Theorem clwwlknonex2lem2 16223

Description: Lemma 2 for clwwlknonex2 16224: Transformation of a walk and two edges into a walk extended by two vertices/edges. (Contributed by AV, 22-Sep-2018.) (Revised by AV, 27-Jan-2022.)

Hypotheses

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

Assertion

Ref

Expression

clwwlknonex2lem2

|- ( ( ( ( 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 )

Distinct variable groups:   i, E   i, V   i, W   i, X   i, Y

Allowed substitution hints:   G( i)   N( i)

Proof of Theorem clwwlknonex2lem2

Step	Hyp	Ref	Expression
1		simpl 109	. . . . . . . . . . . . . . . 16 |- ( ( W e. Word V /\ ( X e. V /\ Y e. V ) ) -> W e. Word V )
2	1	adantr 276	. . . . . . . . . . . . . . 15 |- ( ( ( W e. Word V /\ ( X e. V /\ Y e. V ) ) /\ i e. ( 0..^ ( (♯ ` W ) - 1 ) ) ) -> W e. Word V )
3		elfzonn0 10413	. . . . . . . . . . . . . . . 16 |- ( i e. ( 0..^ ( (♯ ` W ) - 1 ) ) -> i e. NN0 )
4	3	adantl 277	. . . . . . . . . . . . . . 15 |- ( ( ( W e. Word V /\ ( X e. V /\ Y e. V ) ) /\ i e. ( 0..^ ( (♯ ` W ) - 1 ) ) ) -> i e. NN0 )
5		lencl 11104	. . . . . . . . . . . . . . . . . 18 |- ( W e. Word V -> (♯ ` W ) e. NN0 )
6		elfzo0 10409	. . . . . . . . . . . . . . . . . . 19 |- ( i e. ( 0..^ ( (♯ ` W ) - 1 ) ) <-> ( i e. NN0 /\ ( (♯ ` W ) - 1 ) e. NN /\ i < ( (♯ ` W ) - 1 ) ) )
7		nn0re 9399	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( i e. NN0 -> i e. RR )
8	7	adantr 276	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( i e. NN0 /\ (♯ ` W ) e. NN0 ) -> i e. RR )
9		nn0re 9399	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( (♯ ` W ) e. NN0 -> (♯ ` W ) e. RR )
10		peano2rem 8434	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( (♯ ` W ) e. RR -> ( (♯ ` W ) - 1 ) e. RR )
11	9, 10	syl 14	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( (♯ ` W ) e. NN0 -> ( (♯ ` W ) - 1 ) e. RR )
12	11	adantl 277	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( i e. NN0 /\ (♯ ` W ) e. NN0 ) -> ( (♯ ` W ) - 1 ) e. RR )
13	9	adantl 277	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( i e. NN0 /\ (♯ ` W ) e. NN0 ) -> (♯ ` W ) e. RR )
14	8, 12, 13	3jca 1201	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( i e. NN0 /\ (♯ ` W ) e. NN0 ) -> ( i e. RR /\ ( (♯ ` W ) - 1 ) e. RR /\ (♯ ` W ) e. RR ) )
15	9	ltm1d 9100	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( (♯ ` W ) e. NN0 -> ( (♯ ` W ) - 1 ) < (♯ ` W ) )
16	15	adantl 277	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( i e. NN0 /\ (♯ ` W ) e. NN0 ) -> ( (♯ ` W ) - 1 ) < (♯ ` W ) )
17		lttr 8241	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( i e. RR /\ ( (♯ ` W ) - 1 ) e. RR /\ (♯ ` W ) e. RR ) -> ( ( i < ( (♯ ` W ) - 1 ) /\ ( (♯ ` W ) - 1 ) < (♯ ` W ) ) -> i < (♯ ` W ) ) )
18	17	expcomd 1484	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( i e. RR /\ ( (♯ ` W ) - 1 ) e. RR /\ (♯ ` W ) e. RR ) -> ( ( (♯ ` W ) - 1 ) < (♯ ` W ) -> ( i < ( (♯ ` W ) - 1 ) -> i < (♯ ` W ) ) ) )
19	14, 16, 18	sylc 62	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( i e. NN0 /\ (♯ ` W ) e. NN0 ) -> ( i < ( (♯ ` W ) - 1 ) -> i < (♯ ` W ) ) )
20	19	impancom 260	. . . . . . . . . . . . . . . . . . . 20 |- ( ( i e. NN0 /\ i < ( (♯ ` W ) - 1 ) ) -> ( (♯ ` W ) e. NN0 -> i < (♯ ` W ) ) )
21	20	3adant2 1040	. . . . . . . . . . . . . . . . . . 19 |- ( ( i e. NN0 /\ ( (♯ ` W ) - 1 ) e. NN /\ i < ( (♯ ` W ) - 1 ) ) -> ( (♯ ` W ) e. NN0 -> i < (♯ ` W ) ) )
22	6, 21	sylbi 121	. . . . . . . . . . . . . . . . . 18 |- ( i e. ( 0..^ ( (♯ ` W ) - 1 ) ) -> ( (♯ ` W ) e. NN0 -> i < (♯ ` W ) ) )
23	5, 22	syl5com 29	. . . . . . . . . . . . . . . . 17 |- ( W e. Word V -> ( i e. ( 0..^ ( (♯ ` W ) - 1 ) ) -> i < (♯ ` W ) ) )
24	23	adantr 276	. . . . . . . . . . . . . . . 16 |- ( ( W e. Word V /\ ( X e. V /\ Y e. V ) ) -> ( i e. ( 0..^ ( (♯ ` W ) - 1 ) ) -> i < (♯ ` W ) ) )
25	24	imp 124	. . . . . . . . . . . . . . 15 |- ( ( ( W e. Word V /\ ( X e. V /\ Y e. V ) ) /\ i e. ( 0..^ ( (♯ ` W ) - 1 ) ) ) -> i < (♯ ` W ) )
26		simplrl 535	. . . . . . . . . . . . . . 15 |- ( ( ( W e. Word V /\ ( X e. V /\ Y e. V ) ) /\ i e. ( 0..^ ( (♯ ` W ) - 1 ) ) ) -> X e. V )
27		simplrr 536	. . . . . . . . . . . . . . 15 |- ( ( ( W e. Word V /\ ( X e. V /\ Y e. V ) ) /\ i e. ( 0..^ ( (♯ ` W ) - 1 ) ) ) -> Y e. V )
