Theorem clwwlknonex2 16680

Description: Extending a closed walk W on vertex X by an additional edge (forth and back) results in a closed walk. (Contributed by AV, 22-Sep-2018.) (Revised by AV, 25-Feb-2022.) (Proof shortened by AV, 28-Mar-2022.)

Hypotheses

Ref	Expression
clwwlknonex2.v	|- V = (Vtx ` G )
clwwlknonex2.e	|- E = (Edg ` G )

Assertion

Ref	Expression
clwwlknonex2	|- ( ( ( X e. V /\ Y e. V /\ N e. ( ZZ>= ` 3 ) ) /\ { X , Y } e. E /\ W e. ( X (ClWWalksNOn ` G ) ( N - 2 ) ) ) -> ( ( W ++ <" X "> ) ++ <" Y "> ) e. ( N ClWWalksN G ) )

Proof of Theorem clwwlknonex2

Dummy variable i is distinct from all other variables.

Step	Hyp	Ref	Expression
1		uz3m2nn 9973	. . . . . . . 8 |- ( N e. ( ZZ>= ` 3 ) -> ( N - 2 ) e. NN )
2	1	nnne0d 9349	. . . . . . 7 |- ( N e. ( ZZ>= ` 3 ) -> ( N - 2 ) =/= 0 )
3	2	3ad2ant3 1051	. . . . . 6 |- ( ( X e. V /\ Y e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( N - 2 ) =/= 0 )
4		clwwlknonex2.v	. . . . . . 7 |- V = (Vtx ` G )
5		clwwlknonex2.e	. . . . . . 7 |- E = (Edg ` G )
6	4, 5	clwwlknonel 16673	. . . . . 6 |- ( ( N - 2 ) =/= 0 -> ( W e. ( X (ClWWalksNOn ` G ) ( 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 ) ) )
7	3, 6	syl 14	. . . . 5 |- ( ( X e. V /\ Y e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( W e. ( X (ClWWalksNOn ` G ) ( 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 ) ) )
8		simpr11 1112	. . . . . . . . . 10 |- ( ( ( X e. V /\ Y 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 )
9	8	adantr 276	. . . . . . . . 9 |- ( ( ( ( X e. V /\ Y 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 ) ) /\ { X , Y } e. E ) -> W e. Word V )
10		simpll1 1067	. . . . . . . . 9 |- ( ( ( ( X e. V /\ Y 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 ) ) /\ { X , Y } e. E ) -> X e. V )
11		simpll2 1068	. . . . . . . . 9 |- ( ( ( ( X e. V /\ Y 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 ) ) /\ { X , Y } e. E ) -> Y e. V )
12		ccatw2s1cl 11405	. . . . . . . . 9 |- ( ( W e. Word V /\ X e. V /\ Y e. V ) -> ( ( W ++ <" X "> ) ++ <" Y "> ) e. Word V )
13	9, 10, 11, 12	syl3anc 1278	. . . . . . . 8 |- ( ( ( ( X e. V /\ Y 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 ) ) /\ { X , Y } e. E ) -> ( ( W ++ <" X "> ) ++ <" Y "> ) e. Word V )
14	4, 5	clwwlknonex2lem2 16679	. . . . . . . . 9 |- ( ( ( ( X e. V /\ Y 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 ) ) /\ { X , Y } e. E ) -> A. i e. ( ( 0..^ ( (♯ ` W ) - 1 ) ) u. { ( (♯ ` W ) - 1 ) , (♯ ` W ) } ) { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` i ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( i + 1 ) ) } e. E )
15		ccatw2s1leng 11406	. . . . . . . . . . . . 13 |- ( ( W e. Word V /\ X e. V /\ Y e. V ) -> (♯ ` ( ( W ++ <" X "> ) ++ <" Y "> ) ) = ( (♯ ` W ) + 2 ) )
16	9, 10, 11, 15	syl3anc 1278	. . . . . . . . . . . 12 |- ( ( ( ( X e. V /\ Y 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 ) ) /\ { X , Y } e. E ) -> (♯ ` ( ( W ++ <" X "> ) ++ <" Y "> ) ) = ( (♯ ` W ) + 2 ) )
17	16	oveq1d 6100	. . . . . . . . . . 11 |- ( ( ( ( X e. V /\ Y 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 ) ) /\ { X , Y } e. E ) -> ( (♯ ` ( ( W ++ <" X "> ) ++ <" Y "> ) ) - 1 ) = ( ( (♯ ` W ) + 2 ) - 1 ) )
