Theorem clwwlknonex2 16224

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 9795	. . . . . . . 8 |- ( N e. ( ZZ>= ` 3 ) -> ( N - 2 ) e. NN )
2	1	nnne0d 9176	. . . . . . 7 |- ( N e. ( ZZ>= ` 3 ) -> ( N - 2 ) =/= 0 )
3	2	3ad2ant3 1044	. . . . . 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 16217	. . . . . 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 1105	. . . . . . . . . 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 1060	. . . . . . . . 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 1061	. . . . . . . . 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 11201	. . . . . . . . 9 |- ( ( W e. Word V /\ X e. V /\ Y e. V ) -> ( ( W ++ <" X "> ) ++ <" Y "> ) e. Word V )
13	9, 10, 11, 12	syl3anc 1271	. . . . . . . 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 16223	. . . . . . . . 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 11202	. . . . . . . . . . . . 13 |- ( ( W e. Word V /\ X e. V /\ Y e. V ) -> (♯ ` ( ( W ++ <" X "> ) ++ <" Y "> ) ) = ( (♯ ` W ) + 2 ) )
16	9, 10, 11, 15	syl3anc 1271	. . . . . . . . . . . 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 6026	. . . . . . . . . . 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 6027	. . . . . . . . . 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 1023	. . . . . . . . . . . . 13 |- ( ( X e. V /\ Y e. V /\ N e. ( ZZ>= ` 3 ) ) -> N e. ( ZZ>= ` 3 ) )
20		simp2 1022	. . . . . . . . . . . . 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 16222	. . . . . . . . . . 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 2262	. . . . . . . . 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 2738	. . . . . . . 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 1051	. . . . . . . . . . . 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 489	. . . . . . . . . . 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 11196	. . . . . . . . . . . 12 |- ( ( W e. Word V /\ X e. V ) -> ( W ++ <" X "> ) e. Word V )
30		lswccats1 11207	. . . . . . . . . . . 12 |- ( ( ( W ++ <" X "> ) e. Word V /\ Y e. V ) -> (lastS ` ( ( W ++ <" X "> ) ++ <" Y "> ) ) = Y )
31	29, 30	stoic3 1473	. . . . . . . . . . 11 |- ( ( W e. Word V /\ X e. V /\ Y e. V ) -> (lastS ` ( ( W ++ <" X "> ) ++ <" Y "> ) ) = Y )
32	28, 10, 11, 31	syl3anc 1271	. . . . . . . . . 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 9175	. . . . . . . . . . . . . . . . 17 |- ( N e. ( ZZ>= ` 3 ) -> 0 < ( N - 2 ) )
34		breq2 4088	. . . . . . . . . . . . . . . . 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 1043	. . . . . . . . . . . . . . 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 1044	. . . . . . . . . . . . 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 11212	. . . . . . . . . . 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 1272	. . . . . . . . . 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 3752	. . . . . . . . 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 3743	. . . . . . . . . . . . 13 |- { X , Y } = { Y , X }
45	44	eleq1i 2295	. . . . . . . . . . . 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 3745	. . . . . . . . . . . . 13 |- ( ( W ` 0 ) = X -> { Y , ( W ` 0 ) } = { Y , X } )
49	48	eleq1d 2298	. . . . . . . . . . . 12 |- ( ( W ` 0 ) = X -> ( { Y , ( W ` 0 ) } e. E <-> { Y , X } e. E ) )
50	49	3ad2ant3 1044	. . . . . . . . . . 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 489	. . . . . . . . . 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 2306	. . . . . . . 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 1201	. . . . . . 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 6018	. . . . . . . . . . . . . . 15 |- ( (♯ ` W ) = ( N - 2 ) -> ( (♯ ` W ) + 2 ) = ( ( N - 2 ) + 2 ) )
56		eluzelcn 9755	. . . . . . . . . . . . . . . 16 |- ( N e. ( ZZ>= ` 3 ) -> N e. CC )
57		2cn 9202	. . . . . . . . . . . . . . . 16 |- 2 e. CC
58		npcan 8376	. . . . . . . . . . . . . . . 16 |- ( ( N e. CC /\ 2 e. CC ) -> ( ( N - 2 ) + 2 ) = N )
59	56, 57, 58	sylancl 413	. . . . . . . . . . . . . . 15 |- ( N e. ( ZZ>= ` 3 ) -> ( ( N - 2 ) + 2 ) = N )
60	55, 59	sylan9eq 2282	. . . . . . . . . . . . . 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 1043	. . . . . . . . . . . 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 1044	. . . . . . . . . 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 2262	. . . . . . 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 1217	. 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 9789	. . . . 5 |- ( N e. ( ZZ>= ` 3 ) -> N e. NN )
74	4, 5	isclwwlknx 16201	. . . . 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 1044	. . 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 1042	. 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 ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   =/= wne 2400   A.wral 2508   u. cun 3196   {cpr 3668   class class class wbr 4084   ` cfv 5322  (class class class)co 6011   CCcc 8018   0cc0 8020   1c1 8021   + caddc 8023   < clt 8202   - cmin 8338   NNcn 9131   2c2 9182   3c3 9183   ZZ>=cuz 9743  ..^cfzo 10365  ♯chash 11025  Word cword 11100  lastSclsw 11145   ++ cconcat 11154   <"cs1 11179  Vtxcvtx 15850  Edgcedg 15895   ClWWalksN cclwwlkn 16188  ClWWalksNOncclwwlknon 16211