28	2, 4, 25, 26, 27	ccat2s1fvwd 11211	. . . . . . . . . . . . . 14 |- ( ( ( W e. Word V /\ ( X e. V /\ Y e. V ) ) /\ i e. ( 0..^ ( (♯ ` W ) - 1 ) ) ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` i ) = ( W ` i ) )
29	28	eqcomd 2235	. . . . . . . . . . . . 13 |- ( ( ( W e. Word V /\ ( X e. V /\ Y e. V ) ) /\ i e. ( 0..^ ( (♯ ` W ) - 1 ) ) ) -> ( W ` i ) = ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` i ) )
30		peano2nn0 9430	. . . . . . . . . . . . . . . 16 |- ( i e. NN0 -> ( i + 1 ) e. NN0 )
31	4, 30	syl 14	. . . . . . . . . . . . . . 15 |- ( ( ( W e. Word V /\ ( X e. V /\ Y e. V ) ) /\ i e. ( 0..^ ( (♯ ` W ) - 1 ) ) ) -> ( i + 1 ) e. NN0 )
32		1red 8182	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( i e. NN0 /\ (♯ ` W ) e. NN0 ) -> 1 e. RR )
33	8, 32, 13	ltaddsubd 8713	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( i e. NN0 /\ (♯ ` W ) e. NN0 ) -> ( ( i + 1 ) < (♯ ` W ) <-> i < ( (♯ ` W ) - 1 ) ) )
34	33	biimprd 158	. . . . . . . . . . . . . . . . . . . 20 |- ( ( i e. NN0 /\ (♯ ` W ) e. NN0 ) -> ( i < ( (♯ ` W ) - 1 ) -> ( i + 1 ) < (♯ ` W ) ) )
35	34	impancom 260	. . . . . . . . . . . . . . . . . . 19 |- ( ( i e. NN0 /\ i < ( (♯ ` W ) - 1 ) ) -> ( (♯ ` W ) e. NN0 -> ( i + 1 ) < (♯ ` W ) ) )
36	35	3adant2 1040	. . . . . . . . . . . . . . . . . 18 |- ( ( i e. NN0 /\ ( (♯ ` W ) - 1 ) e. NN /\ i < ( (♯ ` W ) - 1 ) ) -> ( (♯ ` W ) e. NN0 -> ( i + 1 ) < (♯ ` W ) ) )
37	6, 36	sylbi 121	. . . . . . . . . . . . . . . . 17 |- ( i e. ( 0..^ ( (♯ ` W ) - 1 ) ) -> ( (♯ ` W ) e. NN0 -> ( i + 1 ) < (♯ ` W ) ) )
38	5, 37	mpan9 281	. . . . . . . . . . . . . . . 16 |- ( ( W e. Word V /\ i e. ( 0..^ ( (♯ ` W ) - 1 ) ) ) -> ( i + 1 ) < (♯ ` W ) )
39	38	adantlr 477	. . . . . . . . . . . . . . 15 |- ( ( ( W e. Word V /\ ( X e. V /\ Y e. V ) ) /\ i e. ( 0..^ ( (♯ ` W ) - 1 ) ) ) -> ( i + 1 ) < (♯ ` W ) )
40	2, 31, 39, 26, 27	ccat2s1fvwd 11211	. . . . . . . . . . . . . 14 |- ( ( ( W e. Word V /\ ( X e. V /\ Y e. V ) ) /\ i e. ( 0..^ ( (♯ ` W ) - 1 ) ) ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( i + 1 ) ) = ( W ` ( i + 1 ) ) )
41	40	eqcomd 2235	. . . . . . . . . . . . 13 |- ( ( ( W e. Word V /\ ( X e. V /\ Y e. V ) ) /\ i e. ( 0..^ ( (♯ ` W ) - 1 ) ) ) -> ( W ` ( i + 1 ) ) = ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( i + 1 ) ) )
42	29, 41	preq12d 3752	. . . . . . . . . . . 12 |- ( ( ( W e. Word V /\ ( X e. V /\ Y e. V ) ) /\ i e. ( 0..^ ( (♯ ` W ) - 1 ) ) ) -> { ( W ` i ) , ( W ` ( i + 1 ) ) } = { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` i ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( i + 1 ) ) } )
43	42	eleq1d 2298	. . . . . . . . . . 11 |- ( ( ( W e. Word V /\ ( X e. V /\ Y e. V ) ) /\ i e. ( 0..^ ( (♯ ` W ) - 1 ) ) ) -> ( { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E <-> { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` i ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( i + 1 ) ) } e. E ) )
44	43	ralbidva 2526	. . . . . . . . . 10 |- ( ( W e. Word V /\ ( X e. V /\ Y e. V ) ) -> ( A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E <-> A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` i ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( i + 1 ) ) } e. E ) )
45	44	biimpd 144	. . . . . . . . 9 |- ( ( W e. Word V /\ ( X e. V /\ Y e. V ) ) -> ( A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E -> A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` i ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( i + 1 ) ) } e. E ) )
46	45	impancom 260	. . . . . . . 8 |- ( ( W e. Word V /\ A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E ) -> ( ( X e. V /\ Y e. V ) -> A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` i ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( i + 1 ) ) } e. E ) )
47	46	3adant3 1041	. . . . . . 7 |- ( ( W e. Word V /\ A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) -> ( ( X e. V /\ Y e. V ) -> A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` i ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( i + 1 ) ) } e. E ) )
48	47	3ad2ant1 1042	. . . . . 6 |- ( ( ( 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 e. V /\ Y e. V ) -> A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` i ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( i + 1 ) ) } e. E ) )
49	48	com12 30	. . . . 5 |- ( ( X e. V /\ Y e. V ) -> ( ( ( 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 ) -> A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` i ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( i + 1 ) ) } e. E ) )