18	17	oveq2d 6101	. . . . . . . . . 10 |- ( ( ( ( X e. V /\ Y 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 ) ) /\ { X , Y } e. E ) -> ( 0..^ ( (♯ ` ( ( W ++ <" X "> ) ++ <" Y "> ) ) - 1 ) ) = ( 0..^ ( ( (♯ ` W ) + 2 ) - 1 ) ) )
19		simp3 1030	. . . . . . . . . . . . 13 |- ( ( X e. V /\ Y e. V /\ N e. ( ZZ>= ` 3 ) ) -> N e. ( ZZ>= ` 3 ) )
20		simp2 1029	. . . . . . . . . . . . 13 |- ( ( ( 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 ) )
21	19, 20	anim12i 338	. . . . . . . . . . . 12 |- ( ( ( X e. V /\ Y 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 ) ) )
22	21	adantr 276	. . . . . . . . . . 11 |- ( ( ( ( X e. V /\ Y 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 ) ) /\ { X , Y } e. E ) -> ( N e. ( ZZ>= ` 3 ) /\ (♯ ` W ) = ( N - 2 ) ) )
23		clwwlknonex2lem1 16678	. . . . . . . . . . 11 |- ( ( N e. ( ZZ>= ` 3 ) /\ (♯ ` W ) = ( N - 2 ) ) -> ( 0..^ ( ( (♯ ` W ) + 2 ) - 1 ) ) = ( ( 0..^ ( (♯ ` W ) - 1 ) ) u. { ( (♯ ` W ) - 1 ) , (♯ ` W ) } ) )
24	22, 23	syl 14	. . . . . . . . . 10 |- ( ( ( ( X e. V /\ Y 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 ) ) /\ { X , Y } e. E ) -> ( 0..^ ( ( (♯ ` W ) + 2 ) - 1 ) ) = ( ( 0..^ ( (♯ ` W ) - 1 ) ) u. { ( (♯ ` W ) - 1 ) , (♯ ` W ) } ) )
25	18, 24	eqtrd 2271	. . . . . . . . 9 |- ( ( ( ( X e. V /\ Y 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 ) ) /\ { X , Y } e. E ) -> ( 0..^ ( (♯ ` ( ( W ++ <" X "> ) ++ <" Y "> ) ) - 1 ) ) = ( ( 0..^ ( (♯ ` W ) - 1 ) ) u. { ( (♯ ` W ) - 1 ) , (♯ ` W ) } ) )
26	14, 25	raleqtrrdv 2759	. . . . . . . 8 |- ( ( ( ( X e. V /\ Y 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 ) ) /\ { X , Y } e. E ) -> A. i e. ( 0..^ ( (♯ ` ( ( W ++ <" X "> ) ++ <" Y "> ) ) - 1 ) ) { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` i ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( i + 1 ) ) } e. E )
27		simp11 1058	. . . . . . . . . . . 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 ) -> W e. Word V )
28	27	ad2antlr 493	. . . . . . . . . . 11 |- ( ( ( ( X e. V /\ Y 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 ) ) /\ { X , Y } e. E ) -> W e. Word V )
29		ccatws1cl 11400	. . . . . . . . . . . 12 |- ( ( W e. Word V /\ X e. V ) -> ( W ++ <" X "> ) e. Word V )
30		lswccats1 11411	. . . . . . . . . . . 12 |- ( ( ( W ++ <" X "> ) e. Word V /\ Y e. V ) -> (lastS ` ( ( W ++ <" X "> ) ++ <" Y "> ) ) = Y )
31	29, 30	stoic3 1480	. . . . . . . . . . 11 |- ( ( W e. Word V /\ X e. V /\ Y e. V ) -> (lastS ` ( ( W ++ <" X "> ) ++ <" Y "> ) ) = Y )
32	28, 10, 11, 31	syl3anc 1278	. . . . . . . . . 10 |- ( ( ( ( X e. V /\ Y 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 ) ) /\ { X , Y } e. E ) -> (lastS ` ( ( W ++ <" X "> ) ++ <" Y "> ) ) = Y )
33	1	nngt0d 9348	. . . . . . . . . . . . . . . . 17 |- ( N e. ( ZZ>= ` 3 ) -> 0 < ( N - 2 ) )
34		breq2 4134	. . . . . . . . . . . . . . . . 17 |- ( (♯ ` W ) = ( N - 2 ) -> ( 0 < (♯ ` W ) <-> 0 < ( N - 2 ) ) )
35	33, 34	imbitrrid 156	. . . . . . . . . . . . . . . 16 |- ( (♯ ` W ) = ( N - 2 ) -> ( N e. ( ZZ>= ` 3 ) -> 0 < (♯ ` W ) ) )
36	35	3ad2ant2 1050	. . . . . . . . . . . . . . 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 ) ) )
37	36	com12 30	. . . . . . . . . . . . . 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 ) ) )
38	37	3ad2ant3 1051	. . . . . . . . . . . . 13 |- ( ( X e. V /\ Y 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 ) ) )