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4200  ax-sep 4203  ax-nul 4211  ax-pow 4260  ax-pr 4295  ax-un 4526  ax-setind 4631  ax-iinf 4682  ax-cnex 8111  ax-resscn 8112  ax-1cn 8113  ax-1re 8114  ax-icn 8115  ax-addcl 8116  ax-addrcl 8117  ax-mulcl 8118  ax-mulrcl 8119  ax-addcom 8120  ax-mulcom 8121  ax-addass 8122  ax-mulass 8123  ax-distr 8124  ax-i2m1 8125  ax-0lt1 8126  ax-1rid 8127  ax-0id 8128  ax-rnegex 8129  ax-precex 8130  ax-cnre 8131  ax-pre-ltirr 8132  ax-pre-ltwlin 8133  ax-pre-lttrn 8134  ax-pre-apti 8135  ax-pre-ltadd 8136  ax-pre-mulgt0 8137

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3890  df-int 3925  df-iun 3968  df-br 4085  df-opab 4147  df-mpt 4148  df-tr 4184  df-id 4386  df-iord 4459  df-on 4461  df-ilim 4462  df-suc 4464  df-iom 4685  df-xp 4727  df-rel 4728  df-cnv 4729  df-co 4730  df-dm 4731  df-rn 4732  df-res 4733  df-ima 4734  df-iota 5282  df-fun 5324  df-fn 5325  df-f 5326  df-f1 5327  df-fo 5328  df-f1o 5329  df-fv 5330  df-riota 5964  df-ov 6014  df-oprab 6015  df-mpo 6016  df-1st 6296  df-2nd 6297  df-recs 6464  df-frec 6550  df-1o 6575  df-er 6695  df-map 6812  df-en 6903  df-dom 6904  df-fin 6905  df-pnf 8204  df-mnf 8205  df-xr 8206  df-ltxr 8207  df-le 8208  df-sub 8340  df-neg 8341  df-reap 8743  df-ap 8750  df-inn 9132  df-2 9190  df-3 9191  df-n0 9391  df-z 9468  df-uz 9744  df-fz 10232  df-fzo 10366  df-ihash 11026  df-word 11101  df-lsw 11146  df-concat 11155  df-s1 11180  df-ndx 13072  df-slot 13073  df-base 13075  df-vtx 15852  df-clwwlk 16177  df-clwwlkn 16189  df-clwwlknon 16212

This theorem is referenced by:  clwwlknonex2e  16225