50	49	a1dd 48	. . . 4 |- ( ( X e. V /\ Y e. V ) -> ( ( ( 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 ) ) { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` i ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( i + 1 ) ) } e. E ) ) )
51	50	3adant3 1041	. . 3 |- ( ( 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 ) ) { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` i ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( i + 1 ) ) } e. E ) ) )
52	51	imp31 256	. 2 |- ( ( ( ( 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 ) ) { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` i ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( i + 1 ) ) } e. E )
53		ax-1 6	. . . . . . . . . . 11 |- ( ( X e. V /\ Y e. V ) -> ( { X , Y } e. E -> ( X e. V /\ Y e. V ) ) )
54	53	3adant3 1041	. . . . . . . . . 10 |- ( ( X e. V /\ Y e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( { X , Y } e. E -> ( X e. V /\ Y e. V ) ) )
55		simpl1l 1072	. . . . . . . . . . . . . . . 16 |- ( ( ( ( W e. Word V /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) /\ ( X e. V /\ Y e. V ) /\ ( W ` 0 ) = X ) /\ ( N e. ( ZZ>= ` 3 ) /\ (♯ ` W ) = ( N - 2 ) ) ) -> W e. Word V )
56		oveq1 6018	. . . . . . . . . . . . . . . . . . . 20 |- ( (♯ ` W ) = ( N - 2 ) -> ( (♯ ` W ) - 1 ) = ( ( N - 2 ) - 1 ) )
57	56	adantr 276	. . . . . . . . . . . . . . . . . . 19 |- ( ( (♯ ` W ) = ( N - 2 ) /\ N e. ( ZZ>= ` 3 ) ) -> ( (♯ ` W ) - 1 ) = ( ( N - 2 ) - 1 ) )
58		eluzelcn 9755	. . . . . . . . . . . . . . . . . . . . . 22 |- ( N e. ( ZZ>= ` 3 ) -> N e. CC )
59		2cnd 9204	. . . . . . . . . . . . . . . . . . . . . 22 |- ( N e. ( ZZ>= ` 3 ) -> 2 e. CC )
60		1cnd 8183	. . . . . . . . . . . . . . . . . . . . . 22 |- ( N e. ( ZZ>= ` 3 ) -> 1 e. CC )
61	58, 59, 60	subsub4d 8509	. . . . . . . . . . . . . . . . . . . . 21 |- ( N e. ( ZZ>= ` 3 ) -> ( ( N - 2 ) - 1 ) = ( N - ( 2 + 1 ) ) )
62		2p1e3 9265	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( 2 + 1 ) = 3
63	62	a1i 9	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( N e. ( ZZ>= ` 3 ) -> ( 2 + 1 ) = 3 )
64	63	oveq2d 6027	. . . . . . . . . . . . . . . . . . . . . 22 |- ( N e. ( ZZ>= ` 3 ) -> ( N - ( 2 + 1 ) ) = ( N - 3 ) )
65		uznn0sub 9776	. . . . . . . . . . . . . . . . . . . . . 22 |- ( N e. ( ZZ>= ` 3 ) -> ( N - 3 ) e. NN0 )
66	64, 65	eqeltrd 2306	. . . . . . . . . . . . . . . . . . . . 21 |- ( N e. ( ZZ>= ` 3 ) -> ( N - ( 2 + 1 ) ) e. NN0 )
67	61, 66	eqeltrd 2306	. . . . . . . . . . . . . . . . . . . 20 |- ( N e. ( ZZ>= ` 3 ) -> ( ( N - 2 ) - 1 ) e. NN0 )
68	67	adantl 277	. . . . . . . . . . . . . . . . . . 19 |- ( ( (♯ ` W ) = ( N - 2 ) /\ N e. ( ZZ>= ` 3 ) ) -> ( ( N - 2 ) - 1 ) e. NN0 )
69	57, 68	eqeltrd 2306	. . . . . . . . . . . . . . . . . 18 |- ( ( (♯ ` W ) = ( N - 2 ) /\ N e. ( ZZ>= ` 3 ) ) -> ( (♯ ` W ) - 1 ) e. NN0 )
70	69	ancoms 268	. . . . . . . . . . . . . . . . 17 |- ( ( N e. ( ZZ>= ` 3 ) /\ (♯ ` W ) = ( N - 2 ) ) -> ( (♯ ` W ) - 1 ) e. NN0 )
71	70	adantl 277	. . . . . . . . . . . . . . . 16 |- ( ( ( ( W e. Word V /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) /\ ( X e. V /\ Y e. V ) /\ ( W ` 0 ) = X ) /\ ( N e. ( ZZ>= ` 3 ) /\ (♯ ` W ) = ( N - 2 ) ) ) -> ( (♯ ` W ) - 1 ) e. NN0 )
72		simpl 109	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( W e. Word V /\ ( N e. ( ZZ>= ` 3 ) /\ (♯ ` W ) = ( N - 2 ) ) ) -> W e. Word V )
73	70	adantl 277	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( W e. Word V /\ ( N e. ( ZZ>= ` 3 ) /\ (♯ ` W ) = ( N - 2 ) ) ) -> ( (♯ ` W ) - 1 ) e. NN0 )
74	5, 9	syl 14	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( W e. Word V -> (♯ ` W ) e. RR )
75	74	adantr 276	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( W e. Word V /\ ( N e. ( ZZ>= ` 3 ) /\ (♯ ` W ) = ( N - 2 ) ) ) -> (♯ ` W ) e. RR )
76	75	ltm1d 9100	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( W e. Word V /\ ( N e. ( ZZ>= ` 3 ) /\ (♯ ` W ) = ( N - 2 ) ) ) -> ( (♯ ` W ) - 1 ) < (♯ ` W ) )
77	72, 73, 76	3jca 1201	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( W e. Word V /\ ( N e. ( ZZ>= ` 3 ) /\ (♯ ` W ) = ( N - 2 ) ) ) -> ( W e. Word V /\ ( (♯ ` W ) - 1 ) e. NN0 /\ ( (♯ ` W ) - 1 ) < (♯ ` W ) ) )