39	38	imp 124	. . . . . . . . . . . 12 |- ( ( ( X e. V /\ Y 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 ) )
40	39	adantr 276	. . . . . . . . . . 11 |- ( ( ( ( X e. V /\ Y 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 ) ) /\ { X , Y } e. E ) -> 0 < (♯ ` W ) )
41		ccat2s1fstg 11416	. . . . . . . . . . 11 |- ( ( ( W e. Word V /\ 0 < (♯ ` W ) ) /\ ( X e. V /\ Y e. V ) ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` 0 ) = ( W ` 0 ) )
42	28, 40, 10, 11, 41	syl22anc 1279	. . . . . . . . . 10 |- ( ( ( ( X e. V /\ Y 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 ) ) /\ { X , Y } e. E ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` 0 ) = ( W ` 0 ) )
43	32, 42	preq12d 3796	. . . . . . . . 9 |- ( ( ( ( X e. V /\ Y 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 ) ) /\ { X , Y } e. E ) -> { (lastS ` ( ( W ++ <" X "> ) ++ <" Y "> ) ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` 0 ) } = { Y , ( W ` 0 ) } )
44		prcom 3787	. . . . . . . . . . . . 13 |- { X , Y } = { Y , X }
45	44	eleq1i 2304	. . . . . . . . . . . 12 |- ( { X , Y } e. E <-> { Y , X } e. E )
46	45	biimpi 120	. . . . . . . . . . 11 |- ( { X , Y } e. E -> { Y , X } e. E )
47	46	adantl 277	. . . . . . . . . 10 |- ( ( ( ( X e. V /\ Y 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 ) ) /\ { X , Y } e. E ) -> { Y , X } e. E )
48		preq2 3789	. . . . . . . . . . . . 13 |- ( ( W ` 0 ) = X -> { Y , ( W ` 0 ) } = { Y , X } )
49	48	eleq1d 2307	. . . . . . . . . . . 12 |- ( ( W ` 0 ) = X -> ( { Y , ( W ` 0 ) } e. E <-> { Y , X } e. E ) )
50	49	3ad2ant3 1051	. . . . . . . . . . 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 ) )
51	50	ad2antlr 493	. . . . . . . . . 10 |- ( ( ( ( X e. V /\ Y 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 ) ) /\ { X , Y } e. E ) -> ( { Y , ( W ` 0 ) } e. E <-> { Y , X } e. E ) )
52	47, 51	mpbird 167	. . . . . . . . 9 |- ( ( ( ( X e. V /\ Y 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 ) ) /\ { X , Y } e. E ) -> { Y , ( W ` 0 ) } e. E )
53	43, 52	eqeltrd 2315	. . . . . . . 8 |- ( ( ( ( X e. V /\ Y 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 ) ) /\ { X , Y } e. E ) -> { (lastS ` ( ( W ++ <" X "> ) ++ <" Y "> ) ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` 0 ) } e. E )
54	13, 26, 53	3jca 1208	. . . . . . 7 |- ( ( ( ( X e. V /\ Y 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 ) ) /\ { X , Y } e. E ) -> ( ( ( 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 ) )
55		oveq1 6092	. . . . . . . . . . . . . . 15 |- ( (♯ ` W ) = ( N - 2 ) -> ( (♯ ` W ) + 2 ) = ( ( N - 2 ) + 2 ) )
56		eluzelcn 9933	. . . . . . . . . . . . . . . 16 |- ( N e. ( ZZ>= ` 3 ) -> N e. CC )
57		2cn 9375	. . . . . . . . . . . . . . . 16 |- 2 e. CC
58		npcan 8535	. . . . . . . . . . . . . . . 16 |- ( ( N e. CC /\ 2 e. CC ) -> ( ( N - 2 ) + 2 ) = N )
59	56, 57, 58	sylancl 417	. . . . . . . . . . . . . . 15 |- ( N e. ( ZZ>= ` 3 ) -> ( ( N - 2 ) + 2 ) = N )
60	55, 59	sylan9eq 2291	. . . . . . . . . . . . . 14 |- ( ( (♯ ` W ) = ( N - 2 ) /\ N e. ( ZZ>= ` 3 ) ) -> ( (♯ ` W ) + 2 ) = N )
61	60	ex 115	. . . . . . . . . . . . 13 |- ( (♯ ` W ) = ( N - 2 ) -> ( N e. ( ZZ>= ` 3 ) -> ( (♯ ` W ) + 2 ) = N ) )
62	61	3ad2ant2 1050	. . . . . . . . . . . 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 ) )
63	62	com12 30	. . . . . . . . . . 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 ) )
