ` W ) - 1 ) ) u. { ( ( # ` W ) - 1 ) , ( # ` W ) } ) )
32	30, 31	syl 17	. . . . . . . . . . 11 |- ( ( ( ( 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. E /\ { ( lastS ` W ) , ( W ` 0 ) } e. E ) /\ ( # ` W ) = ( N - 2 ) /\ ( W ` 0 ) = X ) ) /\ Y e. ( G NeighbVtx X ) ) -> ( 0..^ ( ( ( # ` W ) + 2 ) - 1 ) ) = ( ( 0..^ ( ( # ` W ) - 1 ) ) u. { ( ( # ` W ) - 1 ) , ( # ` W ) } ) )
33	26, 32	eqtrd 2656	. . . . . . . . . 10 |- ( ( ( ( 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. E /\ { ( lastS ` W ) , ( W ` 0 ) } e. E ) /\ ( # ` W ) = ( N - 2 ) /\ ( W ` 0 ) = X ) ) /\ Y e. ( G NeighbVtx X ) ) -> ( 0..^ ( ( # ` ( ( W ++ <" X "> ) ++ <" Y "> ) ) - 1 ) ) = ( ( 0..^ ( ( # ` W ) - 1 ) ) u. { ( ( # ` W ) - 1 ) , ( # ` W ) } ) )
34	33	raleqdv 3144	. . . . . . . . 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. E /\ { ( lastS ` W ) , ( W ` 0 ) } e. E ) /\ ( # ` W ) = ( N - 2 ) /\ ( W ` 0 ) = X ) ) /\ Y e. ( G NeighbVtx X ) ) -> ( A. i e. ( 0..^ ( ( # ` ( ( W ++ <" X "> ) ++ <" Y "> ) ) - 1 ) ) { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` i ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( i + 1 ) ) } e. E <-> A. i e. ( ( 0..^ ( ( # ` W ) - 1 ) ) u. { ( ( # ` W ) - 1 ) , ( # ` W ) } ) { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` i ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( i + 1 ) ) } e. E ) )
35	20, 34	mpbird 247	. . . . . . . 8 |- ( ( ( ( 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. E /\ { ( lastS ` W ) , ( W ` 0 ) } e. E ) /\ ( # ` W ) = ( N - 2 ) /\ ( W ` 0 ) = X ) ) /\ Y e. ( G NeighbVtx X ) ) -> A. i e. ( 0..^ ( ( # ` ( ( W ++ <" X "> ) ++ <" Y "> ) ) - 1 ) ) { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` i ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( i + 1 ) ) } e. E )
36		ccatws1cl 13396	. . . . . . . . . . . 12 |- ( ( W e. Word V /\ X e. V ) -> ( W ++ <" X "> ) e. Word V )
37		lswccats1 13411	. . . . . . . . . . . 12 |- ( ( ( W ++ <" X "> ) e. Word V /\ Y e. V ) -> ( lastS ` ( ( W ++ <" X "> ) ++ <" Y "> ) ) = Y )
38	36, 37	stoic3 1701	. . . . . . . . . . 11 |- ( ( W e. Word V /\ X e. V /\ Y e. V ) -> ( lastS ` ( ( W ++ <" X "> ) ++ <" Y "> ) ) = Y )
39	22, 12, 17, 38	syl3anc 1326	. . . . . . . . . 10 |- ( ( ( ( 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. E /\ { ( lastS ` W ) , ( W ` 0 ) } e. E ) /\ ( # ` W ) = ( N - 2 ) /\ ( W ` 0 ) = X ) ) /\ Y e. ( G NeighbVtx X ) ) -> ( lastS ` ( ( W ++ <" X "> ) ++ <" Y "> ) ) = Y )
40	2	nngt0d 11064	. . . . . . . . . . . . . . . . 17 |- ( N e. ( ZZ>= ` 3 ) -> 0 < ( N - 2 ) )
41		breq2 4657	. . . . . . . . . . . . . . . . 17 |- ( ( # ` W ) = ( N - 2 ) -> ( 0 < ( # ` W ) <-> 0 < ( N - 2 ) ) )
42	40, 41	syl5ibr 236	. . . . . . . . . . . . . . . 16 |- ( ( # ` W ) = ( N - 2 ) -> ( N e. ( ZZ>= ` 3 ) -> 0 < ( # ` W ) ) )
43	42	3ad2ant2 1083	. . . . . . . . . . . . . . 15 |- ( ( ( W e. Word V /\ A. i e. ( 0..^ ( ( # ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { ( lastS ` W ) , ( W ` 0 ) } e. E ) /\ ( # ` W ) = ( N - 2 ) /\ ( W ` 0 ) = X ) -> ( N e. ( ZZ>= ` 3 ) -> 0 < ( # ` W ) ) )