78	77	ex 115	. . . . . . . . . . . . . . . . . . . 20 |- ( W e. Word V -> ( ( N e. ( ZZ>= ` 3 ) /\ (♯ ` W ) = ( N - 2 ) ) -> ( W e. Word V /\ ( (♯ ` W ) - 1 ) e. NN0 /\ ( (♯ ` W ) - 1 ) < (♯ ` W ) ) ) )
79	78	adantr 276	. . . . . . . . . . . . . . . . . . 19 |- ( ( W e. Word V /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) -> ( ( N e. ( ZZ>= ` 3 ) /\ (♯ ` W ) = ( N - 2 ) ) -> ( W e. Word V /\ ( (♯ ` W ) - 1 ) e. NN0 /\ ( (♯ ` W ) - 1 ) < (♯ ` W ) ) ) )
80	79	3ad2ant1 1042	. . . . . . . . . . . . . . . . . 18 |- ( ( ( W e. Word V /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) /\ ( X e. V /\ Y e. V ) /\ ( W ` 0 ) = X ) -> ( ( N e. ( ZZ>= ` 3 ) /\ (♯ ` W ) = ( N - 2 ) ) -> ( W e. Word V /\ ( (♯ ` W ) - 1 ) e. NN0 /\ ( (♯ ` W ) - 1 ) < (♯ ` W ) ) ) )
81	80	imp 124	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( W e. Word V /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) /\ ( X e. V /\ Y e. V ) /\ ( W ` 0 ) = X ) /\ ( N e. ( ZZ>= ` 3 ) /\ (♯ ` W ) = ( N - 2 ) ) ) -> ( W e. Word V /\ ( (♯ ` W ) - 1 ) e. NN0 /\ ( (♯ ` W ) - 1 ) < (♯ ` W ) ) )
82	81	simp3d 1035	. . . . . . . . . . . . . . . 16 |- ( ( ( ( W e. Word V /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) /\ ( X e. V /\ Y e. V ) /\ ( W ` 0 ) = X ) /\ ( N e. ( ZZ>= ` 3 ) /\ (♯ ` W ) = ( N - 2 ) ) ) -> ( (♯ ` W ) - 1 ) < (♯ ` W ) )
83		simpl2l 1074	. . . . . . . . . . . . . . . 16 |- ( ( ( ( W e. Word V /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) /\ ( X e. V /\ Y e. V ) /\ ( W ` 0 ) = X ) /\ ( N e. ( ZZ>= ` 3 ) /\ (♯ ` W ) = ( N - 2 ) ) ) -> X e. V )
84		simpl2r 1075	. . . . . . . . . . . . . . . 16 |- ( ( ( ( W e. Word V /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) /\ ( X e. V /\ Y e. V ) /\ ( W ` 0 ) = X ) /\ ( N e. ( ZZ>= ` 3 ) /\ (♯ ` W ) = ( N - 2 ) ) ) -> Y e. V )
85	55, 71, 82, 83, 84	ccat2s1fvwd 11211	. . . . . . . . . . . . . . 15 |- ( ( ( ( W e. Word V /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) /\ ( X e. V /\ Y e. V ) /\ ( W ` 0 ) = X ) /\ ( N e. ( ZZ>= ` 3 ) /\ (♯ ` W ) = ( N - 2 ) ) ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( (♯ ` W ) - 1 ) ) = ( W ` ( (♯ ` W ) - 1 ) ) )
86		nn0cn 9400	. . . . . . . . . . . . . . . . . . . . . 22 |- ( (♯ ` W ) e. NN0 -> (♯ ` W ) e. CC )
87		ax-1cn 8113	. . . . . . . . . . . . . . . . . . . . . 22 |- 1 e. CC
88		npcan 8376	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( (♯ ` W ) e. CC /\ 1 e. CC ) -> ( ( (♯ ` W ) - 1 ) + 1 ) = (♯ ` W ) )
89	86, 87, 88	sylancl 413	. . . . . . . . . . . . . . . . . . . . 21 |- ( (♯ ` W ) e. NN0 -> ( ( (♯ ` W ) - 1 ) + 1 ) = (♯ ` W ) )
90	5, 89	syl 14	. . . . . . . . . . . . . . . . . . . 20 |- ( W e. Word V -> ( ( (♯ ` W ) - 1 ) + 1 ) = (♯ ` W ) )
91	90	adantr 276	. . . . . . . . . . . . . . . . . . 19 |- ( ( W e. Word V /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) -> ( ( (♯ ` W ) - 1 ) + 1 ) = (♯ ` W ) )
92	91	3ad2ant1 1042	. . . . . . . . . . . . . . . . . 18 |- ( ( ( W e. Word V /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) /\ ( X e. V /\ Y e. V ) /\ ( W ` 0 ) = X ) -> ( ( (♯ ` W ) - 1 ) + 1 ) = (♯ ` W ) )
93	92	fveq2d 5637	. . . . . . . . . . . . . . . . 17 |- ( ( ( W e. Word V /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) /\ ( X e. V /\ Y e. V ) /\ ( W ` 0 ) = X ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( ( (♯ ` W ) - 1 ) + 1 ) ) = ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` (♯ ` W ) ) )
94		eqid 2229	. . . . . . . . . . . . . . . . . . . 20 |- (♯ ` W ) = (♯ ` W )
95		ccatw2s1p1g 11209	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( W e. Word V /\ (♯ ` W ) = (♯ ` W ) ) /\ ( X e. V /\ Y e. V ) ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` (♯ ` W ) ) = X )
96	94, 95	mpanl2 435	. . . . . . . . . . . . . . . . . . 19 |- ( ( W e. Word V /\ ( X e. V /\ Y e. V ) ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` (♯ ` W ) ) = X )
97	96	adantlr 477	. . . . . . . . . . . . . . . . . 18 |- ( ( ( W e. Word V /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) /\ ( X e. V /\ Y e. V ) ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` (♯ ` W ) ) = X )
98	97	3adant3 1041	. . . . . . . . . . . . . . . . 17 |- ( ( ( W e. Word V /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) /\ ( X e. V /\ Y e. V ) /\ ( W ` 0 ) = X ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` (♯ ` W ) ) = X )
