Theorem clwwlknonex2lem2 16559

Description: Lemma 2 for clwwlknonex2 16560: 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 10547	. . . . . . . . . . . . . . . 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 11253	. . . . . . . . . . . . . . . . . 18 |- ( W e. Word V -> (♯ ` W ) e. NN0 )
6		elfzo0 10542	. . . . . . . . . . . . . . . . . . 19 |- ( i e. ( 0..^ ( (♯ ` W ) - 1 ) ) <-> ( i e. NN0 /\ ( (♯ ` W ) - 1 ) e. NN /\ i < ( (♯ ` W ) - 1 ) ) )
7		nn0re 9522	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( i e. NN0 -> i e. RR )
8	7	adantr 276	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( i e. NN0 /\ (♯ ` W ) e. NN0 ) -> i e. RR )
9		nn0re 9522	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( (♯ ` W ) e. NN0 -> (♯ ` W ) e. RR )
10		peano2rem 8556	. . . . . . . . . . . . . . . . . . . . . . . . 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 1204	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( i e. NN0 /\ (♯ ` W ) e. NN0 ) -> ( i e. RR /\ ( (♯ ` W ) - 1 ) e. RR /\ (♯ ` W ) e. RR ) )
15	9	ltm1d 9223	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( (♯ ` W ) e. NN0 -> ( (♯ ` W ) - 1 ) < (♯ ` W ) )
16	15	adantl 277	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( i e. NN0 /\ (♯ ` W ) e. NN0 ) -> ( (♯ ` W ) - 1 ) < (♯ ` W ) )
17		lttr 8363	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( i e. RR /\ ( (♯ ` W ) - 1 ) e. RR /\ (♯ ` W ) e. RR ) -> ( ( i < ( (♯ ` W ) - 1 ) /\ ( (♯ ` W ) - 1 ) < (♯ ` W ) ) -> i < (♯ ` W ) ) )
18	17	expcomd 1487	. . . . . . . . . . . . . . . . . . . . . 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 1043	. . . . . . . . . . . . . . . . . . 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 537	. . . . . . . . . . . . . . 15 |- ( ( ( W e. Word V /\ ( X e. V /\ Y e. V ) ) /\ i e. ( 0..^ ( (♯ ` W ) - 1 ) ) ) -> X e. V )
27		simplrr 538	. . . . . . . . . . . . . . 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 11360	. . . . . . . . . . . . . 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 2240	. . . . . . . . . . . . 13 |- ( ( ( W e. Word V /\ ( X e. V /\ Y e. V ) ) /\ i e. ( 0..^ ( (♯ ` W ) - 1 ) ) ) -> ( W ` i ) = ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` i ) )
30		peano2nn0 9553	. . . . . . . . . . . . . . . 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 8305	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( i e. NN0 /\ (♯ ` W ) e. NN0 ) -> 1 e. RR )
33	8, 32, 13	ltaddsubd 8836	. . . . . . . . . . . . . . . . . . . . 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 1043	. . . . . . . . . . . . . . . . . 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 11360	. . . . . . . . . . . . . 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 2240	. . . . . . . . . . . . 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 3781	. . . . . . . . . . . 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 2303	. . . . . . . . . . 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 2540	. . . . . . . . . 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 1044	. . . . . . 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 1045	. . . . . 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 1044	. . 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 1044	. . . . . . . . . 10 |- ( ( X e. V /\ Y e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( { X , Y } e. E -> ( X e. V /\ Y e. V ) ) )
55		simpl1l 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 ) ) ) -> W e. Word V )
56		oveq1 6065	. . . . . . . . . . . . . . . . . . . 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 9883	. . . . . . . . . . . . . . . . . . . . . 22 |- ( N e. ( ZZ>= ` 3 ) -> N e. CC )
59		2cnd 9327	. . . . . . . . . . . . . . . . . . . . . 22 |- ( N e. ( ZZ>= ` 3 ) -> 2 e. CC )
60		1cnd 8306	. . . . . . . . . . . . . . . . . . . . . 22 |- ( N e. ( ZZ>= ` 3 ) -> 1 e. CC )
61	58, 59, 60	subsub4d 8631	. . . . . . . . . . . . . . . . . . . . 21 |- ( N e. ( ZZ>= ` 3 ) -> ( ( N - 2 ) - 1 ) = ( N - ( 2 + 1 ) ) )
62		2p1e3 9388	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( 2 + 1 ) = 3
63	62	a1i 9	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( N e. ( ZZ>= ` 3 ) -> ( 2 + 1 ) = 3 )
64	63	oveq2d 6074	. . . . . . . . . . . . . . . . . . . . . 22 |- ( N e. ( ZZ>= ` 3 ) -> ( N - ( 2 + 1 ) ) = ( N - 3 ) )
65		uznn0sub 9904	. . . . . . . . . . . . . . . . . . . . . 22 |- ( N e. ( ZZ>= ` 3 ) -> ( N - 3 ) e. NN0 )
66	64, 65	eqeltrd 2311	. . . . . . . . . . . . . . . . . . . . 21 |- ( N e. ( ZZ>= ` 3 ) -> ( N - ( 2 + 1 ) ) e. NN0 )
67	61, 66	eqeltrd 2311	. . . . . . . . . . . . . . . . . . . 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 2311	. . . . . . . . . . . . . . . . . 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 9223	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( W e. Word V /\ ( N e. ( ZZ>= ` 3 ) /\ (♯ ` W ) = ( N - 2 ) ) ) -> ( (♯ ` W ) - 1 ) < (♯ ` W ) )