44	43	com12 32	. . . . . . . . . . . . . 14 |- ( N e. ( ZZ>= ` 3 ) -> ( ( ( W e. Word V /\ A. i e. ( 0..^ ( ( # ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { ( lastS ` W ) , ( W ` 0 ) } e. E ) /\ ( # ` W ) = ( N - 2 ) /\ ( W ` 0 ) = X ) -> 0 < ( # ` W ) ) )
45	44	3ad2ant3 1084	. . . . . . . . . . . . 13 |- ( ( 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. E /\ { ( lastS ` W ) , ( W ` 0 ) } e. E ) /\ ( # ` W ) = ( N - 2 ) /\ ( W ` 0 ) = X ) -> 0 < ( # ` W ) ) )
46	45	imp 445	. . . . . . . . . . . 12 |- ( ( ( 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. E /\ { ( lastS ` W ) , ( W ` 0 ) } e. E ) /\ ( # ` W ) = ( N - 2 ) /\ ( W ` 0 ) = X ) ) -> 0 < ( # ` W ) )
47	46	adantr 481	. . . . . . . . . . 11 |- ( ( ( ( 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. E /\ { ( lastS ` W ) , ( W ` 0 ) } e. E ) /\ ( # ` W ) = ( N - 2 ) /\ ( W ` 0 ) = X ) ) /\ Y e. ( G NeighbVtx X ) ) -> 0 < ( # ` W ) )
48		ccat2s1fst 13416	. . . . . . . . . . 11 |- ( ( ( W e. Word V /\ 0 < ( # ` W ) ) /\ ( X e. V /\ Y e. V ) ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` 0 ) = ( W ` 0 ) )
49	22, 47, 12, 17, 48	syl22anc 1327	. . . . . . . . . 10 |- ( ( ( ( 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. E /\ { ( lastS ` W ) , ( W ` 0 ) } e. E ) /\ ( # ` W ) = ( N - 2 ) /\ ( W ` 0 ) = X ) ) /\ Y e. ( G NeighbVtx X ) ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` 0 ) = ( W ` 0 ) )
50	39, 49	preq12d 4276	. . . . . . . . 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. E /\ { ( lastS ` W ) , ( W ` 0 ) } e. E ) /\ ( # ` W ) = ( N - 2 ) /\ ( W ` 0 ) = X ) ) /\ Y e. ( G NeighbVtx X ) ) -> { ( lastS ` ( ( W ++ <" X "> ) ++ <" Y "> ) ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` 0 ) } = { Y , ( W ` 0 ) } )
51	7	nbusgreledg 26249	. . . . . . . . . . . . 13 |- ( G e. USGraph -> ( Y e. ( G NeighbVtx X ) <-> { Y , X } e. E ) )
52	51	3ad2ant1 1082	. . . . . . . . . . . 12 |- ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( Y e. ( G NeighbVtx X ) <-> { Y , X } e. E ) )
53	52	adantr 481	. . . . . . . . . . 11 |- ( ( ( 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. E /\ { ( lastS ` W ) , ( W ` 0 ) } e. E ) /\ ( # ` W ) = ( N - 2 ) /\ ( W ` 0 ) = X ) ) -> ( Y e. ( G NeighbVtx X ) <-> { Y , X } e. E ) )
54	53	biimpa 501	. . . . . . . . . 10 |- ( ( ( ( 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. E /\ { ( lastS ` W ) , ( W ` 0 ) } e. E ) /\ ( # ` W ) = ( N - 2 ) /\ ( W ` 0 ) = X ) ) /\ Y e. ( G NeighbVtx X ) ) -> { Y , X } e. E )
55		preq2 4269	. . . . . . . . . . . . 13 |- ( ( W ` 0 ) = X -> { Y , ( W ` 0 ) } = { Y , X } )
56	55	eleq1d 2686	. . . . . . . . . . . 12 |- ( ( W ` 0 ) = X -> ( { Y , ( W ` 0 ) } e. E <-> { Y , X } e. E ) )
57	56	3ad2ant3 1084	. . . . . . . . . . 11 |- ( ( ( W e. Word V /\ A. i e. ( 0..^ ( ( # ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { ( lastS ` W ) , ( W ` 0 ) } e. E ) /\ ( # ` W ) = ( N - 2 ) /\ ( W ` 0 ) = X ) -> ( { Y , ( W ` 0 ) } e. E <-> { Y , X } e. E ) )