99	93, 98	eqtrd 2262	. . . . . . . . . . . . . . . 16 |- ( ( ( W e. Word V /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) /\ ( X e. V /\ Y e. V ) /\ ( W ` 0 ) = X ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( ( (♯ ` W ) - 1 ) + 1 ) ) = X )
100	99	adantr 276	. . . . . . . . . . . . . . 15 |- ( ( ( ( W e. Word V /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) /\ ( X e. V /\ Y e. V ) /\ ( W ` 0 ) = X ) /\ ( N e. ( ZZ>= ` 3 ) /\ (♯ ` W ) = ( N - 2 ) ) ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( ( (♯ ` W ) - 1 ) + 1 ) ) = X )
101	85, 100	preq12d 3752	. . . . . . . . . . . . . 14 |- ( ( ( ( W e. Word V /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) /\ ( X e. V /\ Y e. V ) /\ ( W ` 0 ) = X ) /\ ( N e. ( ZZ>= ` 3 ) /\ (♯ ` W ) = ( N - 2 ) ) ) -> { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( (♯ ` W ) - 1 ) ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( ( (♯ ` W ) - 1 ) + 1 ) ) } = { ( W ` ( (♯ ` W ) - 1 ) ) , X } )
102		lswwrd 11147	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( W e. Word V -> (lastS ` W ) = ( W ` ( (♯ ` W ) - 1 ) ) )
103	102	adantl 277	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( W ` 0 ) = X /\ W e. Word V ) -> (lastS ` W ) = ( W ` ( (♯ ` W ) - 1 ) ) )
104		simpl 109	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( W ` 0 ) = X /\ W e. Word V ) -> ( W ` 0 ) = X )
105	103, 104	preq12d 3752	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( W ` 0 ) = X /\ W e. Word V ) -> { (lastS ` W ) , ( W ` 0 ) } = { ( W ` ( (♯ ` W ) - 1 ) ) , X } )
106	105	eleq1d 2298	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( W ` 0 ) = X /\ W e. Word V ) -> ( { (lastS ` W ) , ( W ` 0 ) } e. E <-> { ( W ` ( (♯ ` W ) - 1 ) ) , X } e. E ) )
107	106	biimpd 144	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( W ` 0 ) = X /\ W e. Word V ) -> ( { (lastS ` W ) , ( W ` 0 ) } e. E -> { ( W ` ( (♯ ` W ) - 1 ) ) , X } e. E ) )
108	107	expcom 116	. . . . . . . . . . . . . . . . . 18 |- ( W e. Word V -> ( ( W ` 0 ) = X -> ( { (lastS ` W ) , ( W ` 0 ) } e. E -> { ( W ` ( (♯ ` W ) - 1 ) ) , X } e. E ) ) )
109	108	com23 78	. . . . . . . . . . . . . . . . 17 |- ( W e. Word V -> ( { (lastS ` W ) , ( W ` 0 ) } e. E -> ( ( W ` 0 ) = X -> { ( W ` ( (♯ ` W ) - 1 ) ) , X } e. E ) ) )
110	109	imp31 256	. . . . . . . . . . . . . . . 16 |- ( ( ( W e. Word V /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) /\ ( W ` 0 ) = X ) -> { ( W ` ( (♯ ` W ) - 1 ) ) , X } e. E )
111	110	3adant2 1040	. . . . . . . . . . . . . . 15 |- ( ( ( W e. Word V /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) /\ ( X e. V /\ Y e. V ) /\ ( W ` 0 ) = X ) -> { ( W ` ( (♯ ` W ) - 1 ) ) , X } e. E )
112	111	adantr 276	. . . . . . . . . . . . . 14 |- ( ( ( ( W e. Word V /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) /\ ( X e. V /\ Y e. V ) /\ ( W ` 0 ) = X ) /\ ( N e. ( ZZ>= ` 3 ) /\ (♯ ` W ) = ( N - 2 ) ) ) -> { ( W ` ( (♯ ` W ) - 1 ) ) , X } e. E )
113	101, 112	eqeltrd 2306	. . . . . . . . . . . . 13 |- ( ( ( ( W e. Word V /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) /\ ( X e. V /\ Y e. V ) /\ ( W ` 0 ) = X ) /\ ( N e. ( ZZ>= ` 3 ) /\ (♯ ` W ) = ( N - 2 ) ) ) -> { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( (♯ ` W ) - 1 ) ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( ( (♯ ` W ) - 1 ) + 1 ) ) } e. E )
114	113	exp520 1252	. . . . . . . . . . . 12 |- ( ( W e. Word V /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) -> ( ( X e. V /\ Y e. V ) -> ( ( W ` 0 ) = X -> ( N e. ( ZZ>= ` 3 ) -> ( (♯ ` W ) = ( N - 2 ) -> { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( (♯ ` W ) - 1 ) ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( ( (♯ ` W ) - 1 ) + 1 ) ) } e. E ) ) ) ) )
115	114	com14 88	. . . . . . . . . . 11 |- ( N e. ( ZZ>= ` 3 ) -> ( ( X e. V /\ Y e. V ) -> ( ( W ` 0 ) = X -> ( ( W e. Word V /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) -> ( (♯ ` W ) = ( N - 2 ) -> { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( (♯ ` W ) - 1 ) ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( ( (♯ ` W ) - 1 ) + 1 ) ) } e. E ) ) ) ) )
116	115	3ad2ant3 1044	. . . . . . . . . 10 |- ( ( X e. V /\ Y e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( ( X e. V /\ Y e. V ) -> ( ( W ` 0 ) = X -> ( ( W e. Word V /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) -> ( (♯ ` W ) = ( N - 2 ) -> { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( (♯ ` W ) - 1 ) ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( ( (♯ ` W ) - 1 ) + 1 ) ) } e. E ) ) ) ) )
