Theorem clwwlknonex2 16434

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 9905	. . . . . . . 8 |- ( N e. ( ZZ>= ` 3 ) -> ( N - 2 ) e. NN )
2	1	nnne0d 9282	. . . . . . 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 16427	. . . . . 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 11325	. . . . . . . . 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 16433	. . . . . . . . 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 11326	. . . . . . . . . . . . 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 6065	. . . . . . . . . . 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 6066	. . . . . . . . . 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 16432	. . . . . . . . . . 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 2265	. . . . . . . . 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 2751	. . . . . . . 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 11320	. . . . . . . . . . . 12 |- ( ( W e. Word V /\ X e. V ) -> ( W ++ <" X "> ) e. Word V )
30		lswccats1 11331	. . . . . . . . . . . 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 9281	. . . . . . . . . . . . . . . . 17 |- ( N e. ( ZZ>= ` 3 ) -> 0 < ( N - 2 ) )
34		breq2 4113	. . . . . . . . . . . . . . . . 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 11336	. . . . . . . . . . 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 3776	. . . . . . . . 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 3767	. . . . . . . . . . . . 13 |- { X , Y } = { Y , X }
45	44	eleq1i 2298	. . . . . . . . . . . 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 3769	. . . . . . . . . . . . 13 |- ( ( W ` 0 ) = X -> { Y , ( W ` 0 ) } = { Y , X } )
49	48	eleq1d 2301	. . . . . . . . . . . 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 2309	. . . . . . . 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 6057	. . . . . . . . . . . . . . 15 |- ( (♯ ` W ) = ( N - 2 ) -> ( (♯ ` W ) + 2 ) = ( ( N - 2 ) + 2 ) )
56		eluzelcn 9865	. . . . . . . . . . . . . . . 16 |- ( N e. ( ZZ>= ` 3 ) -> N e. CC )
57		2cn 9308	. . . . . . . . . . . . . . . 16 |- 2 e. CC
58		npcan 8482	. . . . . . . . . . . . . . . 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 2285	. . . . . . . . . . . . . 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 2265	. . . . . . 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 9899	. . . . 5 |- ( N e. ( ZZ>= ` 3 ) -> N e. NN )
74	4, 5	isclwwlknx 16411	. . . . 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 2203   =/= wne 2412   A.wral 2520   u. cun 3209   {cpr 3690   class class class wbr 4109   ` cfv 5352  (class class class)co 6050   CCcc 8125   0cc0 8127   1c1 8128   + caddc 8130   < clt 8308   - cmin 8444   NNcn 9237   2c2 9288   3c3 9289   ZZ>=cuz 9853  ..^cfzo 10476  ♯chash 11138  Word cword 11224  lastSclsw 11269   ++ cconcat 11278   <"cs1 11303  Vtxcvtx 16007  Edgcedg 16052   ClWWalksN cclwwlkn 16398  ClWWalksNOncclwwlknon 16421

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244

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 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-frec 6622  df-1o 6647  df-er 6767  df-map 6884  df-en 6976  df-dom 6977  df-fin 6978  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-reap 8849  df-ap 8856  df-inn 9238  df-2 9296  df-3 9297  df-n0 9497  df-z 9578  df-uz 9854  df-fz 10343  df-fzo 10477  df-ihash 11139  df-word 11225  df-lsw 11270  df-concat 11279  df-s1 11304  df-ndx 13215  df-slot 13216  df-base 13218  df-vtx 16009  df-clwwlk 16387  df-clwwlkn 16399  df-clwwlknon 16422

This theorem is referenced by:  clwwlknonex2e  16435