58	57	ad2antlr 763	. . . . . . . . . 10 |- ( ( ( ( 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. E /\ { ( lastS ` W ) , ( W ` 0 ) } e. E ) /\ ( # ` W ) = ( N - 2 ) /\ ( W ` 0 ) = X ) ) /\ Y e. ( G NeighbVtx X ) ) -> ( { Y , ( W ` 0 ) } e. E <-> { Y , X } e. E ) )
59	54, 58	mpbird 247	. . . . . . . . 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. E /\ { ( lastS ` W ) , ( W ` 0 ) } e. E ) /\ ( # ` W ) = ( N - 2 ) /\ ( W ` 0 ) = X ) ) /\ Y e. ( G NeighbVtx X ) ) -> { Y , ( W ` 0 ) } e. E )
60	50, 59	eqeltrd 2701	. . . . . . . 8 |- ( ( ( ( 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. E /\ { ( lastS ` W ) , ( W ` 0 ) } e. E ) /\ ( # ` W ) = ( N - 2 ) /\ ( W ` 0 ) = X ) ) /\ Y e. ( G NeighbVtx X ) ) -> { ( lastS ` ( ( W ++ <" X "> ) ++ <" Y "> ) ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` 0 ) } e. E )
61	19, 35, 60	3jca 1242	. . . . . . 7 |- ( ( ( ( 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. E /\ { ( lastS ` W ) , ( W ` 0 ) } e. E ) /\ ( # ` W ) = ( N - 2 ) /\ ( W ` 0 ) = X ) ) /\ Y e. ( G NeighbVtx X ) ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) e. Word V /\ A. i e. ( 0..^ ( ( # ` ( ( W ++ <" X "> ) ++ <" Y "> ) ) - 1 ) ) { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` i ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( i + 1 ) ) } e. E /\ { ( lastS ` ( ( W ++ <" X "> ) ++ <" Y "> ) ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` 0 ) } e. E ) )
62		oveq1 6657	. . . . . . . . . . . . . . 15 |- ( ( # ` W ) = ( N - 2 ) -> ( ( # ` W ) + 2 ) = ( ( N - 2 ) + 2 ) )
63		eluzelcn 11699	. . . . . . . . . . . . . . . 16 |- ( N e. ( ZZ>= ` 3 ) -> N e. CC )
64		2cn 11091	. . . . . . . . . . . . . . . 16 |- 2 e. CC
65		npcan 10290	. . . . . . . . . . . . . . . 16 |- ( ( N e. CC /\ 2 e. CC ) -> ( ( N - 2 ) + 2 ) = N )
66	63, 64, 65	sylancl 694	. . . . . . . . . . . . . . 15 |- ( N e. ( ZZ>= ` 3 ) -> ( ( N - 2 ) + 2 ) = N )
67	62, 66	sylan9eq 2676	. . . . . . . . . . . . . 14 |- ( ( ( # ` W ) = ( N - 2 ) /\ N e. ( ZZ>= ` 3 ) ) -> ( ( # ` W ) + 2 ) = N )
68	67	ex 450	. . . . . . . . . . . . 13 |- ( ( # ` W ) = ( N - 2 ) -> ( N e. ( ZZ>= ` 3 ) -> ( ( # ` W ) + 2 ) = N ) )
69	68	3ad2ant2 1083	. . . . . . . . . . . 12 |- ( ( ( W e. Word V /\ A. i e. ( 0..^ ( ( # ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { ( lastS ` W ) , ( W ` 0 ) } e. E ) /\ ( # ` W ) = ( N - 2 ) /\ ( W ` 0 ) = X ) -> ( N e. ( ZZ>= ` 3 ) -> ( ( # ` W ) + 2 ) = N ) )
70	69	com12 32	. . . . . . . . . . 11 |- ( N e. ( ZZ>= ` 3 ) -> ( ( ( W e. Word V /\ A. i e. ( 0..^ ( ( # ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { ( lastS ` W ) , ( W ` 0 ) } e. E ) /\ ( # ` W ) = ( N - 2 ) /\ ( W ` 0 ) = X ) -> ( ( # ` W ) + 2 ) = N ) )
71	70	3ad2ant3 1084	. . . . . . . . . 10 |- ( ( 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. E /\ { ( lastS ` W ) , ( W ` 0 ) } e. E ) /\ ( # ` W ) = ( N - 2 ) /\ ( W ` 0 ) = X ) -> ( ( # ` W ) + 2 ) = N ) )