117	54, 116	syld 45	. . . . . . . . 9 |- ( ( X e. V /\ Y e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( { X , Y } e. E -> ( ( W ` 0 ) = X -> ( ( W e. Word V /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) -> ( (♯ ` W ) = ( N - 2 ) -> { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( (♯ ` W ) - 1 ) ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( ( (♯ ` W ) - 1 ) + 1 ) ) } e. E ) ) ) ) )
118	117	com25 91	. . . . . . . 8 |- ( ( X e. V /\ Y e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( (♯ ` W ) = ( N - 2 ) -> ( ( W ` 0 ) = X -> ( ( W e. Word V /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) -> ( { X , Y } e. E -> { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( (♯ ` W ) - 1 ) ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( ( (♯ ` W ) - 1 ) + 1 ) ) } e. E ) ) ) ) )
119	118	com14 88	. . . . . . 7 |- ( ( W e. Word V /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) -> ( (♯ ` W ) = ( N - 2 ) -> ( ( W ` 0 ) = X -> ( ( X e. V /\ Y e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( { X , Y } e. E -> { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( (♯ ` W ) - 1 ) ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( ( (♯ ` W ) - 1 ) + 1 ) ) } e. E ) ) ) ) )
120	119	3adant2 1040	. . . . . 6 |- ( ( 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 e. V /\ Y e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( { X , Y } e. E -> { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( (♯ ` W ) - 1 ) ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( ( (♯ ` W ) - 1 ) + 1 ) ) } e. E ) ) ) ) )
121	120	3imp 1217	. . . . 5 |- ( ( ( 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 e. V /\ Y e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( { X , Y } e. E -> { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( (♯ ` W ) - 1 ) ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( ( (♯ ` W ) - 1 ) + 1 ) ) } e. E ) ) )
122	121	impcom 125	. . . 4 |- ( ( ( 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 ) - 1 ) ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( ( (♯ ` W ) - 1 ) + 1 ) ) } e. E ) )
123	122	imp 124	. . 3 |- ( ( ( ( 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 ) - 1 ) ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( ( (♯ ` W ) - 1 ) + 1 ) ) } e. E )
124		ccatw2s1p2 11210	. . . . . . . . . . . . . 14 |- ( ( ( W e. Word V /\ (♯ ` W ) = (♯ ` W ) ) /\ ( X e. V /\ Y e. V ) ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( (♯ ` W ) + 1 ) ) = Y )
125	94, 124	mpanl2 435	. . . . . . . . . . . . 13 |- ( ( W e. Word V /\ ( X e. V /\ Y e. V ) ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( (♯ ` W ) + 1 ) ) = Y )
126	96, 125	preq12d 3752	. . . . . . . . . . . 12 |- ( ( W e. Word V /\ ( X e. V /\ Y e. V ) ) -> { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` (♯ ` W ) ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( (♯ ` W ) + 1 ) ) } = { X , Y } )
127	126	expcom 116	. . . . . . . . . . 11 |- ( ( X e. V /\ Y e. V ) -> ( W e. Word V -> { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` (♯ ` W ) ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( (♯ ` W ) + 1 ) ) } = { X , Y } ) )
128	127	a1i 9	. . . . . . . . . 10 |- ( { X , Y } e. E -> ( ( X e. V /\ Y e. V ) -> ( W e. Word V -> { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` (♯ ` W ) ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( (♯ ` W ) + 1 ) ) } = { X , Y } ) ) )
129	128	com13 80	. . . . . . . . 9 |- ( W e. Word V -> ( ( X e. V /\ Y e. V ) -> ( { X , Y } e. E -> { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` (♯ ` W ) ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( (♯ ` W ) + 1 ) ) } = { X , Y } ) ) )
130	129	3ad2ant1 1042	. . . . . . . 8 |- ( ( W e. Word V /\ A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) -> ( ( X e. V /\ Y e. V ) -> ( { X , Y } e. E -> { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` (♯ ` W ) ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( (♯ ` W ) + 1 ) ) } = { X , Y } ) ) )
131	130	3ad2ant1 1042	. . . . . . 7 |- ( ( ( 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 e. V /\ Y e. V ) -> ( { X , Y } e. E -> { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` (♯ ` W ) ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( (♯ ` W ) + 1 ) ) } = { X , Y } ) ) )