77	72, 73, 76	3jca 1204	. . . . . . . . . . . . . . . . . . . . 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 1045	. . . . . . . . . . . . . . . . . 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 1038	. . . . . . . . . . . . . . . 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 1077	. . . . . . . . . . . . . . . 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 1078	. . . . . . . . . . . . . . . 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 11360	. . . . . . . . . . . . . . 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 9523	. . . . . . . . . . . . . . . . . . . . . 22 |- ( (♯ ` W ) e. NN0 -> (♯ ` W ) e. CC )
87		ax-1cn 8236	. . . . . . . . . . . . . . . . . . . . . 22 |- 1 e. CC
88		npcan 8498	. . . . . . . . . . . . . . . . . . . . . 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 1045	. . . . . . . . . . . . . . . . . 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 5679	. . . . . . . . . . . . . . . . 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 2234	. . . . . . . . . . . . . . . . . . . 20 |- (♯ ` W ) = (♯ ` W )
95		ccatw2s1p1g 11358	. . . . . . . . . . . . . . . . . . . 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 1044	. . . . . . . . . . . . . . . . 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 2267	. . . . . . . . . . . . . . . 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 3781	. . . . . . . . . . . . . 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 11296	. . . . . . . . . . . . . . . . . . . . . . 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 3781	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( W ` 0 ) = X /\ W e. Word V ) -> { (lastS ` W ) , ( W ` 0 ) } = { ( W ` ( (♯ ` W ) - 1 ) ) , X } )
106	105	eleq1d 2303	. . . . . . . . . . . . . . . . . . . 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 1043	. . . . . . . . . . . . . . 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 2311	. . . . . . . . . . . . 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 1255	. . . . . . . . . . . 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 1047	. . . . . . . . . 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 1043	. . . . . 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 1220	. . . . 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 11359	. . . . . . . . . . . . . 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 3781	. . . . . . . . . . . 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 1045	. . . . . . . 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 1045	. . . . . . 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 1044	. . . . 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 2311	. . 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 9716	. . . . . . . . 9 |- ( W e. Word V -> (♯ ` W ) e. ZZ )
138		1zzd 9621	. . . . . . . . 9 |- ( W e. Word V -> 1 e. ZZ )
139	137, 138	zsubcld 9723	. . . . . . . 8 |- ( W e. Word V -> ( (♯ ` W ) - 1 ) e. ZZ )
140		fveq2 5675	. . . . . . . . . . 11 |- ( i = ( (♯ ` W ) - 1 ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` i ) = ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( (♯ ` W ) - 1 ) ) )
141		fvoveq1 6081	. . . . . . . . . . 11 |- ( i = ( (♯ ` W ) - 1 ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( i + 1 ) ) = ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( ( (♯ ` W ) - 1 ) + 1 ) ) )
142	140, 141	preq12d 3781	. . . . . . . . . 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 2303	. . . . . . . . 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 5675	. . . . . . . . . . 11 |- ( i = (♯ ` W ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` i ) = ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` (♯ ` W ) ) )
145		fvoveq1 6081	. . . . . . . . . . 11 |- ( i = (♯ ` W ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( i + 1 ) ) = ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( (♯ ` W ) + 1 ) ) )
146	144, 145	preq12d 3781	. . . . . . . . . 10 |- ( i = (♯ ` W ) -> { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` i ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( i + 1 ) ) } = { ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` (♯ ` W ) ) , ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( (♯ ` W ) + 1 ) ) } )
147	146	eleq1d 2303	. . . . . . . . 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 3745	. . . . . . . 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 1045	. . . . . 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 1045	. . . . 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 953	. 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 3404	. 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 1005   = wceq 1398   e. wcel 2205   A.wral 2522   u. cun 3212   {cpr 3695   class class class wbr 4114   ` cfv 5357  (class class class)co 6058   CCcc 8141   RRcr 8142   0cc0 8143   1c1 8144   + caddc 8146   < clt 8324   - cmin 8460   NNcn 9254   2c2 9305   3c3 9306   NN0cn0 9513   ZZcz 9594   ZZ>=cuz 9871  ..^cfzo 10498  ♯chash 11163  Word cword 11249  lastSclsw 11294   ++ cconcat 11303   <"cs1 11328  Vtxcvtx 16133  Edgcedg 16178

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-1o 6660  df-er 6780  df-en 6989  df-dom 6990  df-fin 6991  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-inn 9255  df-2 9313  df-3 9314  df-n0 9514  df-z 9595  df-uz 9872  df-fz 10362  df-fzo 10499  df-ihash 11164  df-word 11250  df-lsw 11295  df-concat 11304  df-s1 11329

This theorem is referenced by:  clwwlknonex2  16560