72	71	imp 445	. . . . . . . . 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. E /\ { ( lastS ` W ) , ( W ` 0 ) } e. E ) /\ ( # ` W ) = ( N - 2 ) /\ ( W ` 0 ) = X ) ) -> ( ( # ` W ) + 2 ) = N )
73	72	adantr 481	. . . . . . . 8 |- ( ( ( ( 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. E /\ { ( lastS ` W ) , ( W ` 0 ) } e. E ) /\ ( # ` W ) = ( N - 2 ) /\ ( W ` 0 ) = X ) ) /\ Y e. ( G NeighbVtx X ) ) -> ( ( # ` W ) + 2 ) = N )
74	24, 73	eqtrd 2656	. . . . . . 7 |- ( ( ( ( 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. E /\ { ( lastS ` W ) , ( W ` 0 ) } e. E ) /\ ( # ` W ) = ( N - 2 ) /\ ( W ` 0 ) = X ) ) /\ Y e. ( G NeighbVtx X ) ) -> ( # ` ( ( W ++ <" X "> ) ++ <" Y "> ) ) = N )
75	61, 74	jca 554	. . . . . 6 |- ( ( ( ( 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. E /\ { ( lastS ` W ) , ( W ` 0 ) } e. E ) /\ ( # ` W ) = ( N - 2 ) /\ ( W ` 0 ) = X ) ) /\ Y e. ( G NeighbVtx X ) ) -> ( ( ( ( W ++ <" X "> ) ++ <" Y "> ) e. Word V /\ A. i e. ( 0..^ ( ( # ` ( ( W ++ <" X "> ) ++ <" Y "> ) ) - 1 ) ) { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` i ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( i + 1 ) ) } e. E /\ { ( lastS ` ( ( W ++ <" X "> ) ++ <" Y "> ) ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` 0 ) } e. E ) /\ ( # ` ( ( W ++ <" X "> ) ++ <" Y "> ) ) = N ) )
76	75	exp31 630	. . . . 5 |- ( ( 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. E /\ { ( lastS ` W ) , ( W ` 0 ) } e. E ) /\ ( # ` W ) = ( N - 2 ) /\ ( W ` 0 ) = X ) -> ( Y e. ( G NeighbVtx X ) -> ( ( ( ( W ++ <" X "> ) ++ <" Y "> ) e. Word V /\ A. i e. ( 0..^ ( ( # ` ( ( W ++ <" X "> ) ++ <" Y "> ) ) - 1 ) ) { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` i ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( i + 1 ) ) } e. E /\ { ( lastS ` ( ( W ++ <" X "> ) ++ <" Y "> ) ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` 0 ) } e. E ) /\ ( # ` ( ( W ++ <" X "> ) ++ <" Y "> ) ) = N ) ) ) )
77	9, 76	sylbid 230	. . . 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. Word V /\ A. i e. ( 0..^ ( ( # ` ( ( W ++ <" X "> ) ++ <" Y "> ) ) - 1 ) ) { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` i ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( i + 1 ) ) } e. E /\ { ( lastS ` ( ( W ++ <" X "> ) ++ <" Y "> ) ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` 0 ) } e. E ) /\ ( # ` ( ( W ++ <" X "> ) ++ <" Y "> ) ) = N ) ) ) )
78	77	com23 86	. . 3 |- ( ( 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. Word V /\ A. i e. ( 0..^ ( ( # ` ( ( W ++ <" X "> ) ++ <" Y "> ) ) - 1 ) ) { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` i ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( i + 1 ) ) } e. E /\ { ( lastS ` ( ( W ++ <" X "> ) ++ <" Y "> ) ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` 0 ) } e. E ) /\ ( # ` ( ( W ++ <" X "> ) ++ <" Y "> ) ) = N ) ) ) )