132	131	com12 30	. . . . . 6 |- ( ( X e. V /\ Y e. V ) -> ( ( ( 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 ) ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( (♯ ` W ) + 1 ) ) } = { X , Y } ) ) )
133	132	3adant3 1041	. . . . 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 "> ) ` (♯ ` W ) ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( (♯ ` W ) + 1 ) ) } = { X , Y } ) ) )
134	133	imp31 256	. . . 4 |- ( ( ( ( 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 ) ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( (♯ ` W ) + 1 ) ) } = { X , Y } )
135		simpr 110	. . . 4 |- ( ( ( ( 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 , Y } e. E )
136	134, 135	eqeltrd 2306	. . 3 |- ( ( ( ( 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 ) ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( (♯ ` W ) + 1 ) ) } e. E )
137	5	nn0zd 9588	. . . . . . . . 9 |- ( W e. Word V -> (♯ ` W ) e. ZZ )
138		1zzd 9494	. . . . . . . . 9 |- ( W e. Word V -> 1 e. ZZ )
139	137, 138	zsubcld 9595	. . . . . . . 8 |- ( W e. Word V -> ( (♯ ` W ) - 1 ) e. ZZ )
140		fveq2 5633	. . . . . . . . . . 11 |- ( i = ( (♯ ` W ) - 1 ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` i ) = ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( (♯ ` W ) - 1 ) ) )
141		fvoveq1 6034	. . . . . . . . . . 11 |- ( i = ( (♯ ` W ) - 1 ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( i + 1 ) ) = ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( ( (♯ ` W ) - 1 ) + 1 ) ) )
142	140, 141	preq12d 3752	. . . . . . . . . 10 |- ( i = ( (♯ ` W ) - 1 ) -> { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` i ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( i + 1 ) ) } = { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( (♯ ` W ) - 1 ) ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( ( (♯ ` W ) - 1 ) + 1 ) ) } )
143	142	eleq1d 2298	. . . . . . . . 9 |- ( i = ( (♯ ` W ) - 1 ) -> ( { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` i ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( i + 1 ) ) } e. E <-> { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( (♯ ` W ) - 1 ) ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( ( (♯ ` W ) - 1 ) + 1 ) ) } e. E ) )
144		fveq2 5633	. . . . . . . . . . 11 |- ( i = (♯ ` W ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` i ) = ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` (♯ ` W ) ) )
145		fvoveq1 6034	. . . . . . . . . . 11 |- ( i = (♯ ` W ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( i + 1 ) ) = ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( (♯ ` W ) + 1 ) ) )
146	144, 145	preq12d 3752	. . . . . . . . . 10 |- ( i = (♯ ` W ) -> { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` i ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( i + 1 ) ) } = { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` (♯ ` W ) ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( (♯ ` W ) + 1 ) ) } )
147	146	eleq1d 2298	. . . . . . . . 9 |- ( i = (♯ ` W ) -> ( { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` i ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( i + 1 ) ) } e. E <-> { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` (♯ ` W ) ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( (♯ ` W ) + 1 ) ) } e. E ) )
148	143, 147	ralprg 3718	. . . . . . . 8 |- ( ( ( (♯ ` W ) - 1 ) e. ZZ /\ (♯ ` W ) e. NN0 ) -> ( A. i e. { ( (♯ ` W ) - 1 ) , (♯ ` W ) } { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` i ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( i + 1 ) ) } e. E <-> ( { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( (♯ ` W ) - 1 ) ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( ( (♯ ` W ) - 1 ) + 1 ) ) } e. E /\ { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` (♯ ` W ) ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( (♯ ` W ) + 1 ) ) } e. E ) ) )
149	139, 5, 148	syl2anc 411	. . . . . . 7 |- ( W e. Word V -> ( A. i e. { ( (♯ ` W ) - 1 ) , (♯ ` W ) } { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` i ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( i + 1 ) ) } e. E <-> ( { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( (♯ ` W ) - 1 ) ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( ( (♯ ` W ) - 1 ) + 1 ) ) } e. E /\ { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` (♯ ` W ) ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( (♯ ` W ) + 1 ) ) } e. E ) ) )