64	63	3ad2ant3 1051	. . . . . . . . . 10 |- ( ( X e. V /\ Y 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 ) )
65	64	imp 124	. . . . . . . . 9 |- ( ( ( X e. V /\ Y 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 )
66	65	adantr 276	. . . . . . . 8 |- ( ( ( ( X e. V /\ Y 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 ) ) /\ { X , Y } e. E ) -> ( (♯ ` W ) + 2 ) = N )
67	16, 66	eqtrd 2271	. . . . . . 7 |- ( ( ( ( X e. V /\ Y 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 ) ) /\ { X , Y } e. E ) -> (♯ ` ( ( W ++ <" X "> ) ++ <" Y "> ) ) = N )
68	54, 67	jca 306	. . . . . 6 |- ( ( ( ( X e. V /\ Y 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 ) ) /\ { X , Y } e. E ) -> ( ( ( ( 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 ) )
69	68	exp31 364	. . . . 5 |- ( ( X e. V /\ Y 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 ) -> ( { X , Y } e. E -> ( ( ( ( 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 ) ) ) )
70	7, 69	sylbid 150	. . . 4 |- ( ( X e. V /\ Y e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( W e. ( X (ClWWalksNOn ` G ) ( N - 2 ) ) -> ( { X , Y } e. E -> ( ( ( ( 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 ) ) ) )
71	70	com23 78	. . 3 |- ( ( X e. V /\ Y e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( { X , Y } e. E -> ( W e. ( X (ClWWalksNOn ` G ) ( 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 ) ) ) )
72	71	3imp 1224	. 2 |- ( ( ( X e. V /\ Y e. V /\ N e. ( ZZ>= ` 3 ) ) /\ { X , Y } e. E /\ W e. ( X (ClWWalksNOn ` G ) ( 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 ) )
73		eluz3nn 9967	. . . . 5 |- ( N e. ( ZZ>= ` 3 ) -> N e. NN )
74	4, 5	isclwwlknx 16657	. . . . 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 ) ) )
75	73, 74	syl 14	. . . 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 ) ) )
76	75	3ad2ant3 1051	. . 3 |- ( ( X e. V /\ Y 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 ) ) )
77	76	3ad2ant1 1049	. 2 |- ( ( ( X e. V /\ Y e. V /\ N e. ( ZZ>= ` 3 ) ) /\ { X , Y } e. E /\ W e. ( X (ClWWalksNOn ` G ) ( 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 ) ) )
78	72, 77	mpbird 167	1 |- ( ( ( X e. V /\ Y e. V /\ N e. ( ZZ>= ` 3 ) ) /\ { X , Y } e. E /\ W e. ( X (ClWWalksNOn ` G ) ( N - 2 ) ) ) -> ( ( W ++ <" X "> ) ++ <" Y "> ) e. ( N ClWWalksN G ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   =/= wne 2420   A.wral 2528   u. cun 3218   {cpr 3710   class class class wbr 4130   ` cfv 5377  (class class class)co 6085   CCcc 8177   0cc0 8179   1c1 8180   + caddc 8182   < clt 8360   - cmin 8497   NNcn 9304   2c2 9355   3c3 9356   ZZ>=cuz 9921  ..^cfzo 10549  ♯chash 11214  Word cword 11304  lastSclsw 11349   ++ cconcat 11358   <"cs1 11383  Vtxcvtx 16253  Edgcedg 16298   ClWWalksN cclwwlkn 16644  ClWWalksNOncclwwlknon 16667

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-1o 6687  df-er 6807  df-map 6924  df-en 7023  df-dom 7024  df-fin 7025  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-inn 9305  df-2 9363  df-3 9364  df-n0 9564  df-z 9645  df-uz 9922  df-fz 10412  df-fzo 10550  df-ihash 11215  df-word 11305  df-lsw 11350  df-concat 11359  df-s1 11384  df-ndx 13355  df-slot 13356  df-base 13358  df-vtx 16255  df-clwwlk 16633  df-clwwlkn 16645  df-clwwlknon 16668

This theorem is used by:  clwwlknonex2e  16681