79	78	3imp 1256	. 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. Word V /\ A. i e. ( 0..^ ( ( # ` ( ( W ++ <" X "> ) ++ <" Y "> ) ) - 1 ) ) { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` i ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( i + 1 ) ) } e. E /\ { ( lastS ` ( ( W ++ <" X "> ) ++ <" Y "> ) ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` 0 ) } e. E ) /\ ( # ` ( ( W ++ <" X "> ) ++ <" Y "> ) ) = N ) )
80		eluzge3nn 11730	. . . . 5 |- ( N e. ( ZZ>= ` 3 ) -> N e. NN )
81	6, 7	isclwwlksnx 26889	. . . . 5 |- ( N e. NN -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) e. ( N ClWWalksN G ) <-> ( ( ( ( W ++ <" X "> ) ++ <" Y "> ) e. Word V /\ A. i e. ( 0..^ ( ( # ` ( ( W ++ <" X "> ) ++ <" Y "> ) ) - 1 ) ) { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` i ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( i + 1 ) ) } e. E /\ { ( lastS ` ( ( W ++ <" X "> ) ++ <" Y "> ) ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` 0 ) } e. E ) /\ ( # ` ( ( W ++ <" X "> ) ++ <" Y "> ) ) = N ) ) )
82	80, 81	syl 17	. . . 4 |- ( N e. ( ZZ>= ` 3 ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) e. ( N ClWWalksN G ) <-> ( ( ( ( W ++ <" X "> ) ++ <" Y "> ) e. Word V /\ A. i e. ( 0..^ ( ( # ` ( ( W ++ <" X "> ) ++ <" Y "> ) ) - 1 ) ) { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` i ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( i + 1 ) ) } e. E /\ { ( lastS ` ( ( W ++ <" X "> ) ++ <" Y "> ) ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` 0 ) } e. E ) /\ ( # ` ( ( W ++ <" X "> ) ++ <" Y "> ) ) = N ) ) )
83	82	3ad2ant3 1084	. . 3 |- ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) e. ( N ClWWalksN G ) <-> ( ( ( ( W ++ <" X "> ) ++ <" Y "> ) e. Word V /\ A. i e. ( 0..^ ( ( # ` ( ( W ++ <" X "> ) ++ <" Y "> ) ) - 1 ) ) { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` i ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( i + 1 ) ) } e. E /\ { ( lastS ` ( ( W ++ <" X "> ) ++ <" Y "> ) ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` 0 ) } e. E ) /\ ( # ` ( ( W ++ <" X "> ) ++ <" Y "> ) ) = N ) ) )
84	83	3ad2ant1 1082	. 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 "> ) e. Word V /\ A. i e. ( 0..^ ( ( # ` ( ( W ++ <" X "> ) ++ <" Y "> ) ) - 1 ) ) { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` i ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( i + 1 ) ) } e. E /\ { ( lastS ` ( ( W ++ <" X "> ) ++ <" Y "> ) ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` 0 ) } e. E ) /\ ( # ` ( ( W ++ <" X "> ) ++ <" Y "> ) ) = N ) ) )
85	79, 84	mpbird 247	1 |- ( ( ( 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 ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   u. cun 3572   {cpr 4179   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   < clt 10074   - 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  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-edg 25940  df-upgr 25977  df-umgr 25978  df-usgr 26046  df-nbgr 26228  df-clwwlks 26877  df-clwwlksn 26878

This theorem is referenced by:  numclwlk1lem2foa  27224