Theorem clwwlknonex2 16363

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 9851	. . . . . . . 8 |- ( N e. ( ZZ>= ` 3 ) -> ( N - 2 ) e. NN )
2	1	nnne0d 9230	. . . . . . 7 |- ( N e. ( ZZ>= ` 3 ) -> ( N - 2 ) =/= 0 )
3	2	3ad2ant3 1047	. . . . . 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 16356	. . . . . 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 1108	. . . . . . . . . 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 1063	. . . . . . . . 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 1064	. . . . . . . . 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 11263	. . . . . . . . 9 |- ( ( W e. Word V /\ X e. V /\ Y e. V ) -> ( ( W ++ <" X "> ) ++ <" Y "> ) e. Word V )
13	9, 10, 11, 12	syl3anc 1274	. . . . . . . 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 16362	. . . . . . . . 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 11264	. . . . . . . . . . . . 13 |- ( ( W e. Word V /\ X e. V /\ Y e. V ) -> (♯ ` ( ( W ++ <" X "> ) ++ <" Y "> ) ) = ( (♯ ` W ) + 2 ) )
16	9, 10, 11, 15	syl3anc 1274	. . . . . . . . . . . 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 6043	. . . . . . . . . . 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 6044	. . . . . . . . . 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 1026	. . . . . . . . . . . . 13 |- ( ( X e. V /\ Y e. V /\ N e. ( ZZ>= ` 3 ) ) -> N e. ( ZZ>= ` 3 ) )
20		simp2 1025	. . . . . . . . . . . . 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 16361	. . . . . . . . . . 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 2264	. . . . . . . . 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 2741	. . . . . . . 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 1054	. . . . . . . . . . . 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 11258	. . . . . . . . . . . 12 |- ( ( W e. Word V /\ X e. V ) -> ( W ++ <" X "> ) e. Word V )
30		lswccats1 11269	. . . . . . . . . . . 12 |- ( ( ( W ++ <" X "> ) e. Word V /\ Y e. V ) -> (lastS ` ( ( W ++ <" X "> ) ++ <" Y "> ) ) = Y )
31	29, 30	stoic3 1476	. . . . . . . . . . 11 |- ( ( W e. Word V /\ X e. V /\ Y e. V ) -> (lastS ` ( ( W ++ <" X "> ) ++ <" Y "> ) ) = Y )
32	28, 10, 11, 31	syl3anc 1274	. . . . . . . . . 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 9229	. . . . . . . . . . . . . . . . 17 |- ( N e. ( ZZ>= ` 3 ) -> 0 < ( N - 2 ) )
34		breq2 4097	. . . . . . . . . . . . . . . . 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 1046	. . . . . . . . . . . . . . 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 1047	. . . . . . . . . . . . 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 11274	. . . . . . . . . . 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 1275	. . . . . . . . . 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 3760	. . . . . . . . 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 3751	. . . . . . . . . . . . 13 |- { X , Y } = { Y , X }
45	44	eleq1i 2297	. . . . . . . . . . . 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 3753	. . . . . . . . . . . . 13 |- ( ( W ` 0 ) = X -> { Y , ( W ` 0 ) } = { Y , X } )
49	48	eleq1d 2300	. . . . . . . . . . . 12 |- ( ( W ` 0 ) = X -> ( { Y , ( W ` 0 ) } e. E <-> { Y , X } e. E ) )
50	49	3ad2ant3 1047	. . . . . . . . . . 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 2308	. . . . . . . 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 1204	. . . . . . 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 6035	. . . . . . . . . . . . . . 15 |- ( (♯ ` W ) = ( N - 2 ) -> ( (♯ ` W ) + 2 ) = ( ( N - 2 ) + 2 ) )
56		eluzelcn 9811	. . . . . . . . . . . . . . . 16 |- ( N e. ( ZZ>= ` 3 ) -> N e. CC )
57		2cn 9256	. . . . . . . . . . . . . . . 16 |- 2 e. CC
58		npcan 8430	. . . . . . . . . . . . . . . 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 2284	. . . . . . . . . . . . . 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 1046	. . . . . . . . . . . 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 1047	. . . . . . . . . 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 2264	. . . . . . 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 1220	. 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 9845	. . . . 5 |- ( N e. ( ZZ>= ` 3 ) -> N e. NN )
74	4, 5	isclwwlknx 16340	. . . . 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 1047	. . 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 1045	. 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 1005   = wceq 1398   e. wcel 2202   =/= wne 2403   A.wral 2511   u. cun 3199   {cpr 3674   class class class wbr 4093   ` cfv 5333  (class class class)co 6028   CCcc 8073   0cc0 8075   1c1 8076   + caddc 8078   < clt 8256   - cmin 8392   NNcn 9185   2c2 9236   3c3 9237   ZZ>=cuz 9799  ..^cfzo 10422  ♯chash 11083  Word cword 11162  lastSclsw 11207   ++ cconcat 11216   <"cs1 11241  Vtxcvtx 15936  Edgcedg 15981   ClWWalksN cclwwlkn 16327  ClWWalksNOncclwwlknon 16350

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-mulrcl 8174  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-precex 8185  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191  ax-pre-mulgt0 8192

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-1o 6625  df-er 6745  df-map 6862  df-en 6953  df-dom 6954  df-fin 6955  df-pnf 8258  df-mnf 8259  df-xr 8260  df-ltxr 8261  df-le 8262  df-sub 8394  df-neg 8395  df-reap 8797  df-ap 8804  df-inn 9186  df-2 9244  df-3 9245  df-n0 9445  df-z 9524  df-uz 9800  df-fz 10289  df-fzo 10423  df-ihash 11084  df-word 11163  df-lsw 11208  df-concat 11217  df-s1 11242  df-ndx 13148  df-slot 13149  df-base 13151  df-vtx 15938  df-clwwlk 16316  df-clwwlkn 16328  df-clwwlknon 16351

This theorem is referenced by:  clwwlknonex2e  16364