150	149	3ad2ant1 1042	. . . . . 6 |- ( ( W e. Word V /\ A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) -> ( A. i e. { ( (♯ ` W ) - 1 ) , (♯ ` W ) } { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` i ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( i + 1 ) ) } e. E <-> ( { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( (♯ ` W ) - 1 ) ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( ( (♯ ` W ) - 1 ) + 1 ) ) } e. E /\ { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` (♯ ` W ) ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( (♯ ` W ) + 1 ) ) } e. E ) ) )
151	150	3ad2ant1 1042	. . . . 5 |- ( ( ( 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 ) -> ( A. i e. { ( (♯ ` W ) - 1 ) , (♯ ` W ) } { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` i ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( i + 1 ) ) } e. E <-> ( { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( (♯ ` W ) - 1 ) ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( ( (♯ ` W ) - 1 ) + 1 ) ) } e. E /\ { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` (♯ ` W ) ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( (♯ ` W ) + 1 ) ) } e. E ) ) )
152	151	adantl 277	. . . 4 |- ( ( ( 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 ) ) -> ( A. i e. { ( (♯ ` W ) - 1 ) , (♯ ` W ) } { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` i ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( i + 1 ) ) } e. E <-> ( { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( (♯ ` W ) - 1 ) ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( ( (♯ ` W ) - 1 ) + 1 ) ) } e. E /\ { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` (♯ ` W ) ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( (♯ ` W ) + 1 ) ) } e. E ) ) )
153	152	adantr 276	. . 3 |- ( ( ( ( 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. { ( (♯ ` W ) - 1 ) , (♯ ` W ) } { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` i ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( i + 1 ) ) } e. E <-> ( { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( (♯ ` W ) - 1 ) ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( ( (♯ ` W ) - 1 ) + 1 ) ) } e. E /\ { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` (♯ ` W ) ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( (♯ ` W ) + 1 ) ) } e. E ) ) )
154	123, 136, 153	mpbir2and 950	. 2 |- ( ( ( ( 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. { ( (♯ ` W ) - 1 ) , (♯ ` W ) } { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` i ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( i + 1 ) ) } e. E )
155		ralunb 3386	. 2 |- ( A. i e. ( ( 0..^ ( (♯ ` W ) - 1 ) ) u. { ( (♯ ` W ) - 1 ) , (♯ ` W ) } ) { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` i ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( i + 1 ) ) } e. E <-> ( A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` i ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( i + 1 ) ) } e. E /\ A. i e. { ( (♯ ` W ) - 1 ) , (♯ ` W ) } { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` i ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( i + 1 ) ) } e. E ) )
156	52, 154, 155	sylanbrc 417	1 |- ( ( ( ( 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 )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   A.wral 2508   u. cun 3196   {cpr 3668   class class class wbr 4084   ` cfv 5322  (class class class)co 6011   CCcc 8018   RRcr 8019   0cc0 8020   1c1 8021   + caddc 8023   < clt 8202   - cmin 8338   NNcn 9131   2c2 9182   3c3 9183   NN0cn0 9390   ZZcz 9467   ZZ>=cuz 9743  ..^cfzo 10365  ♯chash 11025  Word cword 11100  lastSclsw 11145   ++ cconcat 11154   <"cs1 11179  Vtxcvtx 15850  Edgcedg 15895

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-addcom 8120  ax-addass 8122  ax-distr 8124  ax-i2m1 8125  ax-0lt1 8126  ax-0id 8128  ax-rnegex 8129  ax-cnre 8131  ax-pre-ltirr 8132  ax-pre-ltwlin 8133  ax-pre-lttrn 8134  ax-pre-apti 8135  ax-pre-ltadd 8136

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-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-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

This theorem is referenced by:  clwwlknonex2  16224